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# CorporateFinanceTheory FRL367

CSU Pomona

GPA 3.91

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This 550 page Class Notes was uploaded by Dominic Erdman on Saturday October 3, 2015. The Class Notes belongs to FRL367 at California State Polytechnic University taught by AhmadSohrabian in Fall. Since its upload, it has received 29 views. For similar materials see /class/218183/frl367-california-state-polytechnic-university in Finance,Real Estate&Law at California State Polytechnic University.

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Date Created: 10/03/15

Solutions Manual Corporate Finance Ross Wester eld and Jaffe 9th edition CHAPTER 1 INTRODUCTION TO CORPORATE FINANCE Answers to Concept Questions 1 In the corporate form of ownership the shareholders are the owners of the rm The shareholders elect the directors of the corporation who in turn appoint the rm s management This separation of ownership from control in the corporate form of organization is what causes agency problems to exist Management may act in its own or someone else s best interests rather than those of the shareholders If such events occur they may contradict the goal of maximizing the share price of the equity of the rm Such organizations frequently pursue social or political missions so many different goals are conceivable One goal that is often cited is revenue minimization ie provide whatever goods and services are offered at the lowest possible cost to society A better approach might be to observe that even a notforpro t business has equity Thus one answer is that the appropriate goal is to maximize the value of the equity Presumably the current stock value re ects the risk timing and magnitude of all future cash ows both shortterm and longterm If this is correct then the statement is false An argument can be made either way At the one extreme we could argue that in a market economy all of these things are priced There is thus an optimal level of for example ethical andor illegal behavior and the framework of stock valuation explicitly includes these At the other extreme we could argue that these are noneconomic phenomena and are best handled through the political process A classic and highly relevant thought question that illustrates this debate goes something like this A rm has estimated that the cost of improving the safety of one of its products is 30 million However the rm believes that improving the safety of the product will only save 20 million in product liability claims What should the rm do The goal will be the same but the best course of action toward that goal may be different because of differing social political and economic institutions The goal of management should be to maximize the share price for the current shareholders If management believes that it can improve the pro tability of the rm so that the share price will exceed 35 then they should ght the offer from the outside company If management believes that this bidder or other unidenti ed bidders will actually pay more than 35 per share to acquire the company then they should still ght the offer However if the current management cannot increase the value of the rm beyond the bid price and no other higher bids come in then management is not acting in the interests of the shareholders by ghting the offer Since current managers often lose their jobs when the corporation is acquired poorly monitored managers have an incentive to ght corporate takeovers in situations such as this O We would expect agency problems to be less severe in other countries primarily due to the relatively small percentage of individual ownership Fewer individual owners should reduce the number of diverse opinions concerning corporate goals The high percentage of institutional ownership might lead to a higher degree of agreement between owners and managers on decisions concerning risky projects In addition institutions may be better able to implement effective monitoring mechanisms on managers than can individual owners based on the institutions deeper resources and experiences with their own management The increase in institutional ownership of stock in the United States and the growing activism of these large shareholder groups may lead to a reduction in agency problems for US corporations and a more ef cient market for corporate control However this may not always be the case If the managers of the mutual fund or pension plan are not concerned with the interests of the investors the agency problem could potentially remain the same or even increase since there is the possibility of agency problems between the fund and its investors How much is too much Who is worth more Ray Irani or Tiger Woods The simplest answer is that there is a market for executives just as there is for all types of labor Executive compensation is the price that clears the market The same is true for athletes and performers Having said that one aspect of executive compensation deserves comment A primary reason executive compensation has grown so dramatically is that companies have increasingly moved to stockbased compensation Such movement is obviously consistent with the attempt to better align stockholder and management interests In recent years stock prices have soared so management has cleaned up It is sometimes argued that much of this reward is simply due to rising stock prices in general not managerial performance Perhaps in the future executive compensation will be designed to reward only differential performance ie stock price increases in excess of general market increases Maximizing the current share price is the same as maximizing the future share price at any future period The value of a share of stock depends on all of the future cash ows of company Another way to look at this is that barring large cash payments to shareholders the expected price of the stock must be higher in the future than it is today Who would buy a stock for 100 today when the share price in one year is expected to be 80 CHAPTER 2 FINANCIAL STATEMENTS AND CASH FLOW Answers to Concepts Review and Critical Thinking Questions 1 True Every asset can be converted to cash at some price However when we are referring to a liquid asset the added assumption that the asset can be quickly converted to cash at or near market value is important 2 The recognition and matching principles in nancial accounting call for revenues and the costs associated with producing those revenues to be booked when the revenue process is essentially complete not necessarily when the cash is collected or bills are paid Note that this way is not necessarily correct it s the way accountants have chosen to do it 3 The bottom line number shows the change in the cash balance on the balance sheet As such it is not a useful number for analyzing a company 4 The major difference is the treatment of interest expense The accounting statement of cash ows treats interest as an operating cash ow While the nancial cash ows treat interest as a nancing cash ow The logic of the accounting statement of cash ows is that since interest appears on the income statement which shows the operations for the period it is an operating cash ow In reality interest is a nancing expense which results from the company s choice of debt and equity We will have more to say about this in a later chapter When comparing the two cash ow statements the nancial statement of cash ows is a more appropriate measure of the company s performance because of its treatment of interest 5 Market values can never be negative Imagine a share of stock selling for 720 This would mean that if you placed an order for 100 shares you would get the stock along with a check for 2000 How many shares do you want to buy More generally because of corporate and individual bankruptcy laws net worth for a person or a corporation cannot be negative implying that liabilities cannot exceed assets in market value 6 For a successful company that is rapidly expanding for example capital outlays will be large possibly leading to negative cash ow from assets In general what matters is whether the money is spent wisely not whether cash ow from assets is positive or negative 7 It s probably not a good sign for an established company to have negative cash ow from operations but it would be fairly ordinary for a startup so it depends 8 For example if a company were to become more efficient in inventory management the amount of inventory needed would decline The same might be true if the company becomes better at collecting its receivables In general anything that leads to a decline in ending NWC relative to beginning would have this effect Negative net capital spending would mean more longlived assets were liquidated than purchased 9 If a company raises more money from selling stock than it pays in dividends in a particular period its cash ow to stockholders will be negative If a company borrows more than it pays in interest and principal its cash ow to creditors will be negative 10 The adjustments discussed were purely accounting changes they had no cash ow or market value consequences unless the new accounting information caused stockholders to revalue the derivatives Solutions to Questions and Problems NOTE All end of chapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answerfor eachproblem is found without rounding during any step in theproblem Basic 1 To nd owners equity we must construct a balance sheet as follows Balance Sheet CA 3 5300 CL 3 3900 NFA 26 000 LTD 14200 OE TA 31300 TL amp OE 31300 We know that total liabilities and owners equity TL amp OE must equal total assets of 31300 We also know that TL amp OE is equal to current liabilities plus longterm debt plus owner s equity so owner s equity is OE 31300 714200 7 3900 13200 NWC CA 7 CL 5300 7 3900 1400 2 The income statement for the company is Income Statement Sales 493000 Costs 210000 Depreciation 3 5000 248000 Interest 19000 229000 Taxes 80150 Net income 148850 One equation for net income is Net income Dividends Addition to retained earnings Rearranging we get Addition to retained earnings Net income 7 Dividends Addition to retained earnings 148850 7 50000 Addition to retained earnings 98850 To nd the book value of current assets we use NWC CA 7 CL Rearranging to solve for current assets we get CA 7 NWC CL 7 800000 2100000 7 2900000 The market value of current assets and net xed assets is given so Book value CA 2900000 Market value CA 2800000 Book value NFA 5 000 000 Market value NFA 6 300 000 Book value assets 7900000 Market value assets 9100000 Taxes 7 01550K 02525K 03425K 039246K 7 100K Taxes 79190 The average tax rate is the total tax paid divided by net income so Average tax rate 79190 246000 Average tax rate 3219 The marginal tax rate is the tax rate on the next 1 of earnings so the marginal tax rate 39 To calculate OCF we rst need the income statement Income Statement Sales 14900 Costs 5800 Depreciation 1 300 7 800 Interest 780 Taxable income 7020 Taxes 2808 Net income 4212 OCF EBIT Depreciation 7 Taxes OCF 7800 1300 7 2808 OCF 6292 Net capital spending NFAend 7 NFAbe g Depreciation Net capital spending 1730000 7 1650000 284000 Net capital spending 364000 O The longterm debt account will increase by 10 million the amount of the new longterm debt issue Since the company sold 10 million new shares of stock with a 1 par value the common stock account will increase by 10 million The capital surplus account will increase by 33 million the value of the new stock sold above its par value Since the company had a net income of 9 million and paid 2 million in dividends the addition to retained earnings was 7 million which will increase the accumulated retained earnings account So the new longterm debt and stockholders equity portion of the balance sheet will be Longterm debt 82000000 Total longterm debt 82000000 Shareholders equity Preferred stock 9000000 Common stock 1 par value 30000000 Accumulated retained earnings 104000000 Capital surplus 76000000 Total equity 219000000 Total Liabilities amp Equity 301000000 Cash ow to creditors Interest paid 7 Net new borrowing Cash ow to creditors 118000 7 LTDend 7 LTDbeg Cash ow to creditors 118000 7 1390000 7 1340000 Cash ow to creditors 118000 7 50000 Cash ow to creditors 68000 Cash ow to stockholders Dividends paid 7 Net new equity Cash ow to stockholders 385000 7 Commonend APISe d 7 Commonbe g APISbeg Cash ow to stockholders 7 385000 7 450000 3050000 7 430000 2600000 Cash ow to stockholders 385000 7 3500000 7 3030000 Cash ow to stockholders 785000 Note APIS is the additional paidin surplus Cash ow to creditors Cash ow to stockholders 68000 7 85000 Cash ow from assets 7 17000 Cash ow from assets 717000 OCF 7 Change in NWC 7 Net capital spending 717000 OCF 7 769000 7 875000 Operating cash ow 717000 7 69000 875000 Operating cash ow 789000 Intermediate The accounting statement of cash ows explains the change in cash during the year The accounting statement of cash ows will be Statement of cash ows Operations Net income 105 Depreciation 90 Changes in other current assets 55 Accounts payable E Total cash ow from operations M Investing activities Acquisition of xed assets 1140 Total cash ow from investing activities 140 Financing activities Proceeds of longterm debt 30 Dividends 1 45 1 Total cash ow from nancing activities 151 Change in cash on balance sheet 15 Change in NWC NWCend 7 NWCbe g CAend CLend CAbeg CLbeg 50 155 7 85 7 35 140 7 95 120 7 80 40 To nd the cash ow generated by the rm s assets we need the operating cash ow and the capital spending So calculating each of these we nd Operating cash ow Net income 105 Depreciation amp Operating cash ow 195 Note that we can calculate OCF in this manner since there are no taxes Capital spending Ending xed assets 340 Beginning xed assets 290 Depreciation A Capital spending 140 Now we can calculate the cash ow generated by the rm s assets which is Cash ow from assets Operating cash ow 195 Capital spending 140 Change in NWC Cash ow from assets 15 12 With the information provided the cash ows from the rm are the capital spending and the change in net working capital so Cash owsfrom the rm Capital spending 15000 Additions to NWC 1 1500 Cash ows from the rm 16500 And the cash ows to the investors of the rm are Cash ows to investors of the firm Sale oflongterm debt 19000 Sale of common stock 3000 Dividends paid 19 500 Cash ows to investors of the rm 2500 13 a The interest expense for the company is the amount of debt times the interest rate on the debt So the income statement for the company is Income Statement Sales 1200000 Cost of goods sold 450000 Selling costs 225000 Depreciation 110000 EBIT 415000 Interest 81000 Taxable income 334000 Taxes 116900 Net income 217100 b And the operating cash ow is OCF EBIT Depreciation 7 Taxes OCF 415000 7 110000 7116900 OCF 408100 14 To nd the OCF we rst calculate net income Income Statement Sales 167000 Costs 91000 Depreciation 8000 Other expenses 5400 62600 Interest 11000 Taxable income 51600 Taxes 18060 Net income 33540 Dividends 9500 Additions to RE 24040 a OCF EBIT Depreciation 7 Taxes OCF 62600 8000 7 18060 OCF 52540 b CFC Interest 7 Net new LTD CFC 11000 7 77100 CFC 18100 Note that the net new longterm debt is negative because the company repaid part of its long terrn debt 0 CFS Dividends 7 Net new equity CFS 9500 7 7250 CFS 2250 d We know that CFA CFC CFS so CFA 18100 2250 20350 CFA is also equal to OCF 7 Net capital spending 7 Change in NWC We already know OCF Net capital spending is equal to Net capital spending Increase in NFA Depreciation Net capital spending 22400 8000 Net capital spending 30400 Now we can use CFA OCF 7 Net capital spending 7 Change in NWC 20350 52540 7 30400 7 Change in NWC Solving for the change in NWC gives 1790 meaning the company increased its NWC by 1790 15 The solution to this question works the income statement backwards Starting at the bottom Net income Dividends Addition to ret earnings Net income 1530 5300 Net income 6830 Now looking at the income statement EBT 7 EBT gtlt TaX rate Net income Recognize that EBT gtlt taX rate is simply the calculation for taxes Solving this for EBT yields EBT NI 17 TaX rate EBT 6830 1 7 065 EBT 1050769 Now we can calculate EBIT EBT Interest EBIT 1050769 1900 EBIT 1240769 The last step is to use EBIT Sales 7 Costs 7 Depreciation 1240769 43000 7 27500 7 Depreciation Depreciation 309231 Solving for depreciation we nd that depreciation 309231 16 p A l 18 The balance sheet for the company looks like this Balance Sheet Cash 183000 Accounts payable 465000 Accounts receivable 138000 Notes payable 145 000 Inventory 297000 Current liabilities 610000 Current assets 61 8000 Longterm debt 1550000 Total liabilities 2160000 Tangible net xed assets 3200000 Intangible net xed assets 695000 Common stock Accumulated ret earnings 1960000 Total assets 4513000 Total liab amp owners equity 4513000 Total liabilities and owners equity is TL amp OE Total debt Common stock Accumulated retained earnings Solving for this equation for equity gives us Common stock 4513000 7 1960000 7 2160000 Common stock 393000 The market value of shareholders equity cannot be negative A negative market value in this case would imply that the company would pay you to own the stock The market value of shareholders equity can be stated as Shareholders equity Max TA 7 TL 0 So if TA is 9700 equity is equal to 800 and if TA is 6800 equity is equal to 0 We should note here that while the market value of equity cannot be negative the book value of shareholders equity can be negative a Taxes Growth 01550K 02525K 0343K 14770 Taxes Income 01550K 02525K 03425K 039235K 0347465M 2652000 b Each rm has a marginal tax rate of 34 on the next 10000 of taxable income despite their different average tax rates so both rms will pay an additional 3400 in taxes Income Statement Sales 740000 COGS 610000 AampS expenses 100000 Depreciation 140000 1 15000 Interest 70000 Taxable income 185000 Taxes 3 5 0 a Net income 185000 N O 21 b OCF EBIT Depreciation 7 Taxes OCF 115000 140000 7 0 OCF 25000 0 Net income was negative because of the tax deductibility of depreciation and interest expense However the actual cash ow from operations was positive because depreciation is a noncash expense and interest is a nancing expense not an operating expense A rm can still pay out dividends if net income is negative it just has to be sure there is suf cient cash ow to make the dividend payments Change in NWC Net capital spending Net new equity 0 Given Cash ow from assets OCF 7 Change in NWC 7 Net capital spending Cash ow from assets 25000 7 0 7 0 25000 Cash ow to stockholders Dividends 7 Net new equity Cash ow to stockholders 30000 7 0 30000 Cash ow to creditors Cash ow from assets 7 Cash ow to stockholders Cash ow to creditors 25000 7 30000 Cash ow to creditors 75000 Cash ow to creditors is also Cash ow to creditors Interest 7 Net new LTD So Net new LTD Interest 7 Cash ow to creditors Net new LTD 70000 7 75000 Net new LTD 75000 a The income statement is Income Statement Sales 15300 Cost of good sold 10900 Depreciation 2 100 2300 Interest 520 Taxable income 1780 Taxes 712 Net income 1068 b OCF EBIT Depreciation 7 Taxes OCF 2300 2100 7 712 OCF 3688 0 Change in NWC NWCend 7 NWCbe g 7 CA 7 CL 7 CAR 7 CLbeg 3950 7 1950 7 3400 71900 2000 71500 500 Net capital spending NFAend 7 NFAbe g Depreciation 12900 7 11800 2100 3200 CFA OCF 7 Change in NWC 7 Net capital spending 3688 7 500 7 3200 71 2 The cash ow from assets can be positive or negative since it represents whether the rm raised funds or distributed funds on a net basis In this problem even though net income and OCF are positive the rm invested heavily in both xed assets and net working capital it had to raise a net 12 in funds from its stockholders and creditors to make these investments d Cash ow to creditors Interest 7 Net new LTD 520 Cash ow to stockholders Cash ow from assets 7 Cash ow to creditors 712 7 520 7532 We can also calculate the cash ow to stockholders as Cash ow to stockholders Dividends 7 Net new equity Solving for net new equity we get Net new equity 500 7 7532 7 1032 The rm had positive earnings in an accounting sense NT gt 0 and had positive cash ow from operations The rm invested 500 in new net working capital and 3200 in new xed assets The rm had to raise 12 from its stakeholders to support this new investment It accomplished this by raising 1032 in the form of new equity After paying out 500 of this in the form of dividends to shareholders and 520 in the form of interest to creditors 12 was left to meet the rm s cash ow needs for investment a Total assets 2009 Total liabilities 2009 Owners equity 2009 7 780 3480 7 4260 7 318 18007 2118 7 4260 7 2118 7 2142 Total assets 2010 Total liabilities 2010 Owners equity 2010 7 846 4080 7 4926 7 348 2064 7 2412 7 4926 7 2412 7 2514 NWC 2009 CA09 7 CL09 780 7 318 462 NWC 2010 CA10 7 CL10 846 7 348 498 Change in NWC NWC10 7 NWC09 498 7 462 36 We can calculate net capital spending as Net capital spending Net xed assets 2010 7 Net xed assets 2009 Depreciation Net capital spending 4080 7 3480 960 Net capital spending 1560 So the company had a net capital spending cash ow of 1560 We also know that net capital spending is Net capital spending Fixed assets bought 7 Fixed assets sold 1560 1800 7 Fixed assets sold Fixed assets sold 1800 7 1560 240 To calculate the cash ow from assets we must rst calculate the operating cash ow The operating cash ow is calculated as follows you can also prepare a traditional income statement EBIT Sales 7 Costs 7 Depreciation EBIT 10320 7 4980 7 960 EBIT 4380 EBT EBIT 7 Interest EBT 4380 7 259 EBT 4121 Taxes EBT x 35 Taxes 4121 gtlt 35 Taxes 1442 OCF EBIT Depreciation 7 Taxes OCF 4380 960 7 1442 OCF 3898 Cash ow from assets OCF 7 Change in NWC 7 Net capital spending Cash ow from assets 3898 7 36 71560 Cash ow from assets 2302 Net new borrowing LTD10 7 LTD09 Net new borrowing 2064 7 1800 Net new borrowing 264 Cash ow to creditors Interest 7 Net new LTD Cash ow to creditors 259 7 264 Cash ow to creditors 75 Net new borrowing 264 Debt issued 7 Debt retired Debt retired 360 7 264 96 Cash Accounts receivable Inventory Current assets Net xed assets Total assets Cash Accounts receivable Inventory Current assets Net xed assets Total assets Balance sheet as ofDec 31 2009 2739 3626 6 447 12812 Accounts payable Notes payable Current liabilities Longterm debt Owners equity Total liab amp equity Balance sheet as ofDec 31 2010 2802 4085 6 625 13512 2009 Income Statement Sales COGS Other expenses Depreciation Interest Taxes Net income Dividends Additions to RE 522300 179700 42600 75000 225000 35000 190000 64600 1 25400 63700 61700 24 OCF EBIT Depreciation 7 Taxes OCF 2459 7517 69938 OCF 251062 Change in NWC 7 NWCend 7 NWCbeg 7 CA 7 CL end 7 CA 7 CL be Accounts payable Notes payable Current liabilities Longterm debt Owners equity Total liab amp equity 2877 529 3406 9173 23 203 35 782 2790 497 3287 10702 23 041 37 030 2010 Income Statement Other expenses Depreciation Interest Taxes Net income Dividends Additions to RE Change in NWC 7 13512 7 3287 7 12812 7 3406 Change in NWC 819 Net capital spending NFAend 7 NFAbeg Depreciation Net capital spending 23518 7 22970 751 Net capital spending 1299 560600 204000 35600 75100 245900 40200 205700 69938 1 35762 70100 65662 5 quot Cash ow from assets OCF 7 Change in NWC 7 Net capital spending Cash ow from assets 251062 7 819 7 1299 Cash ow from assets 39662 Cash ow to creditors Interest 7 Net new LTD Net new LTD LTDend 7 LTDbe g Cash ow to creditors 402 7 10702 7 9173 Cash ow to creditors 71 127 Net new equity Common stockend 7 Common stockbe g Common stock Retained earnings Total owners equity Net new equity OE 7 RE end 7 OE 7 RE beg Net new equity OEend 7 OEbe g REbe g 7 REM REend REbe g Additions to RE 39 Net new equity OEend 7 OEbe g REbeg 7 REbe g Additions to RE OEend 7 OEbe g 7 Additions to RE Net new equity 230417 23203 7 65662 781862 Cash ow to stockholders Dividends 7 Net new equity Cash ow to stockholders 701 7 781862 Cash ow to stockholders 151962 As a check cash ow from assets is 39662 Cash ow from assets Cash ow from creditors Cash ow to stockholders Cash ow from assets 71127 151962 Cash ow from assets 39262 Challenge We will begin by calculating the operating cash ow First we need the EBIT which can be calculated as EBIT Net income Current taxes Deferred taxes Interest EBIT 144 82 16 43 EBIT 380 Now we can calculate the operating cash ow as Operating cash ow Earnings before interest and taxes 285 Depreciation 78 Current taxes Operating cash ow 281 The cash ow from assets is found in the investing activities portion of the accounting statement of cash ows so Cash owfrom assets Acquisition of xed assets 148 Sale of xed assets 1191 Capital spending 129 The net working capital cash ows are all found in the operations cash ow section of the accounting statement of cash ows However instead of calculating the net working capital cash ows as the change in net working capital we must calculate each item individually Doing so we nd Net working capital cash ow Cash 42 Accounts receivable 15 Inventories 18 Accounts payable l4 Accrued eXpenses 7 Notes payable 5 Other 42 NWC cash ow 25 Except for the interest eXpense and notes payable the cash ow to creditors is found in the nancing activities of the accounting statement of cash ows The interest eXpense from the income statement is given so Cash ow to creditors Interest 43 Retirement of debt Debt service 178 Proceeds from sale of longterm debt E Total 8 1 And we can nd the cash ow to stockholders in the nancing section of the accounting statement of cash ows The cash ow to stockholders was Cash ow to stockholders Dividends 72 Repurchase of stock Cash to stockholders 83 Proceeds from new stock issue Q Total 46 26 Net capital spending NFAend 7 NFAbe g Depreciation NFAend 7 NFAbeg Depreciation ADbeg 7 ADbe g NFAend T NFAbeg ADend T ADbeg NFAend ADend T NFAbeg ADbeg FAend FAbeg The taX bubble causes average taX rates to catch up to marginal taX rates thus eliminating the taX advantage of low marginal rates for high income corporations Assuming ataxable income of 335000 the taxes will be Taxes 7 01550K 02525K 03425K 039235K 7 1139K Average taX rate 1139K 335K 34 The marginal taX rate on the neXt dollar of income is 34 percent For corporate taxable income levels of 335K to 10M average taX rates are equal to marginal taX rates Taxes 7 03410M 0355M 0383333M 7 6416667 Average taX rate 6416667 18333334 35 The marginal taX rate on the neXt dollar of income is 35 percent For corporate taxable income levels over 18333334 average taX rates are again equal to marginal taX rates Taxes 7 034200K 7 68K 7 01550K 02525K 03425K X100K X100K 7 68K 7 2225K 7 4575K X 7 4575K 100K X 7 4575 CHAPTER 3 FINANCIAL STATEMENTS ANALYSIS AND LONGTERM PLANNING Answers to Concepts Review and Critical Thinking Questions 1 Time trend analysis gives a picture of changes in the company s nancial situation over time Comparing a rm to itself over time allows the nancial manager to evaluate whether some aspects of the rm s operations nances or investment activities have changed Peer group analysis involves comparing the nancial ratios and operating performance of a particular rm to a set of peer group rms in the same industry or line of business Comparing a rm to its peers allows the nancial manager to evaluate whether some aspects of the rm s operations nances or investment activities are out of line with the norm thereby providing some guidance on appropriate actions to take to adjust these ratios if appropriate Both allow an investigation into what is different about a company from a nancial perspective but neither method gives an indication of whether the difference is positive or negative For example suppose a company s current ratio is increasing over time It could mean that the company had been facing liquidity problems in the past and is rectifying those problems or it could mean the company has become less ef cient in managing its current accounts Similar arguments could be made for a peer group comparison A company with a current ratio lower than its peers could be more efficient at managing its current accounts or it could be facing liquidity problems Neither analysis method tells us whether a ratio is good or bad both simply show that something is different and tells us where to look If a company is growing by opening new stores then presumably total revenues would be rising Comparing total sales at two different points in time might be misleading Samestore sales control for this by only looking at revenues of stores open Within a speci c period The reason is that ultimately sales are the driving force behind a business A rm s assets employees and in fact just about every aspect of its operations and nancing eXist to directly or indirectly support sales Put differently a rm s future need for things like capital assets employees inventory and nancing are determined by its future sales level Two assumptions of the sustainable growth formula are that the company does not want to sell new equity and that nancial policy is xed If the company raises outside equity or increases its debt equity ratio it can grow at a higher rate than the sustainable growth rate Of course the company could also grow faster than its pro t margin increases if it changes its dividend policy by increasing the retention ratio or its total asset turnover increases UI p A O p n p A p A N p A DJ The sustainable growth rate is greater than 20 percent because at a 20 percent growth rate the negative EFN indicates that there is excess nancing still available If the rm is 100 percent equity nanced then the sustainable and internal growth rates are equal and the internal growth rate would be greater than 20 percent However when the rm has some debt the internal growth rate is always less than the sustainable growth rate so it is ambiguous whether the internal growth rate would be greater than or less than 20 percent If the retention ratio is increased the rm will have more internal funding sources available and it will have to take on more debt to keep the debtequity ratio constant so the EFN will decline Conversely if the retention ratio is decreased the EFN will rise If the retention rate is zero both the internal and sustainable growth rates are zero and the EFN will rise to the change in total assets Commonsize nancial statements provide the nancial manager with a ratio analysis of the company The commonsize income statement can show for example that cost of goods sold as a percentage of sales is increasing The commonsize balance sheet can show a rm s increasing reliance on debt as a form of nancing Commonsize statements of cash ows are not calculated for a simple reason There is no possible denominator It would reduce the external funds needed If the company is not operating at full capacity it would be able to increase sales without a commensurate increase in xed assets ROE is a better measure of the company s performance ROE shows the percentage return for the year earned on shareholder investment Since the goal of a company is to maximize shareholder wealth this ratio shows the company s performance in achieving this goal over the period The EBITDAssets ratio shows the company s operating performance before interest taxes and depreciation This ratio would show how a company has controlled costs While taxes are a cost and depreciation and amortization can be considered costs they are not as easily controlled by company management Conversely depreciation and amortization can be altered by accounting choices This ratio only uses costs directly related to operations in the numerator As such it gives a better metric to measure management performance over a period than does ROA Longterm liabilities and equity are investments made by investors in the company either in the form of a loan or ownership Return on investment is intended to measure the return the company earned from these investments Return on investment will be higher than the return on assets for a company with current liabilities To see this realize that total assets must equal total debt and equity and total debt and equity is equal to current liabilities plus longterm liabilities plus equity So return on investment could be calculated as net income divided by total assets minus current liabilities Presumably not but of course if the product had been much less popular then a similar fate would have awaited due to lack of sales Since customers did not pay until shipment receivables rose The rm s NWC but not its cash increased At the same time costs were rising faster than cash revenues so operating cash ow declined The rm s capital spending was also rising Thus all three components of cash ow from assets were negatively impacted Financing possibly could have been arranged if the company had taken quick enough action Sometimes it becomes apparent that help is needed only when it is too late again emphasizing the need for planning 14 p A UI All three were important but the lack of cash or more generally nancial resources ultimately spelled doom An inadequate cash resource is usually cited as the most common cause of small business failure Demanding cash upfront increasing prices subcontracting production and improving nancial resources via new owners or new sources of credit are some of the options When orders exceed capacity price increases may be especially bene cial Solutions to Questions and Problems NOTE All end of chapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the nal answerfor each problem is found without rounding during any step in the problem Basic ROE PMTATEM ROE 058215135 1683 or 1683 The equity multiplier is EM1DE EM1090 190 One formula to calculate return on equity is ROE ROAEM ROE 01010190 1919 or 1919 ROE can also be calculated as ROE NT TE So net income is NI ROETE NI 1919645000 12377550 This is a multistep problem involving several ratios The ratios given are all part of the Du Pont Identity The only Du Pont Identity ratio not given is the pro t margin If we know the pro t margin we can nd the net income since sales are given So we begin with the Du Pont Identity ROE 016 PMTATEM PMS TA1 DE Solving the Du Pont Identity for pro t margin we get PM ROETA 1 D ES PM 01611580 1 120 3100 0371 Now that we have the pro t margin we can use this number and the given sales gure to solve for net income PM0371NIs N1 7 03713100 7 11491 An increase of sales to 30960 is an increase of Sales increase 30960 7 25800 25800 Sales increase 20 or 20 Assuming costs and assets increase proportionally the pro forma nancial statements will look like t is Pro forma income statement Pro forma balance sheet Sales 3096000 Assets 135600 Debt 2050000 Costs 1980000 Equity 97 65592 EBIT 1116000 Total 135600 Total 11815592 Taxes 34 379440 Net income 736560 The payout ratio is constant so the dividends paid this year is the payout ratio from last year times net income or Dividends 7 184140 6138736560 Dividends 220968 The addition to retained earnings is Addition to retained earnings 7365 7 220968 Addition to retained earnings 515592 And the new equity balance is Equity 7 92500 515592 Equity 7 9765592 So the EFN is EFN Total assets 7 Total liabilities and equity EFN 135600 711815592 EFN 1744408 The maximum percentage sales increase without issung new equity is the sustainable growth rate To calculate the sustainable growth rate we rst need to calculate the ROE which is ROE NI TE ROE 15312 81000 ROE 1890 The plowback ratio b is one minus the payout ratio so 171730 1770 Now we can use the sustainable growth rate equation to get Sustainable growth rate ROE X b 1 7 ROE X b Sustainable growth rate 189070 1 7 189070 Sustainable growth rate 1525 or 1525 So the maximum dollar increase in sales is Maximum increase in sales 670001525 Maximum increase in sales 1021793 We need to calculate the retention ratio to calculate the sustainable growth rate The retention ratio is 171710 1790 Now we can use the sustainable growth rate equation to get Sustainable growth rate ROE X b 1 7 ROE X b Sustainable growth rate 1590 1 7 1590 Sustainable growth rate 1561 or 1561 We must rst calculate the ROE using the Du Pont ratio to calculate the sustainable growth rate The ROE is ROE 7 PMTATEM ROE 7 081190125 ROE 7 1924 or 1924 The plowback ratio is one minus the diVidend payout ratio so 171730 1770 on Now we can use the sustainable growth rate equation to get Sustainable growth rate ROE X b 1 7 ROE X 17 Sustainable growth rate 192470 1 7 192470 Sustainable growth rate 1556 or 1556 An increase of sales to 6669 is an increase of Sales increase 6669 7 5700 5700 Sales increase 17 or 17 Assuming costs and assets increase proportionally the pro forma nancial statements will look like this Pro forma income statement Pro forma balance sheet Sales 6669 Assets 16497 Debt 6300 Costs 4469 Equ39 10 000 Net income 2200 Total 16497 Total 16300 If no dividends are paid the equity account will increase by the net income so Equity 7 7800 2200 Equity 7 10000 So the EFN is EFN Total assets 7 Total liabilities and equity EFN 16497 716300 197 a First we need to calculate the current sales and change in sales The current sales are neXt year s sales diVided by one plus the growth rate so Current sales NeXt year s sales 1 g Current sales 390000000 1 10 Current sales 354545455 And the change in sales is Change in sales 390000000 7 354545455 Change in sales 35454545 We can now complete the current balance sheet The current assets xed assets and shortterm debt are calculated as a percentage of current sales The longterm debt and par value of stock are given The plug variable is the additions to retained earnings So Assets Liabilities and equitv Current assets 70909091 Shortterm debt 53181818 Longterm debt 130000000 Fixed assets 425454545 Common stock 48000000 Accumulated retained earnings 265181818 Total equity 313181818 Total assets 496363636 Total liabilities and equity 496363636 We can use the equation from the text to answer this question The assetssales and debtsales are the percentages given in the problem so EFN M X ASales 7 Debt gtlt ASales 7 PM gtlt Projected sales X 1 7 d Sales Sales EFN 20 120 x 35454545 7 15 x 35454545 7 12 x 390000000 x1730 EFN 11558182 The current assets xed assets and shortterm debt will all increase at the same percentage as sales The longterm debt and common stock will remain constant The accumulated retained earnings will increase by the addition to retained earnings for the year We can calculate the addition to retained earnings for the year as Net income Pro t margin gtlt Sales Net income 12390000000 Net income 46800000 The addition to retained earnings for the year will be the net income times one minus the dividend payout ratio which is Addition to retained earnings Net income1 7 d Addition to retained earnings 468000001 7 30 Addition to retained earnings 32760000 So the new accumulated retained earnings will be Accumulated retained earnings 265181818 32760000 Accumulated retained earnings 297941818 The pro forma balance sheet will be Assets Liabilities and equitv Current assets 78000000 Shortterm debt 58500000 Longterm debt 130000000 Fixed assets 468000000 Common stock 48000000 297941818 Accumulated retained earnings M Total equity Total assets 546000000 Total liabilities and equity 534441818 The EFN is EFN Total assets 7 Total liabilities and equity EFN 546000000 7 534441818 EFN 11558182 The sustainable growth is ROEx b Sustainable growth rate 1 ROE x b where b Retention ratio l 7 Payout ratio 60 So Sustainable growth rate l 1050 x 60 Sustainable growth rate 0672 or 672 It is possible for the sustainable growth rate and the actual growth rate to differ If any of the actual parameters in the sustainable growth rate equation differs from those used to compute the sustainable growth rate the actual growth rate will differ from the sustainable growth rate Since the sustainable growth rate includes ROE in the calculation this also implies that changes in the pro t margin total asset turnover or equity multiplier will affect the sustainable growth rate The company can increase its growth rate by doing any of the following Increase the debttoequity ratio by selling more debt or repurchasing stock Increase the pro t margin most likely by better controlling costs Decrease its total assetssales ratio in other words utilize its assets more ef ciently Reduce the dividend payout ratio n n 13 Intermediate The solution requires substituting two ratios into a third ratio Rearranging DTA Firm A D TA 40 TA7ETA7 40 TA TA 7E TA 7 40 FirmB D TA 30 TA7ETA7 30 TA TA 7 E TA 7 30 17ETA40 17ETA30 ETA60 ETA70 E 60TA E 70TA Rearranging ROA we find NTTA12 NTTA15 NT 12TA NT 15TA Since ROE NI E we can substitute the above equations into the ROE formula which yields ROE 7 12TA 60TA 7 1260 7 20 ROE 7 15TA 70 TA 7 3060 7 2143 PM NT S 7 15834 167983 70943 or 943 As long as both net income and sales are measured in the same currency there is no problem in fact except for some market value ratios like EPS and BVPS none of the financial ratios discussed in the text are measured in terms of currency This is one reason why financial ratio analysis is widely used in international nance to compare the business operations of firms andor divisions across national economic borders The net income in dollars is NT PM gtlt Sales NT 410943251257 72368337 1 The equation for external funds needed is EFN W X ASales 7 Debt gtlt ASales 7 PM gtlt Projected sales X l 7 d Sales Sales where AssetsSales 7 2480000030400000 7 082 ASales Current sales gtlt Sales growth rate 3040000020 6080000 DebtSales 7 640000030400000 7 2105 p Net incomeSales 239200030400000 0787 Projected sales Current sales X l Sales growth rate 30400000l 20 36480000 d DividendsNet income 9568002392000 40 S0 EFN 7 82 x 6080000 72105 x 6080000 7 0787 x 36480000 x 1 740 EFN 7 1957760 The current assets fixed assets and shortterm debt will all increase at the same percentage as sales The longterm debt and common stock will remain constant The accumulated retained earnings will increase by the addition to retained earnings for the year We can calculate the addition to retained earnings for the year as Net income Pro t margin gtlt Sales Net income 078736480000 Net income 2870400 The addition to retained earnings for the year will be the net income times one minus the dividend payout ratio which is Addition to retained earnings Net income1 7 d Addition to retained earnings 28704001 7 40 Addition to retained earnings 1722240 So the new accumulated retained earnings will be Accumulated retained earnings 10400000 1722240 Accumulated retained earnings 12122240 The pro forma balance sheet will be Assets Liabilities and equity Current assets 8640000 Shortterm debt 7680000 Longterm debt 4800000 Fixed assets 21120000 Common stock 3200000 Accumulated retained earnings 12122240 Total equity 15322240 Total assets 29760000 Total liabilities and equity 27802240 The EFN is EFN Total assets 7 Total liabilities and equity EFN 29760000 7 27802240 EFN 1957760 c The sustainable growth is ROEx b Sustainable growth rate 1 ROE x b where ROE Net incomeTotal equity 2392000 13600000 1759 b Retention ratio Retained eamingsNet income 1435200 2392000 60 So 1759 x 60 ll759gtlt 60 Sustainable growth rate 1180 or 1180 Sustainable growth rate d The company cannot just cut its dividends to achieve the forecast growth rate As shown below even with a zero dividend policy the EFN will still be 809600 Assets Liabilities and equity Current assets 8640000 Shortterm debt 7680000 Longterm debt 4800000 Fixed assets 21120000 Common stock 3200000 Accumulated retained earnings 13270400 Total equity 16470400 Total assets 29760000 Total liabilities and equity 28950400 The EFN is EFN Total assets 7 Total liabilities and equity EFN 29760000 7 28950400 EFN 809600 The company does have several alternatives It can increase its asset utilization andor its pro t margin The company could also increase the debt in its capital structure This will decrease the equity account thereby increasing ROE 14 This is a multistep problem involving several ratios It is often easier to look backward to determine where to start We need receivables turnover to nd days sales in receivables To calculate receivables turnover we need credit sales and to nd credit sales we need total sales Since we are given the profit margin and net income we can use these to calculate total sales as PM 0093 NI Sales 205000 Sales Sales 2204301 Credit sales are 80 percent of total sales so Credit sales 2204301080 1763441 UI Now we can nd receivables turnover by Receivables turnover Credit sales Accounts receivable 1763441 162500 1085 times Days sales in receivables 365 days Receivables turnover 365 1085 3363 days The solution to this problem requires a number of steps First remember that CA NFA TA So if we nd the CA and the TA we can solve for NFA Using the numbers given for the current ratio and the current liabilities we solve for CA CR CA CL CA CRCL 130900 1170 To find the total assets we must rst nd the total debt and equity from the information given So we nd the net income using the profit margin PM NI Sales NT Profit margin gtlt Sales 0945320 50008 We now use the net income gure as an input into ROE to find the total equity ROE NI TE TE NI ROE 50008 182 274769 Next we need to nd the longterm debt The longterm debt ratio is Longterm debt ratio 040 LTD LTD TE Inverting both sides gives 1040 LTD TE LTD 1 TE LTD Substituting the total equity into the equation and solving for longterm debt gives the following 1 274769 LTD 25 LTD 274769 15 183179 Now we can nd the total debt of the company TD CL LTD 900 183179 273179 And with the total debt we can nd the TDampE which is equal to TA TA TD TE 273179 274769 547949 And finally we are ready to solve the balance sheet identity as NFA TAi CA 547949 71170 430949 16 p A 1 This problem requires you to work backward through the income statement First recognize that Net income l 7 tCEBT Plugging in the numbers given and solving for EBT we get EBT 9450 066 1431818 Now we can add interest to EBT to get EBIT as follows EBIT EBT Interest paid 1431818 2360 1667818 To get EBITD earnings before interest taxes and depreciation the numerator in the cash coverage ratio add depreciation to EBIT EBITD EBIT Depreciation 1667818 3480 2015818 Now simply plug the numbers into the cash coverage ratio and calculate Cash coverage ratio EBITD Interest 20l58l8 2360 854 times The only ratio given which includes cost of goods sold is the inventory turnover ratio so it is the last ratio used Since current liabilities are given we start with the current ratio Current ratio 23 CA CL CA 270000 CA 621000 Using the quick ratio we solve for inventory Quick ratio ll CA7 Inventory CL 621000 7Inventory 270000 Inventory CA 7 Quick ratio gtlt CL Inventory 621000 7 11 X 270000 Inventory 324000 Inventory turnover 42 COGS Inventory COGS 324000 COGS 7 1360800 Common Common Common 18 2009 size 2010 size base year Assets Current assets Cash 8436 2 86 10157 313 12040 Accounts receivable 21530 729 23406 721 10871 Inventory 38760 1312 42650 1314 11004 Total 68726 2326 76213 2348 11089 Fixed assets Net plant and equipment 226706 7674 248306 7652 10953 Total assets 295432 100 324519 100 10985 Liabilities and Owners Equity Current liabilities Accounts payable 43050 1457 46821 1443 10876 Notes payable 18384 622 17382 536 09455 Total 61434 2079 64203 1978 10451 Longterm debt 25000 846 32000 986 12800 Owners39 equity Common stock and paidin surplus 40000 1354 40000 1233 10000 Accumulated retained earnings 168998 5720 188316 5803 11143 Total 208998 7074 228316 7036 10924 Total liabilities and owners39 equity 295432 100 324519 100 10985 The commonsize balance sheet answers are found by dividing each category by total assets For example the cash percentage for 2009 is 8436 295432 0286 or 286 This means that cash is 286 of total assets The commonbase year answers for Question 18 are found by dividing each category value for 2010 by the same category value for 2009 For example the cash commonbase year number is found by 10157 8436 12040 This means the cash balance in 2010 is 12040 times as large as the cash balance in 2009 19 N O N p A To determine full capacity sales we divide the current sales by the capacity the company is currently using so Full capacity sales 630000 85 Full capacity sales 741176 So the dollar growth rate in sales is Sales growth 741176 7 630000 Sales growth 111176 To nd the new level of xed assets we need to nd the current percentage of xed assets to full capacity sales Doing so we nd Fixed assets Full capacity sales 580000 741176 Fixed assets Full capacity sales 7825 Next we calculate the total dollar amount of fixed assets needed at the new sales figure Total fixed assets 7825790000 Total fixed assets 61820635 The new xed assets necessary is the total xed assets at the new sales figure minus the current level of xed assets New xed assets 61820635 7 580000 New xed assets 3820635 Assuming costs vary with sales and a 20 percent increase in sales the pro forma income statement will look like this MOOSE TOURS INC Pro Forma Income Statement Sales 1114800 Costs 867600 Other expenses 22800 EBIT 224400 Interest 14000 Taxable income 210400 Taxes3 5 73640 Net income 136760 The payout ratio is constant so the dividends paid this year is the payout ratio from last year times net income or Dividends 33735112450136760 Dividends 41028 And the addition to retained earnings will be Addition to retained earnings 136760 7 41028 Addition to retained earnings 95732 The new retained earnings on the pro forma balance sheet will be New retained earnings 182900 95732 New retained earnings 278632 The pro forma balance sheet will look like this MOOSE TOURS INC Pro Forma Balance Sheet Liabilities and Owners Equity N Assets Current assets Cash 30360 Accounts receivable 48840 Inventory 104280 Total 183480 Fixed assets Net plant and equipment 495600 Total assets 679080 So the EFN is EFN Total assets 7 Total liabilities and equity EFN 679080 7 675232 EFN 3848 Full capacity sales 929000 80 Full capacity sales 1161250 The full capacity ratio at full capacity sales is Full capacity ratio Fixed assets Full capacity sales Full capacity ratio 413000 1161250 Full capacity ratio 35565 Current liabilities Accounts payable 81600 Notes payable 17000 Total 98600 Longterm debt 158000 Owners equity Common stock and paidin surplus 140000 Retained earnings 278632 Total 418632 Total liabilities and owners equity 675232 First we need to calculate full capacity sales which is DJ The xed assets required at full capacity sales is the full capacity ratio times the projected sales level Total fixed assets 355651161250 396480 So EFN is EFN 183480 396480 7 675232 795272 Note that this solution assumes that xed assets are decreased sold so the company has a 100 percent xed asset utilization If we assume xed assets are not sold the answer becomes EFN 183480 413000 7 675232 778752 The DE ratio of the company is DE 85000 158000 322900 DE 7526 So the new total debt amount will be New total debt 7526418632 New total debt 315044 This is the new total debt for the company Given that our calculation for EFN is the amount that must be raised externally and does not increase spontaneously with sales we need to subtract the spontaneous increase in accounts payable The new level of accounts payable will be which is the current accounts payable times the sales growth or Spontaneous increase in accounts payable 6800020 Spontaneous increase in accounts payable 13600 This means that 13600 of the new total debt is not raised externally So the debt raised externally which will be the EFN is EFN New total debt 7 Beginning LTD Beginning CL Spontaneous increase in AP EFN 315044 7 158000 85000 13600 58444 The pro forma balance sheet with the new longterm debt will be MOOSE TOURS INC Pro Forma Balance Sheet Assets Liabilities and Owners Equity Current assets Current liabilities Cash 30360 Accounts payable 81600 Accounts receivable 48840 Notes payable 17000 Inventory 104280 Total 98600 Total 183480 Longterm debt 216444 Fixed assets Net plant and Owners equity equipment 495600 Common stock and paidin surplus 140000 Retained earnings 278632 Total 418632 Total liabilities and owners Total assets g 679080 equity g 733676 The funds raised by the debt issue can be put into an excess cash account to make the balance sheet balance The excess debt will be Excess debt 733676 7 679080 54596 To make the balance sheet balance the company will have to increase its assets We will put this amount in an account called excess cash which will give us the following balance sheet MOOSE TOURS INC Pro Forma Balance Sheet Assets Liabilities and Owners Equity Current assets Current liabilities Cash 30360 Accounts payable 81600 Excess cash 54596 Accounts receivable 48480 Notes payable 17000 Inventory 104280 Total 98600 Total 238076 Longterm debt 216444 Fixed assets Net plant and Owners equity equipment 495600 Common stock and paidin surplus 140000 Retained earnings 278632 Total 418632 Total liabilities and owners Total assets 733676 equity 733676 The excess cash has an opportunity cost that we discussed earlier Increasing xed assets would also not be a good idea since the company already has enough xed assets A likely scenario would be the repurchase of debt and equity in its current capital structure weights The company s debtassets and equity assets are Debtassets 7526 1 7526 43 Equityassets 1 1 7526 57 So the amount of debt and equity needed will be Total debt needed 43679080 291600 Equity needed 57679080 3 87480 So the repurchases of debt and equity will be Debt repurchase 98600 216444 7 291600 23444 Equity repurchase 418632 7 387480 31152 Assuming all of the debt repurchase is from longterm debt and the equity repurchase is entirely from the retained earnings the nal pro forma balance sheet will be MOOSE TOURS INC Pro Forma Balance Sheet Assets Liabilities and Owners Equity Current assets Current liabilities Cash 30360 Accounts payable 81600 Accounts receivable 48840 Notes payable 17000 Inventory 104280 Total 98600 Total 183480 Longterm debt 193000 Fixed assets Net plant and Owners equity equipment 495600 Common stock and paidin surplus 140000 Retained earnings 247480 0 mm Total liabilities and owners Total assets 679080 equity 679080 Challenge 4 The pro forma income statements for all three growth rates will be MOOSE TOURS INC Pro Forrna Income Statement 15 Sales 20 Sales 25 Sales Growth Growth Growth Sales 1068350 1114800 1161250 Costs 831450 867600 903750 Other expenses 21850 22800 23750 EBIT 215050 224400 233750 Interest 14000 14000 14000 Taxable income 201050 210400 219750 Taxes 35 70368 73640 76913 Net income 130683 136760 142838 Dividends 39205 41028 42851 Add to RE 91478 95732 99986 We will calculate the EFN for the 15 percent growth rate rst Assuming the payout ratio is constant the dividends paid will be Dividends 33735112450130683 Dividends 39205 And the addition to retained earnings will be Addition to retained earnings 130683 7 39205 Addition to retained earnings 91478 The new retained earnings on the pro forma balance sheet will be New retained earnings 182900 91478 New retained earnings 274378 The pro forma balance sheet will look like this MOOSE TOURS INC Pro Forma Balance Sheet Liabilities and Owners Equity 15 Sales Growth Assets Current assets Cash 29095 Accounts receivable 46805 Inventory 99935 Total 175835 Fixed assets Net plant and equipment 474950 Total assets g 650785 So the EFN is EFN Total assets 7 Total liabilities and equity EFN 650785 7 667578 EFN 716793 Current liabilities Accounts payable 78200 Notes payable 17000 Total 95200 Longterm debt 158000 Owners equity Common stock and paidin surplus 140000 Retained earnings 274378 Total 414378 Total liabilities and owners equity 667578 At a 20 percent growth rate and assuming the payout ratio is constant the dividends paid will be Dividends 33735112450136760 Dividends 41028 And the addition to retained earnings will be Addition to retained earnings 136760 7 41028 Addition to retained earnings 95732 The new retained earnings on the pro forma balance sheet will be New retained earnings 182900 95732 New retained earnings 278632 The pro forma balance sheet will look like this 20 Sales Growth MOOSE TOURS INC Pro Forma Balance Sheet Assets Liabilities and Owners Equity Current assets Current liabilities Cash 30360 Accounts payable 81600 Accounts receivable 48840 Notes payable 17000 Inventory 104280 Total 98600 Total 183480 Longterm debt 158000 Fixed assets Net plant and Owners equity equipment 495600 Common stock and paidin surplus 140000 Retained earnings 278632 Total 418632 Total liabilities and owners Total assets 679080 equity 675232 So the EFN is EFN Total assets 7 Total liabilities and equity EFN 679080 7 675232 EFN 3848 At a 25 percent growth rate and assuming the payout ratio is constant the dividends paid will be Dividends 7 33735112450142838 Dividends 42851 And the addition to retained earnings will be Addition to retained earnings 142838 7 42851 Addition to retained earnings 99986 The new retained earnings on the pro forma balance sheet will be New retained earnings 182900 99986 New retained earnings 282886 The pro forma balance sheet will look like this 25 Sales Growth MOOSE TOURS INC Pro Forma Balance Sheet Assets Liabilities and Owners Equity Current assets Current liabilities Cash 31625 Accounts payable 85000 Accounts receivable 50875 Notes payable 17000 Inventory 108625 Total 102000 Total 191125 Longterm debt 158000 Fixed assets Net plant and Owners equity equipment 516250 Common stock and paidin surplus 140000 Retained earnings 282886 0 422886 Total liabilities and owners Total assets 707375 equity 682886 So the EFN is EFN Total assets 7 Total liabilities and equity EFN 707375 7 682886 EFN 24489 2 The pro forma income statements for all three growth rates will be MOOSE TOURS INC Pro Forrna Income Statement 20 Sales 30 Sales 35 Sales Growth Growth Growth Sales 1114800 1207700 1254150 Costs 867600 939900 976050 Other expenses 22800 24700 25650 EBIT 224400 243100 252450 Interest 14000 14000 14000 Taxable income 210400 229100 23 8450 Taxes 35 73640 80185 83458 Net income 136760 148915 154993 Dividends 41028 44675 46498 Add to RE 95732 104241 108495 At a 30 percent growth rate and assuming the payout ratio is constant the dividends paid will be Dividends 33735112450148915 Dividends 44675 And the addition to retained earnings will be Addition to retained earnings 148915 7 44675 Addition to retained earnings 104241 The new addition to retained earnings on the pro forma balance sheet will be New addition to retained earnings 182900 104241 New addition to retained earnings 287141 The new total debt will be New total debt 7556427141 New total debt 321447 So the new longterm debt will be the new total debt minus the new shortterm debt or New longterm debt 321447 7 105400 New longterm debt 216047 The pro forma balance sheet will look like this Sales growth rate 30 and debtequity ratio 7526 MOOSE TOURS INC Pro Forma Balance Sheet Assets Liabilities and Owners Equity Current assets Current liabilities Cash 32890 Accounts payable 88400 Accounts receivable 52910 Notes payable 17000 Inventory 1 12970 Total 105400 Total 198770 Longterm debt 216047 Fixed assets Net plant and Owners equity equipment 536900 Common stock and paidin surplus 140000 Retained earnings 287141 Total 427141 Total liabilities and owners Total assets 735670 equity 7485 87 So the excess debt raised is Excess debt 748587 7 735670 Excess debt 12917 At a 35 percent growth rate and assuming the payout ratio is constant the dividends paid will be Dividends 7 33735112450154993 Dividends 46498 And the addition to retained earnings will be Addition to retained earnings 154993 7 46498 Addition to retained earnings 108495 The new retained earnings on the pro forma balance sheet will be New retained earnings 182900 108495 New retained earnings 291395 The new total debt will be New total debt 75255431395 New total debt 324648 So the new longterm debt will be the new total debt minus the new shortterm debt or New longterm debt 324648 7 108800 New longterm debt 215848 Sales growth rate 35 and debtequity ratio 75255 MOOSE TOURS INC Pro Forma Balance Sheet Assets Liabilities and Owners Equity Current assets Current liabilities Cash 34155 Accounts payable 91800 Accounts receivable 54945 Notes payable 17000 Inventory 117315 Total 108800 Total 206415 Longterm debt 215848 Fixed assets Net plant and Owners equity equipment 557550 Common stock and paidin surplus 140000 Retained earnings 291395 Total 431395 Total liabilities and owners Total assets 763965 equity 756043 So the excess debt raised is Excess debt 756043 7 763965 Excess debt 77922 F 1 At a 35 percent growth rate the rm will need funds in the amount of 7922 in addition to the external debt already raised So the EFN will be EFN 7 57848 7922 EFN 7 65770 We must need the ROE to calculate the sustainable growth rate The ROE is ROE 7 PMTATEM ROE 05910851 03 ROE 7 0902 or 902 Now we can use the sustainable growth rate equation to nd the retention ratio as Sustainable growth rate ROE X b 1 7 ROE X b Sustainable growth rate 12 0902b 1 7 0902b b 119 This implies the payout ratio is Payout ratio 1 7 b Payout ratio 1 7 119 Payout ratio 7019 This is a negative dividend payout ratio of negative 19 percent which is impossible The growth rate is not consistent with the other constraints The lowest possible payout rate is 0 which corresponds to retention ratio of 1 or total earnings retention The maximum sustainable growth rate for this company is Maximum sustainable growth rate ROE X b 1 7 ROE X b Maximum sustainable growth rate 09021 1 7 09021 Maximum sustainable growth rate 0992 or 992 We know that EFN is EFN Increase in assets 7 Addition to retained earnings The increase in assets is the beginning assets times the growth rate so Increase in assets A x g The addition to retained earnings next year is the current net income times the retention ratio times one plus the growth rate so Addition to retained earnings N1 x b1 g on 0 And rearranging the pro t margin to solve for net income we get NI PMS Substituting the last three equations into the EFN equation we started with and rearranging we get EFN Ag 7PMSb1 g EFN Ag 7 PMSb 7 PMSbg EFN 7 7PMSb A 7 PMSbg We start with the EFN equation we derived in Problem 27 and set it equal to zero EFN 0 7PMSb A 7PMSbg Substituting the rearranged pro t margin equation into the internal growth rate equation we have Internal growth rate PMSb A 7 PMSb Since ROANIA ROAPMSA We can substitute this into the internal growth rate equation and divide both the numerator and denominator by A This gives Internal growth rate PMSb A A 7PMSb A Internal growth rate bROA l 7 bROA To derive the sustainable growth rate we must realize that to maintain a constant DE ratio with no external equity financing EF N must equal the addition to retained earnings times the DE ratio EFN D EPMSb1 g EFN Ag 7PMSb1 g Solving for g and then dividing numerator and denominator by A Sustainable growth rate PMSbl DE A 7 PMSbl DE Sustainable growth rate ROAl DE b l 7 ROAl DE b Sustainable growth rate bROE l 7 bROE In the following derivations the subscript E refers to end of period numbers and the subscript B refers to beginning of period numbers TE is total equity and TA is total assets For the sustainable growth rate Sustainable growth rate ROEE X b l 7ROEE X b Sustainable growth rate NITEE X b l 7 NITEE X b We multiply this equation by TEE TEE Sustainable growth rate NI TEE X b l 7NI TEE X b X TEE TEE Sustainable growth rate NI X b TEE 7 NI X b Recognize that the denominator is equal to beginning of period equity that is TEE 7N1 X b TEE Substituting this into the previous equation we get Sustainable rate NI X b TEE Which is equivalent to Sustainable rate NI TEE X b Since ROEB NI TEE The sustainable growth rate equation is Sustainable growth rate ROEB X b For the internal growth rate Internal growth rate ROAE X b l 7 ROAE X b Internal growth rate NI TAE X b l 7 NI TAE X b We multiply this equation by TAE TAE Internal growth rate NI TAE X b l 7NI TAE X b X TAE TAE Internal growth rate NI X b TAE 7NI X b Recognize that the denominator is equal to beginning of period assets that is TAE 7NI X b TAB Substituting this into the previous equation we get Internal growth rate NI X b TAB Which is equivalent to Internal growth rate NI TAB X b Since ROAB NI TAB O The internal growth rate equation is Internal growth rate ROAB X b Since the company issued no new equity shareholders equity increased by retained earnings Retained earnings for the year were Retained earnings NT 7DiVidends Retained earnings 95000 7 68000 Retained earnings 27000 So the equity at the end of the year was Ending equity 183000 27000 Ending equity 210000 The ROE based on the end of period equity is ROE 95000 210000 ROE 4524 The plowback ratio is Plowback ratio Addition to retained earningsNI Plowback ratio 27000 95000 Plowback ratio 2842 or 2842 Using the equation presented in the text for the sustainable growth rate we get Sustainable growth rate ROE X b 1 7 ROE X b Sustainable growth rate 45242842 1 7 45242842 Sustainable growth rate 1475 or 1475 The ROE based on the beginning of period equity is ROE 95000 183000 ROE 5191 or 5191 Using the shortened equation for the sustainable growth rate and the beginning of period ROE we get Sustainable growth rate ROE X b Sustainable growth rate 5191 X 2842 Sustainable growth rate 1475 or 1475 Using the shortened equation for the sustainable growth rate and the end of period ROE we get Sustainable growth rate ROE X b Sustainable growth rate 4524 X 2842 Sustainable growth rate 1286 or 1286 Using the end of period ROE in the shortened sustainable growth rate results in a growth rate that is too low This will always occur whenever the equity increases If equity increases the ROE based on end of period equity is lower than the ROE based on the beginning of period equity The ROE and sustainable growth rate in the abbreviated equation is based on equity that did not exist when the net income was earned CHAPTER 4 DISCOUNTED CASH FLOW VALUATION Answers to Concepts Review and Critical Thinking Questions 1 Assuming positive cash ows and interest rates the future value increases and the present value decreases Assuming positive cash ows and interest rates the present value will fall and the future value will rise The better deal is the one with equal installments Yes they should APRs generally don t provide the relevant rate The only advantage is that they are easier to compute but with modern computing equipment that advantage is not very important A freshman does The reason is that the freshman gets to use the money for much longer before interest starts to accrue It s a re ection of the time value of money TMCC gets to use the 24099 immediately If TMCC uses it wisely it will be worth more than 100000 in thirty years This will probably make the security less desirable TMCC will only repurchase the security prior to maturity if it is to its advantage ie interest rates decline Given the drop in interest rates needed to make this viable for TMCC it is unlikely the company will repurchase the security This is an example of a call feature Such features are discussed at length in a later chapter The key considerations would be 1 Is the rate of return implicit in the offer attractive relative to other similar risk investments and 2 How risky is the investment ie how certain are we that we will actually get the 100000 Thus our answer does depend on who is making the promise to repay The Treasury security would have a somewhat higher price because the Treasury is the strongest of all borrowers The price would be higher because as time passes the price of the security will tend to rise toward 100000 This rise is just a re ection of the time value of money As time passes the time until receipt of the 100000 grows shorter and the present value rises In 2019 the price will probably be higher for the same reason We cannot be sure however because interest rates could be much higher or TMCC nancial position could deteriorate Either event would tend to depress the security s price Solutions to Questions and Problems NOTE Allend of chapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic 1 The simple interest per year is 5000 X 09 450 So after 10 years you will have 450 X 10 4500 in interest The total balance will be 5000 4500 9500 With compound interest we use the future value formula FV PV1 r39 FV 500010910 1183682 The difference is 1183682 7 9500 233682 2 To nd the FV ofa lump sum we use FV PV1 r a FV 100010610 179085 b FV 100010910 236736 c FV 100010620 320714 d Because interest compounds on the interest already earned the interest earned in pa1t c is more than twice the interest earned 1n pa1t a With compound interest future values grow exponentially 3 To nd the PV ofa lump sum we use PVFV1r PV 15451 1076 1029565 PV 51557 1159 1465572 PV 886073 11118 13541160 PV 550164 118 1222379 To answer this question we can use either the FV or the PV formula Both will give the same answer since they are the inverse of each other We will use the FV formula that is FV PV1 r Solving for r we get rFVPV L1 FV 307 2421 r2 r 30724212 71 1263 FV 896 4101 r9 r 896 410 9 71 907 FV 162181 517001 r r 162181 5 1700 71 792 FV 483500 187501 r r 483500 18750 30 71 1144 To answer this question we can use either the FV or the PV formula Both will give the same answer since they are the inverse of each other We will use the FV formula that is FV PV1 r Solving for t we get I lnFVPVln1 r FV 1284 625106 t ln1284 625 In 106 1236 years FV 4341 810113 t ln4341 810 In 113 1374 years FV 402662 18400132quot t ln402662 18400 In 132 1111 years FV 173439 21500116 t ln173439 21500 In 116 1407 years To nd the length of time for money to double triple etc the present value and future value are irrelevant as long as the future value is twice the present value for doubling three times as large for tripling etc To answer this question we can use either the FV or the PV formula Both will give the same answer since they are the inverse of each other We will use the FV formula that is FV PV1 r Solving for t we get 2 lnFVPVln1 r The length of time to double your money is FV 2 1109 1 ln 2 In 109 804 years The length of time to quadruple your money is FV 4 1109 1 ln 4ln 109 1609years O p A Notice that the length of time to quadruple your money is twice as long as the time needed to double your money the difference in these answers is due to rounding This is an important concept of time value of money To nd the PV of a lump sum we use PV FV 1 r PV 750000000 108220 15506580854 To answer this question we can use either the FV or the PV formula Both will give the same answer since they are the inverse of each other We will use the FV formula that is FV PV1 r Solving for r we get rFVPV L1 r 10311500 12377500 717 446 Notice that the interest rate is negative This occurs when the FV is less than the PV A consol is a perpetuity To nd the PV of a perpetuity we use the equation PV C r PV 120 057 PV 210526 To nd the future value with continuous compounding we use the equation FV PVeRt a FV 190081 346203 13 FV 190081 256473 0 FV 190080510 313257 d FV 19008078 332628 To solve this problem we must nd the PV of each cash ow and add them To nd the PV of a lump sum we use PvFV1r PV10 1200 110 7301102 9651103 15901104 350523 PV18 1200 118 7301182 9651183 15901184 294866 PV24 1200 124 730 1242 965 1243 1590 1244 262117 12 p n DJ p n 4 To nd the PVA we use the equation PVA C1711 r r At a 5 percent interest rate X5 PVA 5500lt1711059 05 3909302 Y5 PVA 8000lt1711055 05 3463581 And at a 22 percent interest rate X22 PVA 5500lt1711229 22 2082457 Y22 PVA 80001 7 11225 22 2290912 Notice that the PV of Cash ow X has a greater PV at a 5 percent interest rate but a lower PV at a 22 percent interest rate The reason is that X has greater total cash ows At a lower interest rate the total cash ow is more important since the cost of waiting the interest rate is not as great At a higher interest rate Y is more valuable since it has larger cash ows At a higher interest rate these bigger cash ows early are more important since the cost of waiting the interest rate is so much greater To nd the PVA we use the equation PVA C1711 a r PVA15 yrs PVA 7 430017110915 09 7 3466096 PVA40 yrs PVA 7 4300171109 0 09 7 4625665 PVA75 yrs PVA 7 430017110975 09 7 4770326 To nd the PV of a perpetuity we use the equation PV C r PV 4300 09 PV 4777778 Notice that as the length of the annuity payments increases the present value of the annuity approaches the present value of the perpetuity The present value of the 75year annuity and the present value of the perpetuity imply that the value today of all perpetuity payments beyond 75 years is only 7451 This cash ow is a perpetuity To find the PV of a perpetuity we use the equation PV C r PV 20000 065 30769231 UI 5 To nd the interest rate that equates the perpetuity cash ows with the PV of the cash ows Using the PV of a perpetuity equation PV C r 340000 20000 r We can now solve for the interest rate as follows r 20000 340000 0588 or 588 For discrete compounding to nd the EAR we use the equation EAR 1 APRm39quot71 EAR 1 08 44 71 0824 or 824 EAR 1 18 1212 71 1956 or 1956 EAR 1 12 365365 71 1275 or 1275 To nd the EAR with continuous compounding we use the equation EAReq71 EAR e14 71 1503 or 1503 Here we are given the EAR and need to nd the APR Using the equation for discrete compounding EAR 1 APRm39quot71 We can now solve for the APR Doing so we get APR 7 m1 EAR1 quot 71 EAR 1030 1 7 APR 22 71 APR 21103012 71 1005 or 1005 EAR 0940 1 APR 1212 71 APR 1210940112 71 0902 or 902 EAR 0720 1 APR 5252 71 APR 5210720152 71 0696 or 696 Solving the continuous compounding EAR equation EAR eq 7 1 We get APR ln1 EAR APR ln1 1590 APR 1476 or 1476 17 p A on For discrete compounding to nd the EAR we use the equation EAR 1 APRm39quot71 So for each bank the EAR is First National EAR 1 1010 1212 7 1 1058 or 1058 First United EAR 1 1040 22 71 1067 or 1067 A higher APR does not necessarily mean the higher EAR The number of compounding periods within a year will also affect the EAR The cost of a case of wine is 10 percent less than the cost of 12 individual bottles so the cost of a case will be Cost of case 12101 7 10 Cost of case 108 Now we need to nd the interest rate The cash ows are an annuity due so PVA1 r C1711 a r 108 7 1 7 r 101711 r r Solving for the interest rate we get r 0198 or 198 per week So the APR of this investment is APR 7 019852 APR 7 10277 or 10277 And the EAR is EAR101985271 EAR 17668 or 17668 The analysis appears to be correct He really can earn about 177 percent buying wine by the case The only question left is this Can you really nd a ne bottle of Bordeaux for 10 Here we need to nd the length of an annuity We know the interest rate the PV and the payments Using the PVA equation PVA C1711 a r 18400 7 600 1711009 009 N O p A N Now we solve for t 11009z 1718400009600 10091 10724 1381 t In 1381 ln 1009 3605 months Here we are trying to nd the interest rate when we know the PV and FV Using the FV equation FV PV1 r 4 31 r r 43 71 3333 per week The interest rate is 3333 per week To nd the APR we multiply this rate by the number of weeks in a year so APR 523333 173333 And using the equation to find the EAR EAR 1APRmm71 EAR 1 333352 71 31391651569 Intermediate To find the FV of a lump sum with discrete compounding we use FV PV1 r2 a FV 10001087 171382 13 FV 10001082 173168 0 FV 100010812 174742 d To find the future value with continuous compounding we use the equation FV PVeRt FV 100080 175067 e The future value increases when the compounding period is shorter because interest is earned on previously accrued interest The shorter the compounding period the more frequently interest is earned and the greater the future value assuming the same stated interest rate The total interest paid by First Simple Bank is the interest rate per period times the number of periods In other words the interest by First Simple Bank paid over 10 years will be 0610 6 DJ 1quot UI First Complex Bank pays compound interest so the interest paid by this bank will be the FV factor of 1 or 1 r10 Setting the two equal we get 0610 1 r10 71 r 16 10 7 1 0481 or 481 We need to nd the annuity payment in retirement Our retirement savings ends at the same time the retirement withdrawals begin so the PV of the retirement withdrawals will be the FV of the retirement savings So we nd the FV of the stock account and the FV of the bond account and add the two FVs Stock account FVA 7001101236 711012 158234155 Bond account FVA 3001 061236071 0612 30135451 So the total amount saved at retirement is 158234155 30135451 188369606 Solving for the withdrawal amount in retirement using the PVA equation gives us PVA 188369606 C1 7 1 1 0812300 0812 C 188369606 1295645 1453867 withdrawal per month Since we are looking to quadruple our money the PV and FV are irrelevant as long as the FV is four times as large as the PV The number of periods is four the number of quarters per year So FV 4 11 no r 4142 or4142 Here we need to nd the interest rate for two possible investments Each investment is a lump sum S0 G PV 75000 135000 1 r6 1 r6 135000 75000 r 18016 7 1 1029 or 1029 H PV 75000 195000 1 r10 1 r10 195000 75000 r 260 10 7 1 1003 or 1003 26 N l N on N O This is a growing perpetuity The present value of a growing perpetuity is PV C r7 g PV 215000 10 7 04 PV 358333333 It is important to recognize that when dealing with annuities or perpetuities the present value equation calculates the present value one period before the first payment In this case since the rst payment is in two years we have calculated the present value one year from now To nd the value today we simply discount this value as a lump sum Doing so we nd the value of the cash ow stream today is PvFV1rt PV 358333333 1 101 PV 325757576 The dividend payments are made quarterly so we must use the quarterly interest rate The quarterly interest rate is Quarterly rate Stated rate 4 Quarterly rate 07 4 Quarterly rate 0175 Using the present value equation for a perpetuity we nd the value today of the dividends paid must be PV Cr PV 5 0175 PV 28571 We can use the PVA annuity equation to answer this question The annuity has 23 payments not 22 payments Since there is a payment made in Year 3 the annuity actually begins in Year 2 So the value of the annuity in Year 2 is PVA C1711r r PVA 500017110823 08 PVA 5185529 This is the value of the annuity one period before the first payment or Year 2 So the value of the cash ows today is PV FVl r PV 5185529 1 082 PV 4445756 We need to nd the present value of an annuity Using the PVA equation and the 15 percent interest rate we get PVA C1711 a r PVA 75017111515 15 PVA 438553 O 1 N This is the value of the annuity in Year 5 one period before the rst payment Finding the value of this amount today we nd PV FVl r2 PV 438553 1 125 PV 248847 The amount borrowed is the value of the home times one minus the down payment or Amount borrowed 450000l 7 20 Amount borrowed 360000 The monthly payments with a balloon payment loan are calculated assuming a longer amortization schedule in this case 30 years The payments based on a 30year repayment schedule would be PVA 360000 Cl 7 1 1 0751236 07512 C 251717 Now at time 8 we need to nd the PV of the payments which have not been made The balloon payment will be PVA 2517171 7 1 1 07512 lt 07512 PVA 32500173 Here we need to nd the FV of a lump sum with a changing interest rate We must do this problem in two parts After the rst six months the balance will be FV 6000 1 024126 607236 This is the balance in six months The FV in another six months will be FV 607236 1 18126 663978 The problem asks for the interest accrued so to nd the interest we subtract the beginning balance from the FV The interest accrued is Interest 663978 7 6000 63978 The company would be indifferent at the interest rate that makes the present value of the cash ows equal to the cost today Since the cash ows are a perpetuity we can use the PV of a perpetuity equation Doing so we find PV C r 150000 13000 r r 13000 150000 r 0867 or 867 33 03 UI The company will accept the project if the present value of the increased cash ows is greater than the cost The cash ows are a growing perpetuity so the present value is PV Clt1rg11rg1X1g1 r1 PV 7 180001117047111704 x 1041115 PV 7 7147947 The company should accept the project since the cost is less than the increased cash ows Since your salary grows at 4 percent per year your salary next year will be Next year s salary 60000 1 04 Next year s salary 62400 This means your deposit next year will be Next year s deposit 6240005 Next year s deposit 3120 Since your salary grows at 4 percent you deposit will also grow at 4 percent We can use the present value of a growing perpetuity equation to find the value of your deposits today Doing so we nd PV Clt1rg11rg1X1g1 r1 PV 31201097 04 7 109704 X 1 04109quot PV 5286198 Now we can find the future value of this lump sum in 40 years We find FV PV1 r FV 528619810940 FV 166036412 This is the value of your savings in 40 years The relationship between the PVA and the interest rate is PVA falls as r increases and PVA rises as r decreases FVA rises as r increases and FVA falls as r decreases The present values of 7500 per year for 12 years at the various interest rates given are PVA10 75001 7 111012 10 5110269 PVA5 750017110512 05 6647439 PVA15 75001 7 1115 15 7 4065464 36 DJ 0 Here we are given the FVA the interest rate and the amount of the annuity We need to solve for the number of payments Using the FVA equation FVA 30000 2501 1012 71 1012 Solving for t we get 1008332 1 300001012 250 t In 2 ln 100833 8352 payments Here we are given the PVA number of periods and the amount of the annuity We need to solve for the interest rate Using the PVA equation PVA 80000 16501 7 1 1 r6 r To nd the interest rate we need to solve this equation on a financial calculator using a spreadsheet or by trial and error If you use trial and error remember that increasing the interest rate lowers the PVA and decreasing the interest rate increases the PVA Using a spreadsheet we nd r 0727 The APR is the periodic interest rate times the number of periods in the year so APR l20727 872 The amount of principal paid on the loan is the PV of the monthly payments you make So the present value of the 1200 monthly payments is PVA 12001 7 1 1 0681236 06812 18407020 The monthly payments of 1200 will amount to a principal payment of 18407020 The amount of principal you will still owe is 250000 7 18407020 6592980 This remaining principal amount will increase at the interest rate on the loan until the end of the loan period So the balloon payment in 30 years which is the FV of the remaining principal will be Balloon payment 65929801 06812360 50412905 We are given the total PV of all four cash ows If we nd the PV of the three cash ows we know and subtract them from the total PV the amount left over must be the PV of the missing cash ow So the PV of the cash ows we know are PV onear 1 CF 1200 110 109091 PV of Year 3 CF 24001103 180316 PV of Year 4 CF 2600 1104 177583 J O J N So the PV ofthe missing CF is 6453 71090917180316 7177583 178310 The question asks for the value of the cash ow in Year 2 so we must nd the future value of this amount The value of the missing CF is 1783101102 7 215755 To solve this problem we simply need to nd the PV of each lump sum and add them together It is important to note that the rst cash ow of 1 million occurs today so we do not need to discount that cash ow The PV of the lottery winnings is 1000000 1350000109 17000001092 20500001093 2400000109 27500001095 31000001096 34500001097 38000001098 41500001099 450000010910 7 1819430869 Here we are nding interest rate for an annuity cash ow We are given the PVA number of periods and the amount of the annuity We need to solve for the number of payments We should also note that the PV of the annuity is not the amount borrowed since we are making a down payment on the warehouse The amount borrowed is Amount borrowed 0802600000 2080000 Using the PVA equation PVA 2080000 140001 7 1 1 r36 r Unfortunately this equation cannot be solved to nd the interest rate using algebra To nd the interest rate we need to solve this equation on a nancial calculator using a spreadsheet or by trial and error If you use trial and error remember that increasing the interest rate decreases the PVA and decreasing the interest rate increases the PVA Using a spreadsheet we nd r 0593 The APR is the monthly interest rate times the number of months in the year so APR 120593 712 And the EAR is EAR 7 1 0059312 7 1 7 0735 or 735 The profit the firm earns is just the PV of the sales price minus the cost to produce the asset We nd the PV of the sales price as the PV of a lump sum PV 7 1350001133 7 9356177 DJ 1quot UI And the film s profit is Pro t 9356177 7 9600000 7243823 To nd the interest rate at which the rm will break even we need to nd the interest rate using the PV or FV of a lump sum Using the PV equation for a lump sum we get 96000 7 135000 1 r3 r 7 135000 96000 7 1 7 1204 or 1204 We want to nd the value of the cash ows today so we will find the PV of the annuity and then bring the lump sum PV back to today The annuity has 17 payments so the PV of the annuity is PVA 400017110717 07 3905289 Since this is an ordinary annuity equation this is the PV one period before the first payment so it is the PV at t 8 To nd the value today we nd the PV of this lump sum The value today is PV 3905289 1078 2272914 This question is asking for the present value of an annuity but the interest rate changes during the life of the annuity We need to nd the present value of the cash ows for the last eight years first The PV of these cash ows is PVAZ 1500 17 1 1 0912960912 10238766 Note that this is the PV of this annuity exactly seven years from today Now we can discount this lump sum to today The value of this cash ow today is PV 7 10238766 1 7 1312 quot 7 4141570 Now we need to nd the PV of the annuity for the first seven years The value of these cash ows today is PVAI 1500 17 1 1 1312841312 8245399 The value of the cash ows today is the sum of these two cash ows so PV 8245399 4141570 12386999 Here we are trying to find the dollar amount invested today that will equal the FVA with a known interest rate and payments First we need to determine how much we would have in the annuity account Finding the FV of the annuity we get FVA 7 1200 1 09812180 7109812 48832861 Now we need to nd the PV of a lump sum that will give us the same FV So using the FV of a lump sum with continuous compounding we get Fv 7 48832861 7 New PV 7 48832861 7 12659444 64 46 J l J on To nd the value of the perpetuity at t 7 we first need to use the PV of a perpetuity equation Using this equation we nd PV 2100 073 2876712 Remember that the PV of a perpetuity and annuity equations give the PV one period before the rst payment so this is the value of the perpetuity at t 14 To nd the value at t 7 we nd the PV of this lump sum as PV 7 2876712 10737 7 1756703 To nd the APR and EAR we need to use the actual cash ows of the loan In other words the interest rate quoted in the problem is only relevant to determine the total interest under the terms given The interest rate for the cash ows of the loan is PVA 26000 7 2491671 7 1 1 r12 r Again we cannot solve this equation for r so we need to solve this equation on a nancial calculator using a spreadsheet or by trial and error Using a spreadsheet we find r 2219 per month So the APR is APR 122219 2662 And the EAR is EAR 10221912 71 3012 The cash ows in this problem are semiannual so we need the effective semiannual rate The interest rate given is the APR so the monthly interest rate is Monthly rate 12 12 01 To get the semiannual interest rate we can use the EAR equation but instead of using 12 months as the exponent we will use 6 months The effective semiannual rate is Semiannual rate 1016 71 615 We can now use this rate to nd the PV of the annuity The PV of the annuity is PVA t 9 45001 7 1 106151 0615 3288316 Note that this is the value one period six months before the first payment so it is the value at t 9 So the value at the various times the questions asked for uses this value 9 years from now PV t 5 3288316106158 2039612 Note that you can also calculate this present value as well as the remaining present values using the number of years To do this you need the EAR The EAR is EAR101 71 1268 So we can find the PV at t 5 using the following method as well PV t 5 3288316112684 2039612 The value of the annuity at the other times in the problem is PV t 3 32883161061512 1606329 PV t 3 3288316112686 1606329 PV t 0 32883161061518 1122704 PV t 0 3288316112689 1122704 11 If the payments are in the form of an ordinary annuity the present value will be PVA C1711 r r PVA 100001711 11511 PVA 3695897 If the payments are an annuity due the present value will be PVAdue 1 r PVA PVAdue 1 113695897 PVAdue 4102446 We can nd the future value of the ordinary annuity as FVAC1r 7 lr FVA 100001 115 7111 FVA 6227801 If the payments are an annuity due the future value will be FVAdue 1 r FVA FVAdue 1 116227801 FVAdue 6912860 Assuming a positive interest rate the present value of an annuity due will always be larger than the present value of an ordinary annuity Each cash ow in an annuity due is received one period earlier which means there is one period less to discount each cash ow Assuming a positive interest rate the future value of an ordinary due will always higher than the future value of an ordinary annuity Since each cash ow is made one period sooner each cash ow receives one extra period of compounding 50 UI p A UI N We need to use the PVA due equation that is PVAd e 1 r PVA Using this equation PVAdue 65000 1 064512 x C17 1 1 064512 quot 064512 C 153174 Notice to find the payment for the PVA due we simply compound the payment for an ordinary annuity forward one period Challenge The monthly interest rate is the annual interest rate divided by 12 or Monthly interest rate 104 12 Monthly interest rate 00867 Now we can set the present value of the lease payments equal to the cost of the equipment or 3500 The lease payments are in the form of an annuity due so PVAdue 1 r C1711 a r 3500 100867C171100867 00867 C 16076 First we will calculate the present value of the college expenses for each child The expenses are an annuity so the present value of the college expenses is PVAC1711 rt r PVA 350001711 085 085 PVA 11464588 This is the cost of each child s college expenses one year before they enter college So the cost of the oldest child s college expenses today will be PV FV1 r PV 11464588108514 PV 3658829 And the cost of the youngest child s college expenses today will be PV FV1 r PV 11464588108516 PV 3108012 Therefore the total cost today of your children s college expenses is Cost today 3658829 3108012 Cost today 6766841 DJ 1quot This is the present value of your annual savings which are an annuity So the amount you must save each year will be PVAC1711 rt r 6766841 C171108515 085 C 814866 The salary is a growing annuity so using the equation for the present value of a growing annuity The salary growth rate is 35 percent and the discount rate is 12 percent so the value of the salary offer today is PV Clt1reg1rglX1g1 r1 PV 450001 12 7 035 7 1 12 7035 x 1 035112 PV 45581618 The yearly bonuses are 10 percent of the annual salary This means that next year s bonus will be Next year s bonus 1045000 Next year s bonus 4500 Since the salary grows at 35 percent the bonus will grow at 35 percent as well Using the growing annuity equation with a 35 percent growth rate and a 12 percent discount rate the present value of the annual bonuses is PV Clt1reg1reglX1g1 r1 PV 4500112 7 035 7 1 12 7035 x 1035112 PV 4558162 Notice the present value of the bonus is 10 percent of the present value of the salary The present value of the bonus will always be the same percentage of the present value of the salary as the bonus percentage So the total value of the offer is PV PVSalary PVBonus Bonus paid today PV 45581618 4558162 10000 PV 51139780 Here we need to compare to options In order to do so we must get the value of the two cash ow streams to the same time so we will nd the value of each today We must also make sure to use the aftertax cash ows since it is more relevant For Option A the aftertax cash ows are A ertax cash ows Pretax cash ows1 7 tax rate A ertax cash ows 175000l 7 28 A ertax cash ows 126000 The aftertax cash ows from Option A are in the form of an annuity due so the present value of the cash ow today is PVAde 1 r C1711 r2 r PVAde 11012600017111031 10 PVAde 131379122 UI F For Option B the aftertax cash ows are A ertax cash ows PretaX cash owsl 7 tax rate A ertaX cash ows l25000l 7 28 A ertaX cash ows 90000 The aftertax cash ows from Option B are an ordinary annuity plus the cash ow today so the present value Pv C1711 a rCF0 PV 7 900001 7 11 1030 10 530000 Pv7 137842230 You should choose Option B because it has a higher present value on an aftertax basis We need to nd the first payment into the retirement account The present value of the desired amount at retirement is PV 7 FVl r2 PV 7 15000001 1030 PV 7 8596283 This is the value today Since the savings are in the form of a growing annuity we can use the growing annuity equation and solve for the payment Doing so we get PV Clt1reg1reglX1g1 r 8596283 7 C110 7 03 7 1 10 703 x 1031103 C7 698968 This is the amount you need to save next year So the percentage of your salary is Percentage of salary 698968 70000 Percentage of salary 0999 or 999 Note that this is the percentage of your salary you must save each year Since your salary is increasing at 3 percent and the savings are increasing at 3 percent the percentage of salary will remain constant Since she put l000 down the amount borrowed will be Amount borrowed 25000 7 1000 Amount borrowed 24000 So the monthly payments will be PVA C1711 r r 24000 7 C1711 084126 084 12 C 7 49124 l The amount remaining on the loan is the present value of the remaining payments Since the first payment was made on October 1 2007 and she made a payment on October 1 2009 there are 35 payments remaining with the first payment due immediately So we can nd the present value of the remaining 34 payments after November 1 2009 and add the payment made on this date So the remaining principal owed on the loan is PVC1711r rC0 PV 7 491241711084123quot 08412 C 7 1481747 She must also pay a one percent prepayment penalty and the payment due on November 1 2009 so the total amount of the payment is Total payment Balloon amount1 Prepayment penalty Current payment Total payment 14817471 01 49124 Total payment 1545689 The cash ows for this problem occur monthly and the interest rate given is the EAR Since the cash ows occur monthly we must get the effective monthly rate One way to do this is to nd the APR based on monthly compounding and then divide by 12 So the preretirement APR is EAR 111APR1212 71 APR 12111112 71 1048 And the postretirement APR is EAR 08 1 APR 1212 71 APR 12108112 71 772 First we will calculate how much he needs at retirement The amount needed at retirement is the PV of the monthly spending plus the PV of the inheritance The PV of these two cash ows is PVA 200001 7 1 1 077212122 077212 244155461 PV 1000000 1 0820 21454821 So at retirement he needs 244155461 21454821 265610281 He will be saving 1900 per month for the next 10 years until he purchases the cabin The value of his savings after 10 years will be FVA 1900 1 104812121 7 1 104812 40012162 After he purchases the cabin the amount he will have left is 40012162 7 320000 8012162 He still has 20 years until retirement When he is ready to retire this amount will have grown to FV 7 80121621 104812 2 7 64696550 on 0 So when he is ready to retire based on his current savings he will be short 2656102817 64596550 201013731 This amount is the EV of the monthly savings he must make between years 10 and 30 So nding the annuity payment using the EVA equation we nd his monthly savings will need to be FVA 7 201013731 7 C 1 104812 2 7 1 104812 C 7 248612 To answer this question we should find the PV of both options and compare them Since we are purchasing the car the lowest PV is the best option The PV of the leasing is simply the PV of the lease payments plus the 1 The interest rate we would use for the leasing option is the same as the interest rate of the loan The PV of leasing is PV 1 5201 7 1 1 0812 3 0812 1659514 The PV of purchasing the car is the current price of the car minus the PV of the resale price The PV of the resale price is PV 7 26000 1 0812 7 2046862 The PV of the decision to purchase is 38000 7 2046862 1753138 In this case it is cheaper to lease the car than buy it since the PV of the leasing cash ows is lower To nd the breakeven resale price we need to nd the resale price that makes the PV of the two options the same In other words the PV of the decision to buy should be 38000 7 PV of resale price 1659514 PV of resale price 2140486 The resale price that would make the PV of the lease versus buy decision is the FV of this value so Breakeven resale price 7 21404861 0812 7 2718925 To nd the quarterly salary for the player we rst need to find the PV of the current contract The cash ows for the contract are annual and we are given a daily interest rate We need to nd the EAR so the interest compounding is the same as the timing of the cash ows The EAR is EAR 1 05365365 71 513 The PV of the current contract offer is the sum of the PV of the cash ows So the PV is PV 7500000 420000010513 5100000105132 5900000105133 680000010513 7400000105135 8100000105136 PV 7 3851952966 O p A The player wants the contract increased in value by 1000000 so the PV of the new contract will be PV 7 3851952966 750000 7 3926952966 The player has also requested a signing bonus payable today in the amount of 10 million We can simply subtract this amount from the PV of the new contract The remaining amount will be the PV of the future quarterly paychecks 3926952966 7 10000000 2926952966 To nd the quarterly payments rst realize that the interest rate we need is the effective quarterly rate Using the daily interest rate we can nd the quarterly interest rate using the EAR equation with the number of days being 9125 the number of days in a quarter 365 4 The effective quarterly rate is Effective quarterly rate 1 0536591 25 7 1 01258 or 1258 Now we have the interest rate the length of the annuity and the PV Using the PVA equation and solving for the payment we get PVA 7 2926952966 7 C171101258 01258 C 7 142047643 To nd the APR and EAR we need to use the actual cash ows of the loan In other words the interest rate quoted in the problem is only relevant to determine the total interest under the terms given The cash ows of the loan are the 20000 you must repay in one year and the 17200 you borrow today The interest rate of the loan is 20000 7 172001 r r 7 20000 17200 7 1 7 1628 or 1628 Because of the discount you only get the use of 17200 and the interest you pay on that amount is 1628 not 14 Here we have cash ows that would have occurred in the past and cash ows that would occur in the future We need to bring both cash ows to today Before we calculate the value of the cash ows today we must adjust the interest rate so we have the effective monthly interest rate Finding the APR with monthly compounding and dividing by 12 will give us the effective monthly rate The APR with monthly compounding is APR 12109112 7 1 865 To nd the value today of the back pay from two years ago we will nd the FV of the annuity salary and then nd the FV of the lump sum value of the salary Doing so gives us FV 4200012 1 08651212 7 1 086512 1 09 4763905 N Notice we found the FV of the annuity with the effective monthly rate and then found the FV of the lump sum with the EAR Alternatively we could have found the FV of the lump sum with the effective monthly rate as long as we used 12 periods The answer would be the same either way Now we need to nd the value today of last year s back pay FVA 4500012 1 08651212 7 1 086512 4682737 Next we nd the value today of the five year s future salary PVA 49000121 7 1 1 086512 5 086512 19833255 The value today of the jury award is the sum of salaries plus the compensation for pain and suffering and court costs The award should be for the amount of Award 4763905 4682737 19833255 150000 25000 Award 46779897 As the plaintiff you would prefer a lower interest rate In this problem we are calculating both the PV and FV of annuities A lower interest rate will decrease the FVA but increase the PVA So by a lower interest rate we are lowering the value of the back pay But we are also increasing the PV of the future salary Since the future salary is larger and has a longer time this is the more important cash ow to the plaintiff Again to nd the interest rate of a loan we need to look at the cash ows of the loan Since this loan is m the form of a lump sum the amount you will repay is the FV of the principal amount which will be Loan repayment amount 10000109 10900 The amount you will receive today is the principal amount of the loan times one minus the points Amount received 100001 7 03 9700 Now we simply nd the interest rate for this PV and FV 10900 7 97001 r r 10900 9700 7 1 7 1237 or 1237 With a 12 percent quoted interest rate loan and two points the EAR is Loan repayment amount 100001 12 11200 Amount received 100001 7 02 9800 11200 7 98001 r r 11200 9800 7 1 7 1429 or 1429 The effective rate is not affected by the loan amount since it drops out when solving for r 63 5 4 First we will nd the APR and EAR for the loan with the refundable fee Remember we need to use the actual cash ows of the loan to nd the interest rate With the 2100 application fee you will need to borrow 202100 to have 200000 after deducting the fee Solving for the payment under these circumstances we get PVA 7 202100 7 C17110056736 00567 where 00567 7 06812 C 7 131754 We can now use this amount in the PVA equation with the original amount we wished to borrow 200000 Solving for r we nd PVA 200000 1317541 7 1 1 r36 r Solving for r with a spreadsheet on a nancial calculator or by trial and error gives r 05752 per month APR 1205752 690 EAR 1 00575212 71 0713 or 713 With the nonrefundable fee the APR of the loan is simply the quoted APR since the fee is not considered part of the loan So APR 680 EAR 7 1 0681212 71 7 0702 or 702 Be careful of interest rate quotations The actual interest rate of a loan is determined by the cash ows Here we are told that the PV of the loan is 1000 and the payments are 4336 per month for three years so the interest rate on the loan is PVA 1000 4336 1 7 1 1 r36 r Solving for r with a spreadsheet on a nancial calculator or by trial and error gives r 264 per month APR 12264 3165 EAR 1 026412 7 1 3667 or 3667 It s called addon interest because the interest amount of the loan is added to the principal amount of the loan before the loan payments are calculated 65 Here we are solving a twostep time value of money problem Each question asks for a different possible cash ow to fund the same retirement plan Each savings possibility has the same FV that is the PV of the retirement spending when your friend is ready to retire The amount needed when your friend is ready to retire is PVA 11000017110925 09 108048376 This amount is the same for all three parts of this question a If your friend makes equal annual deposits into the account this is an annuity with the FVA equal to the amount needed in retirement The required savings each year will be FVA 108048376 7 C10930 7 1 09 C 7 792681 b Here we need to find a lump sum savings amount Using the FV for a lump sum equation we get FV 7 108048376 7 PV1093 PV 7 8143729 5 1 In this problem we have a lump sum savings in addition to an annual deposit Since we already know the value needed at retirement we can subtract the value of the lump sum savings at retirement to nd out how much your friend is short Doing so gives us Fv oftrust fund deposit 5000010910 7 11836818 So the amount your friend still needs at retirement is FV 108048376 711836818 96211558 Using the FVA equation and solving for the payment we get 96211558 C10930 71 09 C 705842 This is the total annual contribution but your friend s employer will contribute 1500 per year so your friend must contribute Friend39s contribution 705842 7 1500 555842 66 We will calculate the number of periods necessary to repay the balance with no fee rst We simply need to use the PVA equation and solve for the number of payments Without fee and annual rate 186 PVA 9000 20017110155 0155 where 0155 18612 Solving for t we get I ln1l790002000155ln10155 t In 33058 ln 10155 1 7774 months Without fee and annual rate 82 PVA 9000 2001711006833 006833 where 006833 08212 Solving for t we get tln1 179000200006833 ln1006833 t In 14440 ln 1006833 1 5396 months Note that we do not need to calculate the time necessary to repay your current credit card with a fee since no fee will be incurred The time to repay the new card with a transfer fee is With fee and annual rate 820 PVA 9180 200 1711006833 006833 where 006833 09212 Solving for t we get tln1 179180200006833 ln1006833 t In 145698 ln 1006833 1 5527 months 67 We need to nd the FV of the premiums to compare with the cash payment promised at age 65 We have to find the value of the premiums at year 6 first since the interest rate changes at that time So 13v1 7 8001115 7 134805 M 7 800111 7 121446 FV3 7 9001113 7 123087 13v 7 9001112 7 110889 FV5 10001111 111000 on Value at year six 134805 121446 123087 110889 111000 100000 701226 Finding the FV of this lump sum at the child s 65th birthday FV 70122610759 37975276 The policy is not worth buying the future value of the policy is 37975276 but the policy contract will pay off 350000 The premiums are worth 2975276 more than the policy payoff Note we could also compare the PV of the two cash ows The PV of the premiums is PV 800111 8001112 9001113 900111quot 10001115 10001116 374904 And the value today of the 350000 at age 65 is PV 35000010759 646287 PV 6462871116 345531 The premiums still have the higher cash ow At time zero the difference is 29373 Whenever you are comparing two or more cash ow streams the cash ow with the highest value at one time will have the highest value at any other time Here is a question for you Suppose you invest 29373 the difference in the cash ows at time zero for six years at an 11 percent interest rate and then for 59 years at a seven percent interest rate How much will it be worth Without doing calculations you know it will be worth 2975276 the difference in the cash ows at time 65 Since the payments occur at six month intervals we need to get the effective sixmonth interest rate We can calculate the daily interest rate since we have an APR compounded daily so the effective sixmonth interest rate is Effective sixmonth rate 1 Daily rate180 7 1 Effective sixmonth rate 1 09360180 7 1 Effective sixmonth rate 0460 or 460 Now we can use the PVA equation to find the present value of the semiannual payments Doing so we nd PVA C1711 rt r PVA 7500001711 0460 0 0460 PVA 1360215232 This is the value six months from today which is one period six months prior to the first payment So the value today is PV 1360215232 1 0460 PV 1300369650 O O This means the total value of the lottery winnings today is Value ofwinnings today 1300369650 2000000 Value ofwinnings today 1500369650 You should not take the offer since the value of the offer is less than the present value of the payments Here we need to nd the interest rate that makes the PVA the college costs equal to the FVA the savings The PV of the college costs are PVA 200001 7 1 1 r4 rl And the FV of the savings is FVA 80001 r6 7 1 r Setting these two equations equal to each other we get 200001 7 1 1 r4 r 8000 1 r6 7 1 r Reducing the equation gives us 1 r10 7 4001 r4 4000 0 Now we need to nd the roots of this equation We can solve using trial and error a rootsolving calculator routine or a spreadsheet Using a spreadsheet we find r 1057 Here we need to nd the interest rate that makes us indifferent between an annuity and a perpetuity To solve this problem we need to nd the PV of the two options and set them equal to each other The PV of the perpetuity is PV 20000 r And the PV of the annuity is PVA 350001 7 1 1 r10 r Setting them equal and solving for r we get 20000 r 350001 7 1 1 r10 r 20000 35000 17 1 1 r10 5714 10 11 r r 715714 10 71 r 7 0576 or 576 71 The cash ows in this problem occur every two years so we need to nd the effective two year rate One way to find the effective two year rate is to use an equation similar to the EAR except use the number of days in two years as the exponent We use the number of days in two years since it is daily compounding if monthly compounding was assumed we would use the number of months in two years So the effective twoyear interest rate is Effective 2year rate 1 133653652 e 1 2969 We can use this interest rate to nd the PV of the perpetuity Doing so we nd PV 8500 2969 2863206 N 1 03 This is an important point Remember that the PV equation for a perpetuity and an ordinary annuity tells you the PV one period before the rst cash ow In this problem since the cash ows are two years apart we have found the value of the perpetuity one period two years before the rst payment which is one year ago We need to compound this value for one year to find the value today The value of the cash ows today is PV 28632061 13365 3260624 The second part of the question assumes the perpetuity cash ows begin in four years In this case when we use the PV of a perpetuity equation we find the value of the perpetuity two years from today So the value of these cash ows today is PV 2863206 1 133652365 2207781 To solveforthePVAdue C C 1r 1 rZ lr PVAd e CLWL 1r 1rquot1 C C C PVAd e 1 r 1r 1 r2 1r PVAd e 1 r PVA And the FVA due is FVACC1 rC1 r2 C1rH FVAdue C1 r C1 r2 C1r FVAde 1 rC C1 r C1 0H FVAdue 1 rFVA a The APR is the interest rate per week times 52 weeks in a year so APR 529 468 EAR 1 0952 1 873442 or 873442 0 In a discount loan the amount you receive is lowered by the discount and you repay the full principal With a 9 percent discount you would receive 910 for every 10 in principal so the weekly interest rate would be 10 9101 r r 10 910 1 0989 or 989 Note the dollar amount we use is irrelevant In other words we could use 091 and 1 91 and 100 or any other combination and we would get the same interest rate Now we can find the APR and the EAR APR 529 89 51429 EAR 1 098952 71 1338490 or 1338490 0 Using the cash ows from the loan we have the PVA and the annuity payments and need to nd the interest rate so PVA 5884 251 7 1 1 rquot r Using a spreadsheet trial and error or a nancial calculator we nd r 2519 per week APR 5225 19 130992 EAR 1251952 711185150194 or 1185150194 74 To answer this we can diagram the perpetuity cash ows which are Note the subscripts are only to differentiate when the cash ows begin The cash ows are all the same amount C3 C2 C2 C1 C1 C1 Thus each of the increased cash ows is a perpetuity in itself So we can write the cash ows stream as ClR CzR C3R C4R So we can write the cash ows as the present value of a perpetuity with a perpetuity payment of CzR C3R C4R I I I I The present value of this perpetuity is PV 7 CR R 7 CR2 5 quot F So the present value equation of a perpetuity that increases by C each period is PV CR CR2 Since it is only an approximation we know the Rule of 72 is exact for only one interest rate Using the basic future value equation for an amount that doubles in value and solving for t we nd FV PV1 Rt 2 11Rt 1112 11111R 111121111R We also know the Rule of 72 approximation is t 72 R We can set these two equations equal to each other and solve for R We also need to remember that the exact future value equation uses decimals so the equation becomes 72R11121111R 0 72R11121111R It is not possible to solve this equation directly for R but using Solver we nd the interest rate for which the Rule of 72 is exact is 7846894 percent We are only concerned with the time it takes money to double so the dollar amounts are irrelevant So we can write the future value of a lump sum with continuously compounded interest as 2 18M 2 em Rt 1112 Rt 693147 I 693147 R Since we are using percentage interest rates while the equation uses decimal form to make the equation correct with percentages we can multiply by 100 t 693147R Calculator Solutions 9 6 9 6 7 15 11 18 1 Enter 10 Solve for 1183682 7 9500 233682 2 Enter 10 Solve for Enter 10 Solve for Enter 20 Solve for 3 Enter 6 Solve for Enter 9 Solve for Enter 18 Solve for Enter 23 Solve for 4 Enter 2 Solve for 1263 5000 1000 1000 1000 1029565 1465572 13541160 1222379 242 1183682 H 4 179085 F 236736 F 320714 15451 H 51557 H 886073 FV39 550164 i307 Enter Solve for Enter Solve for Enter Solve for 5 Enter Solve for Enter Solve for Enter Solve for Enter Solve for 6 Enter Solve for Enter Solve for 7 Enter Solve for 9 15 30 N 1236 804 N 1609 410 907 51700 W 792 18750 1144 6 625 13 810 Pquot 32 18400 W 16 21500 9 1 9 1 82 IY 15506580854 i896 i162181 4153483500 31284 i4341 i402662 173439 2 4 750000000 8 Enter 4 i12377500 10311500 Solve for 4146 1 1 0 1200 1 730 1 965 1 1590 1 NPV CPT NPV CPT NPV CPT 350523 294866 262117 12 Enter 9 5 5500 V Solve for 3909302 Enter 5 5 8000 Solve for 3463581 Enter 9 22 5500 Solve for 2082457 Enter 5 22 8000 Solve for 2290912 13 Enter 15 9 4300 Solve for 3466096 Enter 40 9 4300 Solve for 4625665 Enter 75 9 Solve for 15 Enter 8 NOM Solve for 824 Enter 18 NOM Solve for 1956 Enter 12 NOM Solve for 1275 16 Enter 103 NOM Solve for 1005 Enter 94 NOM Solve for 902 Enter 72 NOM Solve for 696 17 Enter 101 NOM Solve for 1058 Enter 104 NOM Solve for 1067 18 2quot 1 BGN 2quot 1 SET Enter 12 Solve for 198 APR 198 X 52 10277 4770326 4 12 365 CY 12 52 CY 108 4300 810 Enter 10277 52 Solve for 17668 19 Enter 09 18400 i600 Solve for 3605 20 Enter 173333 52 NOM Solve for 31391651569 21 Enter 7 8 1000 PMT Solve for 171382 Enter 7 X 2 82 1000 N PV PMT Solve for 173168 Enter 7 X 12 80012 1000 PMT Solve for 174742 23 Stock account Enter 360 10 12 700 N PMT Solve for 158234155 Bond account Enter 360 6 12 3 00 PMT Solve for 30135451 Savings at retirement 158234155 30135451 188369606 Enter 300 8 12 188369606 39 PMT Solve for 1453867 24 Enter Solve for 2 5 Enter Solve for Enter Solve for 2 8 Enter Solve for Enter Solve for 29 Enter Solve for Enter Solve for 3 0 Enter Solve for Enter Solve for 3 1 Enter Solve for 6 10 23 2 15 5 360 22x12 6 4142 1029 1003 8 8 15 12 7512 7512 240 12 i1 75000 175000 PV 5000 5185529 4445756 750 438553 248847 80450000 251717 251717 325 0173 6000 4 135000 195000 U j J n 1 g 1 10s 1 1 4 lt11 4 4m 4 4 LII LII 39 N Ln 0 H 4 H 4 607236 Enter 6 18 12 607236 Solve for 663978 7 6000 63978 35 Enter 12 10 Solve for 5110269 Enter 12 5 Solve for 6647439 Enter 12 15 Solve for 4065464 36 Enter 10 12 Solve for 8352 37 Enter 60 80000 Solve for 0727 0727 x 12 872 38 Enter 360 68 12 N 39 Solve for 18407020 250000 7 18407020 6592980 Enter 360 68 12 65929 80 N V V Solve for 89 1200 663978 F I I F F I 30000 F I I F F 50412905 I 10 NPV CPT 466990 PV ofmissing CF 6453 7 466990 178310 Value ofmissing CF Enter 2 10 Solve for 40 1000000 1350000 1 1700000 1 2050000 1 2400000 1 2750000 1 3100000 1 3450000 1 3800000 1 4150000 1 4500000 I 9 NPV CPT 1819430869 178310 215755 41 Enter 360 802 600000 i14000 Solve for 0593 APR 0593 x 12 712 Enter 712 12 NOM Solve for 735 42 Enter 3 13 135000 PMT Solve for 9356177 Pro t 9356177 7 96000 7243823 Enter 3 596000 135000 PMT Solve for 1204 43 Enter 17 7 4000 PMT Solve for 3905289 Enter 8 7 3905289 PMT Solve for 2272914 44 Enter 96 9 12 1500 N IY PMT Solve for 102 8766 Enter 84 13 12 1500 10238766 N IY PMT Solve for 123 6969 45 Enter 15 X 12 9812 1200 N I 39 PMT Solve for 48832861 FV 48832861 PV e090 PV 48832861 8 1 35 12659444 46 PV t 14 2100 0073 2876712 Enter 7 73 2876712 PMT Solve for 1756703 47 Enter 12 26000 i249167 PMT Solve for 2219 APR 2219 x 12 2662 Enter 2662 12 NOM Solve for 3012 48 Monthly rate 12 12 01 semiannual rate 1016 71 615 Enter 10 615 4500 V PMT Solve for 3288316 Enter 8 615 3288316 V PMT Solve for 2039612 Enter 12 615 3288316 V PMT Solve for 1606329 Enter 18 615 3288316 V PMT Solve for 1122704 49 a Enter 5 11 Solve for 3 695897 2quot 1 BGN 2quot 1 SET Enter 5 110 Solve for 4102446 b Enter 5 11 N Solve for 2quot 1 BGN 2quot 1 SET Enter 5 11 IY Solve for 50 2nd BGN 2nd SET Enter 48 645 12 65000 IY Solve for 51 2quot 1 BGN 2quot 1 SET Enter 2 X 12 104 12 3500 N 39 PV Solve for 52 PV of college expenses Enter 4 85 Solve for 11464588 Cost today of oldest child s expenses Enter 14 85 1 Solve for 3658829 10000 10000 10000 PMT 10000 PMT 153174 16076 35000 6227801 6912860 11464588 Cost today of youngest child s expenses 1 814866 Enter 16 85 V Solve for 3108012 Total cost today 3658829 3108012 6766841 Enter 15 85 6766841 V Solve for 54 Option A A eItaX cash ows PretaX cash ows1 7 tax rate A eItaX cash ows 1750001 7 28 A eItaX cash ows 126000 2 BGN 2quot 1 SET Enter 31 10 Solve for 131379122 Option B A eItaX cash ows PretaX cash ows1 7 tax rate A eItaX cash ows 1250001 728 A eItaX cash ows 90000 2ND BGN 2quot 1 SET Enter 30 10 IY Solve for 84842230 84842230 530000 137842230 56 Enter 5 X 12 84 12 24000 N PV Solve for Enter 35 84 12 IY Solve for 1481747 94 126000 PMT 90000 PMT 49124 49124 11464588 Total payment Amount due1 Prepayment penalty Last payment Total payment 14817471 01 49124 Total payment 1545689 57 Preretirement APR Enter 1 1 12 NOM Solve for 1048 Postretirement APR Enter 8 12 NOM Solve for 772 At retirement he needs Enter 240 772 12 20000 1000000 39 PMT Solve for 265610281 In 10 years his savings will be worth Enter 120 1048 12 1900 39 PMT Solve for 40012162 After purchasing the cabin he will have 40012162 7 320000 8012162 Each month between years 10 and 30 he needs to save Enter 240 1048 12 8012162i 265610281 N 39 Solve for 72486 12 58 PV of purchase Enter 6 8 12 26000 Solve for 2046862 3 8000 7 2046862 1753138 PV of lease Enter 36 8 12 520 Solve for 1659414 1659414 1 1659514 Lease the car You would be indifferent when the PV of the two cash flows are equal The present value of the purchase decision must be 1659414 Since the difference in the two cash ows is 38000 7 1659415 2140486 this must be the present value of the future resale price of the car The breakeven resale price of the car is Enter 36 8 12 2140486 PMT Solve for 2718925 59 Enter 5 365 NOM Solve for 513 7500000 4200000 1 5100000 1 5900000 1 6800000 1 7400000 1 8100000 1 I 513 NPV CPT 3851952966 New contract value 3851952966 750000 3926952966 PV of payments 3926952966 7 10000000 2926952966 Effective quarterly rate 1 0536591 25 7 1 125 8 Enter 24 1258 2926952966 PV PMT Solve for 142047643 60 Enter 1 17200 r20000 PMT Solve for 1628 6 1 Enter Solve for Enter Solve for Enter Solve for Enter Solve for Enter Solve for 865 12 9 865 12 IY 9 865 12 IY 865 12 12 4370555 PV PV 19833255 42000 12 PMT 4370555 PMT 4763905 45000 12 PMT 4682737 49000 12 PMT Award 4763905 4663905 19833255 150000 25000 46779897 62 Enter Solve for Enter Solve for 1 1 IY 1237 IY 1429 9700 9800 10900 11200 63 Refundable fee With the 2100 application fee you will need to borrow 202100 to have 200000 after deducting the fee Solve for the payment under these circumstances Enter 30 x 12 680 12 202100 PMT Solve for 131754 Enter 30 x 12 200000 i131754 PMT Solve for 05752 APR 05752 x 12 690 Enter 690 12 NOM CY Solve for 713 Without refundable fee APR 680 Enter 680 12 NOM Solve for 702 64 Enter 36 1000 i4336 PMT Solve for 264 APR 264 x 12 3165 Enter 3165 12 NOM CY Solve for 3667 65 What she needs at age 65 Enter 25 9 110000 PMT Solve for 108048376 a Enter 30 9 108048376 PMT Solve for 7926 81 b Enter 30 9 108048376 PMT Solve for 8143729 0 Enter 10 9 50000 PMT Solve for 11836818 At 65 she is short 108048376 7 11836818 96211558 Enter 30 9 i96211558 PMT Solve for 705842 Her employer will contribute 1500 per year so she must contribute 705842 7 1500 555842 per year 66 Without fee Enter 186 12 9000 r200 Solve for 7774 Enter 82 12 9000 r200 PMT Solve for 5396 With fee Enter 82 12 9180 r200 PMT Solve for 5527 67 Value at Year 6 Enter 5 11 800 Solve for 134805 Enter 4 11 800 PMT Solve for 121446 Enter 3 11 900 PMT Solve for 123087 Enter 2 11 900 PMT Solve for 110889 Enter 1 11 1000 PMT Solve for 111000 So at Year 5 the value is 134805 121446 123087 110889 111000 1000 701226 At Year 65 the value is Enter 59 7 701226 PMT Solve for 37975276 The policy is not worth buying the future value of the policy is 37975276 but the policy contract will pay off 350000 68 Effective sixmonth rate 1 Daily rate180 7 1 Effective sixmonth rate l 09360180 7 1 Effective sixmonth rate 0460 or 460 Enter 40 460 750000 Solve for 1360215232 Enter 1 460 1360215232 Solve for 1300369650 Value ofwinnings today 1300369650 2000000 Value ofwinnings today 1500369650 100 IRR CPT 1057 73 a APR 9 x 52 468 Enter 468 N01 EFF Solve for 873442 b Enter 1 Solve for 9 89 APR 989 x 52 51429 Enter 51429 NOM Solve for 1338490 0 Enter 4 Solve for 2519 APR 2519 x 52 130992 Enter 130992 NOM Solve for 1185150194 52 910 51000 52 5884 825 52 CY 101 CHAPTER 4 APPENDIX NET PRESENT VALUE FIRST PRINCIPLES OF FINANCE Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem 1 The potential consumption for a borrower next year is the salary during the year minus the repayment of the loan and interest to fund the current consumption The amount that must be borrowed to fund this year s consumption is Amount to borrow 100000 7 80000 20000 Interest will be charged the amount borrowed so the repayment of this loan next year will be Loan repayment 20000110 22000 So the consumption potential next year is the salary minus the loan repayment or Consumption potential 90000 7 22000 68000 2 The potential consumption for a saver next year is the salary during the year plus the savings from the current year and the interest earned The amount saved this year is Amount saved 50000 7 35000 15000 The saver will earn interest over the year so the value of the savings next year will be Savings value in one year 150001 12 16800 So the consumption potential next year is the salary plus the value of the savings or Consumption potential 60000 16800 76800 3 Financial markets arise to facilitate borrowing and lending between individuals By borrowing and lending people can adjust their pattern of consumption over time to t their particular preferences This allows corporations to accept all positive NPV projects regardless of the intertemporal l 0 1 102 The present value of labor income is the total of the maximum current consumption So solving for the interest rate we nd 86 40 501 R R 0870 or 870 The NPV of the investment is the difference between the new maximum current consumption minus the old maximum current consumption or NPV 98786 12 The total maximum current consumption amount must be the present value of the equal annual consumption amount If C is the equal annual consumption amount we nd 98 C C1R 98 C C10870 C 5104 The market interest rate must be the increase in the maximum current consumption to the maximum consumption next year which is Market interest rate 9000080000 7 1 01250 or 1250 Harry will invest 10000 in nancial assets and 30000 in productive assets today NPV 4530000 562501125 NPV 20000 103 CHAPTER 5 NET PRESENT VALUE AND OTHER INVESTMENT RULES Answers to Concepts Review and Critical Thinking Questions 1 Assuming conventional cash ows a payback period less than the project s life means that the NPV is positive for a zero discount rate but nothing more de nitive can be said For discount rates greater than zero the payback period will still be less than the project s life but the NPV may be positive zero or negative depending on whether the discount rate is less than equal to or greater than the IRR The discounted payback includes the effect of the relevant discount rate If a project s discounted payback period is less than the project s life it must be the case that NPV is positive Assuming conventional cash ows if a project has a positive NPV for a certain discount rate then it will also have a positive NPV for a zero discount rate39 thus the payback period must be less than the project life Since discounted payback is calculated at the same discount rate as is NPV if NPV is positive the discounted payback period must be less than the project s life If NPV is positive then the present value of future cash in ows is greater than the initial investment cost thus PI must be greater than 1 If NPV is positive for a certain discount rate R then it will be zero for some larger discount rate R thus the IRR must be greater than the required return a Payback period is simply the accounting breakeven point of a series of cash ows To actually compute the payback period it is assumed that any cash ow occurring during a given period is realized continuously throughout the period and not at a single point in time The payback is then the point in time for the series of cash ows when the initial cash outlays are fully recovered Given some predetermined cutoff for the payback period the decision rule is to accept projects that pay back before this cutoff and reject projects that take longer to pay back The worst problem associated with the payback period is that it ignores the time value of money In addition the selection of a hurdle point for the payback period is an arbitrary exercise that lacks any steadfast rule or method The payback period is biased towards short term projects it fully ignores any cash ows that occur after the cutoff point b The IR is the discount rate that causes the NPV of a series of cash ows to be identically zero IRR can thus be interpreted as a nancial breakeven rate of return at the IRR discount rate the net value of the project is zero The acceptance and rejection criteria are If C0 lt 0 and all future cash ows are positive accept the project if the internal rate of return is greater than or equal to the discount rate If C0 lt 0 and all future cash ows are positive reject the project if the internal rate of return is less than the discount rate If C0 gt 0 and all future cash ows are negative accept the project if the intemal rate of return is less than or equal to the discount rate If C0 gt 0 and all future cash ows are negative reject the project if the internal rate of return is greater than the discount rate 104 IRR is the discount rate that causes NPV for a series of cash ows to be zero NPV is preferred in all situations to IR IRR can lead to ambiguous results if there are nonconventional cash ows and it also may ambiguously rank some mutually exclusive projects However for stand alone projects with conventional cash ows IR and NPV are interchangeable techniques 0 The pro tability index is the present value of cash in ows relative to the project cost As such it is a bene tcost ratio providing a measure of the relative pro tability of a project The pro tability index decision rule is to accept projects with a PI greater than one and to reject projects with a PI less than one The profitability index can be expressed as PI NPV costcost 1 NPVcost If a firm has a basket of positive NPV projects and is subject to capital rationing PI may provide a good ranking measure of the projects indicating the bang for the buc of each particular project d NPV is simply the present value of a project s cash ows including the initial outlay NPV speci cally measures after considering the time value of money the net increase or decrease in rm wealth due to the project The decision rule is to accept projects that have a positive NPV and reject projects with a negative NPV NPV is superior to the other methods of analysis presented in the text because it has no serious aws The method unambiguously ranks mutually exclusive projects and it can differentiate between projects of different scale and time horizon The only drawback to NPV is that it relies on cash ow and discount rate values that are often estimates and thus not certain but this is a problem shared by the other performance criteria as well A project with NPV 2500 implies that the total shareholder wealth of the rm will increase by 2500 if the project is accepted For a project with future cash ows that are an annuity Payback I C And the IR is 0 71CIRR Solving the IRR equation for IRR we get IRR C I Notice this is just the reciprocal of the payback So IRR 1 PB For longlived projects with relatively constant cash ows the sooner the project pays back the greater is the IRR and the IR is approximately equal to the reciprocal of the payback period There are a number of reasons Two of the most important have to do with transportation costs and exchange rates Manufacturing in the US places the nished product much closer to the point of sale resulting in significant savings in transportation costs It also reduces inventories because goods spend less time in transit Higher labor costs tend to offset these savings to some degree at least compared to other possible manufacturing locations Of great importance is the fact that manufacturing in the US means that a much higher proportion of the costs are paid in dollars Since sales are in dollars the net effect is to immunize pro ts to a large extent against uctuations in exchange rates This issue is discussed in greater detail in the chapter on international nance 105 p n p A The single biggest dif culty by far is coming up with reliable cash ow estimates Determining an appropriate discount rate is also not a simple task These issues are discussed in greater depth in the next several chapters The payback approach is probably the simplest followed by the AAR but even these require revenue and cost projections The discounted cash ow measures discounted payback NPV IRR and pro tability index are really only slightly more dif cult in practice Yes they are Such entities generally need to allocate available capital ef ciently just as forpro ts do However it is frequently the case that the revenues from notforpro t ventures are not tangible For example charitable giving has real opportunity costs but the benefits are generally hard to measure To the extent that bene ts are measurable the question of an appropriate required return remains Payback rules are commonly used in such cases Finally realistic costbene t analysis along the lines indicated should de nitely be used by the US government and would go a long way toward balancing the budget The statement is false If the cash ows of Project B occur early and the cash ows of Project A occur late then for a low discount rate the NPV of A can exceed the NPV of B Observe the following example C0 C1 C2 IRR NPV 0 Project A 71000000 0 1440000 20 440000 ProjectB 72000000 2400000 0 20 400000 However in one particular case the statement is true for equally risky projects If the lives of the two projects are equal and the cash ows of Project B are twice the cash ows of Project A in every time period the NPV of Project B will be twice the NPV of Project A Although the pro tability index PI is higher for Project B than for Project A Project A should be chosen because it has the greater NPV Confusion arises because Project B requires a smaller investment than Project A Since the denominator of the PI ratio is lower for Project B than for Project A B can have a higher PI yet have a lower NPV Only in the case of capital rationing could the company s decision have been incorrect a Project A would have a higher IRR since initial investment for Project A is less than that of Project B if the cash ows for the two projects are identical b Yes since both the cash ows as well as the initial investment are twice that of Project B Project B s NPV would be more sensitive to changes in the discount rate The reason is the time value of money Cash ows that occur further out in the future are always more sensitive to changes in the interest rate This sensitivity is similar to the interest rate risk of a bond The MR is calculated by nding the present value of all cash out ows the future value of all cash in ows to the end of the project and then calculating the IRR of the two cash ows As a result the cash ows have been discounted or compounded by one interest rate the required return and then the interest rate between the two remaining cash ows is calculated As such the MIRR is not a true interest rate In contrast consider the IRR If you take the initial investment and calculate the future value at the IRR you can replicate the future cash ows of the project exactly 106 13 1quot The statement is incorrect It is true that if you calculate the future value of all intermediate cash ows to the end of the project at the required return then calculate the NPV of this future value and the initial investment you will get the same NPV However NPV says nothing about reinvestment of intermediate cash ows The NPV is the present value of the project cash ows What is actually done with those cash ows once they are generated is not relevant Put differently the value of a project depends on the cash ows generated by the project not on the future value of those cash flows The fact that the reinvestment works only if you use the required return as the reinvestment rate is also irrelevant simply because reinvestment is not relevant in the rst place to the value of the project One caveat Our discussion here assumes that the cash ows are truly available once they are generated meaning that it is up to rm management to decide what to do with the cash ows In certain cases there may be a requirement that the cash ows be reinvested For example in international investing a company may be required to reinvest the cash ows in the country in which they are generated and not repatriate the money Such funds are said to be blocked and reinvestment becomes relevant because the cash ows are not truly available The statement is incorrect It is true that if you calculate the future value of all intermediate cash ows to the end of the project at the IRR then calculate the IRR of this future value and the initial investment you will get the same IRR However as in the previous question what is done with the cash ows once they are generated does not affect the IRR Consider the following example C0 C1 C2 7100 10 110 IR 10 Project A Suppose this 100 is a deposit into a bank account The IRR of the cash ows is 10 percent Does the IRR change if the Year 1 cash ow is reinvested in the account or if it is withdrawn and spent on pizza No Finally consider the yield to maturity calculation on a bond If you think about it the YTM is the IRR on the bond but no mention of a reinvestment assumption for the bond coupons is suggested The reason is that reinvestment is irrelevant to the YTM calculation in the same way reinvestment is irrelevant in the IRR calculation Our caveat about blocked funds applies here as well Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic a The payback period is the time that it takes for the cumulative undiscounted cash in ows to equal the initial investment Project A Cumulative cash ows Year 1 6500 6500 Cumulative cash ows Year 2 6500 4000 10500 107 Companies can calculate a more precise value using fractional years To calculate the fractional payback period nd the fraction of year 2 s cash ows that is needed for the company to have cumulative undiscounted cash ows of 10000 Divide the difference between the initial 39 t and the 39 39 quot 39 cash ows as of year 1 by the undiscounted cash ow of year 2 Payback period 1 10000 7 6500 4000 Payback period 1875 years Project B Cumulative cash ows Year 1 7000 7000 Cumulative cash ows Year 2 7000 4000 11000 Cumulative cash ows Year 3 7000 4000 5000 16000 To calculate the fractional payback period nd the fraction of year 3 s cash ows that is needed for the company to have cumulative undiscounted cash ows of 12000 Divide the difference between the initia 39 an 39 39 quot 39 cash ows as of year 2 by the undiscounted cash ow of year 3 Payback period 2 12000 7 7000 7 4000 5000 Payback period 220 years Since project A has a shorter payback period than project B has the company should choose project A Discount each project s cash ows at 15 percent Choose the project with the highest NPV Project A NPV 710000 6500 115 40001152 18001153 NPV 713972 ProjectB NPV 712000 7000 115 40001152 50001153 NPV 39911 The rm should choose Project B since it has a higher NPV than Project A has To calculate the payback period we need to nd the time that the project has taken to recover its initial investment The cash ows in this problem are an annuity so the calculation is simpler If the initial cost is 4100 the payback period is Payback 4 220 970 423 years There is a shortcut to calculate the payback period if the future cash ows are an annuity Just divide the initial cost by the annual cash ow For the 4100 cost the payback period is Payback 4100 970 423 years 108 For an initial cost of 6200 the payback period is Payback 6200 970 639 years The payback period for an initial cost of 8000 is a little trickier Notice that the total cash in ows after eight years will be Total cash in ows 8970 7760 If the initial cost is 8000 the project never pays back Notice that if you use the shortcut for annuity cash ows you get Payback 8000 970 825 years This answer does not make sense since the cash ows stop after eight years so there is no payback period When we use discounted payback we need to nd the value of all cash ows today The value today of the project cash ows for the first four years is Value today of Year 1 cash ow 6000114 526316 Value today of Year 2 cash ow 6500ll42 500154 Value today of Year 3 cash ow 70001143 472480 Value today of Year 4 cash ow 80001144 473664 To find the discounted payback we use these values to nd the payback period The discounted rst year cash ow is 526316 so the discounted payback for an 8000 initial cost is Discounted payback l 8000 7 526316500154 155 years For an initial cost of 13000 the discounted payback is Discounted payback 2 13000 7 526316 7 500154472480 258 years Notice the calculation of discounted payback We know the payback period is between two and three years so we subtract the discounted values of the Year 1 and Year 2 cash ows from the initial cost This is the numerator which is the discounted amount we still need to make to recover our initial investment We divide this amount by the discounted amount we will earn in Year 3 to get the fractional portion of the discounted payback If the initial cost is 18000 the discounted payback is Discounted payback 3 18000 7 526316 7 5001547 472480 473664 364 years To calculate the discounted payback discount all future cash ows back to the present and use these discounted cash ows to calculate the payback period To nd the fractional year we divide the amount we need to make in the last year to payback the project by the amount we will make Doing so we nd R 0 3 2200 2600 385 years Discounted payback Regular payback 385 years 109 R 10 2600l10 2600l102 2600l103 2600l104 2600l105 985605 26001106 146763 Discounted payback 5 10000 7 985605 146763 510 years R 15 2600115 2600l152 26001153 2600l154 26001155 2600l156 983966 The project never pays back The IR is the interest rate that makes the NPV of the project equal to zero So the equation that de nes the IRR for this project is 000 C11IRRC21IRR2C31IRR3 0 711000 55001 IRR 40001 IRR2 30001 IRR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 746 Since the IR is less than the required return we would reject the project The IR is the interest rate that makes the NPV of the project equal to zero So the equation that de nes the IRR for this Project A is 0C0C11IRRC21IRR2C31IRR3 0 7 3500 18001IRR 24001IRR2 19001IRR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 3337 And the IRR for Project B is 0C0 01 11RRC2 1IRR2C31IRR3 0 7 2300 9001 IRR 16001 1RR2 14001 IRR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 2932 The pro tability index is de ned as the PV of the cash in ows diVided by the PV of the cash out ows The cash ows from this project are an annuity so the equation for the pro tability index is PI CPVIFAm C0 PI 65000PVIFA1577 190000 PI 1423 110 The pro tability index is the present value of the future cash ows divided by the initial cost So for Project Alpha the pro tability index is PIAIPha 800 110 9001102 70011031500 1331 And for Project Beta the pro tability index is PIE a 500 110 19001102 210011032500 1441 According to the pro tability index you would accept Project Beta However remember the pro tability index rule can lead to an incorrect decision when ranking mutually exclusive projects Intermediate To have a payback equal to the project s life given C is a constant cash ow for N years C IN To have a positive NPV I lt C PVIFARW N Thus C gt I PVIFARW N Bene ts C PVIFAW N 2 X costs 21 C 2I PVIFAWK N The IR is the interest rate that makes the NPV of the project equal to zero So the equation that defines the IRR for this project is 0 C11IRRC21IRR2C31IRR3C41IRRquot 0 80007 4400 1 IRR 7 2700 1 1RR2 7 1900 1 IRR3 715001IRR4 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 1481 This problem differs from previous ones because the initial cash ow is positive and all future cash ows are negative In other words this is a nancingtype project while previous projects were investingtype projects For nancing situations accept the project when the IR is less than the discount rate Reject the project when the IR is greater than the discount rate IRR 1481 Discount Rate 10 IR gt Discount Rate Reject the offer when the discount rate is less than the IR 111 Using the same reason as part b we would accept the project if the discount rate is 20 percent IRR 1481 Discount Rate 20 IR lt Discount Rate Accept the offer when the discount rate is greater than the IRR The NPV is the sum of the present value of all cash ows so the NPV of the project if the discount rate is 10 percent will be NPV 8000 7 4400 11 7 2700 112 7 1900 113 7 1500 11 NPV 768342 When the discount rate is 10 percent the NPV of the offer is 76 8342 Reject the offer And the NPV of the project if the discount rate is 20 percent will be NPV 8000 7 4400 12 7 2700 122 7 1900 123 7 1500 12 NPV 63542 When the discount rate is 20 percent the NPV of the offer is 63542 Accept the offer Yes the decisions under the NPV rule are consistent with the choices made under the IRR rule since the signs of the cash ows change only once The IRR is the interest rate that makes the NPV of the project equal to zero So the IRR for each project is Deepwater Fishing IRR 07c0 C11IRRC21IRRZC31IRR3 0 7 7750000 310000 1 IRR 430000 1 1RR2 330000 1 IRR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 1983 Submarine Ride IRR 000 C11IRRC21IRRZC31IRR3 0 72100000 1200000 1 IRR 760000 1 1RR2 850000 1 IRR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 1736 112 Based on the IRR rule the deepwater shing project should be chosen because it has the higher To calculate the incremental IRR we subtract the smaller project s cash ows from the larger project s cash ows In this case we subtract the deepwater shing cash ows from the submarine ride cash ows The incremental IR is the IRR of these incremental cash ows So the incremental cash ows of the submarine ride are Year 0 Year 1 Year 2 Year 3 Submarine Ride 72 100000 1200000 760000 850000 Deepwater Fishing 7750000 310000 43 0000 330000 SubmarineiFishing 71350000 890000 330000 520000 Setting the present value of these incremental cash ows equal to zero we nd the incremental IR is 0 C11IRRC21IRR2C31IRR3 0 71350000 890000 1 IRR 330000 1 IRR2 520000 1 IRR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that Incremental IRR 1578 For investingtype projects accept the larger project when the incremental IR is greater than the discount rate Since the incremental IRR 1578 is greater than the required rate of return of 14 percent choose the submarine ride project Note that this is not the choice when evaluating only the IRR of each project The IR decision rule is awed because there is a scale problem That is the submarine ride has a greater initial investment than does the deepwater shing project This problem is corrected by calculating the IRR of the incremental cash ows or by evaluating the NPV of each project The NPV is the sum of the present value of the cash ows from the project so the NPV of each project will be Deepwater shing NPV 7750000 310000 114 430000 1142 330000 1143 NPV 7554146 Submarine ride NPV 72100000 1200000 114 760000 1142 850000 1143 NPV 11115269 Since the NPV of the submarine ride project is greater than the NPV of the deepwater shing project choose the submarine ride project The incremental IRR rule is always consistent with the NPV rule 113 The profitability index is the PV of the future cash ows divided by the initial investment The cash ows for both projects are an annuity so P11 21000PVIFA1073 40000 1306 PIH 8500PVIFA1073 15000 1409 The profitability index decision rule implies that we accept project 11 since P111 is greater than the P11 The NPV of each project is NPVI 7 40000 21000PVIFA103 1222389 NPVH 7 15000 8500PVIFA103 613824 The NPV decision rule implies accepting Project 1 since the NPVI is greater than the NPVH Using the pro tability index to compare mutually exclusive projects can be ambiguous when the magnitudes of the cash ows for the two projects are of different scales In this problem project I is roughly 3 times as large as project 11 and produces a larger NPV yet the pro t ability index criterion implies that project 11 is more acceptable The equation for the NPV of the project is NPV 7 32000000 5700000011 7 9000000112 1238016529 The NPV is greater than 0 so we would accept the project The equation for the IRR of the project is 0 732000000 570000001IRR 7 9000000171RR2 From Descartes rule of signs we know there are two IRRs since the cash ows change signs twice From trial and error the two IRRs are IRR 6061 78249 When there are multiple IRRs the IRR decision rule is ambiguous Both IRRs are correct that is both interest rates make the NPV of the project equal to zero If we are evaluating whether or not to accept this project we would not want to use the IRR to make our decision The payback period is the time that it takes for the cumulative undiscounted cash in ows to equal the initial investment Board game Cumulative cash ows Year 1 700 700 Paybackperiod 600 700 86 years 114 CDROM Cumulative cash ows Year 1 1400 1400 Cumulative cash ows Year 2 1400 900 2300 Payback period 1 1900 7 1400 900 Payback period 156 years Since the board game has a shorter payback period than the CDROM project the company should choose the board game The NPV is the sum of the present value of the cash ows from the project so the NPV of each project will be Board game NPV 7600 700 110 1501102 1001103 NPV 23546 CDROM NPV 71900 1400110 9001102 4001103 NPV 41705 Since the NPV of the CDROM is greater than the NPV of the board game choose the CD ROM The IRR is the interest rate that makes the NPV of a project equal to zero So the IRR of each project is Board game 0 7600 700 1 IRR 150 1 IRR2 100 1 IRR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 4243 CDROM 0 1900 1400 1 IRR 90011RR2 400 1 IRR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 2503 115 Since the IRR of the board game is greater than the IRR of the CDROlVI IRR implies we choose the board game Note that this is the choice when evaluating only the IRR of each project The IR decision rule is awed because there is a scale problem That is the CDROM has a greater initial investment than does the board game This problem is corrected by calculating the IRR of the incremental cash ows or by evaluating the NPV of each project To calculate the incremental IRR we subtract the smaller project s cash ows from the larger project s cash ows In this case we subtract the board game cash ows from the CDROM cash flows The incremental IR is the IRR of these incremental cash ows So the incremental cash ows of the CDROM are Year 0 Year 1 Year 2 Year 3 CDROM 7 1900 1400 900 400 Board game 7600 700 150 100 CDROMi Board game 71300 700 750 300 Setting the present value of these incremental cash ows equal to zero we nd the incremental IR is 0C0C11IRRCZ1IRRZC31IRR3 0713007001IRR7501IRR23001IRR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that Incremental IRR 1878 For investingtype projects accept the larger project when the incremental IR is greater than the discount rate Since the incremental IRR 1878 is greater than the required rate of return of 10 percent choose the CDROM project The pro tability index is the PV of the future cash ows divided by the initial investment The pro tability index for each project is PICDMA 13000000 110 7000000 1102 2000000 1103 5000000 382 PIm 10000000 110 25000000 1102 20000000 1103 10000000 448 lem 10000000 110 20000000 1102 50000000 1103 15000000 421 The pro tability index implies we accept the G4 project Remember this is not necessarily correct 1sz6 ythe profitability index does not necessarily rank projects with different initial investments The NPV of each project is NPVCDMA 5000000 13000000 110 7000000 1102 2000000 1103 NPVCDMA 1410593539 NPVG4 10000000 10000000 110 25000000 1102 20000000 1103 NPVm 3477836213 116 PIWiFi 15000000 10000000 110 20000000 L102 50000000 L103 PIWiFi 4818557476 NPV implies we accept the WiFi project since it has the highest NPV This is the correct decision if the projects are mutually exclusive We would like to invest in all three projects since each has a positive NPV If the budget is limited to 3 l5million we can only accept the CDMA project and the G4 project or the WiFi project NPV is additive across projects and the company The total NPV of the CDMA project and the G4 project is NPVCDMA and 34 1410593539 3477836213 NPchMA and G4 4888429752 This is greater than the WiFi project so we should accept the CDMA project and the G4 project The payback period is the time that it takes for the cumulative undiscounted cash in ows to equal the initial investment AZM MiniSUV Cumulative cash ows Year 1 270000 270000 Cumulative cash ows Year 2 270000 180000 450000 Payback period 1 30000 180000 1 17 years AZF FullSUV Cumulative cash ows Year 1 250000 250000 Cumulative cash ows Year 2 250000 400000 650000 Payback period 1 350000 400000 188 years Since the AZM has a shorter payback period than the AZF the company should choose the AZM Remember the payback period does not necessarily rank projects correctly The NPV of each project is NPVAZM 7300000 270000 110 180000 1102 150000 1103 NPVAZM 20691210 NPVAZF 7600000 250000 110 400000 1102 300000 1103 NPVAZF 18324568 The NPV criteria implies we accept the AZM because it has the highest NPV 117 The IR is the interest rate that makes the NPV of the project equal to zero So the IRR of the AZM is 0 7300000 270000 1 IRR 180000 1 IRR2 150000 1 IRR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRRAZM 5143 And the IRR of the AZF is 0 7600000 250000 1 IRR 400000 1 IRR2 300000 1 IRR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRRAZF 2604 The IR criteria implies we accept the AZM because it has the highest IRR Remember the IRR does not necessarily rank projects correctly Incremental IRR analysis is not necessary The AZM has the smallest initial investment and the largest NPV so it should be accepted The profitability index is the PV of the future cash ows divided by the initial investment The pro tability index for each project is PIA 140000 112 140000 1122 200000 118 PIE 260000 112 260000 1122 400000 110 PIC 150000 112 120000 1122 200000 115 The NPV of each project is vaA 7200000 140000 112 140000 1122 vaA 3660714 NPvB 7400000 260000 112 260000 1122 vaB 3941327 NPvC 7200000 150000 112 120000 1122 vaC 2959184 Accept projects A B and C Since the projects are independent accept all three projects because the respective profitability index of each is greater than one 118 Accept Project B Since the Projects are mutually exclusive choose the Project with the highest PI while taking into account the scale of the Project Because Projects A and C have the same initial investment the problem of scale does not arise when comparing the pro tability indices Based on the pro tability index rule Project C can be eliminated because its PI is less than the PI of Project A Because of the problem of scale we cannot compare the PIs of Projects A and B However we can calculate the PI of the incremental cash ows of the two projects which are Project C0 B 7 A 7200000 C1 C2 120000 120000 When calculating incremental cash ows remember to subtract the cash ows of the project with the smaller initial cash out ow from those of the project with the larger initial cash out ow This procedure insures that the incremental initial cash out ow will be negative The incremental PI calculation is PIB 7 A 120000 112 120000 1122 200000 PIB 7A 1014 The company should accept Project B since the PI of the incremental cash ows is greater than one Remember that the NPV is additive across projects Since we can spend 600000 we could take two of the projects In this case we should take the two projects with the highest NPVs which are Project B and Project A The payback period is the time that it takes for the cumulative undiscounted cash in ows to equal the initial investment Dry Prepeg Cumulative cash ows Year 1 900000 900000 Cumulative cash ows Year 2 900000 800000 1700000 Payback period l 500000800000 163 years Solvent Prepeg Cumulative cash ows Year 1 300000 300000 Cumulative cash ows Year 2 300000 500000 800000 Payback period l 300000500000 160 years Since the solvent prepeg has a shorter payback period than the dry prepeg the company should choose the solvent prepeg Remember the payback period does not necessarily rank projects correctly 119 The NPV of each project is NPVDylmeloeg 1400000 900000 110 800000 L102 1 700000 L103 NPVDylmeloeg 60525920 NPVG4 600000 300000 110 500000 1102 400000 1103 NPVGA 38647633 The NPV criteria implies accepting the dry prepeg because it has the highest NPV The IR is the interest rate that makes the NPV of the project equal to zero So the IRR of the dry prepeg is 0 1400000 900000 1 IRR 800000 1 1RR2 7000000 1 IRR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRRDry mpeg 3445 And the IRR of the solvent prepeg is 0 7600000 300000 1 IRR 500000 1 IRR2 400000 1 IRR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRRSolvent prepeg 41 87 The IR criteria implies accepting the solvent prepeg because it has the highest IRR Remember the IRR does not necessarily rank projects correctly Incremental IRR analysis is necessary The solvent prepeg has a higher IRR but is relatively smaller in terms of investment and NPV In calculating the incremental cash ows we subtract the cash ows from the project with the smaller initial investment from the cash ows of the project with the large initial investment so the incremental cash ows are Year 0 Year 1 Year 2 Year 3 Dry prepeg 1400000 900000 800000 700000 Solvent prepeg 00000 300000 500000 400000 Dry prepeg 7 Solvent prepeg 7800000 600000 300000 300000 Setting the present value of these incremental cash ows equal to zero we nd the incremental IR is 0 7800000 600000 1 IRR 300000 1 IRR2 300000 1 IRR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that Incremental IRR 2749 120 For investingtype projects we accept the larger project when the incremental IR is greater than the discount rate Since the incremental IRR 2749 is greater than the required rate of return of 10 percent we choose the dry prepeg Note that this is the choice when evaluating only the IRR of each project The IR decision rule is awed because there is a scale problem That is the dry prepeg has a greater initial investment than does the solvent prepeg This problem is corrected by calculating the IRR of the incremental cash ows or by evaluating the NPV of each project The NPV of each project is NPVN39PSO 450000 1600001 711155 15 NPVNP30 8634482 NPVquo 200000 80000 115 920001152 105800 1153 121670 1154 1399211155 NPVNX20 14782634 The NPV criteria implies accepting the NX20 The IRR is the interest rate that makes the NPV of the project equal to zero so the IRR of each project is NP30 0 7450000 1600001711IRR5 IRR Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRRNP30 2285 And the IRR of the NX20 is 0 7200000 80000 1 IRR 92000 1 1RR2 105800 1 IRR3 121670 1 1RR 139921 1 IRR5 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRRNHO 4009 The IR criteria implies accepting the NX20 121 Incremental IRR analysis is not necessary The NX20 has a higher IR and is relatively smaller in terms of investment with a larger NPV Nonetheless we will calculate the incremental IRR In calculating the incremental cash ows we subtract the cash ows from the project with the smaller initial investment from the cash ows of the project with the large initial investment so the incremental cash ows are Incremental Year cash ow 7250000 80000 68000 54200 38330 20079 kIIJkktJNt IO Setting the present value of these incremental cash ows equal to zero we nd the incremental IR is 0 7250000 80000 1 IRR 68000 1 1RR2 54200 1 IRR3 38330 1 IRR 200790 1 IRR5 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that Incremental IRR 174 For investingtype projects accept the larger project when the incremental IR is greater than the discount rate Since the incremental IRR 174 is less than the required rate of return of 15 percent choose the NX20 The pro tability index is the present value of all subsequent cash ows divided by the initial investment so the pro tability index of each project is PIN 0 1600001 7 11155 15 450000 Hm0 1192 PINmo 80000 115 920001152 1058001153 1216701154 139921 1155 200000 PINmo 1739 The PI criteria implies accepting the NX20 The payback period is the time that it takes for the cumulative undiscounted cash in ows to equal the initial investment Project A Cumulative cash ows Year 1 190000 190000 Cumulative cash ows Year 2 190000 170000 360000 Payback period l 90000170000 153 years 122 Project B Cumulative cash ows Year 1 270000 270000 Cumulative cash ows Year 2 270000 240000 510000 Payback period 1 120000240000 150 years Project C Cumulative cash ows Year 1 160000 160000 Cumulative cash ows Year 2 160000 190000 350000 Payback period 1 70000190000 137 years Project C has the shortest payback period so payback implies accepting Project C However the paybackperiod does not necessarily rank projects correctly The IR is the interest rate that makes the NPV of the project equal to zero so the IRR of each project is Project A 0 7280000 190000 1 IRR 170000 1 IRR2 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRRA 1891 And the IRR of the Project B is 0 7390000 270000 1 IRR 240000 1 IRR2 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRRB 2036 And the IRR of the Project C is 0 7230000 160000 1 IRR 190000 1 IRR2 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRRC 3210 The IR criteria implies accepting Project C 123 The pro tability index is the present value of all subsequent cash ows divided by the initial investment We need to discount the cash ows of each project by the required return of each project The pro tability index of each project is PIA 190000 110 1700001102280000 PIA 112 P1B 270000 120 240000 1202 390000 P1B 100 PIC 160000 115 190000 1152 230000 123 C The PI criteria implies accepting Project C We need to discount the cash ows of each project by the required return of each project The NPV of each project is vaA 280000 190000 110 170000 1102 vaA 3322314 NPvB 7390000 270000 120 240000 1202 vaB 166667 NPvC 7230000 160000 115 190000 1152 vaC 5279773 The NPV criteria implies accepting Project C In the final analysis since we can accept only one of these projects We should accept Project C since it has the greatest NPV Challenge 21 Given the sixyear payback the worst case is that the payback occurs at the end of the sixth year N N Thus the worst case V 7574000 5740001126 728319374 The best case has infmite cash ows beyond the payback point Thus the bestcase NPV is infinite The equation for the IRR of the project is 0 7504 28621 IR 7 60701 IRR2 57001 IRR3 7 20001 IRR4 Using Descartes rule of signs from looking at the cash ows we know there are four IRRs for this project Even with most computer spreadsheets we have to do some trial and error From trial and error IRRs of 25 3333 4286 and 6667 are found We would accept the project when the NPV is greater than zero See for yourself that the NPV is greater than zero for required returns between 25 and 3333 or between 4286 and 6667 124 Here the cash in ows of the project go on forever which is a perpetuity Unlike ordinary perpetuity cash ows the cash ows here grow at a constant rate forever which is a growing perpetuity The PV of the future cash ows from the project is PV of cash in ows CJR 7g PV of cash in ows 11500013 7 06 164285714 NPV is the PV of the out ows minus by the PV of the inflows so the NPV is NPV ofthe project 71400000 164285714 24285714 The NPV is positive so we would accept the project Here we want to know the minimum growth rate in cash ows necessary to accept the project The minimum growth rate is the growth rate at which we would have a zero NPV The equation for a zero NPV using the equation for the PV of a growing perpetuity is 0 7 1400000 115000 13 7g Solving for g we get g 479 The project involves three cash ows the initial investment the annual cash in ows and the abandonment costs The mine will generate cash in ows over its 11year economic life To express the PV of the annual cash in ows apply the growing annuity formula discounted at the IRR and growing at eight percent PVCash In ows C 1r 7g 7 1r 7g X 1 g1 r PVCash In ows 1750001IRR7 08 7 1IRR 708 X 1 081 IRR At the end of 11 years the Utah Mining Corporate will abandon the mine incurring a 125000 charge Discounting the abandonment costs back 11 years at the IRR to express its present value we get PVAbandonment C11 1 IRR11 PVAbandonment 7125000 1 IRR11 So the IRR equation for this project is 0 7 7900000 1750001IRR 7 08 7 1IRR7 08 x 1 081 IRR 125000 1 IRR Using a spreadsheet financial calculator or trial and error to find the root of the equation we nd that IRR 2226 125 Yes Since the mine s IRR exceeds the required return of 10 percent the mine should be opened The correct decision rule for an investmenttype project is to accept the project if the discount rate is above the IRR Although it appears there is a sign change at the end of the project because of the abandonment costs the last cash ow is actually positive because of the operating cash in the last year 25 First we need to find the future value of the cash flows for the one year in which they are blocked by the government So reinvesting each cash inflow for one year we find Year 2 cash ow 205000104 213200 Year 3 cash ow 265000104 275600 Year 4 cash ow 346000104 359840 Year 5 cash ow 220000104 228800 So the NPV of the project is NP NPV 7262633 7750000 2132001112 2756001113 3598401114 2288001115 And the IRR of the project is 0 750000 2132001 IRR2 2756001 IRR3 3598401 IRR 2288001 IRR5 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 1089 While this may look like a MIRR calculation it is not a MIRR rather it is a standard IRR calculation Since the cash in ows are blocked by the government they are not available to the company for a period of one year Thus all we are doing is calculating the IRR based on when the cash ows actually occur for the company a We can apply the growing perpetuity formula to nd the PV of stream A The perpetuity formula values the stream as of one year before the rst payment Therefore the growing perpetuity formula values the stream of cash ows as of year 2 Next discount the PV as of the end of year 2 back two years to nd the PV as of today year 0 Doing so we find PVA C3 Rgl 1 t R 2 PVA 8900 012 7 004 1122 PVA 8868782 We can apply the perpetuity formula to find the PV of stream B The perpetuity formula discounts the stream back to year 1 one period prior to the rst cash ow Discount the PV as of the end of year 1 back one year to nd the PV as of today year 0 Doing so we nd PVB C2 R 1 R PVB 710000 012 112 PVB 77440476 126 b If we combine the cash ow streams to form Project C we get Project A C3 R 7G 1 R2 ProjectB C2 R 1 R Project C ProjectA Project B ProjectC C3 Rig 1 R2 C2 R 1 R 0 8900 IRRi 04 1 IRR2 710000 IRR 1 IRR Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 1680 c The correct decision rule for an investingtype project is to accept the project if the discount rate is below the IRR Since there is one IRR a decision can be made At a point in the future the cash ows from stream A will be greater than those from stream B Therefore although there are many cash ows there will be only one change in sign When the sign of the cash ows change more than once over the life of the project there may be multiple intemal rates of return In such cases there is no correct decision rule for accepting and rejecting projects using the intemal rate of return 27 To answer this question we need to examine the incremental cash ows To make the projects equally attractive Project Billion must have a larger initial investment We know this because the subsequent cash ows from Project Billion are larger than the subsequent cash ows from Project lVIillion So subtracting the Project Million cash ows from the Project Billion cash ows we nd the incremental cash ows are Incremental M cash ows 0 7L 1200 1 240 2 240 3 400 Now we can nd the present value of the subsequent incremental cash ows at the discount rate 12 percent The present value of the incremental cash ows is PV 1200 240 112 2401122 4001123 PV 189032 So if 10 is greater than 189032 the incremental cash ows will be negative Since we are subtracting Project Million from Project Billion this implies that for any value over l89032 the NPV of Project Billion will be less than that of Project lVIillion so I0 must be less than 189032 127 28 The IR is the interest rate that makes the NPV of the project equal to zero So the IRR of the project is 0 20000 7 26000 1 IRR 13000 1 1RR2 Even though it appears there are two IRRs a spreadsheet nancial calculator or trial and error will not give an answer The reason is that there is no real IRR for this set of cash ows If you examine the IRR equation what we are really doing is solving for the roots of the equation Going back to high school algebra in this problem we are solving a quadratic equation In case you don t remember the quadratic equation is 7birxlb2 74110 211 X In this case the equation is X 7 726000 17260002 7 42000013000 226000 The square root term works out to be 676000000 7 1040000000 64000000 The square root of a negative number is a complex number so there is no real number solution meaning the project has no real IR 128 Calculator Solutions 1 b Pro 39ectA 7 12000 7000 1 4000 1 5000 1 713972 5 IR CPT 746 6 Pro 39ectA 72300 900 1 1600 1 1400 1 IR CPT 3337 129 CFO 0 C0 1 65000 F0 1 7 15 NPV CPT 27042728 PI 27042728 190000 1423 10 8000 74400 72700 1 71900 1 i1500 1 IR CPT 1481 8000 8000 74400 74400 1 1 72700 72700 1 1 71900 71900 1 1 71500 71500 1 1 I 10 I 20 NPV CPT NPV CPT 768342 63542 11 a D88 ater ishing Submarine ride 775 0000 72 100000 310000 1200000 1 1 430000 760000 1 1 330000 850000 1 1 IR CPT IRR CPT 1983 1736 130 12 451350000 890000 330000 520000 IRR CPT 1 1578 Dee ater ishing Submarine ride 7750 00 72100000 310000 1200000 1 430000 760000 1 330000 850000 1 I 14 I 14 NPV CPT NPV CPT 7554146 11115269 Pro 39ect 0 CFO 740000 21000 C01 21000 3 F01 3 I 10 I 10 NPV CPT NPV CPT 5222389 1222389 PI 5222389 40000 1306 Pro 39ectH 0 CFO 715000 8500 C01 8500 3 F01 3 I 10 I 10 NPV CPT NPV CPT 2113824 613824 PI 2113824 15000 1409 131 732000000 732000000 57000000 57000000 1 1 79000000 79000000 1 1 I 10 IR CPT NPV CPT 6061 1238016529 Financial calculators will only give you one IRR even if there are multiple IRRs Using trial and error or a root solving calculator the other IR is 78249 14 b Board ame CDROM I 10 I 10 NPV CPT NPV CPT 23546 41705 c Board ame CDROM 7600 700 1 150 1 100 1 IR CPT IRR CPT 4243 2503 d 71300 700 1 750 1 300 1 IR CPT 1878 132 Wi Fi 15 a CDMA 0 0 0 13000000 10000000 10000000 1 1 1 7000000 25000000 20000000 1 1 1 2000000 20000000 50000000 1 1 1 110 110 110 NPV CPT NPV CPT NPV CPT 1910593539 4477836213 6318557476 PICDMA 1910593539 50000000 382 P164 4477836213 10000000 448 PIWiFi 6318557476 15000000 421 b CDMA G4 75000000 7 10000000 715000000 13000000 10000000 10000000 1 1 1 7000000 25000000 20000000 1 1 1 2000000 20000000 50000000 1 1 1 I10 I 10 110 NPV CPT NPV CPT NPV CPT 1410593539 3477836213 4818557476 16 b AZAJ AZF 7300000 7600000 270000 250000 1 1 180000 400000 1 1 150000 3000000 1 1 I 10 I 10 NPV CPT NPV CPT 20691210 18324568 0 AZAJ AZF 7300000 7600000 270000 250000 1 1 180000 400000 1 1 150000 3000000 1 1 IR CPT IRR CPT 5143 2604 133 17 a Pro 39ectA Pro 39ect B Pro 39ect C 0 0 0 140000 260000 150000 1 1 1 140000 260000 120000 1 1 1 I12 I 12 112 NPV CPT NPV CPT NPV CPT 23660714 439413 27 22951284 PIA 23660714 200000 118 PIE 43941327 400000 110 PIC 22951284 200000 115 b Pro 39ectA Pro 39ect B Pro 39ect C 7200000 7400000 7200000 140000 260000 150000 1 1 1 140000 260000 120000 1 1 1 112 112 112 NPV CPT NPV CPT NPV CPT 3660714 3941327 2959184 Solvent reeg 71400000 7600000 900000 300000 1 1 800000 500000 1 1 700000 4000000 1 1 I 10 I 10 NPV CPT NPV CPT 60525920 38647633 0 Solvent ree 71400000 7600000 900000 300000 1 1 800000 500000 1 1 700000 4000000 1 1 IR CPT IRR CPT 3445 4187 134 7800000 600000 1 300000 1 300000 1 745 0000 7200000 160000 80000 5 1 92000 1 105800 1 121670 1 139921 1 NPV CPT 8634482 14782634 b NP30 NX20 745 0000 7200000 160000 80000 5 1 92000 1 105800 1 121670 1 139921 1 135 725 0000 80000 1 68000 1 54200 1 38330 20079 1 d 7450000 7200000 160000 80000 5 1 92000 1 105800 1 121670 1 139921 1 I 15 NPV CPT 53634482 34782634 PINP30 53634482 450000 1192 PINK20 34782634 200000 1739 20 b Pro 39ectA Pro 39ectB Pro 39ect C 7280000 7390000 7230000 190000 270000 160000 1 1 1 170000 240000 190000 1 1 1 IR CPT IRR CPT IRR CPT 1891 2036 3210 136 6 Pro 39ectA 0 190000 1 170000 1 I 10 NPV CPT 31322314 PIA 31322314 280000 112 PIE 39166667 390000 1004 PIC 28279773 230000 123 d Pro 39ectA 7280000 190000 1 170000 1 I 10 NPV CPT 3322314 28 20000 726000 1 13000 1 IR CPT ERROR 7 Pro 39ect B Pro 39ect C 0 0 270000 160000 1 1 240000 190000 1 1 I 20 I 15 NPV CPT NPV CPT 39166667 28279773 Pro 39ect B Pro 39ect C 7390000 7230000 270000 160000 1 1 240000 190000 1 1 I 20 I 15 NPV CPT NPV CPT 166667 5279773 137 CHAPTER 6 MAKING CAPITAL INVESTMENT DECISIONS Answers to Concepts Review and Critical Thinking Questions 1 In this context an opportunity cost refers to the value of an asset or other input that will be used in a project The relevant cost is what the asset or input is actually worth today not for example what it cost to acquire a Yes the reduction in the sales of the company s other products referred to as erosion should be treated as an incremental cash ow These lost sales are included because they are a cost a revenue reduction that the rm must bear if it chooses to produce the new product Yes expenditures on plant and equipment should be treated as incremental cash ows These are costs of the new product line However if these expenditures have already occurred and cannot be recaptured through a sale of the plant and equipment they are sunk costs and are not included as incremental cash ows No the research and development costs should not be treated as incremental cash flows The costs of research and development undertaken on the product during the past three years are sunk costs and should not be included in the evaluation of the project Decisions made and costs incurred in the past cannot be changed They should not affect the decision to accept or reject the project Yes the annual depreciation expense must be taken into account when calculating the cash ows related to a given project While depreciation is not a cash expense that directly affects cash ow it decreases a firm s net income and hence lowers its tax bill for the year Because of this depreciation tax shield the rm has more cash on hand at the end of the year than it would have had without expensing depreciation No dividend payments should not be treated as incremental cash ows A film s decision to pay or not pay dividends is independent of the decision to accept or reject any given investment project For this reason dividends are not an incremental cash ow to a given project Dividend policy is discussed in more detail in later chapters Yes the resale value of plant and equipment at the end of a project s life should be treated as an incremental cash ow The price at which the fum sells the equipment is a cash in ow and any difference between the book value of the equipment and its sale price will create accounting gains or losses that result in either a tax credit or liability Yes salary and medical costs for production employees hired for a project should be treated as incremental cash ows The salaries of all personnel connected to the project must be included as costs of that project 138 Item a is a relevant cost because the opportunity to sell the land is lost if the new golf club is produced Item b is also relevant because the firm must take into account the erosion of sales of existing products when a new product is introduced If the rm produces the new club the earnings from the existing clubs will decrease effectively creating a cost that must be included in the decision Item c is not relevant because the costs of research and development are sunk costs Decisions made in the past cannot be changed They are not relevant to the production of the new club For tax purposes a rm would choose MACRS because it provides for larger depreciation deductions earlier These larger deductions reduce taxes but have no other cash consequences Notice that the choice between MACRS and straightline is purely a time value issue the total depreciation is the same only the timing differs It s probably only a mild oversimpli cation Current liabilities will all be paid presumably The cash portion of current assets will be retrieved Some receivables won t be collected and some inventory will not be sold of course Counterbalancing these losses is the fact that inventory sold above cost and not replaced at the end of the project s life acts to increase working capital These effects tend to offset one another Management s discretion to set the lm s capital structure is applicable at the rm level Since any one particular project could be nanced entirely with equity another project could be financed with debt and the film s overall capital structure would remain unchanged Financing costs are not relevant in the analysis of a project s incremental cash ows according to the standalone principle The EAC approach is appropriate when comparing mutually exclusive projects with different lives that will be replaced when they wear out This type of analysis is necessary so that the projects have a common life span over which they can be compared For example if one project has a threeyear life and the other has a veyear life then a 15year horizon is the minimum necessary to place the two projects on an equal footing implying that one project will be repeated ve times and the other will be repeated three times Note the shortest common life may be quite long when there are more than two alternatives andor the individual project lives are relatively long Assuming this type of analysis is valid implies that the project cash ows remain the same over the common life thus ignoring the possible effects of among other things 1 in ation 2 changing economic conditions 3 the increasing unreliability of cash ow estimates that occur far into the future and 4 the possible effects of future technology improvement that could alter the project cash ows Depreciation is a noncash expense but it is taxdeductible on the income statement Thus depreciation causes taxes paid an actual cash outflow to be reduced by an amount equal to the depreciation tax shield th A reduction in taxes that would otherwise be paid is the same thing as a cash inflow so the effects of the depreciation tax shield must be added in to get the total incremental aftertax cash ows There are two particularly important considerations The first is erosion Will the essentialized book simply displace copies of the existing book that would have otherwise been sold This is of special concern given the lower price The second consideration is competition Will other publishers step in and produce such a product If so then any erosion is much less relevant A particular concern to book publishers and producers of a variety of other product types is that the publisher only makes money from the sale of new books Thus it is important to examine whether the new book would displace sales of used books good from the publisher s perspective or new books not good The concern arises any time there is an active market for used product 139 p n O De nitely The damage to Porsche s reputation is a factor the company needed to consider If the reputation was damaged the company would have lost sales of its existing car lines 11 One company may be able to produce at lower incremental cost or market better Also of course one of the two may have made a mistake I N Porsche would recognize that the outsized pro ts would dwindle as more products come to market and competition becomes more intense Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic 1 Using the tax shield approach to calculating OCF we get OCF Sales 7 Costsl 7 tc thepreciation OCF 5 X 1900 7 220 X l900l 7 034 034l20005 OCF 432720 So the NPV of the project is NPV 7l2000 432720PVIFA1475 NPV 285563 2 We will use the bottomup approach to calculate the operating cash ow for each year We also must be sure to include the net working capital cash ows each year So the net income and total cash ow each year will be Year 1 Year 2 Year 3 Year 4 Sales 8500 9000 9500 7000 Costs 1900 2000 2200 1700 Depreciation 4000 4000 4000 4000 EBT 2600 3000 3300 1300 Tax 884 1020 1122 442 Net income 1716 1980 2178 858 OCF 5716 5980 6178 4858 Capital spending 7l6000 NWC 7200 7250 73 00 7200 950 Incremental cash ow 7 16200 5466 5680 5978 5808 140 The NPV for the project is NPV 7716200 5466 112 56801122 5978 1123 5808112quot NPV 7 115453 Using the tax shield approach to calculating OCF we get OCF Sales 7 Costs1 7tc thepreciation OCF 2050000 795000017 035 03524000003 OCF 995000 So the NPV of the project is NPV 72400000 995000PVIFA1273 NPV 71017789 The cash out ow at the beginning of the project will increase because of the spending on NWC At the end of the project the company will recover the NWC so it will be a cash in ow The sale of the equipment will result in a cash in ow but we also must account for the taxes which will be paid on this sale So the cash ows for each year of the project will be Year Cash Flow 0 7 2685000 7 72400000 7 285000 1 995000 2 995000 3 1426250 7 995000 285000 225000 0 7 22500035 And the NPV of the project is NPV 7 72685000 995000PVIFA122 1426250 1123 NPV 7 1177734 First we will calculate the annual depreciation for the equipment necessary for the project The depreciation amount each year will be Year 1 depreciation 240000003330 799200 Year 2 depreciation 240000004440 1065600 Year 3 depreciation 24000000 1480 355200 So the book value of the equipment at the end of three years which will be the initial investment minus the accumulated depreciation is Book value in 3 years 7 2400000 7 799200 1065600 355200 Book value in 3 years 180000 The asset is sold at a gain to book value so this gain is taxable A ertax salvage value 225000 180000 7 225000035 A ertax salvage value 209250 141 To calculate the OCF we will use the tax shield approach so the cash ow each year is OCF Sales 7 Costsl 7tc thepreciation Year Cash Flow 0 7 2685000 7 72400000 7 285000 1 994720 7 110000065 035799200 2 1087960 7 110000065 0351065600 3 1333570 7 110000065 035355200 209250 285000 Remember to include the NWC cost in Year 0 and the recovery of the NWC at the end of the project The NPV of the project with these assumptions is NPV 7 2685000 9947201 12 10879601122 13335701123 NPV 1966669 First we will calculate the annual depreciation of the new equipment It will be Annual depreciation charge 850000 5 Annual depreciation charge 170000 The aftertax salvage value of the equipment is Aftertax salvage value 75000l 7 035 Aftertax salvage value 48750 Using the tax shield approach the OCF is OCF 32000017 035 035170000 OCF 267500 Now we can find the project IRR There is an unusual feature that is a part of this project Accepting this project means that we will reduce NWC This reduction in NWC is a cash in ow at Year 0 This reduction in NWC implies that when the project ends we will have to increase NWC So at the end of the project we will have a cash out ow to restore the NWC to its level before the project We also must include the aftertax salvage value at the end of the project The IR of the project is NPV 0 7850000 105000 267500PVIFAIRR75 48750 7 105000 1IRR5 IRR 2201 First we will calculate the annual depreciation of the new equipment It will be Annual depreciation 420000 5 Annual depreciation 84000 Now we calculate the aftertax salvage value The aftertax salvage value is the market price minus or plus the taxes on the sale of the equipment so Aftertax salvage value MV BV 7 MVtc 142 Very often the book value of the equipment is zero as it is in this case If the book value is zero the equation for the a ertaX salvage value becomes A ertax salvage value MV 0 7 MVtc A ertax salvage value MV1 7 to We will use this equation to nd the aftertax salvage value since we know the book value is zero So the a ertaX salvage value is A ertax salvage value 600001 7 034 A ertax salvage value 39600 Using the tax shield approach we nd the OCF for the project is OCF 13500017 034 03484000 OCF 117660 Now we can nd the project NPV Notice that we include the NWC in the initial cash outlay The recovery of the NWC occurs in Year 5 along with the a ertax salvage value NPV 7420000 7 28000 117660PVIFA1075 39600 28000 115 NPV 3999825 To nd the EV at the end of four years we need to find the accumulated depreciation for the first four years We could calculate a table with the depreciation each year but an easier way is to add the MACRS depreciation amounts for each of the rst four years and multiply this percentage times the cost of the asset We can then subtract this from the asset cost Doing so we get BV4 8400000 7 840000002000 03200 01920 01150 BV4 1453200 The asset is sold at a gain to book value so this gain is taxable A ertax salvage value 1900000 1453200 7 190000035 A ertax salvage value 1743620 We will begin by calculating the initial cash outlay that is the cash ow at Time 0 To undertake the project we will have to purchase the equipment and increase net working capital So the cash outlay today for the project will be Equipment 71800000 NWC 7150000 Total 451950000 143 O Using the bottomup approach to calculating the operating cash ow we nd the operating cash ow each year will be Sales 1100000 Costs 275000 Depreciation 450000 EBT 375000 TaX 131250 Net income 243750 The operating cash ow is OCF Net income Depreciation OCF 243750 7 450000 OCF 693750 To nd the NPV of the project we add the present value of the project cash ows We must be sure to add back the net working capital at the end of the project life since we are assuming the net working capital will be recovered So the project NPV is NPV 71950000 7 693750PVIFA1674 7 150000 1164 NPV 7408148 We will need the aftertax salvage value of the equipment to compute the EAC Even though the equipment for each product has a different initial cost both have the same salvage value The aftertax salvage value for both is Both cases aftertaX salvage value 200001 7 035 13000 To calculate the EAC we first need the OCF and NPV of each option The OCF and NPV for Techron I is OCF 7 4500017 035 0352700003 2250 NPV 7270000 2250PVIFA1173 130001123 725534274 EAC 725534274 PVIFAum 710631169 And the OCF and NPV for Techron II is OCF 7 4800017 035 0353700005 75300 NPV 7370000 7 5300PVIFA1275 130001125 738172876 EAC 738172876 PVIFAum 710589527 The two milling machines have unequal lives so they can only be compared by expressing both on an equivalent annual basis which is what the EAC method does Thus you prefer the Techron 11 because it has the lower less negative annual cost 144 11 p A N Intermediate First we will calculate the depreciation each year which will be D1 53000002000 106000 D2 53000003200 169600 D3 53000001920 101760 D4 53000001150 60950 The book value of the equipment at the end of the project is BV4 530000 7 106000 169600 101760 60950 91690 The asset is sold at a loss to book value so this creates a tax refund Aftertax salvage value 70000 91690 7 70000035 7759150 So the OCF for each year will be OCFI 23000017035 035106000 18660000 OCFZ 23000017 035 035169600 20886000 OCF3 23000017035 035101760 18511600 OCF4 23000017 035 03560950 17083250 Now we have all the necessary information to calculate the project NPV We need to be careful with the NWC in this project Notice the project requires 20000 of NWC at the beginning and 3000 more in NWC each successive year We will subtract the 20000 from the initial cash ow and subtract 3000 each year from the OCF to account for this spending In Year 4 we will add back the total spent on NWC which is 29000 The 3000 spent on NWC capital during Year 4 is irrelevant Why Well during this year the project required an additional 3000 but we would get the money back immediately So the net cash ow for additional NWC would be zero With all this the equation for the NPV of the project is NPV 7 7 530000 7 20000 7 186600 7 30001 14 7 208860 7 30001142 7 185116 7 30001143 7 17083250 7 29000 7 7759150114 NPV 7 5663561 If we are trying to decide between two projects that will not be replaced when they wear out the proper capital budgeting method to use is NPV Both projects only have costs associated with them not sales so we will use these to calculate the NPV of each project Using the tax shield approach to calculate the OCF the NPV of System A is OCFA 710500017 034 0343600004 OCFA 738700 vaA 7 7360000 7 38700Pv1FA11 vaA 7 748006465 145 DJ 1quot And the NPV of System B is OCFB 76500017 034 0344800006 OCFB 715700 NPVB 7480000 7 15700PVIFA1135 NPVB 754641944 If the system will not be replaced when it wears out then System A should be chosen because it has the less negative NPV If the equipment will be replaced at the end of its useful life the correct capital budgeting technique is EAC Using the NPVs we calculated in the previous problem the EAC for each system is EACA 7 48006464 PVIFAIMA EACA 715473749 EACB 7 54641944 PVIFAHWS EACB 7129 16075 If the conveyor belt system will be continually replaced we should choose System B since it has the less negative EAC Since we need to calculate the EAC for each machine sales are irrelevant EAC only uses the costs of operating the equipment not the sales Using the bottom up approach or net income plus depreciation method to calculate OCF we get Machine A lVIachine B Variable costs 73675000 73150000 Fixed costs 7180000 7110000 Depreciation 4100 000 00 000 EBT 7425 5000 73860000 Tax 1489250 1351000 Net income 72765750 72509000 Depreciation 400 000 600000 OCF 72365750 7 4 1909000 The NPV and EAC for Machine A is vaA 7 2400000 7 2365750PVIFA1076 vaA 7 71270345800 EACA 7 7 1270345800 PVIFAIW EACA 7 7291680771 146 p A 5 And the NPV and EAC for Machine B is vaB 5400000 7 1909000PVIFA109 NPvB 71639397647 EACB 7 1639397647 PVIFAmm EACB 284665891 You should choose Machine B since it has a less negative EAC When we are dealing with nominal cash ows we must be careful to discount cash ows at the nominal interest rate and we must discount real cash ows using the real interest rate Project A s cash ows are in real terms so we need to nd the real interest rate Using the Fisher equation the real interest rate is 1R1r1h 115 lrl 04 r 1058 or 1058 So the NPV of Project A s real cash ows discounting at the real interest rate is NPV 750000 30000 11058 25000 110582 20000 110583 NPV 1236889 Project B s cash ow are in nominal terms so the NPV discounted at the nominal interest rate is NPV 65000 29000 115 380001152 41000 1153 NPV 1590902 We should accept Project B if the projects are mutually exclusive since it has the highest NPV To determine the value of a firm we can simply nd the present value of the film s future cash ows No depreciation is given so we can assume depreciation is zero Using the tax shield approach we can nd the present value of the aftertax revenues and the present value of the aftertaX costs The required return growth rates price and costs are all given in real terms Subtracting the costs from the revenues will give us the value of the rm s cash ows We must calculate the present value of each separately since each is growing at a different rate First we will nd the present value of the revenues The revenues in year 1 will be the number of bottles sold times the price per bottle or Aftertax revenue in year 1 in real terms 2100000 gtlt 1251 7 034 Aftertax revenue in year 1 in real terms 1732500 Revenues will grow at six percent per year in real terms forever Apply the growing perpetuity formula we nd the present value of the revenues is PV of revenues C1 R 8 PV ofrevenues 1732500 010 7 006 PV of revenues 43312500 147 gt1 The real aftertax costs in year 1 will be Aftertax costs in year 1 in real terms 2100000 gtlt 07517 034 Aftertax costs in year 1 in real terms 1039500 Costs will grow at ve percent per year in real terms forever Applying the growing perpetuity formula we nd the present value of the costs is PV of costs C1 R 7g PV of costs 1039500 010 7 005 PV of costs 20790000 Now we can find the value of the rm which is Value of the firm PV of revenues 7 PV of costs Value of the rm 43312500 7 20790000 Value of the rm 22522500 To calculate the nominal cash ows we simple increase each item in the income statement by the in ation rate except for depreciation Depreciation is a nominal cash ow so it does not need to be adjusted for in ation in nominal cash ow analysis Since the resale value is given in nominal terms as of the end of year 5 it does not need to be adjusted for in ation Also no in ation adjustment is needed for either the depreciation charge or the recovery of net working capital since these items are already expressed in nominal terms Note that an increase in required net working capital is a negative cash ow whereas a decrease in required net working capital is a positive cash ow We rst need to calculate the taxes on the salvage value Remember to calculate the taxes paid or tax credit on the salvage value we take the book value minus the market value times the tax rate which in this case would be Taxes on salvage value BV 7MVtc Taxes on salvage value 0 7 4000034 Taxes on salvage value 7 13600 So the nominal aftertax salvage value is Market price 40000 Tax on sale 713600 Aftertax salvage value 26400 148 on 0 Now we can find the nominal cash ows each year using the income statement Doing so we nd M Year 1 Year 2 Year 3 Year 4 Year 5 Sales 23 0000 23 6900 244007 251327 25 8867 Expenses 60000 61800 63654 65564 67531 Depreciation 61000 61000 61000 61000 61000 EBT 109000 114100 119353 124764 130336 Tax 37060 38794 405 80 42420 44314 Net income 71940 75306 78773 82344 86022 OCF 132940 136306 139773 143344 147022 Capital spending 7305000 26400 NWC 710000 10000 Total cash ow 7315000 132940 136306 139773 143344 183422 The present value of the company is the present value of the future cash ows generated by the company Here we have real cash ows a real interest rate and a real growth rate The cash ows are a growing perpetuity with a negative growth rate Using the growing perpetuity equation the present value of the cash ows are PV c1 R 7 g PV 155000 11 7 705 PV 96875000 To nd the EAC we first need to calculate the NPV of the incremental cash ows We will begin with the aftertax salvage value which is Taxes on salvage value BV iMVtC Taxes on salvage value 0 7 1500034 Taxes on salvage value 75100 Market price 15000 Tax on sale 75100 A ertax salvage value 9900 Now we can nd the operating cash ows Using the tax shield approach the operating cash ow each year will be OCF 7750017 034 034630003 OCF 2190 So the NPV of the cost of the decision to buy is NPV 763000 2190PVIFA1273 99001123 NPV 5069337 149 O In order to calculate the equivalent annual cost set the NPV of the equipment equal to an annuity with the same economic life Since the project has an economic life of three years and is discounted at 12 percent set the NPV equal to a threeyear annuity discounted at 12 percent EAC 75069337 PVIFAIMJ EAC 72110613 We will calculate the aftertax salvage value first The aftertax salvage value of the equipment will be Taxes on salvage value BV iMVtC Taxes on salvage value 0 7 8000034 Taxes on salvage value 72720 Market price 80000 Tax on sale 727200 A ertax salvage value 52800 Next we will calculate the initial cash outlay that is the cash ow at Time 0 To undertake the project we will have to purchase the equipment The new project will decrease the net working capital so this is a cash in ow at the beginning of the project So the cash outlay today for the project will be Equipment 7450000 NWC 90000 Total 73 60000 Now we can calculate the operating cash ow each year for the project Using the bottom up approach the operating cash ow will be Saved salaries 140000 Depreciation 90000 EBT 50000 Taxes 17000 Net income 33000 And the OCF will be OCF 33000 90000 OCF 123000 Now we can nd the NPV of the project In Year 5 we must replace the saved NWC so NPV 7360000 123000PVIFA1275 52800 7 90000 1125 NPV 6227919 150 21 Replacement decision analysis is the same as the analysis of two competing projects in this case keep the current equipment or purchase the new equipment We will consider the purchase of the new machine first Purchase new machine The initial cash outlay for the new machine is the cost of the new machine plus the increased net working capital So the initial cash outlay will be Purchase new machine 712000000 Net working capital 7250000 Total 12250000 Next we can calculate the operating cash ow created if the company purchases the new machine The saved operating expense is an incremental cash ow Additionally the reduced operating expense is a cash in ow so it should be treated as such in the income statement The pro forma income statement and adding depreciation to net income the operating cash ow created by purchasing the new machine each year will be Operating expense 4500000 Depreciation 3000000 EBT 1500000 Taxes 585000 Net income 915000 OCF 3915000 So the NPV of purchasing the new machine including the recovery of the net working capital is NPV 712250000 3915000PVIFA1074 500000 1104 NPV 33077659 And the IR is 0 712250000 3915000PVIFARR74 250000 1 IRR4 Using a spreadsheet or financial calculator we nd the IR is IRR 1123 151 Now we can calculate the decision to keep the old machine Keep old machine The initial cash outlay for the old machine is the market value of the old machine including any potential tax consequence The decision to keep the old machine has an opportunity cost namely the company could sell the old machine Also if the company sells the old machine at its current value it will receive a tax benefit Both of these cash ows need to be included in the analysis So the initial cash ow of keeping the old machine will be Keep machine 73000000 Taxes 7390000 Total 73390000 Next we can calculate the operating cash ow created if the company keeps the old machine There are no incremental cash ows from keeping the old machine but we need to account for the cash ow effects of depreciation The income statement adding depreciation to net income to calculate the operating cash ow will be Depreciation 1000000 EBT 7 1000000 Taxes 7390000 Net income 7 610000 OCF 390000 So the NPV of the decision to keep the old machine will be NPV 3390000 390000Pv1FA104 NPV 215375248 And the IR is 0 73390000 390000PVIFARR74 Using a spreadsheet or financial calculator we nd the IR is IRR72515 152 There is another way to analyze a replacement decision that is often used It is an incremental cash ow analysis of the change in cash ows from the existing machine to the new machine assuming the new machine is purchased In this type of analysis the initial cash outlay would be the cost of the new machine the increased NWC and the cash in ow including any applicable taxes of selling the old machine In this case the initial cash ow under this method would be Purchase new machine 7l2000000 Net working capital 7250000 Sell old machine 3000000 Taxes on old machine 390000 Total 78860000 The cash ows from purchasing the new machine would be the saved operating expenses We would also need to include the change in depreciation The old machine has a depreciation of 1 million per year and the new machine has a depreciation of 3 million per year so the increased depreciation will be 2 million per year The pro forma income statement and operating cash ow under this approach will be Operating expense savings 4500000 Depreciation 2000000 EBT 2500000 Taxes 975000 Net income 1525000 OCF 3525000 The NPV under this method is NPV 8860000 3525000PV1FA104 250000 1104 NPV 248452906 And the IR is 0 78860000 3525000PVIFARR74 250000 l IRR4 Using a spreadsheet or financial calculator we nd the IR is IRR 2226 So this analysis still tells us the company should purchase the new machine This is really the same type of analysis we originally did Consider this Subtract the NPV of the decision to keep the old machine from the NPV of the decision to purchase the new machine You will get Differential NPV 33077659 7 215375248 248452906 This is the exact same NPV we calculated when using the second analysis method 153 22 We can nd the NPV of a project using nominal cash ows or real cash ows Either method will result in the same NPV For this problem we will calculate the NPV using both nominal and real cash ows The initial investment in either case is 150000 since it will be spent today We will begin with the nominal cash ows The revenues and production costs increase at different rates so we must be careful to increase each at the appropriate growth rate The nominal cash ows for each year will be M m M Year 3 Revenues 7000000 7350000 7717500 Costs 2000000 2120000 2247200 Depreciation 2142857 2142857 2142857 EBT 2857143 3087143 3327443 Taxes 971429 1049629 1131331 Net income 1885714 2037514 2196112 OCF 4028571 4180371 4338969 Capital spending 7150000 Total cash ow 7150000 4028571 4180371 4338969 Year 4 Year 5 Year 6 Year 7 Revenues 8103375 8508544 8933971 9380669 Costs 2382032 2524954 2676451 2837038 D l 39 quot 2142857 2142857 2142857 2142857 EBT 3578486 3840733 4114663 4400774 Taxes 1216685 1305849 1398985 1496263 Net income 2361801 2534884 2715677 2904511 OCF 4504658 4677741 4858534 5047368 Capital spending Totalcash ow 4504658 4677741 4858534 5047368 Now that we have the nominal cash ows we can nd the NPV We must use the nominal required return with nominal cash ows Using the Fisher equation to nd the nominal required return we get 1R1r1h 1 12 1 081 05 R 1340 or 1340 154 03 So the NPV of the project using nominal cash ows is NPV 7150000 4028571 11340 4180371 113402 4338969 113403 4504658 113404 4677741113405 4858534113406 5047368113407 NPV 4374888 We can also nd the NPV using real cash ows and the real required return This will allow us to nd the operating cash ow using the tax shield approach Both the revenues and expenses are growing annuities but growing at different rates This means we must nd the present value of each separately We also need to account for the effect of taxes so we will multiply by one minus the tax rate So the present value of the aftertax revenues using the growing annuity equation is PV ofaftertax revenues C 1r7g 7 1r7g X 1 g1 r 17tc PV ofa ertax revenues 700001134 7 05 7 1 134 705 X 105113471734 PV of aftertax revenues 22908028 And the present value of the aftertax costs will be PV ofaftertax costs C 1r7g 7 1r 7g X 1gl rt1 itc PV ofaftertax costs 200001 134 7 06 7 1 134 706 x 1 061 13471 7 34 PV of aftertax costs 6715607 Now we need to find the present value of the depreciation tax shield The depreciation amount in the rst year is a real value so we can nd the present value of the depreciation tax shield as an ordinary annuity using the real required return So the present value of the depreciation tax shield will be PV of depreciation tax shield 150000734PVIFA13 40 PV of depreciation tax shield 3182467 Using the present value of the real cash ows to nd the NPV we get NPV Initial cost PV of revenues 7 PV of costs PV of depreciation tax shield NPV 7150000 22908028 7 6715607 7 3182467 NPV 4374888 Notice the NPV using nominal cash ows or real cash ows is identical which is what we would expect Here we have a project in which the quantity sold each year increases First we need to calculate the quantity sold each year by increasing the current year s quantity by the growth rate So the quantity sold each year will be Year 1 quantity 6000 Year 2 quantity 60001 08 6480 Year 3 quantity 64801 08 6998 Year 4 quantity 6998108 7558 Year 5 quantity 7558108 8163 155 Now we can calculate the sales revenue and variable costs each year The pro forma income statements and operating cash ow each year will be M Year 1 Year 2 Year 3 Year 4 Year 5 Revenues 28800000 31104000 33592320 36279706 391820 82 Fixed costs 8000000 8000000 8000000 8000000 8000000 Variable costs 12000000 12960000 13996800 15116544 16325868 Depreciation 2900000 2900000 2900000 2900000 2900000 EBT 5900000 7244000 8695520 10263162 11956215 Taxes 2006000 2462960 2956477 3489475 4065113 Net income 3 894000 4781040 5739043 67736 87 7891102 OCF 6794000 7681040 8639043 96736 87 10791102 Capital spending 7145000 NWC 728000 28000 Total cash ow 717300000 6794000 7681040 8639043 9673687 13591102 So the NPV of the project is NPV 7 173000 67940 125 7681040 1252 8639043 1253 9673687 125quot 135911021255 NPV 7 5890130 We could also have calculated the cash ows using the tax shield approach with growing annuities and ordinary annuities The sales and variable costs increase at the same rate as sales so both are growing annuities The xed costs and depreciation are both ordinary annuities Using the growing annuity equation the present value of the revenues is PV ofrevenues C 1r 7g71r7g X 1g1 r 17tc PV ofrevenues 288000125 708 7 125 708 X 1081255 PV of revenues 87845180 And the present value of the variable costs will be PV ofvariable costs C 1r 7g 7 1r 7g X 1 g1 r 17tc PV ofvariable costs 120000125 7 08 7 125 708 X 1 081255 PV ofvariable costs 36602158 The xed costs and depreciation are both ordinary annuities The present value of each is PV of xed costs C1711 r r PV offixed costs 80000PVIFA255 PV offixed costs 21514240 156 1quot PV ofdepreciation Cl 7 ll r r PV of depreciation 29000PVIFA2575 PV of depreciation 7798912 Now we can use the depreciation tax shield approach to nd the NPV of the project which is NPV 7173000 8784518 7 36602158 7 215142401 7 34 779891234 28000 1255 NPV 5890130 We will begin by calculating the aftertax salvage value of the equipment at the end of the project s life The aftertax salvage value is the market value of the equipment minus any taxes paid or refunded so the aftertax salvage value in four years will be Taxes on salvage value BV 7MVtc Taxes on salvage value 0 7 40000034 Taxes on salvage value 7l52000 Market price 400000 Tax on sale 7152000 A ertax salvage value 248000 Now we need to calculate the operating cash ow each year Note we assume that the net working capital cash ow occurs immediately Using the bottom up approach to calculating operating cash ow we nd M m M M M Revenues 2418000 2964000 2808000 1950000 Fixed costs 425000 425000 425000 425000 Variable costs 362700 444600 421200 292500 Depreciation 1398600 1864800 621600 310800 EBT 231700 229600 1340200 921700 Taxes 88046 87248 509276 350246 Net income 143654 142352 830924 571454 OCF 1542254 2007152 1452524 882254 Capital spending 74200000 248000 Land 7800000 900000 NWC 7120000 120000 Total cash ow 75120000 1542254 2007152 1452524 2150254 157 UI Notice the calculation of the cash ow at time 0 The capital spending on equipment and investment in net working capital are cash out ows The aftertax selling price of the land is also a cash out ow Even though no cash is actually spent on the land because the company already owns it the aftertax cash ow from selling the land is an opportunity cost so we need to include it in the analysis With all the project cash ows we can calculate the NPV which is NPV 75120000 1542254 113 2007152 1132 1452524 1133 2150254 1134 NPV 14218402 The company should accept the new product line Replacement decision analysis is the same as the analysis of two competing projects in this case keep the current equipment or purchase the new equipment We will consider the purchase of the new machine first Purchase new machine The initial cash outlay for the new machine is the cost of the new machine We can calculate the operating cash ow created if the company purchases the new machine The maintenance cost is an incremental cash ow so using the pro forma income statement and adding depreciation to net income the operating cash ow created by purchasing the new machine each year will be Maintenance cost 350000 Depreciation 600000 EBT 7950000 Taxes 7323000 Net income 7627000 OCF 727000 Notice the taxes are negative implying a tax credit The new machine also has a salvage value at the end of ve years so we need to include this in the cash ows analysis The aftertax salvage value will be Sell machine 500000 Taxes 7170000 Total 330000 The NPV of purchasing the new machine is NPV 73000000 7 27000Pv1FA125 330000 1125 NPV 291007810 158 Notice the NPV is negative This does not necessarily mean we should not purchase the new machine In this analysis we are only dealing with costs so we would expect a negative NPV The revenue is not included in the analysis since it is not incremental to the machine Similar to an EAC analysis we will use the machine with the least negative NPV Now we can calculate the decision to keep the old machine Keep old machine The initial cash outlay for the keeping the old machine is the market value of the old machine including any potential tax The decision to keep the old machine has an opportunity cost namely the company could sell the old machine Also if the company sells the old machine at its current value it will incur taxes Both of these cash ows need to be included in the analysis So the initial cash ow of keeping the old machine will be Keep machine 71800000 Taxes 204000 Total 451596 000 Next we can calculate the operating cash ow created if the company keeps the old machine We need to account for the cost of maintenance as well as the cash ow effects of depreciation The pro forma income statement adding depreciation to net income to calculate the operating cash ow will be Maintenance cost 520000 Depreciation 240000 EBT 7760000 Taxes 7258400 Net income 7501600 OCF 726l600 The old machine also has a salvage value at the end of ve years so we need to include this in the cash ows analysis The aftertax salvage value will be Sell machine 200000 Taxes 768000 Total 132000 So the NPV of the decision to keep the old machine will be NPV 7l596000 7 261600PVIFA125 132000 1125 NPV 7246410911 The company should keep the old machine since it has a greater NPV There is another way to analyze a replacement decision that is often used It is an incremental cash ow analysis of the change in cash ows from the existing machine to the new machine assuming the new machine is purchased In this type of analysis the initial cash outlay would be the cost of the new machine and the cash in ow including any applicable taxes of selling the old machine In this case the initial cash ow under this method would be 159 Purchase new machine 73000000 Sell old machine 1800000 Taxes on old machine 7204000 Total 71404000 The cash ows from purchasing the new machine would be the difference in the operating expenses We would also need to include the change in depreciation The old machine has a depreciation of 240000 per year and the new machine has a depreciation of 600000 per year so the increased depreciation will be 360000 per year The pro forma income statement and operating cash ow under this approach will be Maintenance cost 7170000 Depreciation 360000 EBT 7190000 Taxes 44600 Net income 7125400 OCF 23 4600 The salvage value of the differential cash ow approach is more complicated The company will sell the new machine and incur taxes on the sale in ve years However we must also include the lost sale of the old machine Since we assumed we sold the old machine in the initial cash outlay we lose the ability to sell the machine in five years This is an opportunity loss that must be accounted for So the salvage value is Sell machine 500000 Taxes 7170000 Lost sale of old 7200000 Taxes on lost sale of old 68000 Total 198000 The NPV under this method is NPV 71404000 234600PVIFA125 198000 1125 NPV 744596899 So this analysis still tells us the company should not purchase the new machine This is really the same type of analysis we originally did Consider this Subtract the NPV of the decision to keep the old machine from the NPV of the decision to purchase the new machine You will get Differential NPV 291007810 7 246410911 44596899 This is the exact same NPV we calculated when using the second analysis method 160 26 Here we are comparing two mutually exclusive assets with in ation Since each will be replaced N 1 when it wears out we need to calculate the EAC for each We have real cash ows Similar to other capital budgeting projects when calculating the EAC we can use real cash ows with the real interest rate or nominal cash ows and the nominal interest rate Using the Fisher equation to nd the real required return we get 1 R1 r1 11 1 14 1 rl 05 r 0857 or 857 This is the interest rate we need to use with real cash ows We are given the real aftertax cash ows for each asset so the NPV for the XX40 is va 781500 7 8120PVIFA85W NPV 7180609 So the EAC for the XX40 is 7180609 EACPVIFA8 573 EAC 770806 And the EAC for the RH45 is va 782300 7 8150Pv1FA85W NPV 48288999 7288999 EACPVIFA8 5775 EAC 773475 The company should choose the XX40 because it has the greater EAC The project has a sales price that increases at 5 percent per year and a variable cost per unit that increases at 6 percent per year First we need to nd the sales price and variable cost for each year The table below shows the price per unit and the variable cost per unit each year Year 1 Year 2 Year 3 Year 4 Year 5 Sales price 4000 4200 4410 4631 4862 Cost per unit 2000 2120 2247 2382 2525 Using the sales price and variable cost we can now construct the pro forma income statement for each year We can use this income statement to calculate the cash ow each year We must also make sure to include the net working capital outlay at the beginning of the project and the recovery of the net working capital at the end of the project The pro forma income statement and cash ows for each year will be 161 M Year 1 Year 2 Year 3 Year 4 Year 5 Revenues 60000000 63000000 66150000 69457500 72930375 Fixed costs 7500000 7500000 7500000 7500000 7500000 Variable costs 30000000 31800000 33708000 35730480 37874309 Depreciation 10600000 10600000 10600000 10600000 10600000 EBT 11900000 13100000 14342000 15627020 16956066 Taxes 4046000 4454000 4876280 5313187 5765063 Net income 7854000 8646000 9465720 10313833 11191004 OCF 18454000 19246000 20065720 20913833 21791004 Capital spending 7530000 NWC 25000 25000 Total cash ow 7555000 18454000 19246000 20065720 20913833 24291004 With these cash flows the NPV of the project is NPV7 24291004 1155 NPV 7 12327708 7555000 184540 115 1924601152 200657201153 209138331154 We could also answer this problem using the depreciation tax shield approach The revenues and variable costs are growing annuities growing at different rates The xed costs and depreciation are ordinary annuities Using the growing annuity equation the present value of the revenues is PV ofrevenues C lr 7g7lr7g X 1 g1 rt1tc PV ofrevenues 7 600000115 705 7 115 705 x 1051155 PV of revenues 219277500 And the present value of the variable costs will be PV ofvariable costs C lr 7g 7 lr 7g X 1 gl r l 7tc PV ofvariable costs 7 300000115 7 06 7 1 15 706 x 1 061 7155 PV ofvariable costs 111555125 The xed costs and depreciation are both ordinary annuities The present value of each is PV of xed costs Cl 7 ll r r PV offixed costs 75000lt171l155 15 PV offixed costs 25141163 PV ofdepreciation C1711 r r PV ofdepreciation 106000l 7l1 155 15 PV of depreciation 35532844 162 on Now we can use the depreciation tax shield approach to nd the NPV of the project which is NPV 7555000 2192775 7111555125 7 251411631 7 34 3553284434 250001155 NPV 12327708 Challenge This is an indepth capital budgeting problem Probably the easiest OCF calculation for this problem is the bottom up approach so we will construct an income statement for each year Beginning with the initial cash ow at time zero the project will require an investment in equipment The project will also require an investment in NWC of 1500000 So the cash ow required for the project today will be Capital spending 718000000 Change in NWC 71500000 Total cash ow 719500000 Now we can begin the remaining calculations Sales gures are given for each year along with the price per unit The variable costs per unit are used to calculate total variable costs and xed costs are given at 700000 per year To calculate depreciation each year we use the initial equipment cost of 18 million times the appropriate MACRS depreciation each year The remainder of each income statement is calculated below Notice at the bottom of the income statement we added back depreciation to get the OCF for each year The section labeled Net cash ows will be discussed below 163 Year 1 2 3 4 5 Ending book value 15426000 11016000 7866000 5616000 4014000 Sales 28275000 30550000 38350000 35425000 30875000 Variable costs 20880000 22560000 28320000 26160000 22800000 Fixed costs 700000 700000 700000 700000 700000 D J 39 quot 2574000 4410000 3150000 2250000 1602000 EBIT 4121000 2880000 6180000 6315000 5773000 Taxes 1442350 1008000 2163000 2210250 2020550 Net income 2678650 1872000 4017000 4104750 3752450 D J 39 quot 2574000 4410000 3150000 2250000 1602000 Operating cash ow 5252650 6282000 7167000 6354750 5354450 Net cash ows Operating cash ow 5252650 6282000 7167000 6354750 5354450 Change in NWC 7341250 71170000 438750 682500 1890000 Capital spending 3744900 Total cash ow 4911400 5112000 7605750 7037250 10989350 After we calculate the OCF for each year we need to account for any other cash ows The other cash ows in this case are NWC cash ows and capital spending which is the aftertax salvage of the equipment The required NWC is 15 percent of the sales increase in the next year We will work through the NWC cash ow for Year 1 The total NWC in Year 1 will be 15 percent of sales increase from Year 1 to Year 2 or Increase in NWC for Year 1 1530550000 7 28275000 Increase in NWC for Year 1 341250 Notice that the NWC cash ow is negative Since the sales are increasing we will have to spend more money to increase NWC In Year 4 the NWC cash ow is positive since sales are declining And in Year 5 the NWC cash ow is the recovery of all NWC the company still has in the project To calculate the aftertax salvage value we first need the book value of the equipment The book value at the end of the five years will be the purchase price minus the total depreciation So the ending book value is Ending book value 18000000 7 2574000 4410000 3150000 2250000 1602000 Ending book value 4014000 164 2 O The market value of the used equipment is 20 percent of the purchase price or 36 million so the aftertax salvage value will be A ertax salvage value 3600000 4014000 7 360000035 A ertax salvage value 3744900 The aftertax salvage value is included in the total cash ows are capital spending Now we have all of the cash ows for the project The NPV of the project is NPV 7 719500000 4911400118 51120001182 76057501183 7037250118 109893501185 NPV 7 139593788 And the IR is NPV 7 0 7 719500000 49114001 IRR 51120001 1RR2 76057501 IRR3 70372501 IRR 109893501 IRR5 IR 7 2072 We should accept the project To nd the initial pretax cost savings necessary to buy the new machine we should use the tax shield approach to nd the OCF We begin by calculating the depreciation each year using the MACRS depreciation schedule The depreciation each year is D1 54000003330 179820 54000004440 237760 54000001480 79920 D4 54000000740 39960 D2 D3 Using the tax shield approach the OCF each year is OCF1 7 s 7 C17 035 035179820 OCF2 7 s 7 C17 035 035237760 OCF3 7 s 7 C17 035 03579920 OCF 7 s 7 C17 035 03539960 OCFs 7 s 7 C17 035 Now we need the aftertax salvage value of the equipment The aftertax salvage value is Aftertax salvage value 50000l 7 035 32500 To find the necessary cost reduction we must realize that we can split the cash ows each year The OCF in any given year is the cost reduction S 7 C times one minus the tax rate which is an annuity for the project life and the depreciation tax shield To calculate the necessary cost reduction we would require a zero NPV The equation for the NPV of the project is NPV 7 0 7 7 540000 7 45000 s 7 C065PVIFA1275 0351798201 12 2377601122 799201123 39960112quot 45000 325001125 165 03 O Solving this equation for the sales minus costs we get s 7 C065Pv1FAu5 7 38913507 s 7 C 7 16607670 To nd the bid price we need to calculate all other cash ows for the project and then solve for the bid price The a ertax salvage value of the equipment is A ertax salvage value 60000l 7 035 39000 Now we can solve for the necessary OCF that will give the project a zero NPV The equation for the NPV of the project is NPV 0 7 830000 7 75000 OCFPVIFA1475 75000 39000 1145 Solving for the OCF we nd the OCF that makes the project NPV equal to zero is OCF 84579197 PVIFAIMYS 24636529 The easiest way to calculate the bid price is the tax shield approach so OCF 24636529 P 7vQ 7FC17tcth 24636529 7 P 7 850130000 7 21000017 035 0358300005 P 7 1234 a This problem is basically the same as the previous problem except that we are given a sales price The cash ow at Time 0 for all three parts of this question will be Capital spending 7830000 Change in NWC 775000 Total cash ow 7905000 We will use the initial cash ow and the salvage value we already found in that problem Using the bottom up approach to calculating the OCF we get Assume price per unit 14 and unitsyear 130 000 Year 3 4 5 Sales 1820000 1820000 1820000 1820000 1820000 Variable costs 1105000 1105000 1105000 1105000 1105000 Fixed costs 210000 210000 210000 210000 210000 I 39 quot 166000 166000 166000 166000 166000 EBIT 339000 339000 339000 339000 339000 Taxes 35 118650 118650 118650 118650 118650 Net Income 220350 220350 2203 50 220350 2203 50 J 39 39 166000 166000 166000 166000 166000 Operating CF 3 86350 386350 3 86350 386350 386350 166 Year 1 2 i A i Operating CF 386350 386350 386350 386350 386350 Change in NWC 0 0 0 0 75000 Capital spending 0 0 0 0 39000 Total CF 386350 386350 386350 386350 500350 With these cash ows the NPV of the project is NPV 7 7 830000 7 75000 386350PVIFA145 75000 39000 1145 NPV 7 48057886 If the actual price is above the bid price that results in a zero NPV the project will have a positive NPV As for the cartons sold if the number of cartons sold increases the NPV will increase and if the costs increase the NPV will decrease b To nd the minimum number of cartons sold to still breakeven we need to use the tax shield approach to calculating OCF and solve the problem similar to finding a bid price Using the initial cash ow and salvage value we already calculated the equation for a zero NPV of the project is NPV 0 7 830000 7 75000 OCFPVIFA1475 75000 39000 1145 So the necessary OCF for a zero NPV is OCF 84579197 PVIFA145 24636529 Now we can use the tax shield approach to solve for the minimum quantity as follows OCF 24636529 P 7vQ 7 FC 1 7tc th 24636529 1400 7 850Q 7 210000 1 7 035 0358300005 Q 90843 As a check we can calculate the NPV of the project with this quantity The calculations are M l 2 i A 2 Sales 1271808 1271808 1271808 1271808 1271808 Variable costs 772169 772169 772169 772169 772169 Fixed costs 210000 210000 210000 210000 210000 1 l 39 739 166000 166000 166000 166000 166000 EBIT 123639 123639 123639 123639 123639 Taxes 35 43274 43274 43274 43274 43274 Net Income 80365 80365 80365 80365 80365 1 J 39 739 166000 166000 166000 166000 166000 Operating CF 246365 246365 246365 246365 246365 167 Year 1 2 i A 2 Operating CF 246365 246365 246365 246365 246365 Change in NWC 0 0 0 0 75000 Capital spending 0 0 0 0 39000 Total CF 246365 246365 246365 246365 360365 NPV 7 830000 7 75000 246365PVIFA145 75000 39000 1145 m 0 Note that the NPV is not exactly equal to zero because we had to round the number of cartons sold you cannot sell onehalf of a carton To nd the highest level of xed costs and still breakeven we need to use the tax shield approach to calculating OCF and solve the problem similar to nding a bid price Using the initial cash ow and salvage value we already calculated the equation for a zero NPV of the project is NPV 0 7 830000 775000 OCFPVIFA1475 75000 39000 1145 OCF 84579197 PVIFA145 24636529 Notice this is the same OCF we calculated in part b Now we can use the tax shield approach to solve for the maximum level of xed costs as follows OCF 24636529 P7vQ 7FC 1 2 tc 4 tCD 24636529 1400 7 850130000 7 FC17 035 0358300005 FC 42536110 As a check we can calculate the NPV of the project with this quantity The calculations are M l 2 i A 2 Sales 1820000 1820000 1820000 1820000 1820000 Variable costs 1105000 1105000 1105000 1105000 1105000 Fixed costs 425361 425361 425361 425361 425361 I 39 39 166000 166000 166000 166000 166000 EBIT 123639 123639 123639 123639 123639 Taxes 35 43274 43274 43274 43274 43274 Net Income 80365 80365 80365 80365 80365 A 39 39 166000 166000 166000 166000 166000 Operating CF 246365 246365 246365 246365 246365 Year A 2 i A i Operating CF 246365 246365 246365 246365 246365 Change in NWC 0 0 0 0 75000 Capital spending 0 0 0 0 39000 Total CF 246365 246365 246365 246365 360365 NPV 7 830000 7 75000 246365PVIFA145 75000 39000 114514 0 168 32 We need to nd the bid price for a project but the project has extra cash ows Since we don t already produce the keyboard the sales of the keyboard outside the contract are relevant cash ows Since we know the extra sales number and price we can calculate the cash ows generated by these sales The cash ow generated from the sale of the keyboard outside the contract is Year 1 M M Year 4 Sales 1100000 3300000 3850000 1925000 Variable costs 660000 1980000 2310000 1155000 EBT 440000 1320000 1540000 770000 Tax 176000 528000 616000 308000 Net income and OCF 264000 792000 924000 462000 So the addition to NPV of these market sales is NPV of market sales 2640001 13 7920001132 9240001133 4620001134 NPV of market sales 177761209 You may have noticed that we did not include the initial cash outlay depreciation or xed costs in the calculation of cash ows from the market sales The reason is that it is irrelevant whether or not we include these here Remember that we are not only trying to determine the bid price but we are also determining whether or not the project is feasible In other words we are trying to calculate the NPV of the project not just the NPV of the bid price We will include these cash ows in the bid price calculation The reason we stated earlier that whether we included these costs in this initial calculation was irrelevant is that you will come up with the same bid price if you include these costs in this calculation or if you include them in the bid price calculation Next we need to calculate the a ertax salvage value which is A ertax salvage value 2000001 7 40 120000 Instead of solving for a zero NPV as is usual in setting a bid price the company president requires an NPV of 100000 so we will solve for a NPV of that amount The NPV equation for this project is remember to include the NWC cash ow at the beginning of the project and the NWC recovery at the end NPV 100000 7 73200000 7 75000 177761209 OCF PVIFABW 120000 75000 1134 Solving for the OCF we get OCF 147779075 PVIFADW 49682468 Now we can solve for the bid price as follows OCF 49682468 P 7vQ 7 FC 17tc th 47125344 7 P 7 1659000 7 6000001 7 040 04032000004 P 7 264 41 169 Since the two computers have unequal lives the correct method to analyze the decision is the EAC We will begin with the EAC of the new computer Using the depreciation tax shield approach the OCF for the new computer system is OCF 1250001 7 38 780000 538 136780 Notice that the costs are positive which represents a cash in ow The costs are positive in this case since the new computer will generate a cost savings The only initial cash ow for the new computer is cost of 780000 We next need to calculate the aftertax salvage value which is A ertax salvage value 140000l 7 38 86800 Now we can calculate the NPV of the new computer as NPV 7780000 136780PVIFA1475 86800 1145 NPV 726534199 And the EAC of the new computer is EAC 7 26534199 PVIFAIMYS 77728975 Analyzing the old computer the only OCF is the depreciation tax shield so OCF 13000038 49400 The initial cost of the old computer is a little trickier You might assume that since we already own the old computer there is no initial cost but we can sell the old computer so there is an opportunity cost We need to account for this opportunity cost To do so we will calculate the aftertax salvage value of the old computer today We need the book value of the old computer to do so The book value is not given directly but we are told that the old computer has depreciation of 130000 per year for the next three years so we can assume the book value is the total amount of depreciation over the remaining life of the system or 390000 So the aftertax salvage value of the old computer is A ertax salvage value 230000 390000 7 23000038 290800 This is the initial cost of the old computer system today because we are forgoing the opportunity to sell it today We next need to calculate the aftertax salvage value of the computer system in two years since we are buying it today The aftertax salvage value in two years is A ertax salvage value 90000 130000 7 9000038 105200 Now we can calculate the NPV of the old computer as NPV 7290800 49400PVIFA1472 105200 1142 NPV 712850699 170 And the EAC of the old computer is EAC 7 12850699 PVIFA142 77804097 If we are going to replace the system in two years no matter what our decision today we should instead replace it today since the EAC is lower b If we are only concerned with whether or not to replace the machine now and are not worrying about what will happen in two years the correct analysis is NPV To calculate the NPV of the decision on the computer system now we need the difference in the total cash ows of the old computer system and the new computer system From our previous calculations we can say the cash ows for each computer system are E New computer Old computer Difference 0 7780000 290800 7489200 1 136780 419400 873 80 2 136780 7154600 717820 3 136780 0 136780 4 136780 0 136780 5 223580 0 223580 Since we are only concerned with marginal cash ows the cash ows of the decision to replace the old computer system with the new computer system are the differential cash ows The NPV of the decision to replace ignoring what will happen in two years is NPV 7489200 873801 14 7 178201142 1367801143 1367801144 2235801145 NPV 713683500 If we are not concerned with what will happen in two years we should not replace the old computer system 34 To answer this question we need to compute the NPV of all three alternatives speci cally continue to rent the building Project A or Project B We would choose the project with the highest NPV If all three of the projects have a positive NPV the project that is more favorable is the one with the highest NPV There are several important cash ows we should not consider in the incremental cash ow analysis The remaining fraction of the value of the building and depreciation are not incremental and should not be included in the analysis of the two alternatives The 850000 purchase price of the building is a same for all three options and should be ignored In effect what we are doing is nding the NPV of the future cash ows of each option so the only cash ow today would be the building modi cations needed for Project A and Project B If we did include these costs the effect would be to lower the NPV of all three options by the same amount thereby leading to the same conclusion The cash ows from renting the building after year 15 are also irrelevant No matter what the company chooses today it will rent the building after year 15 so these cash ows are not incremental to any project 171 We will begin by calculating the NPV of the decision of continuing to rent the building first Continue to rent Rent 36000 Taxes 12240 Net income 23760 Since there is no incremental depreciation the operating cash ow is simply the net income So the NPV of the decision to continue to rent is NPV 23760PVIFA12715 NPV 16182614 Product A Next we will calculate the NPV of the decision to modify the building to produce Product A The income statement for this modification is the same for the rst 14 years and in year 15 the company will have an additional expense to convert the building back to its original form This will be an expense in year 15 so the income statement for that year will be slightly different The cash ow at time zero will be the cost of the equipment and the cost of the initial building modi cations both of which are depreciable on a straightline basis So the pro forma cash ows for Product A are Initial cash outlay Building modi cations 745000 Equipment 7165000 Total cash ow 7210000 Years 114 Year 15 Revenue 135000 135000 Expenditures 60000 60000 Depreciation 14000 14000 Restoration cost 0 29000 EBT 61000 32000 Tax 20740 10880 NI 40260 21120 OCF 54260 35120 The OCF each year is net income plus depreciation So the NPV for modifying the building to manufacture Product A is NPV 7210000 54260PVIFA12714 35120 11215 NPV 15606070 172 Product B Now we will calculate the NPV of the decision to modify the building to produce Product B The income statement for this modification is the same for the first 14 years and in year 15 the company will have an additional expense to convert the building back to its original form This will be an expense in year 15 so the income statement for that year will be slightly different The cash ow at time zero will be the cost of the equipment and the cost of the initial building modi cations both of which are depreciable on a straightline basis So the pro forma cash ows for Product B are Initial cash outlay Building modifications 765000 Equipment 7205000 Total cash ow 7270000 Years 1 14 Year 15 Revenue 165000 165000 Expenditures 75000 75000 Depreciation 18000 18000 Restoration cost 0 35000 EBT 72000 37000 Tax 24480 125 80 NT 47520 24420 OCF 65520 42420 The OCF each year is net income plus depreciation So the NPV for modifying the building to manufacture Product B is NPV 270000 65520PV1FA1214 42420 11215 NPV 17202756 We could have also done the analysis as the incremental cash ows between Product A and continuing to rent the building and the incremental cash ows between Product B and continuing to rent the building The results of this type of analysis would be NPV of differential cash ows between Product A and continuing to rent NPV NPVProductA NPVRent NPV 15606070 716182614 NPV 7576544 NPV of differential cash ows between Product B and continuing to rent NPV NPVProductB NPVR ent NPV 17202756 716182614 NPV 1020142 173 UI Since the differential NPV of Product B and renting is the highest and positive the company should choose Product B which is the same as our original result The discount rate is expressed in real terms and the cash ows are expressed in nominal terms We can answer this question by converting all of the cash ows to real dollars We can then use the real interest rate The real value of each cash ow is the present value of the year 1 nominal cash ows discounted back to the present at the in ation rate So the real value of the revenue and costs will be Revenue in real terms 225000 106 21226415 Labor costs in real terms 175000 106 16509434 Other costs in real terms 45000 106 4245283 Lease payment in real terms 25000 106 2358491 Revenues labor costs and other costs are all growing perpetuities Each has a different growth rate so we must calculate the present value of each separately Using the real required return the present value of each of these is PVRevenue 21226415 010 7 005 424528302 PVLabmcosts 16509434 010 7 003 235849057 PVommosts 4245283 0107 001 47169811 The lease payments are constant in nominal terms so they are declining in real terms by the in ation rate Therefore the lease payments form a growing perpetuity with a negative growth rate The real present value of the lease payments is PVLeasepayments 2358491 010 7 4106 14740566 Now we can use the tax shield approach to calculate the net present value Since there is no investment in equipment there is no depreciation therefore no depreciation tax shield so we will ignore this in our calculation This means the cash ows each year are equal to net income There is also no initial cash outlay so the NPV is the present value of the future aftertax cash ows The NPV of the project is NPV PVRevenue PVLabor costs PVOther costs PVLease payments1 tc NPV 424528302 7 235849057 7 47169811 7 147405661 7 34 NPV 83667453 Alternatively we could have solved this problem by expressing everything in nominal terms This approach yields the same answer as given above However in this case the computation would have been impossible The reason is that we are dealing with growing perpetuities In other problems when calculating the NPV of nominal cash ows we could simply calculate the nominal cash ow each year since the cash ows were nite Because of the perpetual nature of the cash ows in this problem we cannot calculate the nominal cash ows each year until the end of the project When faced with two alternative approaches where both are equally correct always choose the simplest one 174 36 We are given the real revenue and costs and the real growth rates so the simplest way to solve this problem is to calculate the NPV with real values While we could calculate the NPV using nominal values we would need to nd the nominal growth rates and convert all values to nominal terms The real labor costs will increase at a real rate of two percent per year and the real energy costs will increase at a real rate of three percent per year so the real costs each year will be Year 1 Year 2 Year 3 Year 4 Real labor cost each year 1675 1709 1743 1778 Real energy cost each year 435 448 461 475 Remember that the depreciation tax shield also affects a lm s a ertax cash ows The present value of the depreciation tax shield must be added to the present value of a rm s revenues and expenses to find the present value of the cash ows related to the project The depreciation the rm will recognize each year is Annual depreciation Investment Economic Life Annual depreciation 175000000 4 Annual depreciation 43750000 Depreciation is a nominal cash ow so to nd the real value of depreciation each year we discount the real depreciation amount by the in ation rate Doing so we nd the real depreciation each year is Year 1 real depreciation 43750000 105 4166666667 Year 2 real depreciation 43750000 1052 3968253968 Year 3 real depreciation 43750000 1053 3779289494 Year 4 real depreciation 43750000 1054 3599323327 Now we can calculate the pro forma income statement each year in real terms We can then add back depreciation to net income to nd the operating cash ow each year Doing so we nd the cash ow of the project each year is M Year 1 Year 2 Year 3 Year 4 Revenues 8250000000 8800000000 9900000000 9350000000 Laborcost 3015000000 3417000000 3659607000 3199542120 Energy cost 76125000 87369750 94605758 95067249 Depreciation 4166666667 3968253968 3779289494 3599323327 EBT 992208333 1327376282 2366497749 2456067304 Taxes 337350833 451307936 804609235 835062883 Net income 645857500 876068346 1561888514 1621004420 OCF 4821524167 4844322314 5341178008 5220327748 Capital spending 7175000000 Total CF 7175000000 4821524167 4844322314 5341178008 5220327748 175 1 We can use the total cash ows each year to calculate the NPV which is NPV 7175000000 4821524167 108 4844322314 1082 5341178008 1083 5220327748 1084 NPV 7805304150 Here we have the sales price and production costs in real terms The simplest method to calculate the project cash ows is to use the real cash ows In doing so we must be sure to adjust the depreciation which is in nominal terms We could analyze the cash ows using nominal values which would require calculating the nominal discount rate nominal price and nominal production costs This method would be more complicated so we will use the real numbers We will rst calculate the NPV of the headache only pill Headache only We can find the real revenue and production costs by multiplying each by the units sold We must be sure to discount the depreciation which is in nominal terms e can then find the pro forma net income and add back depreciation to find the operating cash ow Discounting the depreciation each year by the in ation rate we nd the following cash ows each year Year 1 Year 2 Year 3 Sales 21000000 21000000 21000000 Production costs 9800000 9800000 9800000 D l 39 quot 4761905 4535147 4319188 EBT 6438095 6664853 6880812 Tax 2188952 2266050 2339476 Net income 4249143 4398803 4541336 OCF 9011048 8933950 8860524 And the NPV of the headache only pill is NPV 15000000 9011048 113 8993950 1132 8860524 1133 NPV 611175936 Headache and arthritis For the headache and arthritis pill project the equipment has a salvage value We will find the aftertaX salvage value of the equipment first which will be Market value 1000000 Taxes 7340000 Total 660000 176 on Remember to calculate the taxes on the equipment salvage value we take the book value minus the market value times the tax rate Using the same method as the headache only pill the cash ows each year for the headache and arthritis pill will be Year 1 Year 2 Year 3 Sales 31500000 31500000 31500000 Production costs 16500000 16500000 16500000 Depreciation 6666667 6349206 6046863 EBT 8333333 8650794 8953137 Tax 2833333 2941270 3044067 Net income 5500000 5709524 5909070 OCF 12166667 12058730 11955933 So the NPV of the headache and arthritis pill is NPV 72l000000 12166667 113 12058730 1132 11955933 660000 1133 NPV 795419093 The company should manufacture the headache and arthritis remedy since the project has a higher NPV This is an indepth capital budgeting problem Since the project requires an initial investment in inventory as a percentage of sales we will calculate the sales gures for each year rst The incremental sales will include the sales of the new table but we also need to include the lost sales of the existing model This is an erosion cost of the new table The lost sales of the existing table are constant for every year but the sales of the new table change every year So the total incremental sales gure for the ve years of the project will be Year 1 Year 2 Year 3 Year 4 Year 5 New 10080000 10920000 14000000 13160000 11760000 Lost sales 71 125 000 71 125 000 71 125 000 71 125 000 71 125 000 Total 8955000 9795000 12875000 12035000 10635000 Now we will calculate the initial cash outlay that will occur today The company has the necessary production capacity to manufacture the new table without adding equipment today So the equipment will not be purchased today but rather in two years The reason is that the existing capacity is not being used If the existing capacity were being used the new equipment would be required so it would be a cash ow today The old equipment would have an opportunity cost if it could be sold As there is no discussion that the existing equipment could be sold we must assume it cannot be sold The only initial cash ow is the cost of the inventory The company will have to spend money for inventory in the new table but will be able to reduce inventory of the existing table So the initial cash ow today is New table 7l00 8000 Old table 112500 Total 7895500 177 In year 2 the company will have a cash out ow to pay for the cost of the new equipment Since the equipment will be purchased in two years rather than now the equipment will have a higher salvage value The book value of the equipment in ve years will be the initial cost minus the accumulated depreciation or Book value 16000000 7 2288000 7 3920000 7 2800000 Book value 6992000 The taxes on the salvage value will be Taxes on salvage 6992000 7 740000040 Taxes on salvage 7l63200 So the aftertax salvage value of the equipment in ve years will be Sell equipment 7400000 Taxes 7163 200 1 Salvage value 7236800 Next we need to calculate the variable costs each year The variable costs of the lost sales are included as a variable cost savings so the variable costs will be Year 1 Year 2 Year 3 Year 4 Year 5 New 4536000 4914000 6300000 5922000 5292000 Lost sales 450000 450000 450000 450000 450000 Variable costs 4086000 4464000 5850000 5472000 4842000 Now we can prepare the rest of the pro forma income statements for each year The project will have no incremental depreciation for the first two years as the equipment is not purchased for two years Adding back depreciation to net income to calculate the operating cash ow we get E Year 2 Year 3 M Year 5 Sales 8955000 9795000 12875000 12035000 10635000 VC 4086000 4464000 5850000 5472000 4842000 Fixed costs 1900000 1900000 1900000 1900000 1900000 Dep 0 0 2288000 3920000 2800000 EBT 2969000 3431000 2837000 743000 1093000 Tax 1187600 1372400 1134800 297200 437200 NT 1781400 2058600 1702200 445800 655800 Dep 0 0 2288000 3920000 2800000 OCF 1781400 2058600 3990200 4365800 3455800 178 Next we need to account for the changes in inventory each year The inventory is a percentage of sales The way we will calculate the change in inventory is the beginning of period inventory minus the end of period inventory The sign of this calculation will tell us whether the inventory change is a cash in ow or a cash out ow The inventory each year and the inventory change will be Year 1 Year 2 Year 3 Year 4 Year 5 Beginning 1008000 1092000 1400000 1316000 1176000 Ending 1092000 1400000 1316000 1176000 0 Change 784000 7308000 84000 140000 1176000 Notice that we recover the remaining inventory at the end of the project The total cash ows for the project will be the sum of the operating cash ow the capital spending and the inventory cash ows so m M Year 3 Year 4 M OCF 1781400 2058600 3990200 4365800 3455800 Equipment 0 716000000 0 0 7236800 Inventory 784000 7308000 84000 140000 1176000 Total 1697400 714249400 4074200 4505800 11868600 The NPV of the project including the inventory cash ow at the beginning of the project will be NPV 895500 1697400 114 7 14249400 1142 4047200 1143 4505800 114 11868600 1145 NPV 121093996 The company should go ahead with the new table b You can perform an IRR analysis and would expect to nd three IRRs since the cash ows change signs three times 0 The pro tability index is intended as a bang for the buck measure that is it shows how much shareholder wealth is created for every dollar of initial investment This is usually a good measure of the investment since most projects have conventional cash ows In this case the largest investment is not at the beginning of the project but later in its life so while the interpretation is the same it really does not measure the bang for the dollar invested 179 CHAPTER 7 RISK ANALYSIS REAL OPTIONS AND CAPITAL BUDGETING Answers to Concepts Review and Critical Thinking Questions 1 Forecasting risk is the risk that a poor decision is made because of errors in projected cash ows The danger is greatest with a new product because the cash ows are probably harder to predict With a sensitivity analysis one variable is examined over a broad range of values With a scenario analysis all variables are examined for a limited range of values It is true that if average revenue is less than average cost the firm is losing money This much of the statement is therefore correct At the margin however accepting a project with marginal revenue in excess of its marginal cost clearly acts to increase operating cash ow From the shareholder perspective the nancial breakeven point is the most important A project can exceed the accounting and cash breakeven points but still be below the nancial breakeven point This causes a reduction in shareholder your wealth The project will reach the cash breakeven first the accounting breakeven next and finally the nancial breakeven For a project with an initial investment and sales afterwards this ordering will always apply The cash breakeven is achieved first since it excludes depreciation The accounting breakeven is next since it includes depreciation Finally the nancial breakeven which includes the time value of money is achieved Traditional NPV analysis is often too conservative because it ignores pro table options such as the ability to expand the project if it is profitable or abandon the project if it is unpro table The option to alter a project when it has already been accepted has a value which increases the NPV of the project The type of option most likely to affect the decision is the option to expand If the country just liberalized its markets there is likely the potential for growth First entry into a market whether an entirely new market or with a new product can give a company name recognition and market share This may make it more difficult for competitors entering the market Sensitivity analysis can determine how the nancial breakeven point changes when some factors such as xed costs variable costs or revenue change There are two sources of value with this decision to wait The price of the timber can potentially increase and the amount of timber will almost de nitely increase barring a natural catastrophe or forest re The option to wait for a logging company is quite valuable and companies in the industry have models to estimate the future growth of a forest depending on its age 180 10 When the additional analysis has a negative NPV Since the additional analysis is likely to occur almost immediately this means when the bene ts of the additional analysis outweigh the costs The bene ts of the additional analysis are the reduction in the possibility of making a bad decision Of course the additional benefits are often difficult if not impossible to measure so much of this decision is based on experience Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic 1 a To calculate the accounting breakeven we first need to nd the depreciation for each year The depreciation is Depreciation 7240008 Depreciation 90500 per year And the accounting breakeven is QA 850000 9050039 7 23 QA 58781 units b We will use the tax shield approach to calculate the OCF The OCF is OCFbase P VQ FC1 to t th OCFbZise 39 72375000 7 850000065 03590500 OCFbZise 259175 Now we can calculate the NPV using our basecase projections There is no salvage value or NWC so the NPV is Nvase 7724000 259175Pv1FA158 Nvase 43900155 To calculate the sensitivity of the NPV to changes in the quantity sold we will calculate the NPV at a different quantity We will use sales of 80000 units The NPV at this sales level is OCFew 39 7 2380000 7 850000065 03590500 OCFew 311175 And the NPV is NPvew 7724000 311175PVIFA1578 NPvew 67234227 181 So the change in NPV for every unit change in sales is ANPVAS 43900155 7 6723422775000 7 80000 ANPVAS 46668 If sales were to drop by 500 units then NPV would drop by NPV drop 46668500 2333407 You may wonder why we chose 80000 units Because it doesn t matter Whatever sales number we use when we calculate the change in NPV per unit sold the ratio will be the same 0 To find out how sensitive OCF is to a change in variable costs we will compute the OCF at a variable cost of 24 Again the number we choose to use here is irrelevant We will get the same ratio of OCF to a one dollar change in variable cost no matter what variable cost we use So using the tax shield approach the OCF at a variable cost of 24 is OCFnew 39 7 2475000 7 850000065 03590500 OCFnew 210425 So the change in OCF for a 1 change in variable costs is AOCFAv 259175 7 21042523 7 24 AOCFAv 748750 If variable costs decrease by 1 then OCF would increase by 48750 We will use the tax shield approach to calculate the OCF for the best and worstcase scenarios For the bestcase scenario the price and quantity increase by 10 percent so we will multiply the base case numbers by l l a 10 percent increase The variable and xed costs both decrease by 10 percent so we will multiply the base case numbers by 9 a 10 percent decrease Doing so we get OCF t 391 1 7 23097500011 7 85000009065 03590500 OCFW 72490000 The bestcase NPV is NPVbes 7724000 724900PVIFA1578 NPVbes 252885936 For the worstcase scenario the price and quantity decrease by 10 percent so we will multiply the base case numbers by 9 a 10 percent decrease The variable and xed costs both increase by 10 percent so we will multiply the base case numbers by l l a 10 percent increase Doing so we get OCFworst 3909 7 23117500009 7 85000011065 03590500 OCFworst 7146100 182 The worstcase NPV is Nvam 7724000 7 146100PVIFA1578 NPVWS 7137959767 We can use the accounting breakeven equation QA FC DP7v to solve for the unknown variable in each case Doing so we nd 1 QA 7 110500 7 820000 D41730 D 7 395500 2 QA 143806 32M 115l1P7 56 P 8625 3 QA 7835 160000 105000105 7v V 7118 When calculating the financial breakeven point we express the initial investment as an equivalent annual cost EAC Dividing the initial investment by the veyear annuity factor discounted at 12 percent the EAC of the initial investment is EAC Initial Investment PVIFAum EAC 250000 360478 EAC 6935243 Note that this calculation solves for the annuity payment with the initial investment as the present value of the annuity In other words PVA 7 C1711R R 250000 7 Clt1711125 12 C 7 6935243 The annual depreciation is the cost of the equipment divided by the economic life or Annual depreciation 250000 5 Annual depreciation 50000 Now we can calculate the nancial breakeven point The financial breakeven point for this project is QF EAC FC1 7 tc 7 Depreciationtc P 7 VC1 7 tc QF 6935243 36000017 034 7 50000034 25 7 617 034 QF 2312220 or about 23122 units If we purchase the machine today the NPV is the cost plus the present value of the increased cash ows so NPVO 71800000 340000PVIFA12710 NPVO 12107583 183 We should not necessarily purchase the machine today We would want to purchase the machine when the NPV is the highest So we need to calculate the NPV each year The NPV each year will be the cost plus the present value of the increased cash savings We must be careful however In order to make the correct decision the NPV for each year must be taken to a common date We will discount all ofthe NPVs to today Doing so we get Year 1 NPVI 71670000 340000PVIFA1279 112 NPVI 12643297 Year 2 NPV2 1540000 340000PVIFA1278 1122 NPV2 11877991 Year 3 NPV3 1410000 340000PV1FA127 1123 NPV3 10084305 Year 4 NPV4 1280000 340000PV1FA126 1124 NPV4 7491391 Year 5 NPV5 1150000 340000PV1FA125 1125 NPV5 4291104 Year 6 NPV6 1150000 340000PV1FA124 1126 NPV5 5942845 The company should purchase the machine one year from now when the NPV is the highest We need to calculate the NPV of the two options go directly to market now or utilize test marketing rst The NPV of going directly to market now is NPV Cs ccess Prob of Success CFailm Prob of Failure NPV 22000000050 9000000050 NPV 15500000 Now we can calculate the NPV of test marketing rst Test marketing requires a 15 million cash outlay Choosing the test marketing option will also delay the launch of the product by one year Thus the expected payoff is delayed by one year and must be discounted back to year 0 NPV C0 Cs ccess Prob of Success CFailme Prob of Failure 1 R NPV 71500000 22000000 080 9000000 020 111 NPV 1597747748 The company should test market rst with the product since that option has the highest expected payoff We need to calculate the NPV of each option and choose the option with the highest NPV So the NPV of going directly to market is NPV CSuccess Prob of Success NPV 1500000 050 NPV 750000 184 The NPV of the focus group is NPV C0 Cs ccess Prob of Success NPV 7135000 1500000 065 NPV 840000 And the NPV of using the consulting rm is NPV C0 CSuccess Prob of Success NPV 7400000 1500000 085 NPV 875000 The rm should use the consulting rm since that option has the highest NPV The company should analyze both options and choose the option with the greatest NPV So if the company goes to market immediately the NPV is NPV Cs ccess Prob of Success CFailm Prob of Failure NPV 2800000055 400000045 NPV 1720000000 Customer segment research requires a 18 million cash outlay Choosing the research option will also delay the launch of the product by one year Thus the expected payoff is delayed by one year and must be discounted back to year 0 So the NPV of the customer segment research is NPV C0 Cs ccess Prob of Success CFailme Prob of Failure l R NPV 71800000 28000000 070 4000000 030 115 NPV 1628695652 Graphically the decision tree for the project is Success 28 million at t l 24348 million at t 0 Research 16287 million at t 0 Start 4 million at t l 3478 million at t 0 Success 28 million at t 0 4 million at t 0 1720 million at t 0 The company should go to market now since it has the largest NPV 185 9 a The accounting breakeven is the aftertaX sum of the xed costs and depreciation charge divided by the aftertaX contribution margin selling price minus variable cost So the accounting breakeven level of sales is Q A FC Depreciation17tc P 7 VC1 7 tc QA 750000 3600007 17 035 80 7 540 17 035 QA 1045576 or about 10456 units b When calculating the nancial breakeven point we express the initial investment as an equivalent annual cost EAC Dividing the initial investment by the sevenyear annuity factor discounted at 15 percent the EAC of the initial investment is EAC Initial Investment PVIFAIMJ EAC 360000 41604 EAC 8652973 Note that this calculation solves for the annuity payment with the initial investment as the present value of the annuity In other words PVA C1711R R 360000 7 Clt1711157 15 C7 8652973 Now we can calculate the nancial breakeven point The nancial breakeven point for this project is QF EAC FC1 7 tc 7 Depreciationtc P 7 VC1 7 tc QF 8652973 75000065 7 360000735 80 7 540 65 QF 1162157 or about 11622 units 10 When calculating the financial breakeven point we express the initial investment as an equivalent annual cost EAC Dividing the initial investment by the veyear annuity factor discounted at 8 percent the EAC of the initial investment is EAC Initial Investment PVIFAmYs EAC 390000 399271 EAC 9767802 Note that this calculation solves for the annuity payment with the initial investment as the present value of the annuity In other words PVA 7 C1711R R 390000 7 C1711085 08 C 7 9767802 The annual depreciation is the cost of the equipment divided by the economic life or Annual depreciation 390000 5 Annual depreciation 78000 186 12 Now we can calculate the nancial breakeven point The nancial breakeven point for this project is QF EAC FC1 7 tc 7Depreciationtc P 7 VC1 7 tc QF 9767802 18500017 034 7 78000034 60 7 14 17 034 QF 636555 or about 6366 units Intermediate a At the accounting breakeven the IR is zero percent since the project recovers the initial investment The payback period is N years the length of the project since the initial investment is exactly recovered over the project life The NPV at the accounting breakeven is NPV 71 INPv1FARN 7 1 b At the cash breakeven level the IR is 7100 percent the payback period is negative and the NPV is negative and equal to the initial cash outlay c The de nition of the nancial breakeven is where the NPV of the project is zero If this is true then the IRR of the project is equal to the required return It is impossible to state the payback period except to say that the payback period must be less than the length of the project Since the discounted cash ows are equal to the initial investment the undiscounted cash ows are greater than the initial investment so the payback must be less than the project life Using the tax shield approach the OCF at 90000 units will be OCF P7vQ 7 FC17tCtCD OCF 54 7 4290000 7 185000066 0343800004 OCF 623000 We will calculate the OCF at 91000 units The choice of the second level of quantity sold is arbitrary and irrelevant No matter what level of units sold we choose we will still get the same sensitivity So the OCF at this level of sales is OCF 7 54 7 42910007185000066 0343 800004 OCF 7 630920 The sensitivity of the OCF to changes in the quantity sold is Sensitivity 7 AOCFAQ 7 623000 7 63092090000 791000 AOCFAQ 7 792 OCF will increase by 792 for every additional unit sold a The basecase bestcase and worstcase values are shown below Remember that in the best case unit sales increase while costs decrease In the worstcase unit sales and costs increase Scenario Unit sales Variable cost Fixed costs Base 240 19500 830000 Best 264 17550 747000 Worst 216 21450 913000 187 Using the tax shield approach the OCF and NPV for the base case estimate are OCFbase 25000 7 19500240 7 830000065 0359600004 OCFbase 402500 NPVbzse 7960000 402500PVIFA1574 NPVbzse 18912879 The OCF and NPV for the worst case estimate are OCFworst 25000 7 21450216 7 913000065 0359600004 OCFworst 711030 NPVWS 7960000 7 1 1030Pv1FA15 NPVWS 799149041 And the OCF and NPV for the best case estimate are OCF t 25000 7 17550264 7 747000065 0359600004 OCF t 876870 Nvaest 7960000 876870Pv1FA15 Nvaest 154344488 To calculate the sensitivity of the NPV to changes in xed costs we choose another level of xed costs We will use xed costs of 840000 The OCF using this level of xed costs and the other base case values with the tax shield approach we get OCF 25000 7 19500240 7 840000065 0359600004 OCF 396000 And the NPV is NPV 7960000 396000PVIFA1574 NPV 17057143 The sensitivity of NPV to changes in xed costs is ANPVAFC 18912879 7 17057143830000 7 840000 ANPVAFC 71856 For every dollar F C increase NPV falls by 186 188 c The accounting breakeven is QA FC DP7v QA 830000 960000425000 7 19500 QA 19455 or about 195 units 14 The marketing study and the research and development are both sunk costs and should be ignored We will calculate the sales and variable costs rst Since we will lose sales of the expensive clubs and gain sales of the cheap clubs these must be accounted for as erosion The total sales for the new project will be Sales New clubs 750 gtlt 55000 41250000 Exp clubs 1100 x 712000 713200000 Cheap clubs 400 x 15000 6000000 34050000 For the variable costs we must include the units gained or lost from the existing clubs Note that the variable costs of the expensive clubs are an in ow If we are not producing the sets any more we will save these variable costs which is an in ow So Var costs New clubs 7390 gtlt 55000 721450000 Exp clubs 7620 x 712000 7440000 Cheap clubs 7210 x 15000 73150000 717160000 The pro forma income statement will be Sales 34050000 Variable costs 17160000 Fixed costs 8100000 Depreciation 2700000 6090000 Taxes 2436000 Net income 3654000 Using the bottom up OCF calculation we get OCF N1 Depreciation 3654000 2700000 OCF 6354000 189 UI So the payback period is Payback period 3 123 80006345000 Payback period 3195 years The NPV is NPV NPV 718900000 7 1400000 6354000Pv1FA17 14000001147 750738120 And the IR is IRR718 IR 2515 Base Case Unit sales new 55000 Price new 750 VC new 390 Fixed costs 8100000 Sales lost expensive 12000 Sales gained cheap 15000 Bestcase The upper and lower bounds for the variables are Best Case 60500 825 3 51 7290000 10800 16500 900000 71400000 6354000PVIFAmR7 14000001 IR7 00 Worst Case 49500 675 429 8910000 13200 13500 We will calculate the sales and variable costs rst Since we will lose sales of the expensive clubs and gain sales of the cheap clubs these must be accounted for as erosion The total sales for the new project will be Sales New clubs Exp Cheap clubs clubs 825 x 60500 49912500 1100 x 710800 711880000 400 gtlt 16500 6600000 44632500 For the variable costs we must include the units gained or lost from the existing clubs Note that the variable costs of the expensive clubs are an in ow If we are not producing the sets any more we will save these variable costs which is an in ow So Var costs New clubs Exp clubs 7620 x 710800 Cheap clubs 7210 x 16500 7351 x 60500 721235500 6696000 73 465 000 718004500 190 The pro forma income statement will be Sales 44632500 Variable costs 18004500 Fixed costs 7290000 Depreciation 2700000 EBT 16638000 Taxes 6655200 Net income 9982800 Using the bottom up OCF calculation we get OCF Net income Depreciation 9982800 2700000 OCF 12682800 And the bestcase NPV is NPV 7l8900000 7 1400000 12682800PVIFA1477 14000001 147 NPV 3464720486 Worstcase We will calculate the sales and variable costs first Since we will lose sales of the expensive clubs and gain sales of the cheap clubs these must be accounted for as erosion The total sales for the new project will be 321 168 New clubs 675 gtlt 49500 33412500 Exp clubs 1100 x 713200 7 14520000 Cheap clubs 400 gtlt 13500 5400000 24292500 For the variable costs we must include the units gained or lost from the existing clubs Note that the variable costs of the expensive clubs are an in ow If we are not producing the sets any more we will save these variable costs which is an in ow So Var costs New clubs 7429 x 49500 72l235500 Exp clubs 7620 x 7 13200 8184000 Cheap clubs 7210 gtlt 13500 72835000 715886500 191 5 The pro forma income statement will be Sales 24292500 Variable costs 15886500 Costs 8910000 Depreciation 2700000 73204000 Taxes 71281600 assumes a tax credit Net income 7 1922400 Using the bottom up OCF calculation we get OCF N1 Depreciation 71922400 2700000 OCF 777600 And the worstcase NPV is NPV 18900000 7 1400000 777600PVIFA1477 14000001147 NPV 71640592191 To calculate the sensitivity of the NPV to changes in the price of the new club we simply need to change the price of the new club We will choose 760 but the choice is irrelevant as the sensitivity will be the same no matter what price we choose We will calculate the sales and variable costs first Since we will lose sales of the expensive clubs and gain sales of the cheap clubs these must be accounted for as erosion The total sales for the new project will be Sales New clubs 760 x 55000 41800000 Exp clubs 1100 x 7 12000 713200000 Cheap clubs 400 x 15000 6000000 34600000 For the variable costs we must include the units gained or lost from the existing clubs Note that the variable costs of the expensive clubs are an in ow If we are not producing the sets any more we will save these variable costs which is an in ow So Var costs New clubs 7390 x 55000 72l450000 Exp clubs 7620 x 712000 7440000 Cheap clubs 7210 gtlt 15000 73150000 717l60000 192 The pro forma income statement will be Sales 3 4600000 Variable costs 17160000 Fixed costs 8100000 Depreciation 2700000 EBT 6640000 Taxes 2656000 Net income 3984000 Using the bottom up OCF calculation we get OCF N1 Depreciation 3984000 2700000 OCF 6684000 And the NPV is NPV 18900000 7 1400000 6684000Pv1FA17 14000001147 NPV 892252180 So the sensitivity of the NPV to changes in the price of the new club is ANPVAP 750738120 7 892252180750 7 760 ANPVAP 14151406 For every dollar increase decrease in the price of the clubs the NPV increases decreases by 14151406 To calculate the sensitivity of the NPV to changes in the quantity sold of the new club we simply need to change the quantity sold We will choose 60000 units but the choice is irrelevant as the sensitivity will be the same no matter what quantity we choose We will calculate the sales and variable costs rst Since we will lose sales of the expensive clubs and gain sales of the cheap clubs these must be accounted for as erosion The total sales for the new project will be Sales New clubs 750 x 60000 45000000 Exp clubs 1100 x 712000 713200000 Cheap clubs 400 x 15000 6000000 37800000 193 17 For the variable costs we must include the units gained or lost from the existing clubs Note that the variable costs of the expensive clubs are an in ow If we are not producing the sets any more we will save these variable costs which is an in ow So Var costs New clubs 7390 x 60000 723400000 Exp clubs 7620 x 712000 7440000 Cheap clubs 7210 x 15000 73150000 719110000 The pro forma income statement will be Sales 37800000 Variable costs 19110000 Fixed costs 8100000 Depreciation 2700000 7890000 Taxes 3156000 Net income 4734000 Using the bottom up OCF calculation we get OCF N1 Depreciation 4734000 2700000 OCF 7434000 The NPV at this quantity is NPV 18900000 7 1400000 7434000PVIFA1477 14000001147 NPV 1213875043 So the sensitivity of the NPV to changes in the quantity sold is ANPVAQ 750738120 7 121387504355000 7 60000 ANPVAQ 92627 For an increase decrease of one set of clubs sold per year the NPV increases decreases by 92627 a The basecase NPV is NPV 71900000 450000PVIFA16310 NPV 27495237 194 We would abandon the project if the cash ow from selling the equipment is greater than the present value of the future cash ows We need to nd the sale quantity where the two are equal so 1300000 50QPVIFA169 Q 13000005046065 Q 5664 Abandon the project if Q lt 5664 units because the NPV of abandoning the project is greater than the NPV of the future cash ows The 1300000 is the market value of the project If you continue with the project in one year you forego the 1300000 that could have been used for something else If the project is a success present value of the future cash ows will be PV future CFs 5011000PVIFA1679 PV future CFs 253359913 From the previous question if the quantity sold is 4000 we would abandon the project and the cash ow would be 1300000 Since the project has an equal likelihood of success or failure in one year the expected value of the project in one year is the average of the success and failure cash ows plus the cash ow in one year so Expected value of project at year 1 253359913 13000002 450000 Expected value of project at year 1 236679957 The NPV is the present value of the expected value in one year plus the cost of the equipment so NPV 71900000 236679947116 NPV 14034445 If we couldn t abandon the project the present value of the future cash ows when the quantity is 4000 will be PV future CFs 504000PVIFA1679 PV future CFs 92130878 The gain from the option to abandon is the abandonment value minus the present value of the cash ows if we cannot abandon the project so Gain from option to abandon 1300000 7 92130878 Gain from option to abandon 37869122 We need to nd the value of the option to abandon times the likelihood of abandonment So the value of the option to abandon today is Option value 5037869122116 Option value 16322898 195 19 If the project is a success present value of the future cash ows will be PV future CFs 5022000PVIFA1679 PV future CFs 506719826 If the sales are only 4000 units from Problem 17 we know we will abandon the project with a value of 1300000 Since the project has an equal likelihood of success or failure in one year the expected value of the project in one year is the average of the success and failure cash ows plus the cash ow in one year so Expected value of project at year 1 506719826 13000002 450000 Expected value of project at year 1 363359913 The NPV is the present value of the expected value in one year plus the cost of the equipment so NPV 7 1900000 363359913116 NPV 7 123241304 The gain from the option to expand is the present value of the cash ows from the additional units sold so Gain from option to expand 5011000PVIFA1579 Gain from option to expand 253359913 We need to nd the value of the option to expand times the likelihood of expansion We also need to nd the value of the option to expand today so Option value 502533599131 16 Option value 109206859 a The accounting breakeven is the aftertax sum of the xed costs and depreciation charge divided by the contribution margin selling price minus variable cost In this case there are no xed costs and the depreciation is the entire price of the press in the first year So the accounting breakeven level of sales is Q A FC Depreciation1 7 tc P 7 VC17tc QA 0 3200 1 7 030 10 7 7 1 7030 QA 106667 or about 1067 units b When calculating the nancial breakeven point we express the initial investment as an equivalent annual cost EAC The initial investment is the 12000 in licensing fees Dividing the initial investment by the threeyear annuity factor discounted at 12 percent the EAC of the initial investment is EAC Initial Investment PVIFAIMQ EAC 12000 24018 EAC 499619 196 Note this calculation solves for the annuity payment with the initial investment as the present value of the annuity in other words PVA Cl 7 11 110t R 12000 7 Clt1711123 12 C7 499619 Now we can calculate the nancial breakeven point Notice that there are no xed costs or depreciation The nancial breakeven point for this project is QF EAC FC1 7 tc 7Depreciationtc P 7 VC1 7 tc QF 499619 0 7 0 10 7 7 70 QF 237914 or about 2379 units 21 The payoff from taking the lump sum is 12000 so we need to compare this to the expected payoff from taking one percent of the pro t The decision tree for the movie project is Bi audience 20000000 Script is ood scri n t audience No pro t Don39t make 90 movie No profit The value of one percent of the pro ts as follows There is a 30 percent probability the movie is good and the audience is big so the expected value of this outcome is Value 7 20000000 x 30 Value 7 6000000 The value if the movie is good and has a big audience assuming the script is good is Value 7 6000000 x 10 Value 600000 197 N This is the expected value for the studio but the screenwriter will only receive one percent of this amount so the payment to the screenwriter will be Payment to screenwriter 600000 X 01 Payment to screenwriter 6000 The screenwriter should take the upfront offer of 12000 We can calculate the value of the option to wait as the difference between the NPV of opening the mine today and the NPV of waiting one year to open the mine The remaining life of the mine is 60000 ounces 7500 ounces per year 8 years This will be true no matter when you open the mine The aftertaX cash flow per year if opened today is CF 7500450 3375000 So the NPV of opening the mine today is NPV e14000000 3375000Pv1FAu8 NPV 276578421 If you open the mine in one year the cash ow will be either CFUp 7500500 3750000 per year CPDquot 7500410 3075000 per year The PV of these cash ows is Price increase CF 3750000Pv1FAu8 1862864913 Price decrease CF 3075000Pv1FAu8 1527549228 So the NPV is one year will be NPV e14000000 601862864913 401527549228 NPV 328738639 And the NPV today is NPV today 328738639 112 NPV today 293516642 So the value of the option to wait is Option value 293516642 e 276578421 Option value 16938221 198 The NPV of the project is sum of the present value of the cash ows generated by the project The cash ows from this project are an annuity so the NPV is NPV 84000000 22000000Pv1FA19m NPV 1145656707 The company should abandon the project if the PV of the revised cash ows for the next nine years is less than the project s aftertax salvage value Since the option to abandon the project occurs in year 1 discount the revised cash ows to year 1 as well To determine the level of expected cash ows below which the company should abandon the project calculate the equivalent annual cash ows the project must earn to equal the aftertax salvage value We will solve for C2 the revised cash ow beginning in year 2 So the revised annual cash ow below which it makes sense to abandon the project is A ertax salvage value C2 PVIFA199 3 0000000 C2PVIFA199 C2 30000000 PVIFAlgm C2 720576607 The NPV of the project is sum of the present value of the cash ows generated by the project The annual cash ow for the project is the number of units sold times the cash ow per unit which is Annual cash ow 15410000 Annual cash ow 6150000 The cash ows from this project are an annuity so the NPV is NPV 17000000 6150000Pv1FA25 NPV 46092800 The company will abandon the project if unit sales are not revised upward If the unit sales are revised upward the aftertax cash ows for the project over the last four years will be New annual cash ow 20410000 New annual cash ow 8200000 The NPV of the project will be the initial cost plus the expected cash ow in year one based on 15 unit sales projection plus the expected value of abandonment plus the expected value of expansion We need to remember that the abandonment value occurs in year 1 and the present value of the expansion cash ows are in year one so each of these must be discounted back to today So the project NPV under the abandonment or expansion scenario is NPV 17000000 6150000 125 5011000000 125 508200000PVIFA2574 125 NPV 6604800 199 25 To calculate the unit sales for each scenario we multiply the market sales times the company s market share We can then use the quantity sold to nd the revenue each year and the variable costs each year After doing these calculations we will construct the pro forma income statement for each scenario We can then nd the operating cash ow using the bottom up approach which is net income plus depreciation Doing so we find Pessimistic Exgected Ogtimistic Units per year 27300 37500 46200 Revenue 382200000 543750000 693000000 Variable costs 278460000 367500000 434280000 Fixed costs 101500000 95000000 90000000 D J 39 quot 36666667 35000000 33333333 EBT 734426667 46250000 135386667 Tax 43770667 18500000 54154667 Net income 720656000 27750000 81232000 OCF 16010667 62750000 114565333 Note that under the pessimistic scenario the taxable income is negative We assumed a tax credit in the case Now we can calculate the NPV under each scenario which will be NPvpessimis c 1600000 16010667Pv1FA136 NPV 155996563 NPvExpected 2100000 627500PVIFA1376 NPV 40846249 NPVOPnmis c 2000000 114565333Pv1FA136 NPV 257980624 The NPV under the pessimistic scenario is negative but the company should probably accept the project Challenge 1 Using the tax shield approach the OCF is OCF 245 7 22055000 7 520000062 03817000005 OCF 65930000 And the NPV is NPV 71700000 7 600000 659300PVIFA1375 600000 3000001 7 381135 NPV 44551988 200 b In the worstcase the OCF is OCFworst 7 24509 7 22055000 7 520000062 03819550005 OCFworst 7 7156770 And the worstcase NPV is NPVWOm 71955000 7 600000105 7156770PVIFA1375 600000105 30000008517381135 NPVwom 7270864724 The bestcase OCF is OCF t 7 2451 1 7 22055000 7 520000062 03814450005 OCF t 7 1475370 And the bestcase NPV is NPVbest 7 1445000 7 600000095 1475370PVIFA135 600000095 30000011517381135 NPVbest 359968700 27 To calculate the sensitivity to changes in quantity sold we will choose a quantity of 56000 The OCF at this level of sales is OCF 7 245 7 22056000 7 520000062 03817000005 OCF 7 674800 The sensitivity of changes in the OCF to quantity sold is AOCFAQ 7 659300 7 67480055000 7 56000 AOCFAQ 7 1550 The NPV at this level of sales is NPV 71700000 7 600000 674800PVIFA1375 600000 3000001 7 381135 NPV 50003696 And the sensitivity of NPV to changes in the quantity sold is ANPVAQ 44551988 7 5000369655000 7 56000 ANPVAQ 5452 You wouldn t want the quantity to fall below the point where the NPV is zero We know the NPV changes 5452 for every unit sale so we can divide the NPV for 55000 units by the sensitivity to get a change in quantity Doing so we get 44551988 7 5452 AQ AQ 7 8172 201 on For a zero NPV sales would have to decrease 8172 units so the minimum quantity is QMin 55000 7 8172 QMin 46828 project for all four years the cash ows are Year Sales Operating costs Depreciation EBT Tax Net income Depreciation Operating CF Change in NWC Capital spending Total cash ow There is no salvage value for the equipment The NPV is We will use the bottom up approach to calculate the operating cash ow Assuming we operate the 0 1 2 3 4 7350000 7350000 7350000 7350000 2400000 2400000 2400000 2400000 2500000 2500000 2500000 2500000 2450000 2450000 2450000 2450000 931000 931000 931000 931000 1519000 1519000 1519000 1519000 2500000 2500000 2500000 2500000 4019000 4019000 4019000 4019000 71300000 0 0 0 1300000 710000000 0 0 0 0 711300000 4019000 4019000 4019000 5319000 NPV 711300000 4019000PVIFA1574 13000001164 NPV 66386641 The cash ows if we abandon the project after one year are Year Sales Operating costs Depreciation EBT Tax Net income Depreciation Operating CF Change in NWC Capital spending Total cash ow 0 71300000 710000000 711300000 1 7350000 2400000 2500000 2450000 931000 1519000 2500000 4019000 1300000 7066000 12385000 202 The book value of the equipment is Book value 10000000 7 1100000004 Book value 7500000 So the taxes on the salvage value will be Taxes 7500000 7 680000038 Taxes 266000 This makes the aftertax salvage value A ertax salvage value 6800000 266000 A ertax salvage value 7066000 The NPV if we abandon the project after one year is NPV 711300000 12385000116 NPV 62327586 If we abandon the project after two years the cash ows are Year 0 l 2 Sales 73 50000 7350000 Operating costs 2400000 2400000 Depreciation 2500000 2500000 EBT 2450000 2450000 Tax 931000 931000 Net income 1519000 1519000 Depreciation 2500000 2500000 Operating CF 4019000 4019000 Change in NWC 7 13 00000 0 13 00000 Capital spending 710000000 0 5744000 Total cash ow 711300000 4019000 11063000 The book value of the equipment is Book value 10000000 7 2100000004 Book value 5000000 So the taxes on the salvage value will be Taxes 5000000 7 620000038 Taxes 7456000 203 This makes the aftertax salvage value A ertax salvage value 6200000 7 456000 A ertax salvage value 5744000 The NPV if we abandon the project after two years is NPV 711300000 4019000116 110630001162 NPV 38626635 If we abandon the project after three years the cash ows are Year 0 1 2 3 Sales 7350000 7350000 7350000 Operating costs 2400000 2400000 2400000 Depreciation 2500000 2500000 2500000 EBT 2450000 245 0000 2450000 Tax 931000 931000 931000 Net income 1519000 1519000 1519000 Depreciation 2450000 2450000 2450000 Operating CF 4019000 4019000 4019000 Change in NWC 7 13 00000 0 0 13 00000 Capital spending 710000000 0 0 33 06000 Total cash ow 711300000 4019000 4019000 8625000 The book value of the equipment is Book value 7 10000000 7 3100000004 Book value 2500000 So the taxes on the salvage value will be Taxes 2500000 7 380000038 Taxes 7494000 This makes the aftertax salvage value A ertax salvage value 3800000 7 494000 A ertax salvage value 3306000 204 03 O The NPV if we abandon the project after two years is NPV 7ll300000 4019000PVIFA1672 86250001163 NPV 67709931 We should abandon the equipment after three years since the NPV of abandoning the project after three years has the highest NPV a The NPV of the project is sum of the present value of the cash ows generated by the project The cash ows from this project are an annuity so the NPV is NPV 75000000 880000PVIFA10710 NPV 40721905 The company will abandon the project if the value of abandoning the project is greater than the value of the future cash ows The present value of the future cash ows if the company revises it sales downward will be PV of downward revision 50290000PVIFA1079l 10 PV of downward revision 75914405 Since this is less than the value of abandoning the project the company should abandon in one year So the revised NPV of the project will be the initial cost plus the expected cash ow in year one based on upward sales projection plus the expected value of abandonment We need to remember that the abandonment value occurs in year 1 and the present value of the expansion cash ows are in year one so each of these must be discounted back to today So the project NPV under the abandonment or expansion scenario is NPV 5000000 880000 110 501300000 110 501750000PVIFA109 110 NPV 97195076 First determine the cash ow from selling the old harvester When calculating the salvage value remember that tax liabilities or credits are generated on the difference between the resale value and the book value of the asset Using the original purchase price of the old harvester to determine annual depreciation the annual depreciation for the old harvester is Depreciationold 50000 15 DepreciationOId 333333 Since the machine is ve years old the rm has accumulated ve annual depreciation charges reducing the book value of the machine The current book value of the machine is equal to the initial purchase price minus the accumulated depreciation so Book value Initial Purchase Price 7 Accumulated Depreciation Book value 50000 7 3333333 x 5 years Book value 3333333 205 Since the rm is able to resell the old harvester for 19000 which is less than the 33333 book value of the machine the rm will generate a tax credit on the sale The aftertax salvage value of the old harvester will be A ertax salvage value Market value tc Book value 7 Market value A ertax salvage value 18000 343333333 7 18000 A ertax salvage value 2321333 Next we need to calculate the incremental depreciation We need to calculate depreciation tax shield generated by the new harvester less the forgone depreciation taX shield from the old harvester Let P be the breakeven purchase price of the new harvester So we nd Depreciation tax shieldNew Initial Investment Economic Life gtlt tC Depreciation tax shieldNew P 10 34 And the depreciation tax shield on the old harvester is Depreciation tax shieldOId 50000 15 34 Depreciation tax shieldom 333333034 So the incremental depreciation tax which is the depreciation tax shield from the new harvester minus the depreciation tax shield from the old harvester is Incremental depreciation tax shield P 1034 7 33333334 Incremental depreciation tax shield P 10 7 33333334 The present value of the incremental depreciation tax shield will be PVDeprecia ontax shield P 1034PV1FA15107 33333334PVIFA1510 The new harvester will generate yearend pretaX cash ow savings of 12000 per year for 10 years We can nd the a ertax present value of the cash ows savings as PVSsavings C11 tcPVIFA1510 PVSsavings 123000 034PVIFA1510 vamg 7 3974865 The breakeven purchase price of the new harvester is the price P which makes the NPV of the machine equal to zero NPV 7P Salvage valueom PVDepmmon m shield PVsmngs 0 7 7P 2321333 7 P 1034PVIFA1510 7 33333334PVIFA15710 3974865 P 7 P 1034PVIFA1510 7 6296198 7 33333334PVIFA15710 P1 7 1 1034PVIFA1510 5727405 P 6905797 206 CHAPTER 8 INTEREST RATES AND BOND VALUATION Answers to Concept Questions 1 p A O No As interest rates uctuate the value of a Treasury security will uctuate Longterm Treasury securities have substantial interest rate risk All else the same the Treasury security will have lower coupons because of its lower default risk so it will have greater interest rate risk No If the bid were higher than the ask the implication would be that a dealer was willing to sell a bond and immediately buy it back at a higher price How many such transactions would you like to do Prices and yields move in opposite directions Since the bid price must be lower the bid yield must be higher Bond issuers look at outstanding bonds of similar maturity and risk The yields on such bonds are used to establish the coupon rate necessary for a particular issue to initially sell for par value Bond issuers also simply ask potential purchasers what coupon rate would be necessary to attract them The coupon rate is xed and simply determines what the bond s coupon payments will be The required return is what investors actually demand on the issue and it will uctuate through time The coupon rate and required return are equal only if the bond sells for exactly at par Yes Some investors have obligations that are denominated in dollars ie they are nominal Their primary concern is that an investment provides the needed nominal dollar amounts Pension funds for example often must plan for pension payments many years in the future If those payments are fixed in dollar terms then it is the nominal return on an investment that is important Companies pay to have their bonds rated simply because unrated bonds can be dif cult to sell many large investors are prohibited from investing in unrated issues Treasury bonds have no credit risk since it is backed by the US government so a rating is not necessary Junk bonds often are not rated because there would be no point in an issuer paying a rating agency to assign its bonds a low rating it s like paying someone to kick you The term structure is based on pure discount bonds The yield curve is based on couponbearing issues Bond ratings have a subjective factor to them Split ratings re ect a difference of opinion among credit agencies 207 p A p A p A N p A UI As a general constitutional principle the federal government cannot tax the states without their consent if doing so would interfere with state government functions At one time this principle was thought to provide for the taxexempt status of municipal interest payments However modern court rulings make it clear that Congress can revoke the municipal exemption so the only basis now appears to be historical precedent The fact that the states and the federal government do not tax each other s securities is referred to as reciprocal immunity Lack of transparency means that a buyer or seller can t see recent transactions so it is much harder to determine what the best bid and ask prices are at any point in time When the bonds are initially issued the coupon rate is set at auction so that the bonds sell at par value The wide range coupon of coupon rates shows the interest rate when the bond was issued Notice that interest rates have evidently declined Why Companies charge that bond rating agencies are pressuring them to pay for bond ratings When a company pays for a rating it has the opportunity to make its case for a particular rating With an unsolicited rating the company has no input A lOOyear bond looks like a share of preferred stock In particular it is a loan with a life that almost certainly exceeds the life of the lender assuming that the lender is an individual With a junk bond the credit risk can be so high that the borrower is almost certain to default meaning that the creditors are very likely to end up as part owners of the business In both cases the equity in disguise has a signi cant tax advantage 1 The bond price is the present value of the cash ows from a bond The YTM is the interest rate used in valuing the cash ows from a bond b If the coupon rate is higher than the required return on a bond the bond will sell at a premium since it provides periodic income in the form of coupon payments in excess of that required by investors on other similar bonds If the coupon rate is lower than the required return on a bond the bond will sell at a discount since it provides insufficient coupon payments compared to that required by investors on other similar bonds For premium bonds the coupon rate exceeds the YTM for discount bonds the YTM exceeds the coupon rate and for bonds selling at par the YTM is equal to the coupon rate 0 Current yield is defined as the annual coupon payment divided by the current bond price For premium bonds the current yield exceeds the YTM for discount bonds the current yield is less than the YTM and for bonds selling at par value the current yield is equal to the YTM In all cases the current yield plus the expected oneperiod capital gains yield of the bond must be equal to the required return A longterm bond has more interest rate risk compared to a shortterm bond all else the same A low coupon bond has more interest rate risk than a high coupon bond all else the same When comparing a high coupon longterm bond to a low coupon shortterm bond we are unsure which has more interest rate risk Generally the maturity of a bond is a more important determinant of the interest rate risk so the longterm high coupon bond probably has more interest rate risk The exception would be if the maturities are close and the coupon rates are vastly different 208 Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem NOT E Most problems do not explicitly list a par value for bonds Even though a bond can have any par value in general corporate bonds in the United States will have a par value of 1000 We will use this par value in all problems unless a di erent par value is explicitly stated Basic 1 The price of a pure discount zero coupon bond is the present value of the par Remember even though there are no coupon payments the periods are semiannual to stay consistent with coupon bond payments So the price of the bond for each YTM is a P 10001 05220 61027 b P 10001 10220 37689 c P 10001 15220 23541 2 The price of any bond is the PV of the interest payment plus the PV of the par value Notice this problem assumes a semiannual coupon The price of the bond at each YTM will be a P 351711 03550 035 10001 1 0355 P 100000 When the YTM and the coupon rate are equal the bond will sell at par b P 35lt171104550 045 100011 0455 P 80238 When the YTM is greater than the coupon rate the bond will sell at a discount c P 351711 02550 025 100011 0255 P 128362 When the YTM is less than the coupon rate the bond will sell at a premium We would like to introduce shorthand notation here Rather than write or type as the case may be the entire equation for the PV of a lump sum or the PVA equation it is common to abbreviate the equations as PVIFR 7 1 1 r which stands for Eresent yalue interest Eactor 209 PVIFAR11711 r r which stands for Eresent Xalue lnterest Eactor of an Annuity These abbreviations are short hand notation for the equations in which the interest rate and the number of periods are substituted into the equation and solved We will use this shorthand notation in the remainder of the solutions key Here we are finding the YTM of a semiannual coupon bond The bond price equation is P 1050 39PVIFAR720 1000PVIFR720 Since we cannot solve the equation directly for R using a spreadsheet a financial calculator or trial and error we nd R 3547 Since the coupon payments are semiannual this is the semiannual interest rate The YTM is the APR of the bond so YTM 2 X 3547 709 Here we need to find the coupon rate of the bond All we need to do is to set up the bond pricing equation and solve for39 the coupon payment as follows P 1175 CPVIFA3 8m7 1000PVIF3 8W7 Solving for the coupon payment we get C 4848 Since this is the semiannual payment the annual coupon payment is 2 X 4848 9696 And the coupon rate is the annual coupon payment divided by par value so Coupon rate 9696 1000 09696 or 970 The price of any bond is the PV of the interest payment plus the PV of the par value The fact that the bond is denominated in euros is irrelevant Notice this problem assumes an annual coupon The price of the bond will be P 841711 07615 076 10001 1 076 P 107018 210 O Here we are nding the YTM of an annual coupon bond The fact that the bond is denominated in yen is irrelevant The bond price equation is P 87000 5400PVIFAR721 100000PVIFR721 Since we cannot solve the equation directly for R using a spreadsheet a financial calculator or trial and error we nd R 656 Since the coupon payments are annual this is the yield to maturity The approximate relationship between nominal interest rates R real interest rates r and in ation is R r h Approximate r 05 7039 011 or 110 The Fisher equation which shows the exact relationship between nominal interest rates real interest rates and in ation is 1R1r1h 1 05 1 r1039 Exactr 1 05 1 039 7 1 0106 or 106 The Fisher equation which shows the exact relationship between nominal interest rates real interest rates and in ation is 1R1r1h R 10251047710732 or 732 The Fisher equation which shows the exact relationship between nominal interest rates real interest rates and in ation is 1R1r1h h 1 171 117 1 0541 or 541 The Fisher equation which shows the exact relationship between nominal interest rates real interest rates and in ation is 1 R1 r1 11 r 1 141 1068 7 1 0684 or 684 211 11 p n N p A DJ The coupon rate located in the rst column of the quote is 6125 The bid price is Bid price 11919 1191932 11959375 gtlt 1000 11959375 The previous day s ask price is found by Previous day s asked price Today s asked price 7 Change 119 2132 7 71732 120 632 The previous day s price in dollars was Previous day s dollar price 120 1875 gtlt 1000 1201875 This is a premium bond because it sells for more than 100 of face value The current yield is Current yield Annual coupon payment Asked price 75l347l875 0557 or 557 The YTM is located under the Asked yield column so the YTM is 448 17 The bidask spread is the difference between the bid price and the ask price so BidAsk spread 13423 7 13422 132 Intermediate Here we are nding the YTM of semiannual coupon bonds for various maturity lengths The bond price equation is P 7 CPVIFARW 1000PVIFR Miller Corporation bond P0 45PVIFA3 5mg 1000PVIF3 5 116890 P1 45PVIFA3 5 1000PVIF3 5 116058 P3 45PVIFA3 5mg 1000PVIF3 5720 114212 P8 45PVIFA3 510 1000PVIF3 5710 108317 P12 45PVIFA35 1000PVIF3572 101900 P13 1000 Modigliani Company bond P0 35PVIFA4 5 1000PVIF4 5 84853 P1 35PVIFA4 5 1000PVIF4 5 85505 P3 35PVIFA4 5mg 1000PVIF4 5720 86992 P8 35PVIFA4 510 1000PVIF4 5710 92087 P12 35PVIFA4 5m 1000PVIF4 52 98127 P13 1000 All else held equal the premium over par value for a premium bond declines as maturity approaches and the discount from par value for a discount bond declines as maturity approaches This is called pull to par In both cases the largest percentage price changes occur at the shortest maturity lengths 212 4 UI Also notice that the price of each bond when no time is left to maturity is the par value even though the purchaser would receive the par value plus the coupon payment immediately This is because we calculate the clean price of the bond Any bond that sells at par has a YTM equal to the coupon rate Both bonds sell at par so the initial YTM on both bonds is the coupon rate 8 percent If the YTM suddenly rises to 10 percent 7 40PVIFA54 1000PVIF54 96454 PLaurel PHmdy 40PVIFA5730 1000PVIF530 84628 The percentage change in price is calculated as Percentage change in price New price 7 Original price Original price APLam1 96454 71000 1000 410355 or 7355 APHady 84628 71000 1000 701537 or 71537 If the YTM suddenly falls to 6 percent PLaurel 40PVIFA34 1000PVIF34 103717 PHmdy 40PVIFA3730 1000PVIF330 119600 APLam1 103717 7 1000 1000 00372 or 372 APHady 1196002 7 1000 1000 01960 or 1960 All else the same the longer the maturity of a bond the greater is its price sensitivity to changes in interest rates Notice also that for the same interest rate change the gain from a decline in interest rates is larger than the loss from the same magnitude change For a plain vanilla bond this is always true Initially at a YTM of 10 percent the prices of the two bonds are PFaulk 30PVIFA5716 1000PVIF5716 78324 PGms 70PVIFA5716 1000PVIF5715 121676 If the YTM rises from 10 percent to 12 percent PFaulk 30PVIFA6716 1000PVIF5715 69682 PGmS 70PVIFA6716 1000PVIF6716 110106 213 5 gt1 The percentage change in price is calculated as Percentage change in price New price 7 Original price Original price APFau1k 69682 7 78324 78324 411103 or 71103 APGOMS 110106 7121676121676 700951 or 7951 If the YTM declines from 10 percent to 8 percent PFaulk 88348 7 30PVIFA4716 1000PVIF4716 7 70PVIFA4716 1000PVIF4316 7 134957 PGonas APFau1k 88348 7 78324 78324 01280 or 1280 APGOMS 134957 7 121676 121676 01092 or 1092 All else the same the lower the coupon rate on a bond the greater is its price sensitivity to changes in interest rates The bond price equation for this bond is P0 960 37PVIFAR718 1000PVIFR718 Using a spreadsheet fmancial calculator or trial and error we find R 4016 This is the semiannual interest rate so the YTM is YTM 2 x 4016 803 The current yield is Current yield Annual coupon payment Price 74 960 0771 or 771 The effective annual yield is the same as the EAR so using the EAR equation from the previous chapter Effective annual yield 1 0040162 7 1 0819 or 819 The company should set the coupon rate on its new bonds equal to the required return The required return can be observed in the market by nding the YTM on outstanding bonds of the company So the YTM on the bonds currently sold in the market is P 1063 50PVIFAR40 1000PVIFR740 214 on O O Using a spreadsheet nancial calculator or trial and error we find R 4650 This is the semiannual interest rate so the YTM is YTM 2 x 4650 930 Accrued interest is the coupon payment for the period times the fraction of the period that has passed since the last coupon payment Since we have a semiannual coupon bond the coupon payment per six months is onehalf of the annual coupon payment There are two months until the next coupon payment so four months have passed since the last coupon payment The accrued interest for the bond is Accrued interest 842 X 46 28 And we calculate the clean price as Clean price Dirty price 7 Accrued interest 1090 7 28 1062 Accrued interest is the coupon payment for the period times the fraction of the period that has passed since the last coupon payment Since we have a semiannual coupon bond the coupon payment per six months is onehalf of the annual coupon payment There are four months until the next coupon payment so two months have passed since the last coupon payment The accrued interest for the bond is Accrued interest 722 X 26 1200 And we calculate the dirty price as Dirty price Clean price Accrued interest 904 12 91600 To nd the number of years to maturity for the bond we need to nd the price of the bond Since we already have the coupon rate we can use the bond price equation and solve for the number of years to maturity We are given the current yield of the bond so we can calculate the price as Current yield 0842 90P0 P0 900842 106888 Now that we have the price of the bond the bond price equation is P 106888 9017110781 0781 100010781 We can solve this equation for t as follows 106888 10781 115237 10781 7 115237 1000 15237 834910781 18251 107812 1 log 18251 log 10781 80004 m 8 years The bond has 8 years to maturity 215 21 N N N 03 The bond has 10 years to maturity so the bond price equation is P 87155 4125PVIFAR20 1000PVIFR710 Using a spreadsheet financial calculator or trial and error we find R 5171 This is the semiannual interest rate so the YTM is YTM 2 x 5171 1034 The current yield is the annual coupon payment diVided by the bond price so Current yield 8250 87l55 0947 or 947 We found the maturity of a bond in Problem 20 However in this case the maturity is indeterminate A bond selling at par can have any length of maturity In other words when we solve the bond pricing equation as we did in Problem 20 the number of periods can be any positive number Challenge To nd the capital gains yield and the current yield we need to find the price of the bond The current price of Bond P and the price of Bond P in one year is P P0 90PVIFA75 1000PVIF75 108200 P1 90PVIFA774 1000PVIF74 106774 Current yield 90 108200 0832 or 832 The capital gains yield is Capital gains yield New price 7 Original price Original price Capital gains yield 106774 7108200108200 410132 or 7132 The current price of Bond D and the price of Bond D in one year is D P0 50PVIFA75 1000PVIF75 91800 P1 50PVIFA774 1000PVIF74 93226 Current yield 50 9l800 00545 or 545 Capital gains yield 93226 7 91800 91800 00155 or 155 All else held constant premium bonds pay a high current income while having price depreciation as maturity nears discount bonds pay a lower current income but have price appreciation as maturity nears For either bond the total return is still 7 but this return is distributed differently between current income and capital gains 216 25 N 5 a The rate of return you expect to earn if you purchase a bond and hold it until maturity is the YTM The bond price equation for this bond is P0 1140 90PVIFAR10 1000PVIF Mylo Using a spreadsheet nancial calculator or trial and error we nd R YTM 701 b To nd our HPY we need to nd the price of the bond in two years The price of the bond in two years at the new interest rate will be P2 90PVIFA6 mm 1000PVIF5 01m 118587 To calculate the HPY we need to find the interest rate that equates the price we paid for the bond with the cash ows we received The cash ows we received were 90 each year for two years and the price of the bond when we sold it The equation to nd our HPY is P0 1140 90PVIFAR72 118587PVIFR72 Solving for R we get R HPY 981 The realized HPY is greater than the expected YTM when the bond was bought because interest rates dropped by 1 percent bond prices rise when yields fall The price of any bond or nancial instrument is the PV of the future cash ows Even though Bond M makes different coupons payments to nd the price of the bond we just nd the PV of the cash ows The PV of the cash ows for Bond M is PM 800PVIFA4716PVIFMIZ SE1000PVIFA4712PVIFMZB 20000PVIF4740 PM 1311788 Notice that for the coupon payments of 800 we found the PVA for the coupon payments and then discounted the lump sum back to today Bond N is a zero coupon bond with a 20000 par value therefore the price of the bond is the PV of the par or PN 20000PVIF4740 416578 To nd the present value we need to find the real weekly interest rate To nd the real return we need to use the effective annual rates in the Fisher equation So we nd the real EAR is 1 R 1 r1 11 1 1071r1 035 r 0696 or 696 217 1 Now to nd the weekly interest rate we need to nd the APR Using the equation for discrete compounding EAR 1 APRmm71 We can solve for the APR Doing so we get APR m1 EAR139quot 71 APR 521 0696 52 7 1 APR 0673 or 673 So the weekly interest rate is Weekly rate APR 52 Weekly rate 0673 52 Weekly rate 0013 or 0 13 Now we can nd the present value of the cost of the roses The real cash ows are an ordinary annuity discounted at the real interest rate So the presentvalue of the cost of the roses is PVA C1711 rt r PVA 81 7 11 00133 0013 PVA 535964 To answer this question we need to nd the monthly interest rate which is the APR divided by 12 We also must be careful to use the real interest rate The Fisher equation uses the effective annual rate so the real effective annual interest rates and the monthly interest rates for each account are Stock account 1 R 1 rl h 1 12 1 r1 04 r 0769 or 769 APR ml EAR 71 APR 121 0769 12 7 1 APR 0743 or 743 Monthly rate APR 12 Monthly rate 0743 12 Monthly rate 0062 or 062 Bond account 1R1 00 10 1 07 1 r1 04 r 0288 or 288 APR ml EAR 71 APR 121 0288 12 7 1 APR 0285 or 285 218 Monthly rate APR 12 Monthly rate 0285 12 Monthly rate 0024 or 024 Now we can nd the future value of the retirement account in real terms The future value of each account will be Stock account FVAClr 7 lr FVA 8001 00623607 1 0062 FVA 106376175 Bond account FVAC1 r 7 lr FVA 4001 002436 71 0024 FVA 22708904 The total future value of the retirement account will be the sum of the two accounts or Account value 106376175 22708904 Account value 129085079 Now we need to nd the monthly interest rate in retirement We can use the same procedure that we used to nd the monthly interest rates for the stock and bond accounts so 1R1r1h 1081 r1 04 r 0385 or385 APR m1 EAR 71 APR 121 0385 12 7 1 APR 0378 or 378 Monthly rate APR 12 Monthly rate 0378 12 Monthly rate 0031 or 031 Now we can nd the real monthly withdrawal in retirement Using the present value of an annuity equation and solving for the payment we find PVA C1711 rt r 129085079 C17110031300 0031 C 665774 219 This is the real dollar amount of the monthly withdrawals The nominal monthly withdrawals will increase by the in ation rate each month To find the nominal dollar amount of the last withdrawal we can increase the real dollar withdrawal by the in ation rate We can increase the real withdrawal by the effective annual in ation rate since we are only interested in the nominal amount of the last withdrawal So the last withdrawal in nominal terms will be FV PV1 r2 FV 6657741 04 0 25 FV 5756530 N on In this problem we need to calculate the future value of the annual savings after the ve years of operations The savings are the revenues minus the costs or Savings Revenue 7 Costs Since the annual fee and the number of members are increasing we need to calculate the effective growth rate for revenues which is Effective growth rate 1 061 03 7 1 Effective growth rate 0918 or 918 The revenue for the current year is the number of members times the annual fee or Current revenue 5005 00 Current revenue 250000 The revenue will grow at 918 percent and the costs will grow at 2 percent so the savings each year for the next ve years will be M Revenue Costs Savings 1 27295000 7650000 19645000 2 29800681 7803000 21997681 3 32536384 7959060 24577324 4 35523224 8118241 27404982 5 38784255 8280606 30503649 Now we can find the value of each year s savings using the future value of a lump sum equation so FV PV1 r2 220 M Future Value 1 196450001 094 27730521 2 219976811 093 28487635 3 245773241 092 29200318 4 274049821 091 29871431 5 30503649 Total future value of savings 145793554 He will spend 500000 on a luxury boat so the value of his account will be Value of account 145793554 7500000 Value of account 95793554 Now we can use the present value of an annuity equation to nd the payment Doing so we nd PVA C1711 r r 95793554 C17110925 09 C 9752383 221 Calculator Solutions 1 a Enter 20 25 Solve for b Enter 20 5 Solve for 0 Enter 20 75 Solve for 2 a Enter 50 35 Solve for b Enter 50 45 Solve for 0 Enter 50 25 Solve for 3 Enter 20 Solve for 3547 3547 x 2 709 4 Enter 27 38 Solve for 4848 x 2 9696 9696 1000 970 61027 37689 23541 100000 80238 128362 i1050 PV 1175 222 35 35 35 39 4848 1000 1000 1000 1000 FV 1000 FV 1000 FV 1000 FV 100 5 Enter 15 760 Solve for 6 Enter 21 Solve for 656 13 Miller Corporation 0 Enter 26 35 Solve for P1 Enter 24 35 Solve for P3 Enter 20 35 Solve for P8 Enter 10 35 Solve for P12 Enter 2 35 Solve for Modigliani Company Po Enter 26 45 Solve for P1 Enter 24 45 Solve for 107018 i 87000 PV 116890 116058 114212 PV 108317 101900 84853 85505 223 84 5400 45 45 45 45 45 35 35 1000 100000 FV39 1000 1000 FV 1000 FV 1000 1000 FV 1000 FV 1000 P3 Enter 20 45 35 1000 Solve for 86992 Pa Enter 10 45 35 1000 Solve for 92087 P12 Enter 2 45 35 1000 Solve for 98172 14 If both bonds sell at par the initial YTM on both bonds is the coupon rate 8 percent If the YTM suddenly rises to 10 percent PLaurel Enter 4 5 40 1000 Solve for 964 54 APLaue1 96454 7 1000 1000 7355 PHardy Enter 30 5 40 1000 FV Solve for 84628 APHardyoU 84628 71000 1000 71537 If the YTM suddenly falls to 6 percent PLaurel Enter 4 3 40 1000 Solve for 103717 APmel 103717 71000 1000 372 PHardy Enter 30 3 40 1000 Solve for 119600 APHamy 119600 710001000 1960 All else the same the longer the maturity of a bond the greater is its price sensitivity to changes in interest rates 224 15 Initially at a YTM of 10 percent the prices of the two bonds are PFaulk Enter 16 5 30 1000 Solve for 78324 PGonas Enter 16 50 70 1000 PV Solve for 121676 If the YTM rises from 10 percent to 12 percent Faulk Enter 16 6 30 1000 Solve for 69682 APFau1k 69682 7 78324 78324 71103 PGonas Enter 16 6 70 1000 FV Solve for 110106 APGonas 110106 7121676 121676 7951 If the YTM declines from 10 percent to 8 percent PFaulk Enter 16 4 30 1000 Solve for 88348 APFau1k 88348 7 78324 78324 1280 PGonas Enter 16 4 70 1000 Solve for 1 34957 APGOMS 7 134957 7121676121676 1092 All else the same the lower the coupon rate on a bond the greater is its price sensitivity to changes in interest rates 16 Enter 18 i960 37 1000 PMT Solve for 4016 YTM 4016 x 2 803 225 17 The company should set the coupon rate on its new bonds equal to the required return the required return can be observed in the market by nding the YTM on outstanding bonds of the company Enter 40 i1063 50 1000 PMT Solve for 4650 4650 x 2 930 20 Current yield 0842 90P0 P0 106888 Enter 781 il06888 90 1000 N IY V Solve for 80004 8 years 2 1 Enter 20 i87l55 4125 1000 Solve for 5171 5171 gtlt 2 1034 23 Bond P Po Enter 5 7 90 1000 FV Solve for 108200 P1 Enter 4 7 90 1000 Solve for 106774 Current yield 90 108200 832 Capital gains yield 106774 7 108200 108200 7132 Bond D Po Enter 5 7 50 1000 Solve for 91800 P1 Enter 4 7 50 1000 Solve for 93226 Current yield 50 91800 545 Capital gains yield 93226 7 91800 91800 155 All else held constant premium bonds pay a higher current income while having price depreciation as maturity nears discount bonds pay a lower current income but have price appreciation as maturity nears For either bond the total return is still 7 but this return is distributed differently between current income and capital gains 226 24 a Enter 10 i1140 90 1000 PMT FV Solve for 701 This is the rate of return you expect to earn on your investment when you purchase the bond b Enter 8 601 90 1000 PMT Solve for 118587 The HPY is Enter 2 i1140 90 118517 PMT Solve for 981 The realized HPY is greater than the expected YTM when the bond was bought because interest rates dropped by 1 percent bond prices rise when yields fall 25 PM I 4 NPV CPT 1311788 PN Enter 40 4 20000 PMT Solve for 416578 29 Real return for stock account 1 12 1 r1 04 r 76923 Enter 76923 12 Solve for 74337 Real return for bond account 1 07 1 r1 04 r 28846 Enter 28846 1 N Solve for 28472 227 Real return postretirement 1 nter Solve for 37800 Stock portfolio value Enter 12 X 30 N Solve for Bond portfolio value Enter 12 X 30 N Solve for 3 8462 EFF 74337 12 2 8472 12 08111 04 1 CY r 38462 2 800 400 Retirement value 106376175 22708904 129085079 Retirement withdrawal Enter 25 X 12 Solve for 3780012 129085079 39 PV The last withdrawal in real terms is Enter 30 25 N Solve for 30 Future value of savings Year 1 Enter 4 Solve for Year 2 Enter 3 Solve for Year 3 Enter 2 Solve for Year 4 Enter 1 Solve for 4 IY 9 9 9 9 665774 196450 21997681 24577324 274049 82 PV39 228 665774 106376175 22708904 H 4 5756530 27730521 28487635 292003 18 7 298 1431 Future value 27730521 28487635 29200318 29871431 30503649 Future value 145793554 He will spend 500000 on a luxury boat so the value of his account will be Value of account 145793554 7 500000 Value of account 95793554 Enter 25 9 95793 5 54 PMT Solve for 97523 83 229 CHAPTER 9 STOCK VALUATION Answers to Concept Questions 1 p A O The value of any investment depends on the present value of its cash ows ie what investors will actually receive The cash ows from a share of stock are the dividends Investors believe the company will eventually start paying dividends or be sold to another company In general companies that need the cash will often forgo dividends since dividends are a cash expense Young growing companies with profitable investment opportunities are one example another example is a company in nancial distress This question is examined in depth in a later chapter The general method for valuing a share of stock is to nd the present value of all expected future dividends The dividend growth model presented in the text is only valid i if dividends are expected to occur forever that is the stock provides dividends in perpetuity and ii if a constant growth rate of dividends occurs forever A violation of the first assumption might be a company that is expected to cease operations and dissolve itself some nite number of years from now The stock of such a company would be valued by applying the general method of valuation explained in this chapter A violation of the second assumption might be a startup firm that isn t currently paying any dividends but is expected to eventually start making dividend payments some number of years from now This stock would also be valued by the general dividend valuation method explained in this chapter The common stock probably has a higher price because the dividend can grow whereas it is xed on the preferred However the preferred is less risky because of the dividend and liquidation preference so it is possible the preferred could be worth more depending on the circumstances The two components are the dividend yield and the capital gains yield For most companies the capital gains yield is larger This is easy to see for companies that pay no dividends For companies that do pay dividends the dividend yields are rarely over ve percent and are often much less Yes If the dividend grows at a steady rate so does the stock price In other words the dividend growth rate and the capital gains yield are the same The three factors are l The company s future growth opportunities 2 The company s level of risk which determines the interest rate used to discount cash ows 3 The accounting method used It wouldn t seem to be Investors who don t like the voting features of a particular class of stock are under no obligation to buy it Presumably the current stock value re ects the risk timing and magnitude of all future cash ows both shortterm and longterm If this is correct then the statement is false 230 Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic 1 The constant dividend growth model is P2Dt X1gRg So the price of the stock today is P0 D0 1 g R 7g 190 105 12 7 05 2850 The dividend at year 4 is the dividend today times the FVIF for the growth rate in dividends and four years so P3 D3 1 g R 7g D0 1 gquot Rig 190 1054 12 7 05 3299 We can do the same thing to nd the dividend in Year 16 which gives us the price in Year 15 so P15 D15 1 g R 7g Do 1 g R 7g 190 105 12705 5925 There is another feature of the constant dividend growth model The stock price grows at the dividend growth rate So if we know the stock price today we can nd the future value for any time in the future we want to calculate the stock price In this problem we want to know the stock price in three years and we have already calculated the stock price today The stock price in three years will be P3 P01 g3 28501 053 3299 And the stock price in 15 years will be P15 P01g15 285010515 5925 2 We need to nd the required return of the stock Using the constant growth model we can solve the equation for R Doing so we nd R D1 P0g 285 58 06 1091 or 1091 231 The dividend yield is the dividend next year divided by the current price so the dividend yield is Dividend yield D1 P0 285 58 0491 or 491 The capital gains yield or percentage increase in the stock price is the same as the dividend growth rate so Capital gains yield 6 Using the constant growth model we nd the price of the stock today is P0 D1 R 7g 305 11 7 0525 5304 The required return of a stock is made up of two parts The dividend yield and the capital gains yield So the required return of this stock is R Dividend yield Capital gains yield 047 058 1050 or 1050 We know the stock has a required return of 13 percent and the dividend and capital gains yield are equal so Dividend yield 12 13 065 Capital gains yield Now we know both the dividend yield and capital gains yield The dividend is simply the stock price times the dividend yield so D1 06564 416 This is the dividend next year The question asks for the dividend this year Using the relationship between the dividend this year and the dividend next year D1 D01 g We can solve for the dividend that was just paid 416 Do 1 065 D0 416 1065 391 The price of any nancial instrument is the PV of the future cash ows The future dividends of this stock are an annuity for 9 years so the price of the stock is the PVA which will be P0 11PVIFA109 6335 The price of a share of preferred stock is the dividend divided by the required return This is the same equation as the constant growth model with a dividend growth rate of zero percent Remember that most preferred stock pays a xed dividend so the growth rate is zero Using this equation we nd the price per share of the preferred stock is R DPO 640103 0621 or 621 232 O 1 The growth rate of earnings is the return on equity times the retention ratio so g ROE x b g 1570 g 1050 or 1050 To nd next year s earnings we simply multiply the current earnings times one plus the growth rate so Next year s earnings Current earnings1 g Next year s earnings 280000001 1050 Next year s earnings 30940000 Intermediate This stock has a constant growth rate of dividends but the required return changes twice To find the value of the stock today we will begin by finding the price of the stock at Year 6 when both the dividend growth rate and the required return are stable forever The price of the stock in Year 6 will be the dividend in Year 7 divided by the required return minus the growth rate in dividends So P6 D6 1 g Rig D0 1 g7 Rig 2751067 11 7 06 8270 Now we can find the price of the stock in Year 3 We need to find the price here since the required return changes at that time The price of the stock in Year 3 is the PV of the dividends in Years 4 5 and 6 plus the PV of the stock price in Year 6 The price of the stock in Year 3 is P3 275106quot 114 27510651142 27510661143 8270 1143 P3 6433 Finally we can find the price of the stock today The price today will be the PV of the dividends in Years 1 2 and 3 plus the PV of the stock in Year 3 The price of the stock today is P0 275106 116 2751062 1162 2751063 1163 6433 1163 PO 4812 Here we have a stock that pays no dividends for 10 years Once the stock begins paying dividends it will have a constant growth rate of dividends We can use the constant growth model at that point It is important to remember that general form of the constant dividend growth formula is P D2 X 1 g R g This means that since we will use the dividend in Year 10 we will be nding the stock price in Year 9 The dividend growth model is similar to the PVA and the PV of a perpetuity The equation gives you the PV one period before the first payment So the price of the stock in Year 9 will be P9 D10 R 7g 900 13 7 055 12000 The price of the stock today is simply the PV of the stock price in the future We simply discount the future stock price at the required return The price of the stock today will be P0 12000 1139 3995 233 12 1 J The price of a stock is the PV of the future dividends This stock is paying ve dividends so the price of the stock is the PV of these dividends using the required return The price of the stock is P0 13111161112 191113 22 1114 251115 6792 With differential dividends we find the price of the stock when the dividends level off at a constant growth rate and then nd the PV of the future stock price plus the PV of all dividends during the differential growth period The stock begins constant growth in Year 5 so we can nd the price of the stock in Year 4 one year before the constant dividend growth begins as R D4 1 g Rig 250105 13 7 05 3281 The price of the stock today is the PV of the first four dividends plus the PV of the Year 4 stock price So the price of the stock today will be P0 9 113 71132 5 1133 250 32811134 3857 With differential dividends we find the price of the stock when the dividends level off at a constant growth rate and then nd the PV of the future stock price plus the PV of all dividends during the differential growth period The stock begins constant growth in Year 4 so we can nd the price of the stock in Year 3 one year before the constant dividend growth begins as P3 D3 1 g R 7g D0 1 g13 1 g2 R 82 2401253107 127 07 10031 The price of the stock today is the PV of the rst three dividends plus the PV of the Year 3 stock price The price of the stock today will be P0 240125 112 24012521122 24012531123 10031 1123 PO 8040 Here we need to nd the dividend next year for a stock experiencing differential growth We know the stock price the dividend growth rates and the required return but not the dividend First we need to realize that the dividend in Year 3 is the current dividend times the FVIF The dividend in Year 3 will be D3 D0 1303 And the dividend in Year 4 will be the dividend in Year 3 times one plus the growth rate or D D0 1303 118 The stock begins constant growth after the 4th dividend is paid so we can nd the price of the stock in Year 4 as the dividend in Year 5 divided by the required return minus the growth rate The equation for the price of the stock in Year 4 is P4 D4 1 gRg 234 1 5 p A l 1 on Now we can substitute the previous dividend in Year 4 into this equation as follows P4 D0 1g131 821 g3R g3 P4 D0 1303 118 108 13 7 08 5600D0 When we solve this equation we nd that the stock price in Year 4 is 5600 times as large as the dividend today Now we need to nd the equation for the stock price today The stock price today is the PV of the dividends in Years 1 2 3 and 4 plus the PV of the Year 4 price So P0 D0130113 D013021132 D013031133 D013031181134 5600D01134 We can factor out Do in the equation and combine the last two terms Doing so we get P0 6500 D0130113 13021132 13031133 1303118 56001134 Reducing the equation even further by solving all of the terms in the braces we get 65 3986D0 D0 65003986 163 This is the dividend today so the projected dividend for the next year will be D1 163130 212 The constant growth model can be applied even if the dividends are declining by a constant percentage just make sure to recognize the negative growth So the price of the stock today will be P0 D0 1 g R 7g 121706117706 6635 We are given the stock price the dividend growth rate and the required return and are asked to nd the dividend Using the constant dividend growth model we get P0 4980 D0 1 g R 7g Solving this equation for the dividend gives us Do 498011 7 05 105 285 The price of a share of preferred stock is the dividend payment divided by the required return We know the dividend payment in Year 5 so we can nd the price of the stock in Year 4 one year before the rst dividend payment Doing so we get P4 700 06 11667 The price of the stock today is the PV of the stock price in the future so the price today will be P0 11667 106 9241 235 19 The annual dividend is the dividend divided by the stock price so N O Dividend yield Dividend Stock price 016 Dividend 1947 Dividend 031 The Net Chg of the stock shows the stock decreased by 012 on this day so the closing stock price yesterday was Yesterday s closing price 1947 7 7012 1959 To nd the net income we need to nd the EPS The stock quote tells us the PE ratio for the stock is 19 Since we know the stock price as well we can use the PE ratio to solve for EPS as follows PE l9 Stock price EPS 1947 EPS EPS 1947 19 1025 We know that EPS is just the total net income divided by the number of shares outstanding so EPS NI Shares 1025 NI 25000000 N1 7 102525000000 7 25618421 To nd the number of shares owned we can divide the amount invested by the stock price The share price of any nancial asset is the present value of the cash ows so to nd the price of the stock we need to find the cash ows The cash ows are the two dividend payments plus the sale price We also need to nd the aftertax dividends since the assumption is all dividends are taxed at the same rate for all investors The aftertax dividends are the dividends times one minus the tax rate so Year 1 aftertax dividend l50l 7 28 Year 1 aftertax dividend 108 Year 2 aftertax dividend 225l 7 28 Year 2 aftertax dividend 162 We can now discount all cash ows from the stock at the required return Doing so we nd the price of the stock is P 7 108115 1621152 601153 P 7 4162 The number of shares owned is the total investment divided by the stock price which is Shares owned 100000 4162 Shares owned 240298 236 21 Here we have a stock paying a constant dividend for a xed period and an increasing dividend thereafter We need to nd the present value of the two different cash ows using the appropriate quarterly interest rate The constant dividend is an annuity so the present value of these dividends is PVA CPVIFAm W A 075Pv1FA2 5 PVA 769 Now we can nd the present value of the dividends beyond the constant dividend phase Using the present value of a growing annuity equation we nd P12 D13 R 7g P12 0751 01 025 7 01 P12 5050 This is the price of the stock immediately after it has paid the last constant dividend So the present value of the future price is PV 5050102512 PV 3755 The price today is the sum of the present value of the two cash ows so P0 769 3755 P0 4524 Here we need to nd the dividend next year for a stock with nonconstant growth We know the stock price the dividend growth rates and the required return but not the dividend First we need to realize that the dividend in Year 3 is the constant dividend times the FVIF The dividend in Year 3 will be D3 Dl05 The equation for the stock price will be the present value of the constant dividends plus the present value of the future stock price or PO D111D1112 D105117051112 38 D111 D 1112 D105117051112 We can factor out Do in the equation Doing so we get 38 Dlt111111112 105117051112 Reducing the equation even further by solving all of the terms in the braces we get 38 D159159 D 38 159159 239 237 23 N J The required return of a stock consists of two components the capital gains yield and the dividend yield In the constant dividend growth model growing perpetuity equation the capital gains yield is the same as the dividend growth rate or algebraically R DlPo g We can find the dividend growth rate by the growth rate equation or gROE X b g 16 X 80 g 1280 or 1280 This is also the growth rate in dividends To nd the current dividend we can use the information provided about the net income shares outstanding and payout ratio The total dividends paid is the net income times the payout ratio To nd the dividend per share we can divide the total dividends paid by the number of shares outstanding So Dividend per share Net income gtlt Payout ratio Shares outstanding Dividend per share 10000000 X 20 2000000 Dividend per share 100 Now we can use the initial equation for the required return We must remember that the equation uses the dividend in one year so R DlPO g R 11 128085 1280 R 1413 or 1413 First we need to nd the annual dividend growth rate over the past four years To do this we can use the future value of a lump sum equation and solve for the interest rate Doing so we nd the dividend growth rate over the past four years was FV PV1R 193 1201 R R 193 120 71 R 1261 or 1261 We know the dividend will grow at this rate for ve years before slowing to a constant rate inde nitely So the dividend amount in seven years will be D7 13014r 8151g22 D7 1931126151 072 D7 400 a We can nd the price of all the outstanding company stock by using the dividends the same way we would value an individual share Since earnings are equal to dividends and there is no growth the value of the company s stock today is the present value of a perpetuity so P 750000 14 P 535714286 238 The priceeamings ratio is the stock price divided by the current earnings so the priceeamings ratio of each company with no growth is PE Price Earnings PE 535714286 750000 PE 714 times Since the earnings have increased the price of the stock will increase The new price of the all the outstanding company stock is P D R P 750000 100000 14 P 607142857 The priceeamings ratio is the stock price divided by the current earnings so the priceeamings with the increased earnings is PE Price Earnings PE 607142857 750000 PE 810 times Since the earnings have increased the price of the stock will increase The new price of the all the outstanding company stock is P D R P 750000 200000 14 P 678571429 The priceeamings ratio is the stock price divided by the current earnings so the priceeamings with the increased earnings is PE Price Earnings PE 678571429 750000 PE 905 times If the company does not make any new investments the stock price will be the present value of the constant perpetual dividends In this case all earnings are paid dividends so applying the perpetuity equation we get P Dividend R P 825 12 P 6875 The investment is a onetime investment that creates an increase in EPS for two years To calculate the new stock price we need the cash cow price plus the NPVGO In this case the NPVGO is simply the present value of the investment plus the present value of the increases in EPS So the NPVGO will be NPVGOC11RC21R2C31R3 NPVGO 7160 112 2101122 245 1123 NPVGO 199 239 So the price of the stock if the company undertakes the investment opportunity will be P 6875 199 P 7074 After the project is over and the earnings increase no longer exists the price of the stock will revert back to 6875 the value of the company as a cash cow The price of the stock is the present value of the dividends Since earnings are equal to dividends we can nd the present value of the earnings to calculate the stock price Also since we are excluding taxes the earnings will be the revenues minus the costs We simply need to nd the present value of all future earnings to nd the price of the stock The present value of the revenues is PVRevenue C1 R 7 g PvRWe 60000001 05 15 7 05 PVRevenue 63000000 And the present value of the costs will be PVCosts C1 R 7g PVCosts 31000001 05 15 705 PVCosts 32550000 Since there are no taxes the present value of the company s earnings and dividends will be PVDividends 63000000 7 32550000 PVDividends 30450000 Note that since revenues and costs increase at the same rate we could have found the present value of future dividends as the present value of current dividends Doing so we nd D0 Revenueo 7 Costso D0 6000000 7 3100000 D0 2900000 NOW aPPlinlg the growing perpetuity equation we nd PVDividends C1 R 7g PVDividends 2900000l 05 15 705 PVDividends 30450000 This is the same answer we found previously The price per share of stock is the total value of the company s stock divided by the shares outstanding or P Value of all stock Shares outstanding P 7 30450000 1000000 P 7 3045 240 The value of a share of stock in a company is the present value of its current operations plus the present value of growth opportunities To nd the present value of the growth opportunities we need to discount the cash outlay in Year 1 back to the present and nd the value today of the increase in earnings The increase in earnings is a perpetuity which we must discount back to today So the value of the growth opportunity is NPVGOC0 C11RC2R1R NPVGO 2000000 7 8000000 1 15 7000000 15 1 15 NPVGO 1162318841 To nd the value of the growth opportunity on a per share basis we must divide this amount by the number of shares outstanding which gives us NPVGOPer 1162318841 1000000 NPVGOPer 1162 The stock price will increase by 1162 per share The new stock price will be New stock price 3045 1162 New stock price 4207 If the company continues its current operations it will not grow so we can value the company as a cash cow The total value of the company as a cash cow is the present value of the future earnings which are a perpetuity so Cash cow value of company C R Cash cow value of company 85000000 12 Cash cow value of company 70833333333 The value per share is the total value of the company divided by the shares outstanding so Share price 70833333333 20000000 Share price 3542 To nd the value of the investment we need to nd the NPV of the growth opportunities The initial cash ow occurs today so it does not need to be discounted The earnings growth is a perpetuity Using the present value of a perpetuity equation will give us the value of the earnings growth one period from today so we need to discount this back to today The NPVGO of the investment opportunity is NPVGOC0C11RC2R1R NPVGO 718000000 77000000 1 12 11000000 12 1 12 NPVGO 5759523810 The price of a share of stock is the cash cow value plus the NPVGO We have already calculated the NPVGO for the entire project so we need to nd the NPVGO on a per share basis The NPVGO on a per share basis is the NPVGO of the project divided by the shares outstanding which is NPVGO per share 5759523810 20000000 241 NPVGO per share 288 This means the per share stock price if the company undertakes the project is Share price Cash cow price NPVGO per share Share price 3542 288 Share price 3830 If the company does not make any new investments the stock price will be the present value of the constant perpetual dividends In this case all earnings are paid as dividends so applying the perpetuity equation we get PDividend R P 7 11 P 6364 The investment occurs every year in the growth opportunity so the opportunity is a growing perpetuity So we first need to find the growth rate The growth rate is g Retention Ratio x Return on Retained Earnings g 030 X 020 g 006 or 6 Next we need to calculate the NPV of the investment During year 3 30 percent of the earnings will be reinvested Therefore 210 is invested 7 x 30 One year later the shareholders receive a 20 percent return on the investment or 042 210 X 20 in perpetuity The perpetuity formula values that stream as of year 3 Since the investment opportunity will continue inde nitely and grows at 6 percent apply the growing perpetuity formula to calculate the NPV of the investment as of year 2 Discount that value back two years to today NPVGO Investment Return R R 7 g 1 R2 NPVGO 7210 042 11 0117006 1112 NPVGO 2789 The value of the stock is the PV of the firm without making the investment plus the NPV of the investment or P PVEPS NPVGO P 6364 2789 P 9153 242 Challenge 30 We are asked to nd the dividend yield and capital gains yield for each of the stocks All of the 31 stocks have a 20 percent required return which is the sum of the dividend yield and the capital gains yield To nd the components of the total return we need to nd the stock price for each stock Using this stock price and the dividend we can calculate the dividend yield The capital gains yield for the stock will be the total return required return minus the dividend yield w P0 7 D01 g R7g 7 45011020 7 10 7 4950 Dividend yield DlPo 4501104950 10 or 10 Capital gains yield 20 710 10 or 10 P0 D01 g R 7g 45020 7 0 2250 Dividend yield DlPo 4502250 20 or 20 Capital gains yield 20 720 0 P0 D01gR7g 450170520 05 1710 Dividend yield DlPo 4500951710 25 or 25 Capital gains yield 20 7 25 705 or 75 P2 7 D21 g R 7g 7 D01 g2l g2R 7gz 450130210820 708 P2 7 6845 P0 7 450 130 120 450 1302 1202 6845 1202 PO 7 5769 Dividend yield DlPo 4501305769 1014 or 1014 Capital gains yield 20 7 1014 0986 or 986 In all cases the required return is 20 percent but the return is distributed differently between current income and capital gains Highgrowth stocks have an appreciable capital gains component but a relatively small current income yield conversely mature negativegrowth stocks provide a high current income but also price depreciation over time Using the constant growth model the price of the stock paying annual dividends will be P0 7 D01 g R7g 7 360105147 05 7 4200 243 If the company pays quarterly dividends instead of annual dividends the quarterly dividend will be onefourth of annual dividend or Quarterly dividend 3601054 09450 To find the equivalent annual dividend we must assume that the quarterly dividends are reinvested at the required return We can then use this interest rate to find the equivalent annual dividend In other words when we receive the quarterly dividend we reinvest it at the required return on the stock So the effective quarterly rate is Effective quarterly rate 11425 7 1 0333 The effective annual dividend will be the FVA of the quarterly dividend payments at the effective quarterly required return In this case the effective annual dividend will be Effective D1 09450FVIFA3 3374 397 Now we can use the constant growth model to find the current stock price as P0 397147 05 4414 Note that we cannot simply nd the quarterly effective required return and growth rate to nd the value of the stock This would assume the dividends increased each quarter not each year If the company does not make any new investments the stock price will be the present value of the constant perpetual dividends In this case all earnings are paid as dividends so applying the perpetuity equation we get P Dividend R P 625 13 P 4808 The investment occurs every year in the growth opportunity so the opportunity is a growing perpetuity So we rst need to find the growth rate The growth rate is g Retention Ratio x Return on Retained Earnings g 020 X 011 g 0022 or 220 Next we need to calculate the NPV of the investment During year 3 20 percent of the earnings will be reinvested Therefore 125 is invested 625 x 20 One year later the shareholders receive an 11 percent return on the investment or 0138 125 X 11 in perpetuity The perpetuity formula values that stream as of year 3 Since the investment opportunity will continue indefmitely and grows at 22 percent apply the growing perpetuity formula to calculate the NPV of the investment as of year 2 Discount that value back two years to today NPVGO Investment Return R R 7 g 1 R2 NPVGO 125 0138 13 013 7 0022 1132 NPVGO 7139 244 33 03 UI The value of the stock is the PV of the rm without making the investment plus the NPV of the investment or P PVEPS NPVGO P 4808 7139 P 4668 0 Zero percent There is no retention ratio which would make the project pro table for the company If the company retains more earnings the growth rate of the earnings on the investment will increase but the project will still not be pro table Since the return of the project is less than the required return on the company stock the project is never worthwhile In fact the more the company retains and invests in the project the less valuable the stock becomes Here we have a stock with differential growth but the dividend growth changes every year for the rst four years We can find the price of the stock in Year 3 since the dividend growth rate is constant after the third dividend The price of the stock in Year 3 will be the dividend in Year 4 divided by the required return minus the constant dividend growth rate So the price in Year 3 will e P3 420120115110105 12 7 05 9563 The price of the stock today will be the PV of the rst three dividends plus the PV of the stock price in Year 3 so P0 420120112 4201201151122 4201201151101123 95631123 P0 8173 Here we want to nd the required return that makes the PV of the dividends equal to the current stock price The equation for the stock price is P 4201201 R 4201201151 R2 4201201151101 R3 420120115110105R 7 051 R3 9865 We need to nd the roots of this equation Using spreadsheet trial and error or a calculator with a root solving function we nd that R 1081 or 1081 In this problem growth is occurring from two different sources The learning curve and the new project We need to separately compute the value from the two different sources First we will compute the value from the learning curve which will increase at 5 percent All earnings are paid out as dividends so we nd the earnings per share are EPS Earningstotal number of outstanding shares EPS 15000000 x 105 10000000 EPS 158 245 From the NPVGO mode P ER 7g NPVGO P 1580 10 7 005 NPVGO P 3150 NPVGO Now we can compute the NPVGO of the new project to be launched two years from now The earnings per share two years from now will be EPSZ 1581 052 EPSZ 16538 Therefore the initial investment in the new project will be Initial investment 3016538 Initial investment 050 The earnings per share of the new project is a perpetuity with an annual cash ow of Increased EPS from project 6500000 10000000 shares Increased EPS from project 065 So the value of all future earnings in year 2 one year before the company realizes the earnings is PV 065 10 PV 650 Now we can nd the NPVGO per share of the investment opportunity in year 2 which will be NPVGOZ 7050 650 NPVGOZ 600 The value of the NPVGO today will be NPVGO 600 1 102 NPVGO 496 Plugging in the NPVGO model we get P 3150 496 P 3646 Note that you could also value the company and the project with the values given and then divide the nal answer by the shares outstanding The final answer would be the same 246 CHAPTER 10 RISK AND RETURN LESSONS FROM MARKET HISTORY Answers to Concepts Review and Critical Thinking Questions 1 They all wish they had Since they didn t it must have been the case that the stellar performance was not foreseeable at least not by most As in the previous question it s easy to see after the fact that the investment was terrible but it probably wasn t so easy ahead of time No stocks are riskier Some investors are highly risk averse and the extra possible return doesn t attract them relative to the extra risk Unlike gambling the stock market is a positive sum game everybody can win Also speculators provide liquidity to markets and thus help to promote efficiency Tbill rates were highest in the early eighties This was during a period of high in ation and is consistent with the Fisher effect Before the fact for most assets the risk premium will be positive investors demand compensation over and above the riskfree return to invest their money in the risky asset After the fact the observed risk premium can be negative if the asset s nominal return is unexpectedly low the risk free return is unexpectedly high or if some combination of these two events occurs Yes the stock prices are currently the same Below is a diagram that depicts the stocks price movements Two years ago each stock had the same price P0 Over e first year General Materials stock price increased by 10 percent or 11 x P0 Standard Fixtures stock price declined by 10 percent or 09 x P0 Over the second year General Materials stock price decreased by 10 percent or 091 1 x P0 while Standard Fixtures stock price increased by 10 percent or 1109 x P0 Today each of the stocks is worth 99 percent of its original value 2 years ago 1 year ago Today General Materials P0 11P0 1109P0 099P0 Standard Fixtures P0 09P0 0911P0 099P0 The stock prices are not the same The return quoted for each stock is the arithmetic return not the geometric return The geometric return tells you the wealth increase from the beginning of the period to the end of the period assuming the asset had the same return each year As such it is a better measure of ending wealth To see this assuming each stock had a beginning price of 100 per share the ending price for each stock would be Lake Minerals ending price 1001101 10 12100 Small Town Furniture ending price 10012595 11875 247 Whenever there is any variance in returns the asset with the larger variance will always have the greater difference between the arithmetic and geometric return To calculate an arithmetic return you simply sum the returns and divide by the number of returns As such arithmetic returns do not account for the effects of compounding Geometric returns do account for the effects of compounding As an investor the more important return of an asset is the geometric return Risk premiums are about the same whether or not we account for in ation The reason is that risk premiums are the difference between two returns so in ation essentially nets out Returns risk premiums and volatility would all be lower than we estimated because aftertaX returns are smaller than pretax returns Solutions to Questions and Problems NOTE All end of chapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic The return of any asset is the increase in price plus any dividends or cash ows all divided by the initial price The return of this stock is R 104 7 92 145 92 R 1462 or 1462 The dividend yield is the dividend divided by price at the beginning of the period so Dividend yield 145 92 Dividend yield 0158 or 158 And the capital gains yield is the increase in price divided by the initial price so Capital gains yield 104 7 92 92 Capital gains yield 1304 or 1304 Using the equation for total return we nd R 81 792 145 92 R 71038 or 71038 And the dividend yield and capital gains yield are Dividend yield 145 92 Dividend yield 0158 or 158 248 Capital gains yield 81 7 92 92 Capital gains yield 71196 or 71196 Here s a question for you Can the dividend yield ever be negative No that would mean you were paying the company for the privilege of owning the stock It has happened on bonds The total dollar return is the change in price plus the coupon payment so Total dollar return 1056 7 1090 80 Total dollar return 46 The total nominal percentage return of the bond is R 1056 71090 80 1090 R 0422 or 422 Notice here that we could have simply used the total dollar return of 46 in the numerator of this equation Using the Fisher equation the real return was 1 R1r1 h r 10422103071 r 01180r 118 The nominal return is the stated return which is 1170 percent Using the Fisher equation the real return was 1R1r1h r 111701031 71 r 0834 or 834 Using the Fisher equation the real returns for government and corporate bonds were 1 R 1 r1 h rG 1061103171 rG 0291 or 291 rc1062103171 rc 0301 or 301 249 The average return is the sum of the returns divided by the number of returns The average return for each stock was N 7 Nwosgooigo 11 N Y N1829731191109200r920 2y 5 11 We calculate the variance of each stock as 0X2x177c2N71 2 0X 15 70582 237 0582 7 347 0582 167 0582 097 0582 051970 a 18 70922 2970922 7 3170922 1970922 1170922054620 The standard deviation is the square root of the variance so the standard deviation of each stock is 5X 0051970 2 5X 2280 or 2280 cy 054620 2 6y 2337 or 2337 We will calculate the sum of the returns for each asset and the observed risk premium first Doing so we get Year Large co stock return Tbill return Risk premium 1973 71469 729 72198 1974 72647 799 73 446 1975 3723 587 3136 1976 2393 507 1886 1977 7716 545 71261 1978 657 764 7107 1941 3931 71990 a The average return for large company stocks over this period was Large company stock average return 1941 6 Large company stock average return 324 250 And the average return for Tbills over this period was Tbills average return 393 1 6 Tbills average return 655 Using the equation for variance we nd the variance for large company stocks over this period was Variance 1571469 7 03242 72647 7 03242 3723 7 03242 2393 7 03242 70716 7 03242 0657 7 03242 Variance 0058136 And the standard deviation for large company stocks over this period was Standard deviation 7 0058136 2 Standard deviation 02411 or 2411 Using the equation for variance we nd the variance for Tbills over this period was Variance 150729 7 06552 0799 7 06552 0587 7 06552 0507 7 06552 0545 7 06552 0764 7 06552 Variance 0000153 And the standard deviation for Tbills over this period was Standard deviation 0000153 2 Standard deviation 00124 or 124 The average observed risk premium over this period was Average observed risk premium 71990 6 Average observed risk premium 7332 The variance of the observed risk premium was Variance 1572198 7 703322 73446 7703322 3136 7 703322 1886 7 703322 712617703322 70107 7 703322 Variance 0062078 And the standard deviation of the observed risk premium was Standard deviation 7 006278 Standard deviation 02492 or 2492 To nd the average return we sum all the returns and divide by the number of returns so Arithmetic average return 34 16 19 7 21 085 Arithmetic average return 1120 or 1120 251 11 b Using the equation to calculate variance we nd Variance 143471122 16 7 1122 19 7 1122 72171122 08 7 1122 Variance 0041270 So the standard deviation is Standard deviation 0041270 2 Standard deviation 02032 or 2032 a To calculate the average real return we can use the average return of the asset and the average in ation rate in the Fisher equation Doing so we find 1R1r1h 7 111201042 71 7 0672 or 672 HI HI b The average risk premium is simply the average return of the asset minus the average real risk free rate so the average risk premium for this asset would be 7E 7E E 7112070510 RP 0610 or 610 We can find the average real riskfree rate using the Fisher equation The average real riskfree rate was 1R1r1h E17 10511042 71 E17 0086 or 086 And to calculate the average real risk premium we can subtract the average riskfree rate from the average real return So the average real risk premium was 7 7 E1 7 672 7 086 rp 7 585 Apply the veyear holdingperiod return formula to calculate the total return of the stock over the veyear period we find 5year holdingperiod return 1 R11 R2 1 R31 R41 R5 7 1 5year holdingperiod return 1 18431 16821 06831 32191719877 1 5year holdingperiod return 05655 or 5655 252 13 p A J p n UI p n 5 To find the return on the zero coupon bond we rst need to nd the price of the bond today Since one year has elapsed the bond now has 29 years to maturity so the price today is P1 7 100010929 P1 7 8215 There are no intermediate cash ows on a zero coupon bond so the return is the capital gains or R 8215 7 7781 7781 R 0558 or 558 The return of any asset is the increase in price plus any diVidends or cash ows all diVided by the initial price This preferred stock paid a dividend of 5 so the return for the year was R 9463 7 9285 500 9285 R 0730 or 730 The return of any asset is the increase in price plus any diVidends or cash ows all diVided by the initial price This stock paid no dividend so the return was R82017 7515 7515 R 0913 or 913 This is the return for three months so the APR is APR 7 4913 APR 7 3651 And the EAR is EAR1 0913quot71 EAR 4182 or4182 To nd the real return each year we will use the Fisher equation which is 1R1r1h Using this relationship for each year we nd m m W 1926 00330 00112 00447 1927 00315 00226 00554 1928 00405 00116 00527 1929 00447 00058 00387 1930 00227 00640 00926 1931 00115 00932 01155 1932 00088 01027 01243 253 p A l on 0 So the average real return was Average 0447 0554 0527 0387 0926 1155 1243 7 Average 0748 or 748 Notice the real return was higher than the nominal return during this period because of de ation or negative in ation Looking at the longterm corporate bond return history in Table 102 we see that the mean return was 62 percent with a standard deviation of 84 percent The range of returns you would expect to see 68 percent of the time is the mean plus or minus 1 standard deviation or R6 M Di 10 62i 84 7220 to 1460 The range of returns you would expect to see 95 percent of the time is the mean plus or minus 2 standard deviations or R6 M Di 20 62i 284 71060 to 2300 Looking at the largecompany stock return history in Table 102 we see that the mean return was 117 percent with a standard deviation of 206 percent The range of returns you would expect to see 68 percent of the time is the mean plus or minus 1 standard deviation or R6 M Di 10 117i 206 7890 to 3230 The range of returns you would expect to see 95 percent of the time is the mean plus or minus 2 standard deviations or R6 M Di 20 117i 2206 72950 to 5290 Intermediate Here we know the average stock return and four of the ve returns used to compute the average return We can work the average return equation backward to nd the missing return The average return is calculated as 55197270634R R23 or23 The missing return has to be 23 percent Now we can use the equation for the variance to nd Variance 14 19 7 112 727 7 112 06 7 112 34 7 112 23 7 112 Variance 005515 And the standard deviation is Standard deviation 005515 2 Standard deviation 023 48 or 2348 254 20 N p A N N The arithmetic average return is the sum of the known returns divided by the number of returns so Arithmetic average return 34 18 29 706 16 748 6 Arithmetic average return 0717 or 717 Using the equation for the geometric return we find Geometric average return 1 R1 X 1 R2 gtlt X 1 RT 77 1 Geometric average return 1 341 181 2917061 161 7 4816 7 1 Geometric average return 0245 or 245 Remember the geometric average return will always be less than the arithmetic average return if the returns have any variation To calculate the arithmetic and geometric average returns we must rst calculate the return for each year The return for each year is R1 5583 7 4962 068 4962 1389 or 1389 R2 5703 7 5583 073 5583 0346 or 346 R3 5025 7 5703 084 5703 71042 or 71042 R 53827 5025 091 5025 0892 or 892 R5 6418 7 5382 102 5382 2114 or 2114 The arithmetic average return was RA 01389 00346 7 01042 00892 021145 RA 00740 or 740 And the geometric average return was RG 7 1 13891 03461710421 089212114 5 71 RG 7 00685 or 685 To nd the real return we need to use the Fisher equation Rewriting the Fisher equation to solve for the real return we get r1R1h71 255 So the real return each year was M Tbillw w W 1973 00729 00871 410131 1974 00799 01234 410387 1975 00587 00694 410100 1976 00507 00486 00020 1977 00545 00670 410117 1978 00764 00902 410127 1979 01056 01329 410241 1980 01210 01252 410037 06197 07438 411120 a The average return for Tbills over this period was Average return 06197 8 Average return 0775 or 775 And the average in ation rate was Average in ation 07438 8 Average in ation 0930 or 930 b Using the equation for variance we nd the variance for Tbills over this period was Variance 170729 7 07752 0799 7 07752 0587 7 07752 0507 7 07752 0545 7 07752 0764 7 07752 1056 7 07752 1210 7 07752 Variance 0000616 And the standard deviation for Tbills was Standard deviation 7 0000616 2 Standard deviation 00248 or 248 The variance of in ation over this period was Variance 170871709302 1234 7 09302 0694 7 09302 0486 7 09302 0670 7 09302 09027 09302 1329 7 09302 12527 09302 Variance 0000971 And the standard deviation of in ation was Standard deviation 7 0000971 2 Standard deviation 00312 or 312 c The average observed real return over this period was Average observed real return 71122 8 256 23 N 4 Average observed real return 70140 or 7140 d The statement that Tbills have no risk refers to the fact that there is only an extremely small chance of the government defaulting so there is little default risk Since Tbills are short term there is also very limited interest rate risk However as this example shows there is inflation risk ie the purchasing power of the investment can actually decline over time even if the investor is earning a positive return To nd the return on the coupon bond we first need to nd the price of the bond today Since one year has elapsed the bond now has six years to maturity so the price today is P1 70PVIFA876 10001086 P1 95377 You received the coupon payments on the bond so the nominal return was R 95377 794382 70 94382 R 0847 or 847 And using the Fisher equation to find the real return we get r 10847104871 r 0350 or 350 Looking at the longterm government bond return history in Table 102 we see that the mean return was 61 percent with a standard deviation of 94 percent In the normal probability distribution approximately 23 of the observations are within one standard deviation of the mean This means that 13 of the observations are outside one standard deviation away from the mean Or PrRlt 733 or Rgtl5513 But we are only interested in one tail here that is returns less than 733 percent so PrRlt 733 m 16 You can use the Z statistic and the cumulative normal distribution table to nd the answer as well Doing so we nd Z X 7 u6 Z 733 7 6194 7100 Looking at the Ztable this gives a probability of 1587 or PrRlt 733 m 1587 or 1587 The range of returns you would expect to see 95 percent of the time is the mean plus or minus 2 standard deviations or 95 level Re u Di 20 61 i 294 71270 to 2490 257 UI F gt1 on The range of returns you would expect to see 99 percent of the time is the mean plus or minus 3 standard deviations or 99 level Re p at 30 61 i 394 72210 to 3430 The mean return for small company stocks was 164 percent with a standard deviation of 330 percent Doubling your money is a 100 return so if the return distribution is normal we can use the Zstatistic So Z X 7 u6 Z 100 7 164330 2533 standard deviations above the mean This corresponds to a probability of m 0565 or about once every 200 years Tripling your money wou e Z 200 7 164330 5564 standard deviations above the mean This corresponds to a probability of much less than 05 The actual answer is H000001321 or about once every 1 million years It is impossible to lose more than 100 percent of your investment Therefore return distributions are truncated on the lower tail at 7100 percent Challenge Using the Z statistic we find Z X 7 u6 Z 0 7117206 41568 PrR0 m 2850 For each of the questions asked here we need to use the zstatistic which is Z X 7 u6 a 21 10 7 6284 04523 This Zstatistic gives us the probability that the return is less than 10 percent but we are looking for the probability the return is greater than 10 percent Given that the total probability is 100 percent or 1 the probability of a return greater than 10 percent is 1 minus the probability of a return less than 10 percent Using the cumulative normal distribution table we get PrR10 7 17PrR10 7 3255 258 For a return less than 0 percent 22 07 6284 417381 PrRlt10 1 7 PrRgt0 2302 The probability that Tbill returns will be greater than 10 percent is Z3 10 7 383 1 2 PrR710 17 PrR 10 17 9772 m 228 And the probability that Tbill returns will be less than 0 percent is Z4 07 383 1 712258 PrR70 m 1101 The probability that the return on longterm corporate bonds will be less than 7418 percent is 25 74187 6284 712357 PrR74l18 m 1083 And the probability that Tbill returns will be greater than 1056 percent is 26 1056 7 3831 2181 PrR71056 17 PrR 1056 17 9854 e 146 259 CHAPTER 11 RETURN AND RISK THE CAPITAL ASSET PRICING MODEL Answers to Concepts Review and Critical Thinking Questions 1 Some of the risk in holding any asset is unique to the asset in question By investing in a variety of assets this unique portion of the total risk can be eliminated at little cost On the other hand there are some risks that affect all investments This portion of the total risk of an asset cannot be costlessly eliminated In other words systematic risk can be controlled but only by a costly reduction in expected returns a systematic b unsystematic 0 both probably mostly systematic d unsystematic e unsystematic f systematic No to both questions The portfolio expected return is a weighted average of the asset s returns so it must be less than the largest asset return and greater than the smallest asset return False The variance of the individual assets is a measure of the total risk The variance on a well diversified portfolio is a function of systematic risk only Yes the standard deviation can be less than that of every asset in the portfolio However 81 cannot be less than the smallest beta because Bp is a weighted average of the individual asset betas Yes It is possible in theory to construct a zero beta portfolio of risky assets whose return would be equal to the riskfree rate It is also possible to have a negative beta the return would be less than the riskfree rate A negative beta asset would carry a negative risk premium because of its value as a diversification instrument The covariance is a more appropriate measure of a security s risk in a welldiversi ed portfolio because the covariance re ects the effect of the security on the variance of the portfolio Investors are concerned with the variance of their portfolios and not the variance of the individual securities Since covariance measures the impact of an individual security on the variance of the portfolio covariance is the appropriate measure of risk 260 p n O If we assume that the market has not stayed constant during the past three years then the lack in movement of Southern Co s stock price only indicates that the stock either has a standard deviation or a beta that is very near to zero The large amount of movement in Texas Instrument stock price does not imply that the film s beta is high Total volatility the price uctuation is a function of both systematic and unsystematic risk The beta only re ects the systematic risk Observing the standard deviation of price movements does not indicate whether the price changes were due to systematic factors or rm speci c factors Thus if you observe large stock price movements like that of T1 you cannot claim that the beta of the stock is high All you know is that the total risk of TI is high The wide uctuations in the price of oil stocks do not indicate that these stocks are a poor investment If an oil stock is purchased as part of a welldiversi ed portfolio only its contribution to the risk of the entire portfolio matters This contribution is measured by systematic risk or beta Since price uctuations in oil stocks re ect diversi able plus nondiversifiable risk observing the standard deviation of price movements is not an adequate measure of the appropriateness of adding oil stocks to a portfolio The statement is false If a security has a negative beta investors would want to hold the asset to reduce the variability of their portfolios Those assets will have expected returns that are lower than the riskfree rate To see this examine the Capital Asset Pricing Model ERs Rf BSERM Rfl If 35 lt 0 then the ERs lt Rf Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic The portfolio weight of an asset is total investment in that asset divided by the total portfolio value First we will nd the portfolio value which is Total value 9553 12029 8515 The portfolio weight for each stock is WeightA 95538515 5913 WeightB 120298515 4087 261 The expected return of a portfolio is the sum of the weight of each asset times the expected return of each asset The total value of the portfolio is Total value 1900 2300 4200 So the expected return of this portfolio is ERp 19004200010 23004200015 1274 or 1274 The expected return of a portfolio is the sum of the weight of each asset times the expected return of each asset So the expected return of the portfolio is ERp 4011 3517 2514 1385 or 1385 Here we are given the expected return of the portfolio and the expected return of each asset in the portfolio and are asked to nd the weight of each asset We can use the equation for the expected return of a portfolio to solve this problem Since the total weight of a portfolio must equal 1 100 the weight of Stock Y must be one minus the weight of Stock X Mathematically speaking this means ERp 129 16wX 1017wX We can now solve this equation for the weight of Stock X as 129 16wX 10 7 10wX 029 06wX wX 04833 So the dollar amount invested in Stock X is the weight of Stock X times the total portfolio value or Investment in X 0483310000 483333 And the dollar amount invested in Stock Y is Investment in Y 1 7 0483310000 516667 The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring So the expected return of the asset is ER 2009 511 323 1060 or 1060 262 The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring So the expected return of each stock asset is ERA 7 1506 6507 2011 7 0765 or 765 ERB 7 1572 6513 2033 1205 or 1205 To calculate the standard deviation we first need to calculate the variance To nd the variance we nd the squared deviations from the expected return We then multiply each possible squared deviation by its probability and then add all of these up The result is the variance So the variance and standard deviation of each stock are 6A2 1506 7 07652 6507 7 07652 20117 07652 00029 6A 0002912 0171 or 171 632 15727 12052 6513 7 12052 2033 7 12052 02424 63 0242412 1557 or 1557 The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring So the expected return of the stock is ERA 107045 25 044 4512 20207 1019 or 1019 To calculate the standard deviation we first need to calculate the variance To nd the variance we nd the squared deviations from the expected return We then multiply each possible squared deviation by its probability and then add all of these up The result is the variance So the variance and standard deviation are cl 7107045 7 10192 25044 7 10192 45 12 7 10192 20207 7 10192 7 00535 c 7005351 2 7 0732 or 732 The expected return of a portfolio is the sum of the weight of each asset times the expected return of each asset So the expected return of the portfolio is ERp 7 1508 6515 2024 7 1575 or 1575 If we own this portfolio we would expect to get a return of 1575 percent 263 To find the expected return of the portfolio we need to nd the return of the portfolio in each state of the economy This portfolio is a special case since all three assets have the same weight To nd the expected return in an equally weighted portfolio we can sum the returns of each asset and divide by the number of assets so the expected return of the portfolio in each state of the economy is Boom ERp 07 15 333 1833 or 1833 Bust ERp 13 03 7063 0333 or 333 To nd the expected return of the portfolio we multiply the return in each state of the economy by the probability of that state occurring and then sum Doing this we find ERp 801833 200333 1533 or 1533 This portfolio does not have an equal weight in each asset We still need to nd the return of the portfolio in each state of the economy To do this we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy Doing so we get Boom ERp2007 20 15 6033 2420 or 2420 Bust ERp 2013 2003 60706 70040 or 040 And the expected return of the portfolio is ERp 802420 207004 1928 or 1928 To nd the variance we nd the squared deviations from the expected return We then multiply each possible squared deviation by its probability and then add all of these up The result is the variance So the variance of the portfolio is of 802420 7 19282 2070040 7 19282 00968 This portfolio does not have an equal weight in each asset We rst need to nd the return of the portfolio in each state of the economy To do this we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy Doing so we get Boom ERp 303 4045 3033 3690 or 3690 Good ERp 3012 4010 3015 1210 or 1210 Poor ERp 3001 407 15 30705 70720 or 7720 Bust ERp 30706 40730 30709 71650 or 71650 And the expected return of the portfolio is ERp 203690 351210 3070720 1571650 0698 or 698 264 11 p n N p A J b To calculate the standard deviation we rst need to calculate the variance To nd the variance we nd the squared deviations from the expected return We then multiply each possible squared deviation by its probability and then add all of these up The result is the variance So the variance and standard deviation the portfolio is 52 203690 7 06982 351210 7 06982 3070720 7 06982 1571650 7 06982 2 7 c 7 03312 c 03312 2 1820 or 1820 The beta of a portfolio is the sum of the weight of each asset times the beta of each asset So the beta of the portfolio is B 2575 20190 15138 40116 124 The beta of a portfolio is the sum of the weight of each asset times the beta of each asset If the portfolio is as risky as the market it must have the same beta as the market Since the beta of the market is one we know the beta of our portfolio is one We also need to remember that the beta of the riskfree asset is zero It has to be zero since the asset has no risk Setting up the equation for the beta of our portfolio we get Sp 10 1301318513Bx Solving for the beta of Stock X we get BX 115 CAPM states the relationship between the risk of an asset and its expected return CAPM is ERi R1 ERMR1 X Bi Substituting the values we are given we nd ERi 05 127 05125 1375 or 1375 We are given the values for the CAPM except for the B of the stock We need to substitute these values into the CAPM and solve for the B of the stock One important thing we need to realize is that we are given the market risk premium The market risk premium is the expected return of the market minus the riskfree rate We must be careful not to use this value as the expected return of the market Using the CAPM we nd ERi 142 04 076 3 146 265 15 Here we need to find the expected return of the market using the CAPM Substituting the values given and solving for the expected return of the market we find ERi 7 105 7 055 ERM 7 05573 ERM 7 1235 or 1235 Here we need to find the riskfree rate using the CAPM Substituting the values given and solving for the riskfree rate we nd ERi 7 162 7 Rf 117Rf175 162 Rf 1925 7175Rf Rf 0407 or 407 11 Again we have a special case where the portfolio is equally weighted so we can sum the returns of each asset and divide by the number of assets The expected return of the portfolio is ERp 103 052 0765 or 765 We need to find the portfolio weights that result in a portfolio with a B of 050 We know the B of the riskfree asset is zero We also know the weight of the riskfree asset is one minus the weight of the stock since the portfolio weights must sum to one or 100 percent So Bp 050 ws9217ws0 050 92wS 0 70wS ws 050 92 wS 5435 And the weight of the riskfree asset is wa 1 7 5435 4565 We need to nd the portfolio weights that result in a portfolio with an expected return of 9 percent We also know the weight of the riskfree asset is one minus the weight of the stock since the portfolio weights must sum to one or 100 percent So ERp 09 103wS 0517ws 09 103ws 05 7 05ws ws 7547 So the B of the portfolio will be 310 7 75479217754707 0694 266 d Solving for the B of the portfolio as we did in part a we nd Bp 184 ws92 1 iws0 wS 18492 2 wa 1 7 2 71 The portfolio is invested 200 in the stock and 7100 in the riskfree asset This represents borrowing at the riskfree rate to buy more of the stock 18 First we need to find the B of the portfolio The B of the riskfree asset is zero and the weight of the riskfree asset is one minus the weight of the stock the B of the portfolio is 15 wW1317 ww013wW So to nd the B of the portfolio for any weight of the stock we simply multiply the weight of the stock times its B Even though we are solving for the B and expected return of a portfolio of one stock and the riskfree asset for different portfolio weights we are really solving for the SML Any combination of this stock and the riskfree asset will fall on the SML For that matter a portfolio of any stock and the riskfree asset or any portfolio of stocks will fall on the SML We know the slope of the SML line is the market risk premium so using the CAPM and the information concerning this stock the market risk premium is ERW 138 05 MRP130 MRP 08813 0677 or 677 So now we know the CAPM equation for any stock is ERp 05 0677Bp The slope of the SML is equal to the market risk premium which is 00677 Using these equations to ll in the table we get the following results wW ERp 15p 0 0500 25 0720 0 325 50 0940 0 650 75 1160 0 975 100 1380 1 300 125 1600 1 625 150 1820 1 950 267 19 There are two ways to correctly answer this question We will work through both First we can use N O the CAPM Substituting in the value we are given for each stock we nd ERY 055 068135 1468 or 1468 It is given in the problem that the expected return of Stock Y is 14 percent but according to the CAPM the return of the stock based on its level of risk the expected return should be 1468 percent This means the stock return is too low given its level of risk Stock Y plots below the SML and is overvalued In other words its price must decrease to increase the expected return to 1468 percent For Stock Z we nd ERz 055 068085 1128 or 1128 The return given for Stock Z is 115 percent but according to the CAPM the expected return of the stock should be 1128 percent based on its level of risk Stock Z plots above the SML and is undervalued In other words its price must increase to decrease the expected return to 1128 percent We can also answer this question using the rewardtorisk ratio All assets must have the same rewardtorisk ratio that is every asset must have the same ratio of the asset risk premium to its beta This follows from the linearity of the SML in Figure 1111 The rewardtorisk ratio is the risk premium of the asset divided by its B This is also know as the Treynor ratio or Treynor index We are given the market risk premium and we know the B of the market is one so the rewardtorisk ratio for the market is 0068 or 68 percent Calculating the rewardtorisk ratio for Stock Y we find Rewardtorisk ratio Y 14 7 055 135 0630 The rewardtorisk ratio for Stock Y is too low which means the stock plots below the SML and the stock is overvalued Its price must decrease until its rewardtorisk ratio is equal to the market rewardtorisk ratio For Stock Z we find Rewardtorisk ratio Z 115 7 055 85 0706 The rewardtorisk ratio for Stock Z is too high which means the stock plots above the SML and the stock is undervalued Its price must increase until its rewardtorisk ratio is equal to the market rewardtorisk ratio We need to set the rewardtorisk ratios of the two assets equal to each other see the previous problem which is 14 7Rf135 115 7 Rf085 We can cross multiply to get 085 14 7Rf135115 7Rf Solving for the riskfree rate we find 0119 7 085Rf 015525 7135Rf Rf 0725 or 725 268 21 N N Intermediate For a portfolio that is equally invested in largecompany stocks and longterm bonds Return 117 612 895 For a portfolio that is equally invested in small stocks and Treasury bills Return 164 382 1010 We know that the rewardtorisk ratios for all assets must be equal See Question 19 This can be expressed as ERA erBA ERB erBB The numerator of each equation is the risk premium of the asset so 1PABA RPBBB We can rearrange this equation to get BBBA RPBRPA If the rewardtorisk ratios are the same the ratio of the betas of the assets is equal to the ratio of the risk premiums of the assets a We need to nd the return of the portfolio in each state of the economy To do this we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy Doing so we get Boom ERp 420 435 260 3400 or 3400 Normal ERp 415 412 205 1180 or 1180 Bust ERp 401 4725 2750 71960 or 71960 And the expected return of the portfolio is ERp 3534 40118 257196 1172 or 1172 To calculate the standard deviation we first need to calculate the variance To nd the variance we nd the squared deviations from the expected return We then multiply each possible squared deviation by its probability than add all of these up The result is the variance So the variance and standard deviation of the portfolio is 62 3534 7 11722 7 40 118 7 11722 257196 7 11722 62p 04190 c 04190 2 2047 or 2047 269 b The risk premium is the return of a risky asset minus the riskfree rate Tbills are often used as the riskfree rate so RP ERp 7 Rf 1172 7 038 0792 or 792 c The approximate expected real return is the expected nominal return minus the in ation rate so Approximate expected real return 1172 7 035 0822 or 822 To nd the exact real return we will use the Fisher equation Doing so we get 1 ERi 1 h1 er 11172 103501 er eri1117210357 1 0794 or 794 The approximate real risk premium is the expected return minus the in ation rate so Approximate expected real risk premium 0792 7 035 0442 or 442 To nd the exact expected real risk premium we use the Fisher effect Doing do we nd Exact expected real risk premium 10792 1035 7 1 0427 or 427 24 We know the total portfolio value and the investment of two stocks in the portfolio so we can nd the weight of these two stocks The weights of Stock A and Stock B are wA 180000 1000000 18 WE 2900001000000 29 Since the portfolio is as risky as the market the B of the portfolio must be equal to one We also know the B of the riskfree asset is zero We can use the equation for the B of a portfolio to nd the weight of the third stock Doing so we nd Bp 10 wA75 wB130 wc145 wa0 Solving for the weight of Stock C we nd wC 33655172 So the dollar investment in Stock C must be Invest in Stock C 336551721000000 33655172 270 UI 5 We also know the total portfolio weight must be one so the weight of the riskfree asset must be one minus the asset weight we know or 1wAwB wcwkf 1 18 29 33655172 Wm WM 19344828 So the dollar investment in the riskfree asset must be Invest in riskfree asset 19344828l000000 19344828 We are given the expected return and B of a portfolio and the expected return and B of assets in the portfolio We know the B of the riskfree asset is zero We also know the sum of the weights of each asset must be equal to one So the weight of the riskfree asset is one minus the weight of Stock X and the weight of Stock Y Using this relationship we can express the expected return of the portfolio as ERp 1070 wX172 Wy08751 WK 7 Wy055 And the B of the portfolio is Bp 8 wX18 Wy050 1 iwx Wy0 We have two equations and two unknowns Solving these equations we nd that wx 4111111 wy 200000 wa488889 The amount to invest in Stock X is Investment in stock X 411111100000 71111111 A negative portfolio weight means that you short sell the stock If you are not familiar with short selling it means you borrow a stock today and sell it You must then purchase the stock at a later date to repay the borrowed stock If you short sell a stock you make a pro t if the stock decreases in value The negative weight on the riskfree asset means that we borrow money to invest The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring So the expected return of each stock is ERA 33082 33095 33063 0800 or 800 ERB 337065 33124 33185 0813 or 813 271 1 To calculate the standard deviation we first need to calculate the variance To nd the variance we nd the squared deviations from the expected return We then multiply each possible squared deviation by its probability and then add all of these up The result is the variance So the variance and standard deviation of Stock A are 2 33082 7 08002 33095 7 08002 33063 7 08002 00017 c 0001712 0131 or 131 And the standard deviation of Stock B is 2 337065 7 08132 33124 7 08132 33185 7 08132 01133 c 01133 2 1064 or 1064 To nd the covariance we multiply each possible state times the product of each assets deviation from the mean in that state The sum of these products is the covariance So the covariance is CovAB 7 33092 7 08007065 7 0813 33095 7 0800 124 7 0813 33063 7 0800185 7 0813 CovAB 7 7000472 And the correlation is P1813 CovAB 6A 6 B p 7 7000472 013 1 1064 p 7 73373 The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring So the expected return of each stock is ERA 307020 50138 20218 1066 or 1066 ERB 30034 50062 20092 0596 or 596 To calculate the standard deviation we first need to calculate the variance To nd the variance we nd the squared deviations from the expected return We then multiply each possible squared deviation by its probability and then add all of these up The result is the variance So the variance and standard deviation of Stock A are a 307020 7 10662 50138 7 10662 20218 710662 00778 6A 0077812 0882 or 882 And the standard deviation of Stock B is 6123 30034 7 05962 500627 05962 20092 7 05962 00041 GB 7 00041 2 7 0202 or 202 272 To nd the covariance we multiply each possible state times the product of each assets deviation from the mean in that state The sum of these products is the covariance So the covariance is CovAB 307020 7 1066034 7 0596 50138 7 1066062 7 0596 4 20218 710660927 0596 CovAB 001732 And the correlation is CovAB 6A 63 001732 08820202 9701 The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset so ERP WFERF WGERG ERp 3010 7017 ERp 1490 or 1490 The variance of a portfolio of two assets can be expressed as 612 w a szGZG 2waG chGprg of 302262 702582 23070265825 c 18675 So the standard deviation is 5 18675 2 4322 or 4322 The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset so ERP WAERA WBERB ERp 4513 5519 ERp 1630 or 1630 The variance of a portfolio of two assets can be expressed as of wici w 6123 2wAchsACSBpAB of 452382 552622 24555386250 of 20383 So the standard deviation is 5 20383 2 4515 or 4515 273 Z 7 GP W15 W12361232WAWB6A6BPAJ cf 452382 552622 245553862750 cf 08721 So the standard deviation is c 08721 2 2953 or 2953 As Stock A and Stock B become less correlated or more negatively correlated the standard deviation of the portfolio decreases i We can use the equation to calculate beta we nd BA PAM5A 5M 085 pAM027 020 pAM 063 ii Using the equation to calculate beta we nd BB PBMGB 5M 150 50cB 020 GB 060 iii Using the equation to calculate beta we nd BC PQMXGC 5M BC 3570 020 BC 123 iv The market has a correlation of 1 with itself v The beta of the market is 1 vi The riskfree asset has zero standard deviation vii The riskfree asset has zero correlation with the market portfolio viii The beta of the riskfree asset is 0 Using the CAPM to find the expected return of the stock we nd Firm A ERA Rf BA Rf ERA 005 0850127 005 ERA 1095 or 1095 274 31 According to the CAPM the expected return on Firm A s stock should be 1095 percent However the expected return on Firm A s stock given in the table is only 10 percent Therefore Firm A s stock is overpriced and you should sell it Firm B ERB Rf BB Rfl ERB 005 150 12 7 005 ERB 1550 or 1550 According to the CAPM the expected return on Firm B s stock should be 1550 percent However the expected return on Firm B s stock given in the table is 14 percent Therefore Firm B s stock is overpriced and you should sell it Firm C ERc Rf BCERM Rfl ERc 005 123012 7 005 ERc 1358 or 1358 According to the CAPM the expected return on Firm C s stock should be 1358 percent However the expected return on Firm C s stock given in the table is 17 percent Therefore Firm C s stock is underpriced and you should buy it Because a welldiversi ed portfolio has no unsystematic risk this portfolio should lie on the Capital lVIarket Line CML The slope of the CML equals S10PECML Rf cM SlopeCML 0127005 019 SlopeCML 036842 1 The expected return on the portfolio equals ERp Rf SlopeCMLcp ERp 05 3684207 ERp 0758 or 758 b The expected return on the portfolio equals ERp Rf SlopeCMLcp 20 05 368420 5 4071 or 4071 First we can calculate the standard deviation of the market portfolio using the Capital Market Line CML We know that the riskfree rate asset has a return of 5 percent and a standard deviation of zero and the portfolio has an expected return of 9 percent and a standard deviation of 13 percent These two points must lie on the Capital Market Line The slope of the Capital Market Line equals SlopeCML Rise Run SlopeCML Increase in expected return Increase in standard deviation SlopeCML 09 7 05 13 7 0 SlopeCML 31 275 03 According to the Capital Market Line ERI Rf SlopeCMLcI Since we know the expected return on the market portfolio the riskfree rate and the slope of the Capital Market Line we can solve for the standard deviation of the market portfolio which is ERM Rf SlopeCMLcM 12 05 316M 6M 127 05 31 6M 2275 or 2275 Next we can use the standard deviation of the market portfolio to solve for the beta of a security using the beta equation Doing so we nd the beta of the security is BI pIMcI 5M BI 4540 2275 BI 079 Now we can use the beta of the security in the CAPM to nd its expected return which is ERI Rf BIERM Rfl ER1 005 07912 7005 ERI 1054 or 1054 First we need to nd the standard deviation of the market and the portfolio which are 5M 0429 2 GM 2071 or 2071 oz 1783 2 cl 4223 or 4223 Now we can use the equation for beta to nd the beta of the portfolio which is 82 PZM52 5M BZ 394223 2071 82 80 Now we can use the CAPM to nd the expected return of the portfolio which is ERz Rf 32 Rf ERz 048 80114 7 048 ERZ 1005 or 1005 276 Challenge 34 The amount of systematic risk is measured by the B of an asset Since we know the market risk premium and the riskfree rate if we know the expected return of the asset we can use the CAPM to solve for the B of the asset The expected return of StockI is ERI 1509 5542 3026 3225 or 3225 Using the CAPM to find the B of Stock 1 we nd 3225 04 7 0753I BI 377 The total risk of the asset is measured by its standard deviation so we need to calculate the standard deviation of Stock 1 Begirming with the calculation of the stock s variance we nd 02 1509 7 32252 5542 7 32252 3026 7 32252 012 01451 01 0145112 1205 or 1205 Using the same procedure for Stock 11 we nd the expected return to be ERH 15730 5512 3044 1530 Using the CAPM to find the B of Stock 11 we nd 1530 04 075BH BII 151 And the standard deviation of Stock 11 is 6112 15730 7 15302 7 55 12 7 15302 3044 7 15302 512 05609 on 0560912 2368 or 2368 Although Stock 11 has more total risk than I it has much less systematic risk since its beta is much smaller than I s Thus I has more systematic risk and II has more unsystematic and more total risk Since unsystematic risk can be diversi ed away I is actually the riskier stock despite the lack of volatility in its returns StockI will have a higher risk premium and a greater expected return 277 35 Here we have the expected return and beta for two assets We can express the returns of the two assets using CAPM If the CAPM is true then the security market line holds as well which means all assets have the same risk premium Setting the rewardtorisk ratios of the assets equal to each other and solving for the riskfree rate we nd 15 7 Rf14 7 115 7 Rf90 9015 7Rf714115 7R 135 7 9Rf 7 161 714Rf 5Rf 7 026 Rf 052 or 520 Now using CAPM to nd the expected return on the market with both stocks we nd 15 0520 14RM 70520 RM 1220 or 1220 115 0520 9RM 70520 RM 1220 or 1220 a The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring To calculate the standard deviation we rst need to calculate the variance To nd the variance we nd the squared deviations from the expected return We then multiply each possible squared deviation by its probability and then add all of these up The result is the variance So the expected return and standard deviation of each stock are Asset ER1 1525 3520 3515 1510 1750 or 1750 of 71525 717502 3520 7 17502 3515 7 17502 1510 7 17502 7 00213 61 7 00213 2 7 0461 or 461 Asset2 ER2 1525 3515 3520 1510 1750 or 1750 c 1525 717502 3515 7 17502 3520 7 17502 1510 7 17502 00213 62 7 00213 2 7 0461 or 461 Asset 3 ER3 1510 3515 3520 1525 1750 or 1750 a 71510 717502 7 3515 7 17502 3520 7 17502 7 1525 7 17502 7 00213 63 7 00213 2 7 0461 or 461 278 To find the covariance we multiply each possible state times the product of each assets deviation from the mean in that state The sum of these products is the covariance The correlation is the covariance divided by the product of the two standard deviations So the covariance and correlation between each possible set of assets are Asset I and Asset 2 Cov12 1525 7 175025 7 1750 3520 7 175015 71750 3515 7 175020 71750 1510 7 175010 71750 Cov12 000125 P11 C0V12 c1 62 P1 000125 04610461 P11 5882 Asset andAsset 3 Cov13 1525 7 1750 10 7 1750 3520 7 175015 71750 3515 7 175020 71750 1510 7 175025 71750 Cov13 7002125 P13 C0V13 c1 53 P13 7002125 04610461 P13 1 Asset 2 andAsset 3 Cov23 1525 7 175010 7 1750 3515 7 175015 71750 3520 7 175020 71750 1510 7 175025 71750 Cov23 7000125 P23 C0V253 c2 53 P23 7000125 04610461 P23 75882 The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset so for a portfolio ofAsset l and Asset 2 ERP W1ER1 W2ER2 ERp 7 501750 501750 ERp 1750 or 1750 The variance of a portfolio of two assets can be expressed as 2 7 6P W12 612 W 5 T 2W1WzG152Pi2 6i 5020461250204612 25050046l04615882 6i 001688 And the standard deviation of the portfolio is cl 7 001688 2 279 6p 0411 or 411 The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset so for a portfolio ofAsset 1 and Asset 3 ERP W1ER1 W3ER3 ERP 501750 501750 ERp 1750 or 1750 The variance of a portfolio of two assets can be expressed as 2 2 2 2 2 GP WI 61 w3 63 2W1W36163p173 of 50204612 50204612 250500461046171 of 000000 Since the variance is zero the standard deviation is also zero The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset so for a portfolio ofAsset 1 and Asset 3 ERP W2ER2 W3ER3 ERp 501750 501750 ERp 1750 or 1750 The variance of a portfolio of two assets can be expressed as 2 2 2 2 2 GP wz 62 w3 63 2W2W36263p173 of 50204612 50204612 250500461046175882 a 000438 And the standard deviation of the portfolio is 51 000438 2 5 0209 or 209 As long as the correlation between the returns on two securities is below 1 there is a benefit to diversi cation A portfolio with negatively correlated stocks can achieve greater risk reduction than a portfolio with positively correlated stocks holding the expected return on each stock constant Applying proper weights on perfectly negatively correlated stocks can reduce portfolio variance to 0 280 The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring So the expected return of each stock is ERA 7 15708 7013 1548 7 1510 or 1510 ERB 15705 7014 1529 1340 or 1340 We can use the expected returns we calculated to find the slope of the Security Market Line We know that the beta of Stock A is 25 greater than the beta of Stock B Therefore as beta increases by 25 the expected return on a security increases by 017 1510 7 1340 The slope of the security market line SML equals SlopeSML Rise Run SlopeSML Increase in expected return Increase in beta SlopeSML 1510 7 1340 25 SlopeSML 0680 or 680 Since the market s beta is 1 and the riskfree rate has a beta of zero the slope of the Security lVIarket Line equals the expected market risk premium So the expected market risk premium must be 68 percent We could also solve this problem using CAPM The equations for the expected returns of the two stocks are 151 Rf 3 25MRP 134 7 R BBMRP We can rewrite the CAPM equation for Stock A as 151 Rf BBMRP 25MRP Subtracting the CAPM equation for Stock B from this equation yields 017 25MRP lVIRP 068 or 68 which is the same answer as our previous result A typical riskaverse investor seeks high returns and low risks For a riskaverse investor holding a welldiversified portfolio beta is the appropriate measure of the risk of an individual security To assess the two stocks we need to nd the expected return and beta of each of the two securities StockA Since Stock A pays no dividends the return on Stock A is simply P1 7P0 P0 So the return for each state of the economy is RRecession 63 7 75 75 7160 or 7160 RNomal 83 7 75 75 107 or 107 RExpmding 96 7 75 75 280 or 280 281 The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring So the expected return of the stock is ERA 207 160 60107 20280 0880 or 880 And the variance of the stock is a 2070 160 7 00882 601077 0882 20280 7 0882 cg 00199 Which means the standard deViation is 5 00199 2 5 1410 or 1410 Now we can calculate the stock s beta which is BA PAM5A 5M BA 80l410 18 BA 627 For Stock B we can directly calculate the beta from the information proVided So the beta for Stock B is Stock B BB PBMGB 5M BB 2534 18 BB 472 The expected return on Stock B is higher than the expected return on Stock A The risk of Stock B as measured by its beta is lower than the risk of Stock A Thus a typical riskaverse investor holding a welldiversified portfolio will prefer Stock B Note this situation implies that at least one of the stocks is mispriced since the higher risk beta stock has a lower return than the lower risk beta stock The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset so ERP WAERA WBERB ERp 70088 3013 ERp 1006 or 1006 282 To nd the standard deviation of the portfolio we first need to calculate the variance The variance of the portfolio is of wici w c 2wAchAchAYB of 7021412 302342 270301413448 of 02981 And the standard deviation of the portfolio is 5 7 002981 2 5 7 1727 or 1727 The beta of a portfolio is the weighted average of the betas of its individual securities So the beta of the portfolio is B 7 70627 300472 Bl 580 The variance of a portfolio of two assets equals 2 7 Z Z Z 2 GP 7 wAcA wB GB 2wAchAcBCovAB Since the weights of the assets must sum to one we can write the variance of the portfolio as c 7 Wis 17wm 2WA17WAcAcBCovAB To find the minimum for any function we nd the derivative and set the derivative equal to zero Finding the derivative of the variance function with respect to the weight of Asset A setting the derivative equal to zero and solving for the weight of Asset A we find wA c5123 7 CovAB 6 6123 7 2CovAB Using this expression we find the weight ofAsset A must be wA 452 7 001 222 4527 2001 wA 7 8096 This implies the weight of Stock B is W l 7 wA wB 1 7 8096 WE 1904 Using the weights calculated in part 1 determine the expected return of the portfolio we find ERP WAERA WBERB ERp 809609 19040 15 283 ERp 01014 or 1014 Using the derivative from part a with the new covariance the weight of each stock in the minimum variance portfolio is wA 5123 CovAB 6 6123 7 2CovAB wA 452 705 222 4527 2905 1 wA 7 7 96 This implies the weight of Stock B is wB liwA The variance of the portfolio with the weights on part c is of wici w 6123 2wAchAcBCovAB of 71962222 28042452 2719628042245705 of 0208 And the standard deviation of the portfolio is 51 00208 2 up 1442 or 1442 284 CHAFHHLU AN ALTERNATIVE VIEW OF RISK AND RETURN THE ARBITRAGE PRICING THEORY Answers to Concept Questions 1 Systematic risk is risk that cannot be diversified away through formation of a portfolio Generally systematic risk factors are those factors that affect a large number of rms in the market however those factors will not necessarily affect all rms equally Unsystematic risk is the type of risk that can be diversi ed away through portfolio formation Unsystematic risk factors are speci c to the rm or industry Surprises in these factors will affect the returns of the rm in which you are interested but they will have no effect on the returns of firms in a different industry and perhaps little effect on other rms in the same industry Any return can be explained with a large enough number of systematic risk factors However for a factor model to be useful as a practical matter the number of factors that explain the returns on an asset must be relatively limited The market risk premium and in ation rates are probably good choices The price of wheat while a risk factor for Ultra Products is not a market risk factor and will not likely be priced as a risk factor common to all stocks In this case wheat would be a firm speci c risk factor not a market risk factor A better model would employ macroeconomic risk factors such as interest rates GDP energy prices and industrial production among others a Real GNP was higher than anticipated Since returns are positively related to the level of GNP returns should rise based on this factor b In ation was exactly the amount anticipated Since there was no surprise in this announcement it will not affect LewisStriden returns 0 Interest rates are lower than anticipated Since returns are negatively related to interest rates the lower than expected rate is good news Returns should rise due to interest rates d The President s death is bad news Although the president was expected to retire his retirement would not be effective for six months During that period he would still contribute to the rm His untimely death means that those contributions will not be made Since he was generally considered an asset to the firm his death will cause returns to fall However since his departure was expected soon the drop might not be very large 8 The poor research results are also bad news Since LewisStriden must continue to test the drug it will not go into production as early as expected The delay will affect expected future earnings and thus it will dampen returns now f The research breakthrough is positive news for Lewis Striden Since it was unexpected it will cause returns to rise 285 p n O The l quot s is also 39 but it is not a welcome surprise This announcement will lower the returns on LewisStriden The systematic factors in the list are real GNP in ation and interest rates The unsystematic risk factors are the president s ability to contribute to the firm the research results and the competitor The main difference is that the market model assumes that only one factor usually a stock market aggregate is enough to explain stock returns while a k factor model relies on k factors to explain returns The fact that APT does not give any guidance about the factors that in uence stock returns is a commonlycited criticism However in choosing factors we should choose factors that have an economically valid reason for potentially affecting stock returns For example a smaller company has more risk than a large company Therefore the size of a company can affect the returns of the company stock Assuming the market portfolio is properly scaled it can be shown that the onefactor model is identical to the CAPM It is the weighted average of expected returns plus the weighted average of each security39s beta times a factor F plus the weighted average of the unsystematic risks of the individual securities Choosing variables because they have been shown to be related to returns is data mining The relation found between some attribute and returns can be accidental thus overstated For example the occurrence of sunbums and ice cream consumption are related however sunbums do not necessarily cause ice cream consumption or vice versa For a factor to truly be related to asset returns there should be sound economic reasoning for the relationship not just a statistical one Using a benchmark composed of English stocks is wrong because the stocks included are not of the same style as those in a US growth stock fund Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic Since we have the expected return of the stock the revised expected return can be determined using the innovation or surprise in the risk factors So the revised expected return is R11123273570834729 R 1024 a If m is the systematic risk portion of return then m BGNP AGNP Bln a on AIn ation BAInterest rates m 0000479136017132757 130320 7 390 7 67470 7 520 m 281 286 b The unsystematic return is the return that occurs because of a rm speci c factor such as the bad news about the company So the unsystematic return of the stock is 726 percent The total return is the expected return plus the two components of unexpected return the systematic risk portion of return and the unsystematic portion So the total return of the stock is R E m 8 R1080281726 R 1101 a If m is the systematic risk portion of return then m BGNPAGNP BAInterest rates m 20426 718711548 743 m 106 b The unsystematic is the return that occurs because of a firm speci c factor such as the increase in market share If 8 is the unsystematic risk portion of the return then s 045277 23 s 180 c The total return is the expected return plus the two components of unexpected return the systematic risk portion of return and the unsystematic portion So the total return of the stock is R E m 8 R 1050 106 180 R 1336 The beta for a particular risk factor in a portfolio is the weighted average of the betas of the assets This is true whether the betas are from a single factor model or a multifactor model So the betas of the portfolio are F1 20145 20073 60089 F1 097 F2 20080 20125 60414 F2 033 F3 20005 207020 60124 F3 071 So the expression for the return of the portfolio is R 5 097F1 033F2 7071F3 Which means the return of the portfolio is R 5 097550 0334207 071490 R 821 287 Intermediate We can express the multifactor model for each portfolio as ERP RF BIFI 3ze where F1 and F 2 are the respective risk premiums for each factor Expressing the return equation for each portfolio we get 164085F1 115F2 124145F1 7025F2 We can solve the system of two equations with two unknowns Multiplying each equation by the respective F 2 factor for the other equation we get 400 10 2125F1 02875F2 138 46 16675F1 7 02875F2 Summing the equations and solving F1 for gives us 178 7 56 188 F1 F1 7 649 And now using the equation for portfolio A we can solve for F 2 which is 16 4 0856490 115F2 F2 564 a The market model is specified by R7 BRM 7 EM s so applying that to each Stock Stock A RA RA BARM7 RM 8A RA 105 12RM 7142 8A Stock B RB RB BBRM 7 RM SB RB 130 098RM 7142 SB Stock C RC Rc BCRM 7 RM 8c RC 157 137RM 7142 8c 288 Since we don t have the actual market return or unsystematic risk we will get a formula with those values as unknowns RP 30RA 45RB 25Rc RP 30105 12RM 7142 8A 45130 098RM 7142 83 25157 137RM 7142 8c RP 30105 4513 25157 3012 4598 25137RM 7142 308A 4583 258c RP 12925 11435RM 7142308A 4583 258c Using the market model if the return on the market is 15 percent and the systematic risk is zero the return for each individual stock is RA 7 105 120157142 RA 71146 RB 13 098157142 RB 1378 RC 1570 137157142 RC 1680 To calculate the return on the portfolio we can use the equation from part b so RP 12925 11435157142 RP 1384 Alternatively to nd the portfolio return we can use the return of each asset and its portfolio weight or RP 7 lel sz2 x3113 RP 7 301146 451378 251680 RP 7 1384 Since ve stocks have the same expected returns and the same betas the portfolio also has the same expected return and beta However the unsystematic risks might be different so the expected return of the portfolio is RP 711 084F1169F2 1581 82 83 84 85 289 b Consider the expected return equation of a portfolio of ve assets we calculated in part 1 Since we now have a very large number of stocks in the portfolio as N ooi 0 N But the sjs are in nite so lN81 82 t 83 t 84 8N 0 Thus R111084F1169F2 Challenge To determine which investment an investor would prefer you must compute the variance of portfolios created by many stocks from either market Because you know that diversi cation is good it is reasonable to assume that once an investor has chosen the market in which she will invest she will buy many stocks in that market Known EF0andc010 EE 0 and SS 020 for all i If we assume the stocks in the portfolio are equallyweighted the weight of each stock is that is If a portfolio is composed of N stocks each forming lN proportion of the portfolio the return on the portfolio is lN times the sum of the returns on the N stocks To nd the variance of the respective portfolios in the 2 markets we need to use the de nition of variance from Statistics VarX EX 7 EX2 In our case Vamp ERP 7 ERp2 290 Note however to use this first we must find Rp and ERp So using the assumption about equal weights and then substituting in the known equation for R 1 RP EZRi 1 RP EX 010 BF 81 1 RP 010 BF E25 Also recall from Statistics a property of expected value that is If Z a 3 where a is a constant and Z X and 37 are random variables then 132 Emma 130 and Ea 1 Now use the above to find ERp ERp E010 3117 281 1 ERp 010 3135 E2mg ERP 010 30 20 ERp 010 Next substitute both of these results into the original equation for variance VarRp ERp 7ERp2 VarRpE010BFZsl p10 VarRp EBF a VarRp Ebze mix 57 Z VarRp 3262 628 1Covel 8 291 Finally since we can have as many stocks in each market as we want in the limit as N a no i a 0 so we get N VarRp 3262 Cov81sj and since C0V818j clcjpslsj and the problem states that 61 62 010 so Varmp 0ch 616mm VarRp 02001 004p818j So now summarize what we have so far R1 010 15F 81 R2 010 05F 82 ER1p ER2p 010 VarR1p 00225 004psh81j VarR2p 00025 004p82182j Finally we can begin answering the questions a b amp c for various values of the correlations 1 Substitute p81181j p82182j 0 into the respective variance formulas VarR1p 00225 VarR2p 00025 Since VarR1p gt VarR2p and expected returns are equal a risk averse investor will prefer to invest in the second market b If we assume p8h81j 09 and p82182j 0 the variance of each portfolio is VarR1p 00225 004psh81j VarR1p 00225 00409 VarR1p 00585 VarR2p 00025 004p82182j VarR2p 00025 0040 VarR2p 00025 Since VarR1p gt VarR2p and expected returns are equal a risk averse investor will prefer to invest in the second market 292 If we assume p81181j 0 and p82182j 5 the variance of each portfolio is VarR1p 00225 004psh81j VarR1p 00225 0040 VarR1p 00225 VarR2p 00025 004p82182j VarR2p 00025 00405 VarR2p 00225 Since VarR1p VarR2p and expected returns are equal a risk averse investor will be indifferent between the two markets Since the expected returns are equal indifference implies that the variances of the portfolios in the two markets are also equal So set the variance equations equal and solve for the correlation of one market in terms of the other VarR1p VarR2p 00225 004psh81j 00025 004p82182j 821821 811811 05 Therefore for any set of correlations that have this relationship as found in part c a risk adverse investor will be indifferent between the two markets In order to nd standard deviation 6 you must rst find the Variance since 6 VVar Recall from Statistics a property of Variance If Z a 3 where a is a constant and Z X and 37 are random variables then VarZ aware Vac and Vara 0 The problem states that retumgeneration can be described by R1 0H BzRM 812 293 Realize that RM RM and 81 are random variables and 00 and B are constants Then applying the above properties to this model we get VarR1 VarRM Var81 and now we can find the standard deviation for each asset a 07200121 001 0015929 GA J0015929 1262 or 1262 a 12200121 00144 0031824 a 40031824 1784 or 1784 oz 15200121 00225 0049725 cc 4004972 2230 or 2230 From the above formula for variance note that as N a 00 W a 0 so you get VarR1 VarRM So the variances for the assets are c 0720121 0005929 6123 1220121 0017424 6 1520121 0027225 We can use the model E RF 131 M RF which is the CAPM or APT Model when there is one factor and that factor is the Market So the expected return of each asset is A 33 071067 33 841 B 33121067 33 1206 c 33151067 33 1425 m WI WI We can compare these results for expected asset returns as per CAPM or APT with the expected returns given in the table This shows that assets A amp B are accurately priced but asset C is overpriced the model shows the return should be higher Thus rational investors will not hold asset C 294 If short selling is allowed rational investors will sell short asset C causing the price of asset C to decrease until no arbitrage opportunity exists In other words the price of asset C should decrease until the return becomes 1425 percent Let X1 the proportion of Security 1 in the portfolio and X2 the proportion of Security 2 in the portfolio and note that since the weights must sum to 10 X1 1 7 X2 Recall from Chapter 10 that the beta for a portfolio or in this case the beta for a factor is the weighted average of the security betas so BPI X13114r X2821 BPI X18111 X1le Now apply the condition given in the hint that the return of the portfolio does not depend on F1 This means that the portfolio beta for that factor will be 0 so BPI 0 X18111X1le pp 0 X11017X105 and solving for X1 and X2 X17l X22 Thus sell short Security 1 and buy Security 2 To nd the expected return on that portfolio use RP XIRI XZRZ so applying the above ERP 7120 220 ERp 20 Bpl 711 205 P1 295 u runeth the same lug as m part a we have Bm X3B311TX3BM BM XZUV 1 TX1 5 and X3 Thus seu shun Semnty 4 and buy Seemcy 3 Then ERP1 1uez1u E01 Wu pm 3m 5 72m 75 By I Nuts Lhatsmcebuth pm and pm are n thsxs ansk eepunfulm The punfulm m pm h prumdes ansk ee return uf lEI Va wheh Ts Ingmar than the 5 Tatum rate uf 5 and invest the funds m a purtfulm hum hy seumg shurt secunty fuur and huymg seeumy three wnh weights 372 as m part h e decreasE buy Wm merease Hehee the return uf secunty fuur Wm mereese and the return uf security three Wm decrease Th altemanvexs Lhatthe pnces ufsecunuesthree and fuurwxll remam the same and the pnce quhe nskr ee security amps uhhl its return Ts lEI many h and the nskr 39ee secunty Wm decrease and the pnce uf security three Wm mereese hm the uppurtumty mssppears W1 ZU 512 U I CHAPTER 13 RISK COST OF CAPITAL AND CAPITAL BUDGETING Answers to Concepts Review and Critical Thinking Questions 1 2 No The cost of capital depends on the risk of the project not the source of the money Interest expense is taxdeductible There is no difference between pretax and a ertax equity costs You are assuming that the new project s risk is the same as the risk of the rm as a whole and that the rm is nanced entirely with equity Two primary advantages of the SML approach are that the model explicitly incorporates the relevant risk of the stock and the method is more widely applicable than is the DCF model since the SML doesn t make any assumptions about the film s dividends The primary disadvantages of the SML method are 1 three parameters the riskfree rate the expected return on the market and beta must be estimated and 2 the method essentially uses historical information to estimate these parameters The riskfree rate is usually estimated to be the yield on very short maturity Tbills and is hence observable the market risk premium is usually estimated from historical risk premiums and hence is not observable The stock beta which is unobservable is usually estimated either by determining some average historical beta from the firm and the market s return data or by using beta estimates provided by analysts and investment rms The appropriate a ertax cost of debt to the company is the interest rate it would have to pay if it were to issue new debt today Hence if the YTM on outstanding bonds of the company is observed the company has an accurate estimate of its cost of debt If the debt is privatelyplaced the film could still estimate its cost of debt by 1 looking at the cost of debt for similar firms in similar risk classes 2 looking at the average debt cost for rms with the same credit rating assuming the rm s private debt is rated or 3 consulting analysts and investment bankers Even if the debt is publicly traded an additional complication arises when the rm has more than one issue outstanding these issues rarely have the same yield because no two issues are ever completely homogeneous a This only considers the dividend yield component of the required return on equity b This is the current yield only not the promised yield to maturity In addition it is based on the book value of the liability and it ignores taxes 0 Equity is inherently riskier than debt except perhaps in the unusual case where a rm s assets have a negative beta For this reason the cost of equity exceeds the cost of debt If taxes are considered in this case it can be seen that at reasonable tax rates the cost of equity does exceed the cost of debt 297 O RS p 12 7508 1800 or 1800 Both should proceed The appropriate discount rate does not depend on which company is investing it depends on the risk of the project Since Superior is in the business it is closer to a pure play Therefore its cost of capital should be used With an 18 cost of capital the project has an NPV of 1 million regardless of who takes it If the different operating divisions were in much different risk classes then separate cost of capital gures should be used for the different divisions the use of a single overall cost of capital would be inappropriate If the single hurdle rate were used riskier divisions would tend to receive more funds for investment projects since their return would exceed the hurdle rate despite the fact that they may actually plot below the SML and hence be unpro table projects on a riskadjusted basis The typical problem encountered in estimating the cost of capital for a division is that it rarely has its own securities traded on the market so it is dif cult to observe the market s valuation of the risk of the division Two typical ways around this are to use a pure play proxy for the division or to use subjective adjustments of the overall firm hurdle rate based on the perceived risk of the division The discount rate for the projects should be lower that the rate implied by the security market line The security market line is used to calculate the cost of equity The appropriate discount rate for projects is the rm s weighted average cost of capital Since the rm s cost of debt is generally less that the rm s cost of equity the rate implied by the security market line will be too high Beta measures the responsiveness of a security39s returns to movements in the market Beta is determined by the cyclicality of a rm39s revenues This cyclicality is magni ed by the rm39s operating and nancial leverage The following three factors will impact the rm s beta 1 Revenues The cyclicality of a rm s sales is an important factor in determining beta In general stock prices will rise when the economy expands and will fall when the economy contracts As we said above beta measures the responsiveness of a security39s returns to movements in the market Therefore rms whose revenues are more responsive to movements in the economy will generally have higher betas than firms with lesscyclical revenues 2 Operating leverage Operating leverage is the percentage change in earnings before interest and taxes EBIT for a percentage change in sales A rm with high operating leverage will have greater uctuations in EBIT for a change in sales than a firm with low operating leverage In this way operating leverage magnifies the cyclicality of a firm39s revenues leading to a higher beta 3 Financial leverage Financial leverage arises from the use of debt in the film39s capital structure A levered rm must make fixed interest payments regardless of its revenues The effect of financial leverage on beta is analogous to the effect of operating leverage on beta Fixed interest payments cause the percentage change in net income to be greater than the percentage change in EBIT magnifying the cyclicality of a rm39s revenues Thus returns on highlylevered stocks should be more responsive to movements in the market than the returns on stocks with little or no debt in their capital structure 298 Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic 1 With the information given we can find the cost of equity using the CAPM The cost of equity is RE 045 11511 7 045 1198 or 1198 2 With the information given we can nd the cost of equity using the dividend growth model Using this model the cost of equity is RE 240105552 055 1037 or 1037 3 We have the information available to calculate the cost of equity using the CAPM and the dividend growth model Using the CAPM we nd RE 05 08508 1180 or 1180 And using the dividend growth model the cost of equity is RE 16010637 06 1058 or 1058 Both estimates of the cost of equity seem reasonable If we remember the historical return on large capitalization stocks the estimate from the CAPM model is about the same as the historical average and the estimate from the dividend growth model is about one percent lower than the historical average so we cannot de nitively say one of the estimates is incorrect Given this we would use the average of the two so RE 1180 10582 1119 or 1119 4 The pretax cost of debt is the YTM of the company s bonds so P0 950 40PVIFAR724 1000PVIFR724 R 4339 YTM 2 X 4339 868 And the a ertax cost of debt is RD 0868 1 7 35 0564 or 564 5 a The pretax cost of debt is the YTM of the company s bonds so P0 1080 35PVIFAR746 1000PVIFR746 3 16 0 R o YTM 2 X 3167 633 299 b The aftertaX cost of debt is RD 063317 35 0412 or 412 c The aftertax rate is more relevant because that is the actual cost to the company The book value of debt is the total par value of all outstanding debt so BVD 60000000 80000000 140000000 To nd the market value of debt we nd the price of the bonds and multiply by the number of bonds Alternatively we can multiply the price quote of the bond times the par value of the bonds Doing so we nd MVD 10860000000 7380000000 123200000 The YTM of the zero coupon bonds is Pz 730 1000PVIFR714 R 2273 YTM 2 X 2273 455 So the aftertaX cost of the zero coupon bonds is Rz 04551 7 35 0296 or 296 The aftertax cost of debt for the company is the weighted average of the aftertax cost of debt for all outstanding bond issues We need to use the market value weights of the bonds The total aftertax cost of debt for the company is RD 04126481232 02965841232 0357 or 357 Using the equation to calculate the WACC we nd WACC 7015 30081735 1206 or 1206 Here we need to use the debtequity ratio to calculate the WACC Doing so we nd WACC 7 171145 10451451 735 7 1374 or 1374 Here we have the WACC and need to nd the debtequity ratio of the company Setting up the WACC equation we find WACC 7 0980 7 15EN 0750DN1735 Rearranging the equation we nd 0980VE 15 075065DE Now we must realize that the VE is just the equity multiplier which is equal to 300 VE1D 3 0980DE 1 15 04875D 3 Now we can solve for DE as 04925DE 052 DE 10558 C The book value of equity is the book value per share times the number of shares and the book value of debt is the face value of the company s debt so BVE 75000004 30000000 BVD 60000000 50000000 110000000 So the total value of the company is V 30000000 110000000 140000000 And the book value weights of equity and debt are EV 30000000140000000 2143 DV 17 EN 7857 The market value of equity is the share price times the number of shares so MVE 750000049 367500000 Using the relationship that the total market value of debt is the price quote times the par value of the bond we find the market value of debt is MVD 9360000000 96550000000 104050000 This makes the total market value of the company V 367500000 104050000 471550000 And the market value weights of equity and debt are EV 367500000471550000 7793 DV 17 EN 2207 The market value weights are more relevant 11 First we will find the cost of equity for the company The information provided allows us to solve for the cost of equity using the CAPM so RE 052 1207 1360 or 1360 301 p A 03 Next we need to nd the YTM on both bond issues Doing so we find P1 930 35PVIFAR720 1000PVIFR720 R 4016 YTM 4016 X 2 803 P2 965 325PVIFAR712 1000PVIFR712 R 3496 YTM 3496 X 2 723 To nd the weighted average aftertaX cost of debt we need the weight of each bond as a percentage of the total debt We nd le 9360000000104050000 536 tz 96550000000104050000 464 Now we can multiply the weighted average cost of debt times one minus the tax rate to nd the weighted average aftertax cost of debt This gives us RD 173553608034640723 0498 or 498 Using these costs and the weight of debt we calculated earlier the WACC is WACC 77931360 22070498 1170 or 1170 1 Using the equation to calculate WACC we nd WACC 112 116515 651651735RD RD 0824 or 824 b Using the equation to calculate WACC we nd WACC 112 1165RE 65165064 RE 1432 or 1432 We will begin by nding the market value of each type of nancing We nd MvD 50001000103 5150000 MvE 16000057 9120000 And the total market value of the rm is V 5150000 9120000 14270000 Now we can find the cost of equity using the CAPM The cost of equity is RE 06 11007 1370 or 1370 302 The cost of debt is the YTM of the bonds so P0 1030 840Pv1FAWO 1000Pv1FR40 R 3851 YTM 3851 x 2 770 And the a ertax cost of debt is RD 17350770 0501 or 501 Now we have all of the components to calculate the WACC The WACC is WACC 05015151427 13709121427 1056 or 1056 Notice that we didn t include the 1 7 tc term in the WACC equation We simply used the a ertax cost of debt in the equation so the term is not needed here a We will begin by nding the market value of each type of financing We nd MvD 2000001000093 186000000 MvE 850000034 289000000 And the total market value of the rm is V 186000000 289000000 475000000 So the market value weights of the company s nancing is Dv 186000000475000000 3916 EN 289000000475000000 6084 For projects equally as risky as the rm itself the WACC should be used as the discount rate First we can nd the cost of equity using the CAPM The cost of equity is RE 05 12007 1340 or 1340 The cost of debt is the YTM of the bonds so P0 930 375PVIFAR30 1000PVIFR730 R 4163 YTM 4163 X 2 833 And the a ertaX cost of debt is RD 1 7 350833 0541 or 541 Now we can calculate the WACC as WACC 13406084 05413916 1027 or 1027 303 p A l p A on 1 Projects Y and Z 9 Using the CAPM to consider the projects we need to calculate the expected return of each project given its level of risk This expected return should then be compared to the expected return of the project If the return calculated using the CAPM is lower than the project expected return we should accept the project if not we reject the project After considering risk via the CAPM EW 05 7511705 0950 lt 10 so acceptW EX 05 9011705 1040 gt 102 so rejectX EY 05 12011705 1220 gt 12 so rejectY EZ 05 15011705 1400 lt 15 so acceptZ 5 1 Project W would be incorrectly rejected Project Y would be incorrectly accepted a He should look at the weighted average otation cost not just the debt cost b The weighted average otation cost is the weighted average of the otation costs for debt and equity so fT 0575175 081175 0671 or 671 c The total cost of the equipment including otation costs is Amount raised1 7 0671 20000000 Amount raised 200000001 7 0671 21439510 Even if the specific funds are actually being raised completely from debt the otation costs and hence true investment cost should be valued as if the rm s target capital structure is used We rst need to find the weighted average otation cost Doing so we find fT 6509 0506 3003 071 or 71 And the total cost of the equipment including otation costs is Amount raised1 7 071 45000000 Amount raised 4500000017071 48413125 Intermediate Using the debtequity ratio to calculate the WACC we find WACC 7 65165055 116515 7 1126 or 1126 Since the project is riskier than the company we need to adjust the project discount rate for the additional risk Using the subjective risk factor given we nd Project discount rate 1126 200 1326 304 0 We would accept the project if the NPV is positive The NPV is the PV of the cash out ows plus the PV of the cash in ows Since we have the costs we just need to nd the PV of in ows The cash in ows are a growing perpetuity If you remember the equation for the PV of a growing perpetuity is the same as the dividend growth equation so PV offuture CF 35000001326 7 05 42385321 The project should only be undertaken if its cost is less than 42385321 since costs less than this amount will result in a positive NPV We will begin by finding the market value of each type of nancing We will use D1 to represent the coupon bond and D2 to represent the zero coupon bond So the market value of the rm s nancing is MVD1 40000100011980 47920000 MVDZ 15000010001820 27300000 MVP 10000078 7800000 MVE 180000065 117000000 And the total market value of the rm is V 47920000 27300000 7800000 117000000 200020000 Now we can find the cost of equity using the CAPM The cost of equity is RE 04 11007 1170 or 1170 The cost of debt is the YTM of the bonds so P0 1198 35PVIFAR750 1000PVIFR750 R 2765 YTM 2765 X 2 553 And the a ertax cost of debt is RD1 17400553 0332 or 332 And the a ertax cost of the zero coupon bonds is P0 182 1000PVIFR760 R 2880 YTM 288 X 2 576 RD2 17400576 0346 or 346 Even though the zero coupon bonds make no payments the calculation for the YTM or price still assumes semiannual compounding consistent with a coupon bond Also remember that even though the company does not make interest payments the accrued interest is still tax deductible for the company 305 N O 21 To nd the required return on preferred stock we can use the preferred stock pricing equation which is the level perpetuity equation so the required return on the company s preferred stock is RP D1 P0 Rp 4 78 RP 0513 or 513 Notice that the required return in the preferred stock is lower than the required on the bonds This result is not consistent with the risk levels of the two instruments but is a common occurrence There is a practical reason for this Assume Company A owns stock in Company B The tax code allows Company A to exclude at least 70 percent of the dividends received from Company B meaning Company A does not pay taxes on this amount In practice much of the outstanding preferred stock is owned by other companies who are willing to take the lower return since it is effectively tax exempt Now we have all of the components to calculate the WACC The WACC is WACC 0332479220002 034627320002 117011720002 05137820002 WACC 0831 or 831 The total cost of the equipment including otation costs was Total costs 15000000 850000 15850000 Using the equation to calculate the total cost including otation costs we get Amount raisedl 7 fr Amount needed after otation costs 1585000017 fT 15000000 fT 0536 or 536 Now we know the weighted average otation cost The equation to calculate the percentage otation costs is fT 0536 07EV 03DV We can solve this equation to find the debtequity ratio as follows 0536VE 07 03DE We must recognize that the VE term is the equity multiplier which is l DE so 0536DE 1 07 03D 3 DE 06929 1 Using the dividend discount model the cost of equity is 306 RE 08010561 05 RE 0638 or 638 b Using the CAPM the cost of equity is RE 055 1501200 7 0550 RE 1525 or 1525 c When using the dividend growth model or the CAPM you must remember that both are estimates for the cost of equity Additionally and perhaps more importantly each method of estimating the cost of equity depends upon different assumptions Challenge 22 We can use the debtequity ratio to calculate the weights of equity and debt The debt of the company has a weight for longterm debt and a weight for accounts payable We can use the weight given for accounts payable to calculate the weight of accounts payable and the weight of longterm debt The weight of each will be Accounts payable weight 20 120 17 Longterm debt weight l 120 83 Since the accounts payable has the same cost as the overall WACC we can write the equation for the WACC as WACC 117 14 071720l2WACC 112081735 Solving for WACC we nd WACC 0824 41182012WACC 0433 WACC 0824 0686WACC 0178 9314WACC 1002 WACC 1076 or 1076 We will use basically the same equation to calculate the weighted average otation cost except we will use the otation cost for each form of nancing Doing so we get Flotation costs 1170807172012011204 0608 or 608 The total amount we need to raise to fund the new equipment will be Amount raised cost 450000001 7 0608 Amount raised 479123 17 Since the cash ows go to perpetuity we can calculate the present value using the equation for the PV of a perpetuity The NPV is NPV 747912317 6200000 1076 NPV 9719777 307 23 We can use the debtequity ratio to calculate the weights of equity and debt The weight of debt in the capital structure is wD 120 220 5455 or 5455 And the weight of equity is wE 175455 4545 or 4545 Now we can calculate the weighted average otation costs for the various percentages of internally raised equity To nd the portion of equity otation costs we can multiply the equity costs by the percentage of equity raised externally which is one minus the percentage raised internally So if the company raises all equity externally the otation costs are fT 7 0454508170 05455035 f 7 555or 555 T The initial cash out ow for the project needs to be adjusted for the otation costs To account for the otation costs Amount raised1 7 0555 145000000 Amount raised 145000000l 70555 Amount raised 153512993 If the company uses 60 percent intemally generated equity the otation cost is fT 7 045450817 060 05455035 fT 7 0336 or 336 And the initial cash ow will be Amount raised1 7 0336 145000000 Amount raised 1450000001 7 0336 Amount raised 150047037 If the company uses 100 percent internally generated equity the otation cost is fT 7 0454508171 05455035 fT 7 0191 or 191 And the initial cash ow will be Amount raised1 7 0191 145000000 Amount raised 145000000l 70191 Amount raised 147822057 308 24 The 4 million cost of the land 3 years ago is a sunk cost and irrelevant the 51 million appraised value of the land is an opportunity cost and is relevant The 6 million land value in 5 years is a relevant cash ow as well The fact that the company is keeping the land rather than selling it is unimportant The land is an opportunity cost in 5 years and is a relevant cash ow for this project The market value capitalization weights are MVD 240000l000094 225600000 MVE 900000071 639000000 MVP 40000081 32400000 The total market value of the company is v 7 225600000 639000000 32400000 7 897000000 Next we need to nd the cost of funds We have the information available to calculate the cost of equity using the CAPM so RE 05 12008 1460 or 1460 The cost of debt is the YTM of the company s outstanding bonds so P0 7 940 7 3750PVIFAR740 1000Pv1FR40 R 7 4056 YTM 4056 X 2 811 And the aftertax cost of debt is RD 17350811 0527 or 527 The cost of preferred stock is Rp 55081 0679 or 679 a The weighted average otation cost is the sum of the weight of each source of funds in the capital structure of the company times the otation costs so fT 63989708 32489706 225689704 0692 or 692 The initial cash outflow for the project needs to be adjusted for the otation costs To account for the otation costs Amount raised1 7 0692 35000000 Amount raised 35000000l 70692 37602765 So the cash ow at time zero will be CF0 7 75 100000 7 37602765 7 13000000 7 744002765 309 There is an important caveat to this solution This solution assumes that the increase in net working capital does not require the company to raise outside funds therefore the otation costs are not included However this is an assumption and the company could need to raise outside funds for the NWC If this is true the initial cash outlay includes these otation costs so Total cost of NWC including otation costs 13000001 7 0692 1396674 This would make the total initial cash ow CF0 7 45100000 4 37602765 4 1396674 7 444099439 To nd the required return on this project we first need to calculate the WACC for the company The company s WACC 1s WACC 6398971460 3248970679 22568970527 1197 The company wants to use the subjective approach to this project because it is located overseas The adjustment factor is 2 percent so the required return on this project is Project required return 1197 02 1397 The annual depreciation for the equipment will be 350000008 4375000 So the book value of the equipment at the end of five years will be BV5 35000000 7 54375000 13125000 So the aftertaX salvage value will be Aftertax salvage value 6000000 3513125000 7 6000000 8493750 Using the tax shield approach the OCF for this project is OCF 7 P 4 vQ 4 FC1 4 t 4 th OCF 7 10900 4 940018000 4 70000001 4 35 35350000008 7 14531250 The accounting breakeven sales gure for this project is QA 7 FC 7 DP 7v 7 7000000 7 437500010900 49400 7 7583 units 310 f We have calculated all cash ows of the project We just need to make sure that in Year 5 we add back the aftertaX salvage value and the recovery of the initial NWC The cash ows for the project are M Flow Cash 0 744002765 1 14531250 2 14531250 3 14531250 4 14531250 5 30325000 Using the required return of 1397 percent the NPV of the project is NPV 744002765 14531250PVIFA13974 fl 30325000113975 NPV 1413071381 And the IR is NPV 0 44002765 14531250PVIFARR4 303250001 1RR5 IRR 2525 If the initial 1 WC is assumed to be nanced from outside sources the cash ows are M Flow Cash 0 744099439 1 14531250 2 14531250 3 14531250 4 14531250 5 30325000 With this assumption and the required return of 1397 percent the NPV of the project is NPV 44099439 14531250Pv1FA13 9m 30325000113975 NPV 1403403967 And the IR is NPV 0 44099439 14531250PVIFARR4 303250001 1RR5 IRR 2515 311 CHAPTER 14 EFFICIENT CAPITAL MARKETS AND BEHAVIORAL CHALLENGES Answers to Concepts Review and Critical Thinking Questions 1 To create value firms should accept nancing proposals with positive net present values Firms can create valuable nancing opportunities in three ways 1 F001 investors A rm can issue a complex security to receive more than the fair market value Financial managers attempt to package securities to receive the greatest value 2 Reduce costs or increase subsidies A rm can package securities to reduce taxes Such a security will increase the value of the rm In addition nancing techniques involve many costs such as accountants lawyers and investment bankers Packaging securities in a way to reduce these costs will also increase the value of the rm 3 Create a new security A previously unsatis ed investor may pay extra for a specialized security catering to his or her needs Corporations gain from developing unique securities by issuing these securities at premium prices The three forms of the efficient markets hypothesis are 1 Weak form Market prices re ect information contained in historical prices Investors are unable to earn abnormal returns using historical prices to predict future price movements 2 Semistrong form In addition to historical data market prices re ect all publiclyavailable information Investors with insider or private information are able to earn abnormal returns 3 Strong form Market prices re ect all information public or private Investors are unable to earn abnormal returns using insider information or historical prices to predict future price movements 1 False Market efficiency implies that prices re ect all available information but it does not imply certain knowledge Many pieces of information that are available and re ected in prices are fairly uncertain Efficiency of markets does not eliminate that uncertainty and therefore does not imply perfect forecasting ability b True Market efficiency exists when prices re ect all available information To be efficient in the weak form the market must incorporate all historical data into prices Under the semi strong form of the hypothesis the market incorporates all publiclyavailable information in addition to the historical data In strong form ef cient markets prices re ect all publicly and privately available information 0 False Market ef ciency implies that market participants are rational Rational people will immediately act upon new information and will bid prices up or down to re ect that information d False In efficient markets prices re ect all available information Thus prices will uctuate whenever new information becomes available 312 8 True Competition among investors results in the rapid transmission of new market information In efficient markets prices immediately re ect new information as investors bid the stock price up or down On average the only return that is earned is the required retumiinvestors buy assets with returns in excess of the required return positive NPV bidding up the price and thus causing the return to fall to the required return zero NPV investors sell assets with returns less than the required return negative NPV driving the price lower and thus causing the return to rise to the required return zero NPV The market is not weak form efficient Yes historical information is also public information weak form efficiency is a subset of semi strong form efficiency Ignoring trading costs on average such investors merely earn what the market offers the trades all have zero NPV If trading costs exist then these investors lose by the amount of the costs Unlike gambling the stock market is a positive sum game everybody can win Also speculators provide liquidity to markets and thus help to promote efficiency The EMH only says within the bounds of increasingly strong assumptions about the information processing of investors that assets are fairly priced An implication of this is that on average the typical market participant cannot earn excessive pro ts from a particular trading strategy However that does not mean that a few particular investors cannot outperform the market over a particular investment horizon Certain investors who do well for a period of time get a lot of attention from the nancial press but the scores of investors who do not do well over the same period of time generally get considerably less attention from the nancial press a If the market is not weak form efficient then this information could be acted on and a pro t earned from following the price trend Under 2 3 and 4 this information is fully impounded in the current price and no abnormal pro t opportunity exists b Under 2 if the market is not semistrong form efficient then this information could be used to buy the stock cheap before the rest of the market discovers the financial statement anomaly Since 2 is stronger than 1 both imply that a pro t opportunity exists under 3 and 4 this information is fully impounded in the current price and no pro t opportunity exists 0 Under 3 if the market is not strong form efficient then this information could be used as a pro table trading strategy by noting the buying activity of the insiders as a signal that the stock is underpriced or that good news is imminent Since 1 and 2 are weaker than 3 all three imply that a profit opportunity exists Note that this assumes the individual who sees the insider trading is the only one who sees the trading If the information about the trades made by company management is public information it will be discounted in the stock price and no pro t opportunity exists Under 4 this information does not signal any pro t opportunity for traders any pertinent information the managerinsiders may have is fully re ected in the current share price A technical analyst would argue that the market is not ef cient Since a technical analyst examines past prices the market cannot be weak form ef cient for technical analysis to work If the market is not weak form efficient it cannot be efficient under stronger assumptions about the information available 313 p n N p A DJ p n J F p A l Investor sentiment captures the mood of the investing public If investors are bearish in general it may be that the market is headed down in the future since investors are less likely to invest If the sentiment is bullish it would be taken as a positive signal to the market To use investor sentiment in technical analysis you would probably want to construct a ratio such as a bullsbears ratio To use the ratio simply compare the historical ratio to the market to determine if a certain level on the ratio indicates a market upturn or downturn Of course there is a group of investors called contrarians who view the market signals as reversed That is if the number of bearish investors reaches a certain level the market will head up For a contrarian these signals are reversed Taken at face value this fact suggests that markets have become more ef cient The increasing ease with which information is available over the Internet lends strength to this conclusion On the other hand during this particular period largecapitalization growth stocks were the top performers Valueweighted indexes such as the SampP 500 are naturally concentrated in such stocks thus making them especially hard to beat during this period So it may be that the dismal record compiled by the pros is just a matter of bad luck or benchmark error It is likely the market has a better estimate of the stock price assuming it is semistrong form ef cient However semistrong form ef ciency only states that you cannot easily pro t from publicly available information If nancial statements are not available the market can still price stocks based upon the available public information limited though it may be Therefore it may have been as dif cult to examine the limited public information and make an extra return a Aerotech s stock price should rise immediately after the announcement of the positive news b Only scenario it indicates market ef ciency In that case the price of the stock rises immediately to the level that re ects the new information eliminating all possibility of abnormal returns In the other two scenarios there are periods of time during which an investor could trade on the information and earn abnormal returns False The stock price would have adjusted before the founder s death only if investors had perfect forecasting ability The 125 percent increase in the stock price after the founder s death indicates that either the market did not anticipate the death or that the market had anticipated it imperfectly However the market reacted immediately to the new information implying ef ciency It is interesting that the stock price rose after the announcement of the founder s death This price behavior indicates that the market felt he was a liability to the firm The announcement should not deter investors from buying UPC s stock If the market is semistrong form ef cient the stock price will have already re ected the present value of the payments that UPC must make The expected return after the announcement should still be equal to the expected return before the announcement UPC s current stockholders bear the burden of the loss since the stock price falls on the announcement After the announcement the expected return moves back to its original level The market is often considered to be relatively ef cient up to the semistrong form If so no systematic pro t can be made by trading on publiclyavailable information Although illegal the lead engineer of the device can pro t from purchasing the rm s stock before the news release on the implementation of the new technology The price should immediately and fully adjust to the new information in the article Thus no abnormal return can be expected from purchasing after the publication of the article 314 N O N p A N N N J N UI Under the semistrong form of market efficiency the stock price should stay the same The accounting system changes are publicly available information Investors would identify no changes in either the rm s current or its future cash ows Thus the stock price will not change after the announcement of increased earnings Because the number of subscribers has increased dramatically the time it takes for information in the newsletter to be re ected in prices has shortened With shorter adjustment periods it becomes impossible to earn abnormal returns with the information provided by Durkin If Durkin is using only publiclyavailable information in its newsletter its ability to pick stocks is inconsistent with the ef cient markets hypothesis Under the semistrong form of market ef ciency all publiclyavailable information should be re ected in stock prices The use of private information for trading purposes is illegal You should not agree with your broker The performance ratings of the small manufacturing firms were published and became public information Prices should adjust immediately to the information thus preventing future abnormal returns Stock prices should immediately and fully rise to re ect the announcement Thus one cannot expect abnormal returns following the announcement 11 No Earnings information is in the public domain and re ected in the current stock price b Possibly If the rumors were publicly disseminated the prices would have already adjusted for the possibility of a merger If the rumor is information that you received from an insider you could earn excess returns although trading on that information is illegal c No The information is already public and thus already re ected in the stock price Serial correlation occurs when the current value of a variable is related to the future value of the variable If the market is ef cient the information about the serial correlation in the macroeconomic variable and its relationship to net earnings should already be re ected in the stock price In other words although there is serial correlation in the variable there will not be serial correlation in stock returns Therefore 39 39 39 of the 39 quot in the 39 variable will not lead to abnormal returns for investors The statement is false because every investor has a different risk preference Although the expected return from every welldiversi ed portfolio is the same after adjusting for risk investors still need to choose funds that are consistent with their particular risk level The share price will decrease immediately to re ect the new information At the time of the announcement the price of the stock should immediately decrease to re ect the negative information 315 27 N on In an ef cient market the cumulative abnormal return CAR for Prospectors would rise substantially at the announcement of a new discovery The CAR falls slightly on any day when no discovery is announced There is a small positive probability that there will be a discovery on any given day If there is no discovery on a particular day the price should fall slightly because the good event did not occur The substantial price increases on the rare days of discovery should balance the small declines on the other days leaving CARs that are horizontal over time Behavioral finance attempts to explain both the 1987 stock market crash and the Internet bubble by changes in investor sentiment and psychology These changes can lead to nonrandom price behavior Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic To nd the cumulative abnormal returns we chart the abnormal returns for each of the three airlines for the days preceding and following the announcement The abnormal return is calculated by subtracting the market return from a stock s return on a particular day R 7 RM Group the returns by the number of days before or after the announcement for each respective airline Calculate the cumulative average abnormal return by adding each abnormal return to the previous day s abnormal return Abnormal returns 1RL 713M Days from Average Cumulative Delta United American Sum abnormal return average residual 74 412 702 412 706 702 70 73 02 701 02 03 01 701 72 02 702 00 00 00 701 71 02 02 414 00 00 701 0 33 02 19 54 18 17 1 02 01 00 03 01 18 2 411 00 01 00 00 18 3 412 01 412 413 411 17 4 411 411 411 413 411 16 316 Cumulative Abnormal Returns 15 16 a 1 U 05 0 01 05 4 3 2 1 0 1 2 3 4 Days from announcement The market reacts favorably to the announcements Moreover the market reacts only on the day of the announcement Before and after the event the cumulative abnormal returns are relatively at This behavior is consistent with market ef ciency The diagram does not support the efficient markets hypothesis The CAR should remain relatively at following the announcements The diagram reveals that the CAR rose in the rst month only to drift down to lower levels during later months Such movement violates the semistrong form of the efficient markets hypothesis because an investor could earn abnormal pro ts while the stock price gradually decreased 1 Supports The CAR remained constant after the event at time 0 This result is consistent with market efficiency because prices adjust immediately to re ect the new information Drops in CAR prior to an event can easily occur in an efficient capital market For example consider a sample of forced removals of the CEO Since any CEO is more likely to be red following bad rather than good stock performance CARs are likely to be negative prior to removal Because the ring of the CEO is announced at time 0 one cannot use this information to trade pro tably before the announcement Thus price drops prior to an event are neither consistent nor inconsistent with the efficient markets hypothesis Rejects Because the CAR increases after the event date one can pro t by buying after the event This possibility is inconsistent with the efficient markets hypothesis Supports The CAR does not uctuate after the announcement at time 0 While the CAR was rising before the event insider information would be needed for pro table trading Thus the graph is consistent with the semistrong form of efficient markets 317 d Supports The diagram indicates that the information announced at time 0 was of no value Similar to part a such movement is neither consistent nor inconsistent with the efficient markets hypothesis ElVIH Movements at the event date are neither consistent nor inconsistent with the ef cient markets hypothesis Once the verdict is reached the diagram shows that the CAR continues to decline after the court decision allowing investors to earn abnormal returns The CAR should remain constant on average even if an appeal is in progress because no new information about the company is being revealed Thus the diagram is not consistent with the efficient markets hypothesis EMH 318 CHAPTER 15 LONGTERM FINANCING AN INTRODUCTION Answers to Concepts Review and Critical Thinking Questions 1 The indenture is a legal contract and can run into 100 pages or more Bond features which would be included are the basic terms of the bond the total amount of the bonds issued description of the property used as security repayment arrangements call provisions convertibility provisions and details of protective covenants The differences between preferred stock and debt are a The dividends on preferred stock cannot be deducted as interest expense when determining taxable corporate income From the individual investor s point of View preferred dividends are ordinary income for tax purposes For corporate investors 70 of the amount they receive as dividends from preferred stock are exempt from income taxes b In case of liquidation at bankruptcy preferred stock is junior to debt and senior to common stock 0 There is no legal obligation for firms to pay out preferred dividends as opposed to the obligated payment of interest on bonds Therefore firms cannot be forced into default if a preferred stock dividend is not paid in a given year Preferred dividends can be cumulative or noncumulative and they can also be deferred inde nitely of course inde nitely deferring the dividends might have an undesirable effect on the market value of the stock Some rms can bene t from issuing preferred stock The reasons can be a Public utilities can pass the tax disadvantage of issuing preferred stock on to their customers so there is a substantial amount of straight preferred stock issued by utilities b Firms reporting losses to the IRS already don t have positive income for any tax deductions so they are not affected by the tax disadvantage of dividends versus interest payments They may be willing to issue preferred stock 0 Firms that issue preferred stock can avoid the threat of bankruptcy that exists with debt nancing because preferred dividends are not a legal obligation like interest payments on corporate debt The return on nonconvertible preferred stock is lower than the return on corporate bonds for two reasons 1 Corporate investors receive 70 percent tax deductibility on dividends if they hold the stock Therefore they are willing to pay more for the stock that lowers its return 2 Issuing corporations are willing and able to offer higher returns on debt since the interest on the debt reduces their tax liabilities Preferred dividends are paid out of net income hence they provide no tax shield Corporate investors are the primary holders of preferred stock since unlike individual investors they can deduct 70 percent of the dividend when computing their tax liabilities Therefore they are willing to accept the lower return that the stock generates 319 p A O p A p A p A DJ The following table summarizes the main difference between debt and equity Debt Equity Repayment is an obligation of the firm Yes No Grants ownership of the rm No Yes Provides a tax shield Yes No Liquidation will result if not paid Yes No Companies often issue hybrid securities because of the potential tax shield and the bankruptcy advantage If the IRS accepts the security as debt the rm can use it as a tax shield If the security maintains the bankruptcy and ownership advantages of equity the rm has the best of both worlds There are two bene ts First the company can take advantage of interest rate declines by calling in an issue and replacing it with a lower coupon issue Second a company might wish to eliminate a covenant for some reason Calling the issue does this The cost to the company is a higher coupon A put provision is desirable from an investor s standpoint so it helps the company by reducing the coupon rate on the bond The cost to the company is that it may have to buy back the bond at an unattractive price It is the grant of authority by a shareholder to someone else to vote his or her shares Preferred stock is similar to both debt and common equity Preferred shareholders receive a stated dividend only and if the corporation is liquidated preferred stockholders get a stated value However unpaid preferred dividends are not debts of a company and preferred dividends are not a tax deductible business expense A company has to issue more debt to replace the old debt that comes due if the company wants to maintain its capital structure There is also the possibility that the market value of a company continues to increase we hope This also means that to maintain a specific capital structure on a market value basis the company has to issue new debt since the market value of existing debt generally does not increase as the value of the company increases at least by not as much Internal financing comes from internally generated cash ows and does not require issuing securities In contrast external nancing requires the rm to issue new securit1es The three basic factors that affect the decision to issue external equity are l The general economic environment speci cally business cycles 2 The level of stock prices and 3 The availability of positive NPV projects When a company has dual class stock the difference in the share classes are the voting rights Dual share classes allow minority shareholders to retain control of the company even though they do not own a majority of the total shares outstanding Often dual share companies were started by a family taken public but the founders want to retain control of the company The statement is true In an efficient market the callable bonds will be sold at a lower price than that of the noncallable bonds other things being equal This is because the holder of callable bonds effectively sold a call option to the bond issuer Since the issuer holds the right to call the bonds the price of the bonds will re ect the disadvantage to the bondholders and the advantage to the bond issuer ie the bondholder has the obligation to surrender their bonds when the call option is exercised by the bond issuer 320 14 p n UI As the interest rate falls the call option on the callable bonds is more likely to be exercised by the bond issuer Since the noncallable bonds do not have such a drawback the value of the bond will go up to re ect the decrease in the market rate of interest Thus the price of noncallable bonds will move higher than that of the callable bonds Sinking funds provide additional security to bonds If a firm is experiencing nancial difficulty it is likely to have trouble making its sinking fund payments Thus the sinking fund provides an early warning system to the bondholders about the quality of the bonds A drawback to sinking funds is that they give the rm an option that the bondholders may nd distasteful If bond prices are low the rm may satisfy its sinking fund obligation by buying bonds in the open market If bond prices are high though the firm may satisfy its obligation by purchasing bonds at face value or other xed price depending on the speci c terms Those bonds being repurchased are chosen through a lottery Solutions to Questions and Problems NOTE All end of chapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic If the company uses straight voting the board of directors is elected one at a time You will need to own onehalf of the shares plus one share in order to guarantee enough votes to win the election So the number of shares needed to guarantee election under straight voting will be Shares needed 600000 shares 2 1 Shares needed 300001 And the total cost to you will be the shares needed times the price per share or Total cost 300001 x 39 Total cost 11700039 If the company uses cumulative voting the board of directors are all elected at once You will need 1N 1 percent of the stock plus one share to guarantee election where N is the number of seats up for election So the percentage of the company s stock you need is Percent of stock needed 1N 1 Percent of stock needed 1 7 1 Percent of stock needed 1250 or 1250 321 So the number of shares you need to purchase is Number of shares to purchase 600000 X 1250 1 Number of shares to purchase 75001 And the total cost to you will be the shares needed times the price per share or Total cost 75001 gtlt 39 Total cost 2925039 If the company uses cumulative voting the board of directors are all elected at once You will need 1N 1 percent of the stock plus one share to guarantee election where N is the number of seats up for election So the percentage of the company s stock you need is Percent of stock needed 1N 1 Percent of stock needed 1 3 1 Percent of stock needed 25 or 25 So the number of shares you need is Number of shares to purchase 5800 X 25 1 Number of shares to purchase 1451 So the number of additional shares you need to purchase is New shares to purchase 1451 7 300 New shares to purchase 1151 If the company uses cumulative voting the board of directors are all elected at once You will need 1N 1 percent of the stock plus one share to guarantee election where N is the number of seats up for election So the percentage of the company s stock you need is Percent of stock needed 1N 1 Percent of stock needed 1 3 1 Percent of stock needed 25 or 25 So the number of shares you need to purchase is Number of shares to purchase 1200000 X 20 1 Number of shares to purchase 300001 And the total cost will be the shares needed times the price per share or Total cost 300001 x 9 Total cost 2700009 322 Under cumulative voting she will need 1N 1 percent of the stock plus one share to guarantee election where N is the number of seats up for election So the percentage of the company s stock she needs is Percent of stock needed 1N 1 Percent of stock needed 1 6 1 Percent of stock needed 1429 or 1429 Her nominee is guaranteed election If the elections are staggered the percentage of the company s stock needed is Percent of stock needed 1N 1 Percent of stock needed 1 3 1 Percent of stock needed 25 or 25 Her nominee is no longer guaranteed election Zero coupon bonds are priced with semiannual compounding to correspond with coupon bonds The price of the bond when purchased was P0 1000 1 03550 P0 17905 And the price at the end of one year is P0 1000 1 03548 P0 19181 So the implied interest which will be taxable as interest income is Implied interest 19181 7 17905 Implied interest 1275 a The price of the bond today is the present value of the expected price in one year So the price of the bond in one year if interest rates increase will be P1 60PVIFA758 t 1000PVIF758 P1 85997 If interest rates fall the price if the bond in one year will be P1 60PVIFA3 558 t 1000PVIF3 558 P1 161716 Now we can find the price of the bond today which will be P0 5085997 50161716 10552 P0 111279 For students who have studied term structure the assumption of riskneutrality implies that the forward rate is equal to the expected future spot rate 323 b If the bond is callable then the bond value will be less than the amount computed in part a If the bond price rises above the call price the company will call it Therefore bondholders will not pay as much for a callable bond The price of the bond today is the present value of the expected price in one year The bond will be called whenever the price of the bond is greater than the call price of 1150 First we need to nd the expected price in one year If interest rates increase next year the price of the bond will be the present value of the perpetual interest payments plus the interest payment made in one year so P1 100 12 100 P1 93333 This is lower than the call price so the bond will not be called If the interest rates fall next year the price of the bond will be P1 100 07 100 P1 152857 This is greater than the call price so the bond will be called The present value of the expected value of the bond price in one year is P0 4093333 601150 110 PO 96667 Intermediate If interest rates rise the price of the bonds will fall If the price of the bonds is low the company will not call them The rm would be foolish to pay the call price for something worth less than the call price In this case the bondholders will receive the coupon payment C plus the present value of the remaining payments So if interest rates rise the price of the bonds in one year will be P1 C C 013 If interest rates fall the assumption is that the bonds will be called In this case the bondholders will receive the call price plus the coupon payment C So the price of the bonds if interest rates fall will be P1 1250 C The selling price today of the bonds is the PV of the expected payoffs to the bondholders To nd the coupon rate we can set the desired issue price equal to present value of the expected value of end of year payoffs and solve for C Doing so we nd P0 1000 60C C 13 401250 C111 C 10863 So the coupon rate necessary to sell the bonds at par value will be Coupon rate 10863 1000 Coupon rate 1086 or 1086 324 The price of the bond today is the present value of the expected price in one year So the price of the bond in one year if interest rates increase will be P1 80 80 09 P1 96889 If interest rates fall the price if the bond in one year will be P1 80 80 06 P1 141333 Now we can find the price of the bond today which will be P0 3596889 65141333 108 PO 116461 If interest rates rise the price of the bonds will fall If the price of the bonds is low the company will not call them The rm would be foolish to pay the call price for something worth less than the call price In this case the bondholders will receive the coupon payment C plus the present value of the remaining payments So if interest rates rise the price of the bonds in one year will be P1 C C 09 If interest rates fall the assumption is that the bonds will be called In this case the bondholders will receive the call price plus the coupon payment C The call premium is not xed but it is the same as the coupon rate so the price of the bonds if interest rates fall will be P1 1000 C C P1 1000 2C The selling price today of the bonds is the PV of the expected payoffs to the bondholders To nd the coupon rate we can set the desired issue price equal to present value of the expected value of end of year payoffs and solve for C Doing so we nd P0 1000 35C C 09 651000 20 108 C 7763 So the coupon rate necessary to sell the bonds at par value will be Coupon rate 77633 1000 Coupon rate 0776 or 776 To the company the value of the call provision will be given by the difference between the value of an outstanding noncallable bond and the call provision So the value of a non callable bond with the same coupon rate would be Noncallable bond value 7763 006 129388 325 10 So the value of the call provision to the company is Value 65129388 7107763 108 Value 13015 The company should refund when the NPV of refunding is greater than zero so we need to nd the interest rate that results in a zero NPV The NPV of the refunding is the difference between the gain from refunding and the refunding costs The gain from refunding is the bond value times the difference in the interest rate discounted to the present value We must also consider that the interest payments are tax deductible so the aftertax gain is NPV PVGain 7 PVCost The present value of the gain will be Gain 25000000008 7 R R Since refunding would cost money today we must determine the aftertax cost of refunding which wi e Aftertax cost 7 250000000121 7 35 Aftertax cost 19500000 So setting the NPV of refunding equal to zero we find 0 7 719500000 25000000008 7 R R R 7 0742 or 742 Any interest rate below this will result in a positive NPV from refunding In this case we need to find the NPV of each alternative and choose the option with the highest NPV assuming either NPV is positive The NPV of each decision is the gain minus the cost So the NPV of refunding the 8 percent perpetual bond is BondA Gain 7 7500000008 7 07 07 Gain 7 1071428571 Assuming the call premium is tax deductible the aftertax cost of refunding this issue is Cost 7 750000000851 7 35 100000001 7 35 Cost 7 1064375000 Note that the gain can be calculated using the pretax or aftertaX cost of debt If we calculate the gain using the aftertax cost of debt we nd Aftertax gain 7 75000000081 7 35 7 071 7 35 071 7 35 Aftertax gain 107142857l 326 Thus the inclusion of the tax rate in the calculation of the gains from refunding is irrelevant The NPV of refunding this bond is NPV 71064375000 1071428571 NPV 7053571 The NPV of refunding the second bond is Band B Gain 8750000009 7 0725 0725 Gain 2112068966 Assuming the call premium is taX deductible the aftertax cost of refunding this issue is Cost 875000000951 7 35 120000001 7 35 Cost 1320312500 The NPV of refunding this bond is NPV 71320312500 2112068966 NPV 791756466 Since the NPV of refunding both bonds is positive both bond issues should be refunded 12 The price of a zero coupon bond is the PV of the par so a P0 1000104550 11071 b In one year the bond will have 24 years to maturity so the price will be P1 1000104548 12090 The interest deduction is the price of the bond at the end of the year minus the price at the beginning of the year so Year 1 interest deduction 12090 7 11071 1019 The price of the bond when it has one year left to maturity will be P24 100010452 91573 Year 24 interest deduction 1000 7 91573 8427 327 p A J 0 Previous IRS regulations required a straightline calculation of interest The total interest received by the bondholder is Total interest 1000 7 11071 88929 The annual interest deduction is simply the total interest divided by the maturity of the bond so the straightline deduction is Annual interest deduction 88929 25 3557 d The company will prefer straightline methods when allowed because the valuable interest deductions occur earlier in the life of the bond a The coupon bonds have an 8 coupon which matches the 8 required return so they will sell at par The number of bonds that must be sold is the amount needed divided by the bond price so Number of coupon bonds to sell 30000000 1000 30000 The number of zero coupon bonds to sell would be Price of zero coupon bonds 100010460 9506 Number of zero coupon bonds to sell 30000000 9506 315589 b The repayment of the coupon bond will be the par value plus the last coupon payment times the number of bonds issued So Coupon bonds repayment 300001080 32400000 The repayment of the zero coupon bond will be the par value times the number of bonds issued so Zeroes repayment 3155891000 315588822 Challenge To calculate this we need to set up an equation with the callable bond equal to a weighted average of the noncallable bonds We will invest X percent of our money in the first noncallable bond which means our investment in Bond 3 the other noncallable bond will be 1 7 X The equation is C2 C1X C317X 825 650X1217X 825 650X 12 7 12X X 068182 328 UI So we invest about 68 percent of our money in Bond 1 and about 32 percent in Bond 3 This combination of bonds should have the same value as the callable bond excluding the value of the call So P2 068182P1 031819P3 P2 068182106375 03181913496875 P2 1154730 The call value is the difference between this implied bond value and the actual bond price So the call value is Call value 1154730 710350 119730 Assuming 1000 par value the call value is 11973 In general this is not likely to happen although it can and did The reason that this bond has a negative YTM is that it is a callable US Treasury bond Market participants know this Given the high coupon rate of the bond it is extremely likely to be called which means the bondholder will not receive all the cash ows promised A better measure of the return on a callable bond is the yield to call YTC The YTC calculation is the basically the same as the YTM calculation but the number of periods is the number of periods until the call date If the YTC were calculated on this bond it would be positive 329 CHAPTER 16 CAPITAL STRUCTURE BASIC CONCEPTS Answers to Concepts Review and Critical Thinking Questions 1 Assumptions of the ModiglianiMiller theory in a world without taxes 1 Individuals can borrow at the same interest rate at which the rm borrows Since investors can purchase securities on margin an individual s effective interest rate is probably no higher than that for a firm Therefore this assumption is reasonable when applying lVllVI s theory to the real world If a rm were able to borrow at a rate lower than individuals the film s value would increase through corporate leverage As MM Proposition I states this is not the case in a world with no taxes 2 There are no taxes In the real world rms do pay taxes In the presence of corporate taxes the value of a rm is positively related to its debt level Since interest payments are deductible increasing debt reduces taxes and raises the value of the film 3 There are no costs of nancial distress In the real world costs of nancial distress can be substantial Since stockholders eventually bear these costs there are incentives for a rm to lower the amount of debt in its capital structure This topic will be discussed in more detail in later chapters False A reduction in leverage will decrease both the risk of the stock and its expected return Modigliani and Miller state that in the absence of taxes these two effects exactly cancel each other out and leave the price of the stock and the overall value of the rm unchanged False ModiglianiMiller Proposition II No Taxes states that the required return on a rm s equity is positively related to the film s debtequity ratio Rs R0 BSR0 7 RB Therefore any increase in the amount of debt in a film s capital structure will increase the required return on the rm s equity Interest payments are tax deductible where payments to shareholders dividends are not tax deductible Business risk is the equity risk arising from the nature of the film s operating activity and is directly related to the systematic risk of the film s assets Financial risk is the equity risk that is due entirely to the rm s chosen capital structure As nancial leverage or the use of debt financing increases so does nancial risk and hence the overall risk of the equity Thus Firm B could have a higher cost of equity if it uses greater leverage No it doesn t follow While it is true that the equity and debt costs are rising the key thing to remember is that the cost of debt is still less than the cost of equity Since we are using more and more debt the WACC does not necessarily rise 330 7 Because many relevant factors such as bankruptcy costs tax asymmetries and agency costs cannot easily be identified or quanti ed it is practically impossible to determine the precise debtequity ratio that maximizes the value of the film However if the film s cost of new debt suddenly becomes much more expensive it s probably true that the firm is too highly leveraged 8 It s called leverage or gearing in the UK because it magni es gains or losses 9 Homemade leverage refers to the use of borrowing on the personal level as opposed to the corporate level 10 The basic goal is to minimize the value of nonmarketed claims Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic 1 a A table outlining the income statement for the three possible states of the economy is shown below The EPS is the net income divided by the 5000 shares outstanding The last row shows the percentage change in EPS the company will experience in a recession or an expansion economy Recession Normal Expansion EBIT 7600 19000 24700 Interest 0 0 0 7 600 19 000 24700 EPS 152 380 494 AEPS 760 7 30 b If the company undergoes the proposed recapitalization it will repurchase Share price Equity Shares outstanding Share price 2250005000 Share price 45 Shares repurchased Debt issued Share price Shares repurchased 90000 45 Shares repurchased 2000 The interest payment each year under all three scenarios will be Interest payment 9000008 7200 331 The last row shows the percentage change in EPS the company will experience in a recession or an expansion economy under the proposed recapitalization Recession Normal Expansion EBIT 7600 19000 24700 Interest 7200 7200 7200 NI 1quot 400 11 800 17500 EPS 013 393 583 AEPS 79661 7 4831 A table outlining the income statement with taxes for the three possible states of the economy is shown below The share price is 45 and there are 5000 shares outstanding The last row shows the percentage change in EPS the company will experience in a recession or an expansion economy Recession Normal Expansion EBIT 7600 19000 24700 Interest 0 0 0 Taxes 2660 6650 8645 NI 4940 12 350 16055 EPS 099 247 321 AEPS 760 7 30 A table outlining the income statement with taxes for the three possible states of the economy and assuming the company undertakes the proposed capitalization is shown below The interest payment and shares repurchased are the same as in part b of Problem 1 Recession Normal Expansion EBIT 7600 19000 24700 Interest 7200 7 200 7 200 Taxes 140 4130 6125 NI 260 7670 11375 EPS 009 256 379 AEPS 79691 7 4831 Notice that the percentage change in EPS is the same both with and without taxes Since the company has a markettobook ratio of 10 the total equity of the firm is equal to the market value of equity Using the equation for ROE ROE NI225000 332 The ROE for each state of the economy under the current capital structure and no taxes is Recession Normal Expansion ROE 338 844 1098 AROE 760 7 30 The second row shows the percentage change in ROE from the normal economy If the company undertakes the proposed recapitalization the new equity value will be Equity 225000 7 90000 Equity 135000 So the ROE for each state of the economy is ROE NI135000 Recession Normal Expansion ROE 030 874 1296 AROE 79661 7 4831 If there are corporate taxes and the company maintains its current capital structure the ROE is ROE 220 549 714 AROE 40 7 30 If the company undertakes the proposed recapitalization and there are corporate taxes the ROE for each state of the economy is ROE 019 568 843 AROE 79661 7 4831 Notice that the percentage change in ROE is the same as the percentage change in EPS The percentage change in ROE is also the same with or without taxes Under Plan I the unlevered company net income is the same as EBIT with no corporate tax The EPS under this capitalization will be EPS 750000240000 shares EPS 313 Under Plan 11 the levered company EBIT will be reduced by the interest payment The interest payment is the amount of debt times the interest rate so NI 750000 7 103100000 NI 440000 333 And the EPS will be EPS 440000 160000 shares EPS 275 Plan I has the higher EPS when EBIT is 750000 b Under Plan I the net income is 1500000 and the EPS is EPS 1500000240000 shares EPS 625 Under Plan II the net income is N1 1500000 7 103100000 N1 1190000 And the EPS is EPS 1190000160000 shares EPS 744 Plan H has the higher EPS when EBIT is 1500000 0 To nd the breakeven EBIT for two different capital structures we simply set the equations for EPS equal to each other and solve for EBIT The breakeven EBIT is EBIT240000 EBIT 7 103100000160000 EBIT 930000 We can nd the price per share by dividing the amount of debt used to repurchase shares by the number of shares repurchased Doing so we nd the share price is Share price 3100000240000 7 160000 Share price 3875 per share The value of the company under the allequity plan is V 3875240000 shares 9300000 And the value of the company under the levered plan is v 3875160000 shares 3100000 debt 9300000 334 6 a The income statement for each capitalization plan is 1 II Allegg EBIT 12000 12000 12000 Interest 2000 3000 0 N1 10000 9000 12000 EPS 667 818 522 Plan 11 has the highest EPS the allequity plan has the lowest EPS b The breakeven level of EBIT occurs when the capitalization plans result in the same EPS The EPS is calculated as EPS EBIT 7 RDDShares outstanding This equation calculates the interest payment RDD and subtracts it from the EBIT which results in the net income Dividing by the shares outstanding gives us the EPS For the all equity capital structure the interest paid is zero To find the breakeven EBIT for two different capital structures we simply set the equations equal to each other and solve for EBIT The breakeven EBIT between the allequity capital structure and Plan I is EBIT2300 EBIT 7 10200001500 EBIT 5750 And the breakeven EBIT between the allequity capital structure and Plan 11 is EBIT2300 EBIT 7 10300001100 EBIT 5750 The breakeven levels of EBIT are the same because of MampM Proposition 1 0 Setting the equations for EPS from Plan I and Plan 11 equal to each other and solving for EBIT we get EBIT 7 10200001500 EBIT 7 10300001100 EBIT 5750 This breakeven level of EBIT is the same as in part b again because of MampM Proposition 1 335 The income statement for each capitalization plan with corporate income taxes is 1 II Allegg EBIT 12000 12000 12000 Interest 2000 3000 0 Taxes 4000 3600 4800 NI 60 0 5400 7 200 EPS 400 491 313 Plan II still has the highest EPS the allequity plan still has the lowest EPS We can calculate the EPS as EPS EBIT 7RDD1 7 tcShares outstanding This is similar to the equation we used before except that now we need to account for taxes Again the interest expense term is zero in the allequity capital structure So the breakeven EBIT between the allequity plan and Plan I is EBIT1 7 402300 EBIT 7 10200001 7 401500 EBIT 5750 The breakeven EBIT between the allequity plan and Plan H is EBIT1 7 402300 EBIT 7 10300001 7 401100 EBIT 5750 And the breakeven between Plan I and Plan H is EBIT 7 10200001 7 401500 EBIT 7 10300001 7 401100 EBIT 5750 The breakeven levels of EBIT do not change because the addition of taxes reduces the income of all three plans by the same percentage therefore they do not change relative to one another 336 8 To nd the value per share of the stock under each capitalization plan we can calculate the price as the value of shares repurchased divided by the number of shares repurchased The dollar value of the shares repurchased is the increase in the value of the debt used to repurchase shares or Dollar value of repurchase 30000 7 20000 10000 The number of shares repurchased is the decrease in shares outstanding or Number of shares repurchased 1500 7 1100 400 So under Plan I the value per share is P 10000400 shares P 25 per share And under Plan 11 the number of shares repurchased from the all equity plan by the 30000 in debt are Shares repurchased 2300 7 1100 1200 So the share price is P 300001200 shares P 25 per share This shows that when there are no corporate taxes the stockholder does not care about the capital structure decision of the firm This is MampM Proposition I without taxes 1 The earnings per share are EPS 375005000 shares EPS 750 So the cash ow for the company is Cash ow 750100 shares Cash ow 750 b To determine the cash ow to the shareholder we need to determine the EPS of the firm under the proposed capital structure The market value of the rm is v 7 655000 v 7 325000 Under the proposed capital structure the rm will raise new debt in the amount of D 7 040325000 D 7 130000 337 This means the number of shares repurchased will be Shares repurchased l30000 65 Shares repurchased 2000 Under the new capital structure the company will have to make an interest payment on the new debt The net income with the interest payment will be NI 37500 7 08130000 NI 27100 This means the EPS under the new capital structure will be EPS 27100 3000 shares EPS 903 Since all earnings are paid as dividends the shareholder will receive Shareholder cash ow 903100 shares Shareholder cash ow 90333 To replicate the proposed capital structure the shareholder should sell 40 percent of their shares or 40 shares and lend the proceeds at 8 percent The shareholder will have an interest cash ow of Interest cash flow 406508 Interest cash ow 20800 The shareholder will receive dividend payments on the remaining 60 shares so the dividends received will be Dividends received 90360 shares Dividends received 54200 The total cash ow for the shareholder under these assumptions will be Total cash ow 208 542 Total cash ow 750 This is the same cash ow we calculated in part a The capital structure is irrelevant because shareholders can create their own leverage or unlever the stock to create the payolf they desire regardless of the capital structure the rm actually chooses The rate of return earned will be the dividend yield The company has debt so it must make an interest payment The net income for the company is NI 95000 7 10400000 NI 55 000 338 The investor will receive dividends in proportion to the percentage of the company s shares they own The total dividends received by the shareholder will be Dividends received 5500030000 400000 Dividends received 4125 So the return the shareholder expects is R 412530000 R 1375 or 1375 To generate exactly the same cash ows in the other company the shareholder needs to match the capital structure of ABC The shareholder should sell all shares in XYZ This will net 30000 The shareholder should then borrow 30000 This will create an interest cash ow of Interest cash ow 10730000 Interest cash ow 73000 The investor should then use the proceeds of the stock sale and the loan to buy shares in ABC The investor will receive dividends in proportion to the percentage of the company s share they own The total dividends received by the shareholder will be Dividends received 9500060000 800000 Dividends received 7125 The total cash ow for the shareholder will be Total cash ow 7300 73000 Total cash ow 4125 The shareholders return in this case will be R 412530000 R 1375 or 1375 ABC is an all equity company so RE RA 95000800000 RE 1188 or 1188 To nd the cost of equity for XYZ we need to use MampM Proposition 11 so RE RA RA RDXDEXI tc RE 1188 1188 7 1011 RE 1375 or 1375 339 10 p A p A d To nd the WACC for each company we need to use the WACC equation WACC EVRE DVRD1 7 tc So for ABC the WACC is WACC 7 11188 0 10 WACC 1188 or 1188 And for XYZ the WACC is WACC 7 12 1375 1210 WACC 7 1188 or 1188 When there are no corporate taxes the cost of capital for the firm is unaffected by the capital structure this is MampM Proposition I without taxes With no taxes the value of an unlevered lm is the interest rate divided by the unlevered cost of equity so v 7 EBITWACC 43000000 7 EBIT11 EBIT 7 1143000000 EBIT 7 4730000 If there are corporate taxes the value of an unlevered rm is VU EBIT1 7 tcRU Using this relationship we can nd EBIT as 43000000 7 EBIT1 7 35 11 EBIT 7 727692308 The WACC remains at 11 percent Due to taxes EBIT for an allequity film would have to be higher for the rm to still be onth 43 million a With the information provided we can use the equation for calculating WACC to nd the cost of equity The equation for WACC is WACC EVRE DVRD1 7 tc The company has a debtequity ratio of 15 which implies the weight of debt is l525 and the weight of equity is l25 so WACC 7 12 7 125RE 1525091 7 35 RE 7 2123 or2123 340 To nd the unlevered cost of equity we need to use MampM Proposition 11 with taxes so RE R0 R0 7 RDDE1tc 2123 R0 R0 7 09151 7 35 R0 1519 or 1519 To nd the cost of equity under different capital structures we can again use MampM Proposition 11 with taxes With a debtequity ratio of 2 the cost of equity is RE R0 R0 7 RDXDEXI tc RE 1519 151970921735 RE 2324 or 2324 With a debtequity ratio of 10 the cost of equity is RE 71519 151970911735 RE 7 1921 or 1921 And with a debtequity ratio of 0 the cost of equity is RE 71519 151970901735 RE 7 R0 7 1519 or 1519 For an allequity financed company WACC R0 RE 11 or 11 To nd the cost of equity for the company with leverage we need to use MampM Proposition H with taxes so RE R0 R0 RDXDEX1 tc RE 11 11 7 0725751 7 35 RE 1187 or 1187 Using MampM Proposition 11 with taxes again we get RE R0 R0 RDXDEX1 tc RE 7 11 1170750501735 RE 7 1360 or 1360 341 15 d The WACC with 25 percent debt is WACC 7 EVRE DVRD1 71c WACC 7 751187 25071735 WACC 7 1004 or 1004 And the WACC with 50 percent debt is WACC EVRE DVRD1 7 tc WACC 7 501360 50071735 WACC 7 0908 or 908 a The value of the unlevered rm is V 7 EBIT1 71am V 7 140000173517 V 7 53529412 b The value of the levered rm is V VU tcB V 53529412 35135000 V 58254412 We can find the cost of equity using MampM Proposition II with taxes First we need to find the market value of equity which is V D E 58254412 135000 E E 44754412 Now we can nd the cost of equity which is RE R0 Re RDXDEXI t RE 17 17 7 09135000447544121 7 35 RE 1857 or 1857 Using this cost of equity the WACC for the rm after recapitalization is WACC EVRE DVRD1 7 tc WACC 44754412582544121857 13500058254412091 7 35 WACC 1562 or 1562 When there are corporate taxes the overall cost of capital for the rm declines the more highly leveraged is the rm s capital structure This is MampM PropositionI with taxes 342 16 p A 1 Since Unlevered is an allequity firm its value is equal to the market value of its outstanding shares Unlevered has 7 million shares of common stock outstanding worth 80 per share Therefore the value of Unlevered vU 700000080 560000000 ModiglianiMiller Proposition I states that in the absence of taxes the value of a levered rm equals the value of an otherwise identical unlevered firm Since Levered is identical to Unlevered in every way except its capital structure and neither firm pays taxes the value of the two firms should be equal Therefore the market value of Levered Inc should be 560 million also Since Levered has 34 million outstanding shares worth 100 per share the market value of Levered s equity is EL 3400000100 340000000 The market value of Levered s debt is 185 million The value of a levered firm equals the market value of its debt plus the market value of its equity Therefore the current market value of Levered is vL B s vL 185000000 340000000 vL 525000000 The market value of Levered s equity needs to be 375 million 35 million higher than its current market value of 340 million for W Proposition I to hold Since Levered s market value is less than Unlevered s market value Levered is relatively underpriced and an investor should buy shares of the rm s stock Intermediate To nd the value of the levered rm we rst need to find the value of an unlevered firm So the value of the unlevered firm is VU EBIT1 7 tcR0 VU 42000173515 VU 182000 Now we can nd the value of the levered rm as vL vU tcB vL 182000 3570000 vL 206500 Applying MampM Proposition I with taxes the rm has increased its value by issuing debt As long as MampM Proposition I holds that is there are no bankruptcy costs and so forth then the company should continue to increase its debtequity ratio to maximize the value of the rm 343 18 p A 0 With no debt we are nding the value of an unlevered rm so V EBIT17tc 0 V 15000173517 V 5735294 With debt we simply need to use the equation for the value of a levered rm With 50 percent debt onehalf of the rm value is debt so the value of the levered rm is V VU tcB V 5735294 3557352942 V 6738971 And with 100 percent debt the value of the firm is V VU tcB V 5735294 355735294 V 7742647 According to MampM Proposition I with taxes the increase in the value of the company will be the present value of the interest tax shield Since the loan will be repaid in equal installments we need to nd the loan interest and the interest tax shield each year The loan schedule will be Year Loan Balance Interest Tax Shield 0 140000000 1 70000000 112000 39200 2 0 56000 19600 So the increase in the value of the company is Value increase 39200l08 l9600l082 Value increase 5310014 1 Since Alpha Corporation is an allequity rm its value is equal to the market value of its outstanding shares Alpha has 10000 shares of common stock outstanding worth 20 per share so the value of Alpha Corporation is V Alpha 1000020 200000 b ModiglianiMiller Proposition I states that in the absence of taxes the value of a levered rm equals the value of an otherwise identical unlevered rm Since Beta Corporation is identical to Alpha Corporation in every way except its capital structure and neither rm pays taxes the value of the two rms should be equal So the value of Beta Corporation is 200000 as well 344 The value of a levered rm equals the market value of its debt plus the market value of its equity So the value of Beta s equity is VL B S 200000 50000 S S 150000 The investor would need to invest 20 percent of the total market value of Alpha s equity which is Amount to invest in Alpha 20200000 40000 Beta has less equity outstanding so to purchase 20 percent of Beta s equity the investor would nee Amount to invest in Beta 20150000 30000 Alpha has no interest payments so the dollar return to an investor who owns 20 percent of the company s equity would be Dollar return on Alpha investment 2055000 11000 Beta Corporation has an interest payment due on its debt in the amount of Interest on Beta s debt 1250000 6000 So the investor who owns 20 percent of the company would receive 20 percent of EBIT minus the interest expense or Dollar return on Beta investment 2055000 7 6000 9800 From part d we know the initial cost of purchasing 20 percent of Alpha Corporation s equity is 40000 but the cost to an investor of purchasing 20 percent of Beta Corporation s equity is only 30000 In order to purchase 40000 worth of Alpha s equity using only 30000 of his own money the investor must borrow 10000 to cover the difference The investor will receive the same dollar return from the Alpha investment but will pay interest on the amount borrowed so the net dollar return to the investment is Net dollar return 11000 7 1210000 9800 Notice that this amount exactly matches the dollar return to an investor who purchases 20 percent of Beta s equity The equity of Beta Corporation is riskier Beta must pay off its debt holders before its equity holders receive any of the rm s earnings If the rm does not do particularly well all of the rm s earnings may be needed to repay its debt holders and equity holders will receive nothing 345 A firm s debtequity ratio is the market value of the rm s debt divided by the market value of a rm s equity So the debtequity ratio of the company is Debtequity ratio MV of debt MV of equity Debtequity ratio 14000000 35000000 Debtequity ratio 40 We rst need to calculate the cost of equity To do this we can use the CAPM which gives us Rs RF BERM RF Rs 06 11513 7 06 Rs 1405 or 1405 We need to remember that an assumption of the ModiglianiMiller theorem is that the company debt is riskfree so we can use the Treasury bill rate as the cost of debt for the company In the absence of taxes a rm s weighted average cost of capital is equal to RWACC B B SRB S B Sle RWACC 140000004900000006 35000000490000001405 RWACC 1175 or 1175 According to ModiglianiMiller Proposition 11 with no taxes Rs R0 BSXRO RB 1405 R0 40R0 7 06 R0 1175 or 1175 This is consistent with ModiglianiMiller s proposition that in the absence of taxes the cost of capital for an allequity firm is equal to the weighted average cost of capital of an otherwise identical levered firm To purchase 5 percent of Knight s equity the investor would need Knight investment 052532000 126600 And to purchase 5 percent of Veblen without borrowing would require Veblen investment 053600000 180000 In order to compare dollar returns the initial net cost of both positions should be the same Therefore the investor will need to borrow the difference between the two amounts or Amount to borrow 180000 7 126600 53400 346 An investor who owns 5 percent of Knight s equity will be entitled to 5 percent of the firm s earnings available to common stock holders at the end of each year While Knight s expected operating income is 400000 it must pay 72000 to debt holders before distributing any of its earnings to stockholders So the amount available to this shareholder will be Cash ow from Knight to shareholder 05400000 7 72000 16400 Veblen will distribute all of its earnings to shareholders so the shareholder will receive Cash ow from Veblen to shareholder 05400000 20000 However to have the same initial cost the investor has borrowed 53400 to invest in Veblen so interest must be paid on the borrowings The net cash ow from the investment in Veblen wi e Net cash ow from Veblen investment 20000 7 0653400 16796 For the same initial cost the investment in Veblen produces a higher dollar return Both of the two strategies have the same initial cost Since the dollar return to the investment in Veblen is higher all investors will choose to invest in Veblen over Knight The process of investors purchasing Veblen s equity rather than Knight s will cause the market value of Veblen s equity to rise and or the market value of Knight s equity to fall Any differences in the dollar returns to the two strategies will be eliminated and the process will cease when the total market values of the two firms are equal Before the announcement of the stock repurchase plan the market value of the outstanding debt is 4300000 Using the debtequity ratio we can find that the value of the outstanding equity must be Debtequity ratio B S 40 4300000 S S 10750000 The value of a levered rm is equal to the sum of the market value of the rm s debt and the market value of the rm s equity so VL B S VL 4300000 10750000 VL 15050000 According to lVllVl Proposition I without taxes changes in a firm s capital structure have no effect on the overall value of the firm Therefore the value of the firm will not change after the announcement of the stock repurchase plan 347 The expected return on a firm s equity is the ratio of annual earnings to the market value of the rm s equity or return on equity Before the restructuring the company was expected to pay interest in the amount of Interest payment 104300000 430000 The return on equity which is equal to Rs will be ROE RS 1680000 7 430000 10750000 RS 1163 or 1163 According to ModiglianiMiller Proposition 11 with no taxes Rs R0 BSXRO RB 1163 R0 40120 7 10 R0 1116 or 1116 This problem can also be solved in the following way R0 Earnings before interest VU According to ModiglianiMiller Proposition 1 in a world with no taxes the value of a levered rm equals the value of an otherwiseidentical unlevered rm Since the value of the company as a levered rm is 15050000 4300000 10750000 and since the rm pays no taxes the value of the company as an unlevered firm is also 15050000 million So 120 1680000 15050000 1120 1116 or 1116 In part c we calculated the cost of an allequity rm We can use ModiglianiMiller Proposition 11 with no taxes again to nd the cost of equity for the rm with the new leverage ratio The cost of equity under the stock repurchase plan will be Rs R0 BSXRO RB RS 1116 501116 710 RS 1174 or 1174 348 The expected return on a fum s equity is the ratio of annual aftertaX earnings to the market value of the rm s equity The amount the rm must pay each year in taxes will be Taxes 401800000 720000 So the return on the unlevered equity will be R0 1800000 7 720000 9500000 R0 1137 or 1137 Notice that perpetual annual earnings of 1080000 discounted at 1137 percent yields the market value of the film s equity The company s market value balance sheet before the announcement of the debt issue is Debt 0 Assets 9500000 Equity 9500000 Total assets 9500000 Total DampE 9500000 The price per share is simply the total market value of the stock divided by the shares outstanding or Price per share 9500000 600000 1583 ModiglianiMiller Proposition I states that in a world with corporate taxes VL VU TCB When Green announces the debt issue the value of the rm will increase by the present value of the tax shield on the debt The present value of the tax shield is PVTaX Shield TCB PVTaX Shield 403000000 PVTaX Shield 1200000 Therefore the value of Green Manufacturing will increase by 1200000 as a result of the debt issue The value of Green f L mg after the 39 1s VL VU TCB VL 9500000 403000000 VL 10700000 Since the firm has not yet issued any debt Green s equity is also worth 10700000 349 Green s market value balance sheet after the announcement of the debt issue is Old assets 9500000 Debt 7 PVtaX shield 1200000 Equity 10700000 Total assets 10700000 Total DampE 10700000 The share price immediately after the announcement of the debt issue will be New share price 10700000 600000 1783 The number of shares repurchased will be the amount of the debt issue divided by the new share price or Shares repurchased 3000000 1783 16822430 The number of shares outstanding will be the current number of shares minus the number of shares repurchased or New shares outstanding 600000 7 16822430 43177570 The share price will remain the same after restructuring takes place The total market value of the outstanding equity in the company will be Market value of equity 178343177570 7700000 The marketvalue balance sheet after the restructuring is Old assets 9500000 Debt 3000000 PVtaX shield 1 200 000 Equity 7 700 000 Total assets 10 700 000 Total DampE 10 700 000 According to ModiglianiMiller Proposition 11 with corporate taxes Rs R0 BSR0 7RB17tc RS 7 1137 3000000 77000001137 e 061 e 40 RS 7 1262 or 1262 350 In a world with corporate taxes a firm s weighted average cost of capital is equal to RWACC B BS1 7tcRB S BSRs We do not have the company s debttovalue ratio or the equitytovalue ratio but we can calculate either from the debttoequity ratio With the given debtequity ratio we know the company has 25 dollars of debt for every dollar of equity Since we only need the ratio of debt tovalue and equitytovalue we can say 333 7 2525 1 7 7143 3 33 7 1 25 1 7 2857 We can now use the weighted average cost of capital equation to find the cost of equity which is 15 7714317035102857RS RS 7 3625 or 3625 We can use ModiglianiMiller Proposition 11 with corporate taxes to nd the unlevered cost of equity Doing so we find Rs R0 BSXRO RBXI tc 3625 R0 25R0 7 101 7 35 R0 2000 or 2000 We first need to find the debttovalue ratio and the equitytovalue ratio We can then use the cost of levered equity equation with taxes and nally the weighted average cost of capital equation So Ifdebt equity 75 3 33 7 75 75 1 7 4286 3 33 7 175 1 7 5714 The cost of levered equity will be Rs R0 BSXRO RBXI tc Rs 20 7520 7 101 7 35 RS 2488 or 2488 And the weighted average cost of capital will be RWACC BBSl1 tcR3 t S BtSRs RWACC 4286l 7 3510 57142488 RWACC 1 351 Ifdebt equity 150 B 33 7 150 150 1 7 6000 EBS1150 1 4000 The cost of levered equity will be Rs R0 BSXRO RBXI tc RS 20 15020 7 101 7 35 RS 2975 or 2975 And the weighted average cost of capital will be RWACC B BS1 7tcRB S BSRs RWACC 6000173510 40002975 RWACC 1580 or 1580 Challenge 26 MampM Proposition 11 states RE R0 R0 RDXDEXI tc And the equation for WACC is WACC EVRE DVRD1 7 tc Substituting the MampM Proposition H equation into the equation for WACC we get WACC EVR0 R0 7 RDDE1 7 tc DVRD1 7 tc Rearranging and reducing the equation we get WACC RoEN t ENXDEXI tc t RD1 tcDV END 3 WACC R0EV DV1 7 tc WACC 7 R0EDN 7tcDN WACC 7 R01 7 tcDN 352 27 The return on equity is net income divided by equity Net income can be expressed as NI EBIT 7 RDD1 itc So ROE is RE EBIT 7RDD1 7 tcE Now we can rearrange and substitute as follows to arrive at MampM Proposition 11 with taxes RE EBIT1 tCE RDDE1 tC RE ROVUE RDDE1 tcl RE RoVL 7 thWE RDDE1 tcl RE R003 D ithE 7 RDDE1 itcl RE R0 R0 7 RDD 31 tc MampM Proposition 11 with no taxes is RE RA RA RfXBS Note that we use the riskfree rate as the return on debt This is an important assumption of MampM Proposition 11 The CAPM to calculate the cost of equity is expressed as RE BERM Rf Rf We can rewrite the CAPM to express the return on an unlevered company as R0 BARM Rf Rf We can now substitute the CAPM for an unlevered company into MampM Proposition H Doing so and rearranging the terms we get RE BARM Rf Rf BARM Rf Rf RfBS RE BARM Rf Rf t BARM RflBS RE 14r BSBARM Rf Rf Now we set this equation equal to the CAPM equation to calculate the cost of equity and reduce BERM Rf t Rf 1 t BSBARM Rf t Rf BERM Rr 1 t BSBARM Rf BE BA1 t 33 353 29 03 O Using the equation we derived in Problem 28 BE BA1 DE The equity beta for the respective asset betas is Debtegui ratio Equity beta 0 11 0 1 1 11 1 2 5 11 5 6 20 11 20 21 The equity risk to the shareholder is composed of both business and nancial risk Even if the assets of the rm are not very risky the risk to the shareholder can still be large if the nancial leverage is high These higher levels of risk will be re ected in the shareholder s required rate of return RE which will increase with higher debtequity ratios We rst need to set the cost of capital equation equal to the cost of capital for an allequity rm so B s R R R 35 8 35 S 0 Multiplying both sides by B SS yields B B S R R R S B S S 0 We can rewrite the righthand side as B B R R R R S B S S 0 0 Moving BSRB to the righthand side and rearranging gives us B Rs R0 5030 RB 354 CHAPTER 1 7 CAPITAL STRUCTURE LIMITS TO THE USE OF DEBT Answers to Concepts Review and Critical Thinking Questions 1 Direct costs are potential legal and administrative costs These are the costs associated with the litigation arising from a liquidation or bankruptcy These costs include lawyer s fees courtroom costs and expert witness fees Indirect costs include the following 1 Impaired ability to conduct business Firms may suffer a loss of sales due to a decrease in consumer con dence and loss of reliable supplies due to a lack of con dence by suppliers 2 Incentive to take large risks When faced with projects of different risk levels managers acting in the stockholders interest have an incentive to undertake highrisk projects Imagine a rm with only one project which pays 100 in an expansion and 60 in a recession If debt payments are 60 the stockholders receive 40 100 7 60 in the expansion but nothing in the recession The bondholders receive 60 for certain Now alternatively imagine that the project pays 110 in an expansion but 50 in a recession Here the stockholders receive 50 110 7 60 in the expansion but nothing in the recession The bondholders receive only 50 in the recession because there is no more money in the firm That is the rm simply declares bankruptcy leaving the bondholders holding the bag Thus an increase in risk can bene t the stockholders The key here is that the bondholders are hurt by risk since the stockholders have limited liability If the rm declares bankruptcy the stockholders are not responsible for the bondholders shortfall 3 Incentive to underinvest If a company is near bankruptcy stockholders may well be hurt if they contribute equity to a new project even if the project has a positive NPV The reason is that some or all of the cash ows will go to the bondholders Suppose a real estate developer owns a building that is likely to go bankrupt with the bondholders receiving the property and the developer receiving nothing Should the developer take 1 million out of his own pocket to add a new wing to a building Perhaps not even if the new wing will generate cash ows with a present value greater than 1 million Since the bondholders are likely to end up with the property anyway why would the developer pay the additional 1 million and likely end up with nothing to show for it 4 Milking the property In the event of bankruptcy bondholders have the rst claim to the assets of the firm When faced with a possible bankruptcy the stockholders have strong incentives to vote for increased dividends or other distributions This will ensure them of getting some of the assets of the rm before the bondholders can lay claim to them 2 The statement is incorrect If a firm has debt it might be advantageous to stockholders for the rm to undertake risky projects even those with negative net present values This incentive results from the fact that most of the risk of failure is borne by bondholders Therefore value is transferred from the bondholders to the shareholders by undertaking risky projects even if the projects have negative NPVs This incentive is even stronger when the probability and costs of bankruptcy are high 3 The rm should issue equity in order to nance the project The taxloss carryforwards make the rm s effective tax rate zero Therefore the company will not bene t from the tax shield that debt provides Moreover since the rm already has a moderate amount of debt in its capital structure additional debt will likely increase the probability that the firm will face nancial distress or bankruptcy As long as there are bankruptcy costs the firm should issue equity in order to nance the project 355 Stockholders can undertake the following measures in order to minimize the costs of debt 1 Use protective covenants Firms can enter into agreements with the bondholders that are designed to decrease the cost of debt There are two types of protective covenants Negative covenants prohibit the company from taking actions that would expose the bondholders to potential losses An example would be prohibiting the payment of dividends in excess of earnings Positive covenants specify an action that the company agrees to take or a condition the company must abide by An example would be agreeing to maintain its working capital at a minimum level 2 Repurchase debt A rm can eliminate the costs of bankruptcy by eliminating debt from its capital structure 3 Consolidate debt If a firm decreases the number of debt holders it may be able to decrease the direct costs of bankruptcy should the firm become insolvent Modigliani and lVIiller s theory with corporate taxes indicates that since there is a positive tax advantage of debt the lm should maximize the amount of debt in its capital structure In reality however no firm adopts an alldebt nancing strategy lVllVI s theory ignores both the nancial distress and agency costs of debt The marginal costs of debt continue to increase with the amount of debt in the lm s capital structure so that at some point the marginal costs of additional debt will outweigh its marginal tax bene ts Therefore there is an optimal level of debt for every film at the point where the marginal tax benefits of the debt equal the marginal increase in nancial distress and agency costs There are two major sources of the agency costs of equity 1 Shirking Managers with small equity holdings have a tendency to reduce their work effort thereby hurting both the debt holders and outside equity holders 2 Perquisites Since management receives all the bene ts of increased perquisites but only shoulder a fraction of the cost managers have an incentive to overspend on luxury items at the expense of debt holders and outside equity holders The more capital intensive industries such as air transport television broadcasting stations and hotels tend to use greater nancial leverage Also industries with less predictable future earnings such as computers or drugs tend to use less nancial leverage Such industries also have a higher concentration of growth and startup firms Overall the general tendency is for firms with identi able tangible assets and relatively more predictable future earnings to use more debt nancing These are typically the firms with the greatest need for external nancing and the greatest likelihood of benefiting from the interest tax shelter One answer is that the right to le for bankruptcy is a valuable asset and the nancial manager acts in shareholders best interest by managing this asset in ways that maximize its value To the extent that a bankruptcy ling prevents a race to the courthouse steps it would seem to be a reasonable use of the process As in the previous question it could be argued that using bankruptcy laws as a sword may simply be the best use of the asset Creditors are aware at the time a loan is made of the possibility of bankruptcy and the interest charged incorporates it 356 10 One side is that Continental was going to go bankrupt because its costs made it uncompetitive The bankruptcy ling enabled Continental to restructure and keep ying The other side is that Continental abused the bankruptcy code Rather than renegotiate labor agreements Continental simply abrogated them to the detriment of its employees In this and the last several questions an important thing to keep in mind is that the bankruptcy code is a creation of law not economics A strong argument can always be made that making the best use of the bankruptcy code is no different from for example minimizing taxes by making best use of the tax code Indeed a strong case can be made that it is the nancial manager s duty to do so As the case of Continental illustrates the code can be changed if socially undesirable outcomes are a problem Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic 1 a Using MampM Proposition I with taxes the value of a levered firm is VL EBIT1 7 tcR0 tcB VL 850000173514 351900000 VL 461142857 b The CFO may be correct The value calculated in part a does not include the costs of any non marketed claims such as bankruptcy or agency costs 2 a Debt issue The company needs a cash infusion of 12 million If the company issues debt the annual interest payments will be Interest 120000008 96000 The cash ow to the owner will be the EBIT minus the interest payments or 40 hour week cash ow 400000 7 96000 304000 50 hour week cash ow 500000 7 96000 404000 Equity issue If the company issues equity the company value will increase by the amount of the issue So the current owner s equity interest in the company will decrease to Tom s ownership percentage 2500000 2500000 1200000 68 357 So Tom s cash ow under an equity issue will be 68 percent of EBIT or 40 hour week cash ow 68400000 270270 50 hour week cash ow 68500000 337838 Tom will work harder under the debt issue since his cash ows will be higher Tom will gain more under this form of financing since the payments to bondholders are xed Under an equity issue new investors share proportionally in his hard work which will reduce his propensity for this additional work The direct cost of both issues is the payments made to new investors The indirect costs to the debt issue include potential bankruptcy and nancial distress costs The indirect costs of an equity issue include shirking and perquisites The interest payments each year will be Interest payment 0870000 5600 This is exactly equal to the EBIT so no cash is available for shareholders Under this scenario the value of equity will be zero since shareholders will never receive a payment Since the market value of the company s debt is 70000 and there is no probability of default the total value of the company is the market value of debt This implies the debt to value ratio is 1 one At a 3 percent growth rate the earnings next year will be Earnings next year 5600l03 5768 So the cash available for shareholders is Payment to shareholders 5768 7 5600 l68 Since there is no risk the required return for shareholders is the same as the required return on the company s debt The payments to stockholders will increase at the growth rate of three percent a growing perpetuity so the value of these payments today is Value of equity l68 08 7 03 336000 And the debt to value ratio now is DebtValue ratio 70000 70000 3360 0954 358 c At a 7 percent growth rate the earnings next year will be Earnings next year 5600l07 599200 So the cash available for shareholders is Payment to shareholders 5992 7 5600 392 Since there is no risk the required return for shareholders is the same as the required return on the company s debt The payments to stockholders will increase at the growth rate of seven percent a growing perpetuity so the value of these payments today is Value of equity 392 08 7 07 39200 And the debt to value ratio now is DebtValue ratio 70000 70000 39200 0641 According to MampM Proposition I with taxes the value of the levered rm is VL VU tcB VL 14500000 355000000 VL 16250000 We can also calculate the market value of the rm by adding the market value of the debt and equity Using this procedure the total market value of the rm is VBS V10mum0300mmw3 V 15500000 With no nonmarketed claims such as bankruptcy costs we would expect the two values to be the same The difference is the value of the nonmarketed claims which are WWW 1150Q0001Q250 0047Vamp Vamp75Q000 The president may be correct but he may also be incorrect It is true the interest tax shield is valuable and adding debt can possibly increase the value of the company However if the company s debt is increased beyond some level the value of the interest tax shield becomes less than the additional costs from financial distress 359 Intermediate The total value of a firm s equity is the discounted expected cash ow to the rm s stockholders If the expansion continues each firm will generate earnings before interest and taxes of 24 million Ifthere is a recession each rm will generate earnings before interest and taxes of only 900000 Since Steinberg owes its bondholders 800000 at the end of the year its stockholders will receive 16 million 2400000 7 800000 if the expansion continues If there is a recession its stockholders will only receive 100000 900000 7 800000 So assuming a discount rate of 15 percent the market value of Steinberg s equity is ssmnberg 7 801600000 20100000 115 7 1130435 Steinberg s bondholders will receive 800000 whether there is a recession or a continuation of the expansion So the market value of Steinberg s debt is Bsmnberg 80800000 20800000 115 695652 Since Dietrich owes its bondholders 11 million at the end of the year its stockholders will receive 13 million 24 million 7 11 million if the expansion continues If there is a recession its stockholders will receive nothing since the rm s bondholders have a more senior claim on all 800000 of the rm s earnings So the market value of Dietrich s equity is SDiemch 801300000 200 115 904348 Dietrich s bondholders will receive 11 million if the expansion continues and 900000 if there is a recession So the market value of Dietrich s debt is 13mm 7 801100000 20900000 115 7 921739 The value of company is the sum of the value of the rm s debt and equity So the value of Steinberg is VSteinberg B S VSteinberg 1130435 695652 Vsmnberg 1826087 And value of Dietrich is VDietrich B S VDietrich 904348 921739 VDietrich 1826087 You should disagree with the CEO s statement The risk of bankruptcy per se does not affect a rm s value It is the actual costs of bankruptcy that decrease the value of a firm Note that this problem assumes that there are no bankruptcy costs 360 The expected value of each project is the sum of the probability of each state of the economy times the value in that state of the economy Since this is the only project for the company the company value will be the same as the project value so Lowvolatility project value 502500 502700 Lowvolatility project value 2600 Highvolatility project value 502100 502800 Highvolatility project value 2450 The lowvolatility project maximizes the expected value of the rm The value of the equity is the residual value of the company after the bondholders are paid off If the lowvolatility project is undertaken the rm s equity will be worth 0 if the economy is bad and 200 if the economy is good Since each of these two scenarios is equally probable the expected value of the rm s equity is Expected value of equity with lowvolatility project 500 50200 Expected value of equity with lowvolatility project 100 And the value of the company if the highvolatility project is undertaken will be Expected value of equity with highvolatility project 500 50300 Expected value of equity with highvolatility project 150 Riskneutral investors prefer the strategy with the highest expected value Thus the company s stockholders prefer the highvolatility project since it maximizes the expected value of the company s equity In order to make stockholders indifferent between the lowvolatility project and the high volatility project the bondholders will need to raise their required debt payment so that the expected value of equity if the highvolatility project is undertaken is equal to the expected value of equity if the lowvolatility project is undertaken As shown in part a the expected value of equity if the lowvolatility project is undertaken is 2600 If the highvolatility project is undertaken the value of the firm will be 2100 if the economy is bad and 2800 if the economy is good If the economy is bad the entire 2100 will go to the bondholders and stockholders will receive nothing If the economy is good stockholders will receive the difference between 2800 the total value of the firm and the required debt payment Let X be the debt payment that bondholders will require if the highvolatility project is undertaken In order for stockholders to be indifferent between the two projects the expected value of equity if the highvolatility project is undertaken must be equal to 2100 so Expected value of equity 100 500 502800 7 X X 2600 361 The expected payoff to bondholders is the face value of debt or the value of the company whichever is less Since the value of the company in a recession is 85 million and the required debt payment in one year is 120 million bondholders will receive the lesser amount or 85 million The promised return on debt is Promised return Face value of debt Market value of debt 7 1 Promised return 120000000 94000000 7 1 Promised return 2766 or 2766 In part a we determined bondholders will receive 85 million in a recession In a boom the bondholders will receive the entire 120 million promised payment since the market value of the company is greater than the payment So the expected value of debt is Expected payment to bondholders 60120000000 4085000000 Expected payment to bondholders 106000000 So the expected return on debt is Expected return Expected value of debt Market value of debt 7 1 Expected return 106000000 94000000 7 1 Expected return 1277 or 1277 Challenge In their no tax model MM assume that to 1 3 and C13 are all zero Under these assumptions VL VU signifying that the capital structure of a firm has no effect on its value There is no optimal debtequity ratio In their model with corporate taxes MlVI assume that re gt 0 and both IB and CB are equal to zero Under these assumptions VL VU tcB implying that raising the amount of debt in a rm s capital structure will increase the overall value of the firm This model implies that the debtequity ratio of every firm should be infmite If the costs of financial distress are zero the value of a levered firm equals VL VU1 1 tc1 IB XB Therefore the change in the value of this allequity firm that issues debt and uses the proceeds to repurchase equity is Change invalue 1717tc17tB X B 1 Change in value 7 17 1 7 34 1 7 20 x 1000000 Change in value 175000 362 If the costs of nancial distress are zero the value of a levered firm equals VL VU t 11tc1IB XB Therefore the change in the value of an allequity firm that issues 1 of perpetual debt instead of 1 of perpetual equity is Change invalue 1 7 1 7tc 1 tBH X 1 If the firm is not able to bene t from interest deductions the rm s taxable income will remain the same regardless of the amount of debt in its capital structure and no tax shield will be created by issuing debt Therefore the rm will receive no tax bene t as a result of issuing debt in place of equity In other words the re ective corporate tax rate when we consider the change in the value of the rm is zero Debt will have no effect on the value of the firm since interest payments will not be tax deductible Since this firm is able to deduct interest payments the change in value is Change invalue 171701720 gtlt 1 Change invalue 7025 The value of the rm will decrease by 025 if it adds 1 of perpetual debt rather than 1 of equity If the company decides to retire all of its debt it will become an unlevered rm The value of an allequity rm is the present value of the aftertax cash ow to equity holders which will be VU EBIT17 tc R0 VU 13000001 7 35 20 VU 4225000 Since there are no bankruptcy costs the value of the company as a levered rm is VL VU t 11 rid1 4131 X3 VL 4225000 1 7 1 7 35 1 7 25 X 2500000 VL 455833333 The bankruptcy costs would not affect the value of the unlevered firm since it could never be forced into bankruptcy So the value of the levered firm with bankruptcy would be VL 7 VU 1717tc17tB XB7CB VL 7 4225000 1 7 1 7 35 1 7 25 x 2500000 7 400000 VL 7 415833333 The company should choose the allequity plan with this bankruptcy cost 363 CHAPTER 18 VALUATION AND CAPITAL BUDGETING FOR THE LEVERED FIRM Answers to Concepts Review and Critical Thinking Questions 1 APV is equal to the NPV of the project ie the value of the project for an unlevered firm plus the NPV of nancing side effects The WACC is based on a target debt level while the APV is based on the amount of debt F TE uses levered cash ow and other methods use unlevered cash ow The WACC method does not explicitly include the interest cash ows but it does implicitly include the interest cost in the WACC If he insists that the interest payments are explicitly shown you should use the FTE method You can estimate the unlevered beta from a levered beta The unlevered beta is the beta of the assets of the firm as such it is a measure of the business risk Note that the unlevered beta will always be lower than the levered beta assuming the betas are positive The difference is due to the leverage of the company Thus the second risk factor measured by a levered beta is the nancial risk of the company Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic a The maximum price that the company should be willing to pay for the eet of cars with all equity funding is the price that makes the NPV of the transaction equal to zero The NPV equation for the project is NPV iPurchase Price PV1 7 tc EBTD PVDepreciation Tax Shield If we let P equal the purchase price of the eet then the NPV is NPV 7P 1 7 35quot 140000PIFA135 35P5PVIFA1375 364 Setting the NPV equal to zero and solving for the purchase price we nd 0 7P 1 7 35140000PVIFA1375 35P5PVIFA1375 P 32006804 P0355PVIFA1375 P 32006804 2462P 7538P 32006804 P 42460954 The adjusted present value APV of a project equals the net present value of the project if it were funded completely by equity plus the net present value of any financing side effects In this case the NPV of nancing side effects equals the aftertax present value of the cash ows resulting from the rm s debt so APV NPVAllEquity NPVFinancing Side Effects So the NPV of each part of the APV equation is NPV AllEquity NPV 7Purchase Price PVl 7 tc EBTD PVDepreciation Tax Shield The company paid 395000 for the eet of cars Because this eet will be fully depreciated over ve years using the straightline method annual depreciation expense equals Depreciation 395000 5 Depreciation 79000 So the NPV of an allequity project is NPV 7395000 1 7 035l40000PVIFA135 03579000PVIFA135 NPV 2231949 NPV Financing Side Effects 1 The net present value of nancing side effects equals the aftertax present value of cash ows resulting from the rm s debt so NPV Proceeds 7 Aftertax PVInterest Payments 7 PVPrincipal Payments Given a known level of debt debt cash ows should be discounted at the pretaX cost of debt RB So the NPV of the financing side effects are NPV 7 260000 7 1 7 035008260000PVIFA85 7 2600001085 NPV 7 2906693 So the APV of the project is APV NPVAllEquity NPVFinancing Side Effects 93 APV 7 2231949 29066 APV 7 5138642 365 The adjusted present value APV of a project equals the net present value of the project if it were funded completely by equity plus the net present value of any nancing side effects In this case the NPV of nancing side effects equals the aftertax present value of the cash ows resulting from the rm s debt so APV NPVAllEquity NPVFinancing Side Effects So the NPV of each part of the APV equation is NPV1 AllEquity NPV 7Purchase Price PV1 7 tc EBTD PVDepreciation Tax Shield Since the initial investment of 19 million will be fully depreciated over four years using the straightline method annual depreciation expense is Depreciation 19000004 Depreciation 475000 NPV 7 71900000 17 030685000PVIFA9 5 030475000Pv1FA134 NPV Allequity 7 7 4987884 NPV Financing Side Effects 1 The net present value of nancing side effects equals the aftertax present value of cash ows resulting from the film s debt So the NPV of the nancing side effects are NPV ProceedsNet of otation 7 Aftertax PVInterest Payments 7 PVPrincipal Payments PVFlotation Cost Tax Shield Given a known level of debt debt cash ows should be discounted at the pretax cost of debt RB Since the otation costs will be amortized over the life of the loan the annual otation costs that will be expensed each year are Annual otation expense 280004 Annual otation expense 7000 NPV 1900000 7 28000 7 1 7 03000951900000PVIFA9 54 7 190000010954 0307000 PVIFAg 54 NPV 15225206 So the APV of the project is APV NPVAllEquity NPVFinancing Side Effects APV 74987884 15225206 APV 10237323 366 In order to value a rm s equity using the owtoequity approach discount the cash ows available to equity holders at the cost of the rm s levered equity The cash ows to equity holders will be the film s net income Remembering that the company has three stores we nd Sales 3600000 COGS 1530000 G amp A costs 1020000 Interest 102000 EBT 948000 Taxes 379200 NI 568800 Since this cash ow will remain the same forever the present value of cash ows available to the rm s equity holders is a perpetuity We can discount at the levered cost of equity so the value of the company s equity is PVFlowtoequity 568800 019 PVFlowtoequity 299368421 The value of a rm is equal to the sum of the market values of its debt and equity or VL B S We calculated the value of the company s equity in part a so now we need to calculate the value of debt The company has a debttoequity ratio of 040 which can be written algebraically as B S 040 We can substitute the value of equity and solve for the value of debt doing so we find B 299368421 040 B 119747368 So the value of the company is v 299368421 119747368 v 419115789 In order to determine the cost of the rm s debt we need to nd the yield to maturity on its current bonds With semiannual coupon payments the yield to maturity in the company s bonds is 975 40PVIFAR40 1000PVIFR740 R 0413 or 413 367 Since the coupon payments are semiannual the YTM on the bonds is YTM 413X 2 YTM 826 We can use the Capital Asset Pricing Model to nd the return on unlevered equity According to the Capital Asset Pricing Model R0 RF UnleveredRM RF R0 7 511127 5 R0 1270 Now we can find the cost of levered equity According to ModiglianiMiller Proposition II with corporate taxes Rs R0 BSR0 7RB17tc RS 1270 401270 7 08261 7 34 Rs 1387 or 1387 In a world with corporate taxes a firm s weighted average cost of capital is equal to RWACC B B S1 tCRB S B Sle The problem does not provide either the debtvalue ratio or equityvalue ratio However the rm s debtequity ratio of is B S 040 Solving for B B 04S Substituting this in the debtvalue ratio we get BV 4S 4S 7 S BV 4 14 BV 29 And the equityvalue ratio is one minus the debtvalue ratio or SV 1729 SV71 So the WACC for the company is RWACC 2917340826 711387 RWACC 1147 or 1147 368 The equity beta of a rm nanced entirely by equity is equal to its unlevered beta Since each rm has an unlevered beta of 125 we can nd the equity beta for each Doing so we nd North Pole Equity 1 1 i thB S113Un1evered 3mm 7 1 173529000003800000125 Equity 1 South Pole Equity 1 1 i thB S113Un1evered 3mm 7 1 173538000002900000125 Equity We can use the Capital Asset Pricing Model to nd the required return on each rm s equity Doing so we nd North Pole Rs RF BEquityRM RF Rs 530 1871240 7 530 Rs 1858 South Pole Rs RF BEquityRM RF Rs 530 2311240 7 530 Rs 2173 If otation costs are not taken into account the net present value of a loan equals NPVLcan Gross Proceeds 7 A ertax present value of interest and principal payments NPVLcan 5350000 7 0853500001 7 40PVIFA8710 7 535000010810 NPVLcan 114876594 The otation costs of the loan will be Flotation costs 53500000125 Flotation costs 66875 So the annual otation expense will be Annual otation expense 66875 10 Annual otation expense 668750 369 If otation costs are taken into account the net present value of a loan equals NPVLcan Proceeds net of otation costs 7 A ertax present value of interest and principal a ments Present value of the otation cost tax shield NPVLoan 5350000 7 66875 7 0853500001 7 40PVIFA8710 7 535000010810 66875040PVIFA8710 NPVLoan 109984040 First we need to nd the aftertax value of the revenues minus expenses The a ertax value is A ertax revenue 38000001 7 40 A ertax revenue 2280000 Next we need to nd the depreciation tax shield The depreciation tax shield each year is Depreciation tax shield Depreciationtc Depreciation tax shield 11400000 640 Depreciation tax shield 760000 Now we can find the NPV of the project which is NPV Initial cost PV of depreciation tax shield PV of aftertax revenue To nd the present value of the depreciation tax shield we should discount at the riskfree rate and we need to discount the a ertax revenues at the cost of equity so NPV 7 711400000 760000PVIFA676 2280000Pv1FA146 NPV 7 120332843 Whether the company issues stock or issues equity to nance the project is irrelevant The company s optimal capital structure determines the WACC In a world with corporate taxes a rm s weighted average cost of capital equals RWACC B B Sl1 tcRB S B le RWACC 7 801734072 201140 RWACC 0608 or 608 Now we can use the weighted average cost of capital to discount NEC s unlevered cash ows Doing so we nd the NPV ofthe project is NPV 7 740000000 2600000 00608 NPV 7 275190739 a The company has a capital structure with three parts longterm debt shortterm debt and equity Since interest payments on both longterm and shortterm debt are taxdeductible multiply the pretax costs by 1 7 tc to determine the aftertax costs to be used in the weighted average cost of capital calculation The WACC using the book value weights is RWACC WSTDXRSTDX1 itc WLTDXRLTDX1 tc WEquity REquity RWACC 3 190351 7 35 10 190681 735 6 19145 RWACC 00726 or 726 370 Using the market value weights the company s WACC is RWACC WSTDXRSTDXI itc WLTDXRLTDX1 tc WEquityXREquity RWAcc 3 400351 7 35 11 400681 7 35 26 40 145 RWACC 01081 or 1081 Using the target debtequity ratio the target debtvalue ratio for the company is BS 060 B 06S Substituting this in the debtvalue ratio we get BV 6S 6S S BV 6 16 BV 375 And the equityvalue ratio is one minus the debtvalue ratio or SV 17 375 SN 625 We can use the ratio of shortterm debt to longterm debt in a similar manner to nd the short term debt to total debt and longterm debt to total debt Using the shortterm debt to longterm debt ratio we get STDLTD 020 STD 02LTD Substituting this in the shortterm debt to total debt ratio we get STDB 2LTD 2LTD LTD STDB 2 12 STDB 167 And the longterm debt to total debt ratio is one minus the shortterm debt to total debt ratio or LTDB 17 167 LTDB 833 Now we can nd the shortterm debt to value ratio and longterm debt to value ratio by multiplying the respective ratio by the debtvalue ratio So STDN STD SXBN STDN 167375 STDN 063 371 And the longterm debt to value ratio is LTDv 7 LTDBBN LTDv 7 833375 LTDv 7 313 So using the target capital structure weights the company s WACC is RWACC WS39139DRS39139D1 itc WLTDXRLTDX1 tc WEquityXREquity RWACC 060351 7 35 310681 7 35 625 145 RWACC 01059 or 1059 d The differences in the WACCs are due to the different weighting schemes The company s WACC will most closely resemble the WACC calculated using target weights since future projects will be nanced at the target ratio Therefore the WACC computed with target weights should be used for project evaluation Intermediate 10 The adjusted present value of a project equals the net present value of the project under allequity nancing plus the net present value of any nancing side effects In the joint venture s case the NPV of nancing side effects equals the aftertax present value of cash ows resulting from the rms debt So the APV is APV NPVAllEquity NPVFinancing Side Effects The NPV for an allequity rm is NPV AllEquity NPV 7Initial Investment PV1 7 tcEBITD PVDepreciation Tax Shield Since the initial investment will be fully depreciated over ve years using the straightline method annual depreciation expense is Annual depreciation 30000000 5 Annual depreciation 6000000 NPV 7 730000000 17 0353800000Pv1FA51320 0356000000PVIFA5713720 NPV 526267795 NPV Financing Side Effects 1 The NPV of nancing side effects equals the aftertax present value of cash ows resulting from the rm s debt The coupon rate on the debt is relevant to determine the interest payments but the resulting cash ows should still be discounted at the pretax cost of debt So the NPV of the nancing effects is NPV Proceeds 7 Aftertax PVInterest Payments 7 PVPrincipal Repayments NPV 7 18000000 7 1 7 03500518000000PVIFA8 515 7 18000000108515 NPV 7 784750356 372 p A N So the APV of the project is APV NPVAllEquity NPVFinancing Side Effects APV 7526267795 784750356 APV 258482561 If the company had to issue debt under the terms it would normally receive the interest rate on the debt would increase to the company s normal cost of debt The NPV of an allequity project would remain unchanged but the NPV of the nancing side effects would change The NPV of the nancing side effects would be NPV Proceeds 7 Aftertax PVInterest Payments 7 PVPrincipal Repayments NPV 18000000 7 1 7 035008518000000PVIFA3515 18000000108515 NPV 444691869 Using the NPV of an allequity project from the previous problem the new APV of the project would be APV NPVAllEquity NPVFinancing Side Effects APV 7526267795 444691869 APV 781575927 The gain to the company from issuing subsidized debt is the difference between the two APVs so Gain from subsidized debt 2584825617781575927 Gain from subsidized debt 340058488 Most of the value of the project is in the form of the subsidized interest rate on the debt issue The adjusted present value of a project equals the net present value of the project under allequity nancing plus the net present value of any financing side effects First we need to calculate the unlevered cost of equity According to ModiglianilVIiller Proposition 11 with corporate taxes Rs R0 BSR0 REX1 itc 16 R0 050R0 7 00917 040 R0 01438 or 1438 Now we can find the NPV of an allequity project which is NPV PVUnlevered Cash Flows NPV 721000000 69000001 1438 11000000114382 9500000114383 NPV 721263889 Next we need to nd the net present value of nancing side effects This is equal the aftertaX present value of cash ows resulting from the rm s debt So NPV Proceeds 7 Aftertax PVInterest Payments 7 PVPrincipal Payments 373 Each year an equal principal payment will be made which will reduce the interest accrued during the year Given a known level of debt debt cash ows should be discounted at the pretaX cost of debt so the NPV of the nancing effects are NPV 7000000 7 1 7 40097000000 109 7 233333333109 7 1 7 40094666666671092 7 2333333331092 7 1 7 40092333333331093 7 2333333331093 NPV 43745831 So the APV ofproject is APV NPVAllequity NPVFinancing side effects APV 721263889 43745831 APV 22481942 a To calculate the NPV of the project we rst need to nd the company s WACC In a world with corporate taxes a rm s weighted average cost of capital equals RWACC 13B t S1 tCRB S B t Sle The market value of the company s equity is lVIarket value of equity 600000020 Market value of equity 120000000 So the debtvalue ratio and equityvalue ratio are Debtvalue 35000000 35000000 120000000 Debtvalue 225 8 Equityvalue 120000000 35000000 120000000 Equityvalue 7742 Since the CEO believes its current capital structure is optimal these values can be used as the target weights in the rm s weighted average cost of capital calculation The yield to maturity of the company s debt is its pretax cost of debt To nd the company s cost of equity we need to calculate the stock beta The stock beta can be calculated as B 633M 64 3 036 202 B 090 Now we can use the Capital Asset Pricing Model to determine the cost of equity The Capital Asset Pricing Model is Rs RF 3RM RF RS 6 090750 RS 1275 374 Now we can calculate the company s WACC which is RWACC B B t Sl1 tcRB t S B t Sle RWACC 7 225817 3508 77421275 RWACC 7 1105 or 1105 Finally we can use the WACC to discount the unlevered cash ows which gives us an NPV of NPV 7 745000000 13500000Pv1FA11 05m NPV 7 483797859 The weighted average cost of capital used in part a will not change if the rm chooses to fund the project entirely with debt The weighted average cost of capital is based on optimal capital structure weights Since the current capital structure is optimal alldebt funding for the project simply implies that the rm will have to use more equity in the future to bring the capital structure back towards the target Challenge The company is currently an allequity rm so the value as an allequity rm equals the present value of aftertaX cash ows discounted at the cost of the rm s unlevered cost of equity So the current value of the company is VU PretaX eamingsl 7 tc R0 vU 280000001 7 35 20 vU 7 91000000 The price per share is the total value of the company divided by the shares outstanding or Price per share 91000000 1500000 Price per share 6067 The adjusted present value of a rm equals its value under allequity nancing plus the net present value of any nancing side effects In this case the NPV of nancing side effects equals the aftertax present value of cash ows resulting from the rm s debt Given a known level of debt debt cash ows can be discounted at the pretax cost of debt so the NPV of the nancing effects are NPV Proceeds 7 Aftertax PVInterest Payments NPV 35000000 7 1 7 350935000000 09 NPV 12250000 So the value of the company after the recapitalization using the APV approach is v 7 91000000 12250000 v 7 103250000 375 Since the company has not yet issued the debt this is also the value of equity after the announcement So the new price per share will be New share price 103250000 1500000 New share price 6883 The company will use the entire proceeds to repurchase equity Using the share price we calculated in part b the number of shares repurchased will be Shares repurchased 35000000 6883 Shares repurchased 508475 And the new number of shares outstanding will be New shares outstanding 1500000 7 508475 New shares outstanding 991525 The value of the company increased but part of that increase will be funded by the new debt The value of equity after recapitalization is the total value of the company minus the value of debt or New value of equity 103250000 7 35000000 New value of equity 68250000 So the price per share of the company after recapitalization will be New share price 68250000 991525 New share price 6883 The price per share is unchanged In order to value a rm s equity using the owtoequity approach we must discount the cash ows available to equity holders at the cost of the rm s levered equity According to ModiglianiMiller Proposition 11 with corporate taxes the required return of levered equity is Rs R0 BSXRO RBXI itc RS 20 35000000 6825000020 7 091 7 35 RS 2367 or 2367 After the recapitalization the net income of the company will be EBIT 28000000 Interest 3150000 EBT 24850000 Taxes 8697500 Net income 16152500 376 The rm pays all of its earnings as dividends so the entire net income is available to shareholders Using the owtoequity approach the value of the equity is S Cash ows available to equity holders RS S 16152500 2367 S 68250000 If the company were financed entirely by equity the value of the rm would be equal to the present value of its unlevered aftertax earnings discounted at its unlevered cost of capital First we need to find the company s unlevered cash ows which are Sales 28900000 Variable costs 17340000 EBT 11560000 Tax 4624000 Net income 6936000 So the value of the unlevered company is vU 6936000 17 vU 40800000 According to ModiglianiMiller Proposition 11 with corporate taxes the value of levered equity is Rs R0 13SRo RBXI tc Rs 17 35177 091 7 40 Rs 1868 or 1868 In a world with corporate taxes a firm s weighted average cost of capital equals RWACC B B t S1 tCRB S B t SRs So we need the debtvalue and equityvalue ratios for the company The debtequity ratio for the company is BS 035 B 035S Substituting this in the debtvalue ratio we get BV 35S 35S S BV 35 135 BV 26 377 And the equityvalue ratio is one minus the debtvalue ratio or SV1726 SV74 So using the capital structure weights the company s WACC is RWAcc B B S17tcRB s B SRS RWACC 261 7 4009 741868 RWACC 1524 or 1524 We can use the weighted average cost of capital to discount the rm s unlevered aftertax earnings to value the company Doing so we nd vL 6936000 1524 vL 4552066116 Now we can use the debtvalue ratio and equityvalue ratio to find the value of debt and equity which are B VL Debtvalue B 455206611626 B 1180165289 S VLEquityvalue S 455206611674 S 3371900826 In order to value a rm s equity using the owtoequity approach we can discount the cash ows available to equity holders at the cost of the rm s levered equity First we need to calculate the levered cash ows available to shareholders which are Sales 28900000 Variable costs 17340000 EBIT 11560000 Interest 1062149 EBT 10497851 Tax 4199140 Net income 6298711 So the value of equity with the owtoequity method is S Cash ows available to equity holders RS 3 6298711 1868 s 3371900826 378 Since the company is currently an allequity rm its value equals the present value of its unlevered aftertax earnings discounted at its unlevered cost of capital The cash ows to shareholders for the unlevered rm are EBIT 83000 Tax 33200 Net income 49800 So the value of the company is VU 49800 15 VU 332000 b The adjusted present value of a firm equals its value under allequity nancing plus the net present value of any nancing side effects In this case the NPV of nancing side effects equals the aftertax present value of cash ows resulting from debt Given a known level of debt debt cash ows should be discounted at the pre taX cost of debt so NPV Proceeds 7 Aftertax PVInterest payments NPV 195000 7 1 7 4009195000 009 NPV 78000 So using the APV method the value of the company is APV VU NPVFinancing side effects APV 332000 78000 APV 410000 The value of the debt is given so the value of equity is the value of the company minus the value of the debt or S V7B S 410000 7 195000 S 215000 According to ModiglianilVIiller Proposition 11 with corporate taxes the required return of levered equity is Rs R0 BSXRO RBXI itc RS 715 195000 215000 15 7 091 7 40 RS 7 1827 or 1827 379 d In order to value a f1rm s equity using the owtoequity approach we can discount the cash ows available to equity holders at the cost of the rm s levered equity First we need to calculate the levered cash ows available to shareholders which are EBIT 83000 Interest 17550 EBT 65450 Tax 26180 Net income 39270 So the value of equity with the owtoequity method is S Cash ows available to equity holders RS S 39270 1827 S 215000 17 Since the company is not publicly traded we need to use the industry numbers to calculate the industry levered return on equity We can then find the industry unlevered return on equity and re lever the industry return on equity to account for the different use of leverage So using the CAPM to calculate the industry levered return on equity we nd RS RF 3MRP RS 5 127 RS 1340 Next to nd the average cost of unlevered equity in the holiday gift industry we can use Modigliani lVIiller Proposition II with corporate taxes so Rs R0 BSXRO RBXI itc 1340 R0 35R0 7 051 7 40 R0 1194 or 1194 Now we can use the ModiglianilVIiller Proposition H with corporate taxes to relever the return on equity to account for this company s debtequity ratio Doing so we nd Rs R0 BSXRO RBXI itc Rs 1194 401194 7 051 7 40 Rs 1361 or 1361 Since the project is nanced at the rm s target debtequity ratio it must be discounted at the company s weighted average cost of capital In a world with corporate taxes a firm s weighted average cost of capital equals RWACC B B S1 tCRB S B SRs 380 So we need the debtvalue and equityvalue ratios for the company The debtequity ratio for the company is BS 040 B 040S Substituting this in the debtvalue ratio we get BV 40S 40S S BV 40 140 BV 29 And the equityvalue ratio is one minus the debtvalue ratio or SV 1729 SV71 So using the capital structure weights the company s WACC is RWACC B B S17tcRB S B SRs RWACC 29174005 711361 RWACC 1058 or 1058 Now we need the project s cash ows The cash ows increase for the rst ve years before leveling off into perpetuity So the cash ows from the project for the next six years are Year 1 cash ow 8000000 Year 2 cash ow 8400000 Year 3 cash ow 8820000 Year 4 cash ow 9261000 Year 5 cash ow 9724050 Year 6 cash ow 9724050 So the NPV of the project is NPV 7475000 800001 1058 84000110582 88200110583 92610110584 9724050110585 97240501058110585 NPV 40812567 381 CHAPTER 19 DIVIDENDS AND OTHER PAYOUTS Answers to Concepts Review and Critical Thinking Questions 1 Dividend policy deals with the timing of dividend payments not the amounts ultimately paid Dividend policy is irrelevant when the timing of dividend payments doesn t affect the present value of all future dividends A stock repurchase reduces equity while leaving debt unchanged The debt ratio rises A rm could if desired use excess cash to reduce debt instead This is a capital structure decision The chief drawback to a strict dividend policy is the variability in dividend payments This is a problem because investors tend to want a somewhat predictable cash ow Also if there is information content to dividend announcements then the rm may be inadvertently telling the market that it is expecting a downturn in earnings prospects when it cuts a dividend when in reality its prospects are very good In a compromise policy the rm maintains a relatively constant dividend It increases dividends only when it expects earnings to remain at a suf ciently high level to pay the larger dividends and it lowers the dividend only if it absolutely has to Friday December 29 is the exdividend day Remember not to count January 1 because it is a holiday and the exchanges are closed Anyone who buys the stock before December 29 is entitled to the dividend assuming they do not sell it again before December 29 No because the money could be better invested in stocks that pay dividends in cash which bene t the fundholders directly The change in price is due to the change in dividends not due to the change in dividend policy Dividend policy can still be irrelevant without a contradiction The stock price dropped because of an expected drop in future dividends Since the stock price is the present value of all future dividend payments if the expected future dividend payments decrease then the stock price will decline The plan will probably have little effect on shareholder wealth The shareholders can reinvest on their own and the shareholders must pay the taxes on the dividends either way However the shareholders who take the option may bene t at the expense of the ones who don t because of the discount Also as a result of the plan the rm will be able to raise equity by paying a 10 otation cost the discount which may be a smaller discount than the market otation costs of a new issue for some companies If these rms just went public they probably did so because they were growing and needed the additional capital Growth rms typically pay very small cash dividends if they pay a dividend at all This is because they have numerous projects available and they reinvest the earnings in the firm instead of paying cash dividends 382 p A O p A p A p n N p A DJ p n J p n UI It would not be irrational to nd lowdividend highgrowth stocks The trust should be indifferent between receiving dividends or capital gains since it does not pay taxes on either one ignoring possible restrictions on invasion of principal etc It would be irrational however to hold municipal bonds Since the trust does not pay taxes on the interest income it receives it does not need the tax break associated with the municipal bonds Therefore it should prefer to hold higher yield taxable bonds The stock price drop on the exdividend date should be lower With taxes stock prices should drop by the amount of the dividend less the taxes investors must pay on the dividends A lower tax rate lowers the investors tax liability With a high tax on dividends and a low tax on capital gains investors in general will prefer capital gains If the dividend tax rate declines the attractiveness of dividends increases Knowing that share price can be expressed as the present value of expected future dividends does not make dividend policy relevant Under the growing perpetuity model if overall corporate cash ows are unchanged then a change in dividend policy only changes the timing of the dividends The PV of those dividends is the same This is true because given that future earnings are held constant dividend policy simply represents a transfer between current and future stockholders In a more realistic context and assuming a nite holding period the value of the shares should represent the future stock price as well as the dividends Any cash ow not paid as a dividend will be re ected in the future stock price As such the PV of the cash ows will not change with shifts in dividend policy dividend policy is still irrelevant The birdinthehand argument is based upon the erroneous assumption that increased dividends make a firm less risky If capital spending and investment spending are unchanged the rm s overall cash ows are not affected by the dividend policy This argument is theoretically correct In the real world with transaction costs of security trading homemade dividends can be more expensive than dividends directly paid out by the rms However the existence of financial intermediaries such as mutual funds reduces the transaction costs for individuals greatly Thus as a whole the desire for current income shouldn t be a major factor favoring highcurrentdividend policy a Cap s past behavior suggests a preference for capital gains while Sarah exhibits a preference for current income b Cap could show the Sarah how to construct homemade dividends through the sale of stock Of course Cap will also have to convince her that she lives in an MM world Remember that homemade dividends can only be constructed under the W assumptions 0 Sarah may still not invest in Neotech because of the transaction costs involved in constructing homemade dividends Also Sarah may desire the uncertainty resolution which comes with high dividend stocks To minimize her tax burden your aunt should divest herself of high dividend yield stocks and invest in low dividend yield stocks Or if possible she should keep her high dividend stocks borrow an equivalent amount of money and invest that money in a taxdeferred account 383 18 p A O N O The capital investment needs of small growing companies are very high Therefore payment of dividends could curtail their investment opportunities Their other option is to issue stock to pay the dividend thereby incurring issuance costs In either case the companies and thus their investors are better off with a zero dividend policy during the rms rapid growth phases This fact makes these rms attractive only to low dividend clienteles This example demonstrates that dividend policy is relevant when there are issuance costs Indeed it may be relevant whenever the assumptions behind the MlVImodel are not met Unless there is an unsatisfied high dividend clientele a rm cannot improve its share price by switching policies If the market is in equilibrium the number of people who desire high dividend payout stocks should exactly equal the number of such stocks available The supplies and demands of each clientele will be exactly met in equilibrium If the market is not in equilibrium the supply of high dividend payout stocks may be less than the demand Only in such a situation could a rm bene t from a policy shift This nding implies that rms use initial dividends to signal their potential growth and positive NPV prospects to the stock market The initiation of regular cash dividends also serves to convince the market that their high current earnings are not temporary Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem 2 m The aftertax dividend is the pretax dividend times one minus the tax rate so Aftertax dividend 560l 7 15 476 The stock price should drop by the aftertax dividend amount or Exdividend price 75 7 476 7024 a The shares outstanding increases by 10 percent so New shares outstanding 20000l10 22000 New shares issued 2000 384 Since the par value of the new shares is 1 the capital surplus per share is 47 The total capital surplus is therefore Capital surplus on new shares 200047 94000 Common stock 1 par value 22000 Capital surplus 304000 Retained earnings 639300 965300 The shares outstanding increases by 25 percent so New shares outstanding 20000125 25000 New shares issued 5000 Since the par value of the new shares is 1 the capital surplus per share is 47 The total capital surplus is therefore Capital surplus on new shares 500047 235000 Common stock 1 par value 25000 Capital surplus 445000 Retained earnings 495300 965300 To nd the new shares outstanding we multiply the current shares outstanding times the ratio of new shares to old shares so New shares outstanding 200004 1 80000 The equity accounts are unchanged except that the par value of the stock is changed by the ratio of new shares to old shares so the new par value is New par value 1 14 025 per share To nd the new shares outstanding we multiply the current shares outstanding times the ratio of new shares to old shares so New shares outstanding 200001 5 4000 The equity accounts are unchanged except that the par value of the stock is changed by the ratio of new shares to old shares so the new par value is New par value 151 500 per share 385 5 To nd the new stock price we multiply the current stock price by the ratio of old shares to new shares so a 7835 4680 13 781115 6783 c 7811425 5474 d 7874 13650 To nd the new shares outstanding we multiply the current shares outstanding times the ratio of new shares to old shares so a 26000053 433333 b 2600001 15 299000 c 2600001425 370500 d 26000047 148571 The stock price is the total market value of equity divided by the shares outstanding so P0 380000 equity8000 shares 4750 per share Ignoring tax effects the stock price will drop by the amount of the dividend so PX 4750 7160 4590 The total dividends paid will be 160 per share8000 shares 12800 The equity and cash accounts will both decline by 12800 Repurchasing the shares will reduce shareholders equity by 12800 The shares repurchased will be the total purchase amount divided by the stock price so Shares bought 128004750 269 And the new shares outstanding will be New shares outstanding 8000 7269 7731 After repurchase the new stock price is Share price 3672007731 shares 4750 386 The repurchase is effectively the same as the cash dividend because you either hold a share worth 4750 or a share worth 4590 and 160 in cash Therefore you participate in the repurchase according to the dividend payout percentage you are unaffected The stock price is the total market value of equity divided by the shares outstanding so P0 455000 equity 20000 shares 2275 per share The shares outstanding will increase by 25 percent so New shares outstanding 20000l25 25000 The new stock price is the market value of equity divided by the new shares outstanding so PX 45500025000 shares 1820 With a stock dividend the shares outstanding will increase by one plus the dividend amount so New shares outstanding 380000112 425600 The capital surplus is the capital paid in excess of par value which is 1 so Capital surplus for new shares 4560044 2006400 The new capital surplus will be the old capital surplus plus the additional capital surplus for the new shares so Capital surplus 1750000 2006400 3756400 The new equity portion of the balance sheet will look like this Common stock 1 par value 425600 Capital surplus 3756400 Retained earnings 2098000 6280000 The only equity account that will be affected is the par value of the stock The par value will change by the ratio of old shares to new shares so New par value 115 020 per share The total dividends paid this year will be the dividend amount times the number of shares outstanding The company had 380000 shares outstanding before the split We must remember to adjust the shares outstanding for the stock split so Total dividends paid this year 060380000 shares51 split 1140000 The dividends increased by 10 percent so the total dividends paid last year were Last year s dividends 1140000110 103636364 387 p n p A And to nd the dividends per share we simply divide this amount by the shares outstanding last year Doing so we get Dividends per share last year 103636364380000 shares 273 a If the dividend is declared the price of the stock will drop on the eXdividend date by the value of the dividend 5 It will then trade for 115 b If it is not declared the price will remain at 120 0 lVIann s outflows for investments are 3000000 These out ows occur immediately One year from now the firm will realize 1400000 in net income and it will pay 750000 in dividends but the need for nancing is immediate Mann must nance 3000000 through the sale of shares worth 120 It must sell 3000000 120 25000 shares d The lVIM model is not realistic since it does not account for taxes brokerage fees uncertainty over future cash ows investors preferences signaling effects and agency costs Intermediate The price of the stock today is the PV of the dividends so P0 095114 451142 3546 To find the equal two year dividends with the same present value as the price of the stock we set up the following equation and solve for the dividend Note The dividend is a two year annuity so we could solve with the annuity factor as well 3546 D114 D1142 D 2153 We now know the cash ow per share we want each of the next two years We can find the price of stock in one year which will be P1 45114 3947 Since you own 1000 shares in one year you want Cash ow in Year one 10002153 2153411 But you ll only get Dividends received in one year 1000095 95000 Thus in one year you will need to sell additional shares in order to increase your cash ow The number of shares to sell in year one is Shares to sell at time one 21534117 9503947 52146 shares 388 p A N 13 At Year 2 your cash ow will be the dividend payment times the number of shares you still own so the Year 2 cash ow is Year 2 cash ow 451000 7 52146 2153411 If you only want 500 in Year 1 you will buy 950 7 5003947 1140 shares at Year 1 Your dividend payment in Year 2 will be Year 2 dividend 1000 114045 4551300 Note that the present value of each cash ow stream is the same Below we show this by nding the present values as PV 500114 455131142 3545937 PV 1000095114 1000451142 3545937 a If the company makes a dividend payment we can calculate the wealth of a shareholder as Dividend per share 3000 600 shares 500 The stock price after the dividend payment will be Px 58 7 5 53 per share The shareholder will have a stock worth 53 and a 5 dividend for a total wealth of 58 If the company makes a repurchase the company will repurchase Shares repurchased 300058 5172 shares If the shareholder lets their shares be repurchased they will have 58 in cash If the shareholder keeps their shares they re still worth 58 b If the company pays dividends the current EPS is 150 and the PE ratio is PE 53150 3533 If the company repurchases stock the number of shares will decrease The total net income is the EPS times the current number of shares outstanding Dividing net income by the new number of shares outstanding we nd the EPS under the repurchase is EPS 150600600 7 5172 164 The stock price will remain at 58 per share so the PE ratio is PE 58164 3533 389 C A share repurchase would seem to be the preferred course of action Only those shareholders who wish to sell will do so giving the shareholder a tax timing option that he or she doesn t get with a dividend payment Since the rm has a 100 percent payout policy the entire net income 45000 will be paid as a dividend The current value of the firm is the discounted value one year from now plus the current income which is Value 7 45000 1635000112 Value 7 1504821 The current stock price is the value of the rm divided by the shares outstanding which is Stock price l50482l20000 Stock price 7524 Since the company has a 100 percent payout policy the current dividend per share will be the company s net income divided by the shares outstanding or Current dividend 4500020000 Current dividend 225 The stock price will fall by the value of the dividend to EXdividend stock price 7524 7 225 EXdividend stock price 7299 139 According to MM it cannot be true that the low dividend is depressing the price Since dividend policy is irrelevant the level of the dividend should not matter Any funds not distributed as dividends add to the value of the firm hence the stock price These directors merely want to change the timing of the dividends more now less in the future As the calculations below indicate the value of the firm is unchanged by their proposal Therefore the share price will be unchanged To show this consider what would happen if the dividend were increased to 460 Since only the existing shareholders will get the dividend the required dollar amount to pay the dividends is Total dividends 46020000 Total dividends 92000 To fund this dividend payment the company must raise Dollars raised Required funds 7 Net income Dollars raised 92000 7 45000 Dollars raised 47000 390 This money can only be raised with the sale of new equity to maintain the allequity nancing Since those new shareholders must also earn 12 percent their share of the rm one year from now is New shareholder value in one year 47000l 12 New shareholder value in one year 52640 This means that the old shareholders39 interest falls to Old shareholder value in one year 1635000 7 52640 Old shareholder value in one year 1582360 Under this scenario the current value of the rm is Value 92000 1582360112 Value 1504821 Since the firm value is the same as in part a the change in dividend policy had no effect The new shareholders are not entitled to receive the current dividend They will receive only the value of the equity one year hence The present value of those ows is Present value 1582360l12 Present value 141282143 And the current share price will be Current share price 14128214320000 Current share price 7064 So the number of new shares the company must sell will be Shares sold 470007064 Shares sold 66534 shares The current price is the current cash ow of the company plus the present value of the expected cash ows divided by the number of shares outstanding So the current stock price is Stock price 1400000 20000000 750000 Stock price 2853 To achieve a zero dividend payout policy he can invest the dividends back into the company s stock The dividends per share will be Dividends per share l40000050750000 Dividends per share 093 And the stockholder in question will receive Dividends paid to shareholder 0931000 391 17 Dividends paid to shareholder 93333 The new stock price after the dividends are paid will be Exdividend stock price 2853 7 093 Exdividend stock price 2760 So the number of shares the investor will buy is Number of shares to buy 93333 2760 Number of shares to buy 3382 1 Using the formula from the text proposed by Lintner Divl Divo st EP31 7 Dive Divl 150 34415 7 150 Divl 1548 b Now we use an adjustment rate of 060 so the dividend next year will be Divl Divo st EP31 7 Dive Div1 150 64415 7 150 Divl 1596 c The lower adjustment factor in part a is more conservative The lower adjustment factor will always result in a lower future dividend Challenge Assuming no capital gains tax the aftertax return for the Gordon Company is the capital gains growth rate plus the dividend yield times one minus the tax rate Using the constant growth dividend model we get Aftertax return g Dl 7t l2 Solving for g we get 12 g 061735 g 0810 The equivalent pretax return for Gecko Company which pays no dividend is Pretax return g D 0810 06 1410 Using the equation for the decline in the stock price exdividend for each of the tax rate policies we get P0 PxD I TP1 TG a P07Px D170170 P07PXD 392 0 13 P0 7Px 7 D1715170 P07Px 7 85D 0 P0 7Px 7 D1 7 151 7 20 P0 7Px 7 10625D d With this tax policy we simply need to multiply the personal tax rate times one minus the dividend exemption percentage so P0 7Px Dl 7 3530l7 35 P0 7Px 13769D e Since different investors have widely varying tax rates on ordinary income and capital gains dividend payments have different aftertax implications for different investors This differential taxation among investors is one aspect of what we have called the clientele effect Since the 3000000 cash is after corporate tax the full amount will be invested So the value of each alternative is Alternative 1 The rm invests in Tbills or in preferred stock and then pays out as a special dividend in 3 years If the firm invests in T Bills If the rm invests in Tbills the aftertax yield of the Tbills will be Aftertax corporate yield 051 7 35 Aftertax corporate yield 0325 or 325 So the future value of the corporate investment in Tbills will be FV of investment in Tbills 3000000l 03253 FV of investment in Tbills 330210923 Since the future value will be paid to shareholders as a dividend the aftertax cash ow will be Aftertax cash ow to shareholders 330210923l 7 l5 Aftertax cash ow to shareholders 280679285 If the firm invests in preferred stock If the rm invests in preferred stock the assumption would be that the dividends received will be reinvested in the same preferred stock The preferred stock will pay a dividend of Preferred dividend 073000000 Preferred dividend 210000 Since 70 percent of the dividends are excluded from tax Taxable preferred dividends l 7 70210000 393 Taxable preferred dividends 63000 And the taxes the company must pay on the preferred dividends will be Taxes on preferred dividends 3563000 Taxes on preferred dividends 22050 So the aftertax dividend for the corporation will be Aftertax corporate dividend 210000 7 22050 Aftertax corporate dividend 187950 This means the aftertax corporate dividend yield is Aftertax corporate dividend yield 187950 3000000 Aftertax corporate dividend yield 0627 or 627 The future value of the company s investment in preferred stock will be FV of investment in preferred stock 30000001 06273 FV of investment in preferred stock 359991291 Since the future value will be paid to shareholders as a dividend the aftertax cash ow will be Aftertax cash ow to shareholders 35999129117 15 Aftertax cash ow to shareholders 305992597 Alternative 2 The firm pays out dividend now and individuals invest on their own The aftertax cash received by shareholders now will be Aftertax cash received today 30000001 7 15 Aftertax cash received today 2550000 The individuals invest in Treasury bills If the shareholders invest the current aftertax dividends in Treasury bills the aftertax individual yield will be Aftertax individual yield on Tbills 051 7 31 Aftertax individual yield on Tbills 0345 or 345 So the future value of the individual investment in Treasury bills will be FV of investment in Tbills 25500001 03453 FV of investment in Tbills 282313512 394 The individuals invest in preferred stock If the individual invests in preferred stock the assumption would be that the dividends received will be reinvested in the same preferred stock The preferred stock will pay a dividend of Preferred dividend 072550000 Preferred dividend 178500 And the taxes on the preferred dividends will be Taxes on preferred dividends 31178500 Taxes on preferred dividends 55335 So the aftertax preferred dividend will be Aftertax preferred dividend 178500 7 55335 Aftertax preferred dividend 123165 This means the aftertax individual dividend yield is Aftertax corporate dividend yield 123165 2550000 Aftertax corporate dividend yield 0483 or 483 The future value of the individual investment in preferred stock will be FV of investment in preferred stock 25500001 04833 FV of investment in preferred stock 293762894 The aftertax cash ow for the shareholders is maximized when the rm invests the cash in the preferred stocks and pays a special dividend later a Let x be the ordinary income tax rate The individual receives an aftertax dividend of Aftertax dividend 10001 7 x which she invests in Treasury bonds The Treasury bond will generate aftertax cash ows to the investor of Aftertax cash ow from Treasury bonds 10001 7x1 081 7x If the rm invests the money its proceeds are Firm proceeds 10001 081735 And the proceeds to the investor when the firm pays a dividend will be Proceeds if rrn invests rst 1 7x10001 081 7 35 395 To be indifferent the investor s proceeds must be the same whether she invests the aftertax dividend or receives the proceeds from the rm s investment and pays taxes on that amount To nd the rate at which the investor would be indifferent we can set the two equations equal and solve for x Doing so we nd 100017x10817x17x10001081735 1 081 7x 1 0817 35 x 35 or 35 Note that this argument does not depend upon the length of time the investment is held Yes this is a reasonable answer She is only indifferent if the aftertax proceeds from the 1000 investment in identical securities are identical That occurs only when the tax rates are identical Since both investors will receive the same pretaX return you would expect the same answer as in part a Yet because the company enjoys a tax bene t from investing in stock 70 percent of income from stock is exempt from corporate taxes the tax rate on ordinary income which induces indifference is much lower Again set the two equations equal and solve for x 100017x11217x17x10001 1270 17701735 11217x11270 17701735 x 1050 or 1050 It is a compelling argument but there are legal constraints which deter firms from investing large sums in stock of other companies 396 CHAPTER 20 ISSUING SECURITIES TO THE PUBLIC Answers to Concepts Review and Critical Thinking Questions 1 A company s internally generated cash ow provides a source of equity nancing For a pro table company outside equity may never be needed Debt issues are larger because large companies have the greatest access to public debt markets small companies tend to borrow more from private lenders Equity issuers are frequently small companies going public such issues are often quite small Additionally to maintain a debtequity ratio a company must issue new bonds when the current bonds mature From the previous question economies of scale are part of the answer Beyond this debt issues are simply easier and less risky to sell from an investment bank s perspective The two main reasons are that very large amounts of debt securities can be sold to a relatively small number of buyers particularly large institutional buyers such as pension funds and insurance companies and debt securities are much easier to price They are riskier and harder to market from an investment bank s perspective Yields on comparable bonds can usually be readily observed so pricing a bond issue accurately is much less difficult It is clear that the stock was sold too cheaply so Eyetech had reason to be unhappy No but in fairness pricing the stock in such a situation is extremely difficult It s an important factor Only 65 million of the shares were underpriced The other 32 million were in effect priced completely correctly The evidence suggests that a nonunderwritten rights offering might be substantially cheaper than a cash offer However such offerings are rare and there may be hidden costs or other factors not yet identi ed or well understood by researchers He could have done worse since his access to the oversubscribed and presumably underpriced issues was restricted while the bulk of his funds were allocated to stocks from the undersubscribed and quite possibly overpriced issues a The price will probably go up because IPOs are generally underpriced This is especially true for smaller issues such as this one b It is probably safe to assume that they are having trouble moving the issue and it is likely that the issue is not substantially underpriced 397 p A p A p n J p A UI Competitive offer and negotiated offer are two methods to select investment bankers for underwriting Under the competitive offers the issuing rm can award its securities to the underwriter with the highest bid which in turn implies the lowest cost On the other hand in negotiated deals the underwriter gains much information about the issuing firm through negotiation which helps increase the possibility of a successful offering There are two possible reasons for stock price drops on the announcement of a new equity issue 1 Management may attempt to issue new shares of stock when the stock is overvalued that is the intrinsic value is lower than the market price The price drop is the result of the downward adjustment of the overvaluation 2 When there is an increase in the possibility of nancial distress a rm is more likely to raise capital through equity than debt The market price drops because the market interprets the equity issue announcement as bad news If the interest of management is to increase the wealth of the current shareholders a rights offering may be preferable because issuing costs as a percentage of capital raised are lower for rights offerings Management does not have to worry about underpricing because shareholders get the rights which are worth something Rights offerings also prevent existing shareholders from losing proportionate ownership control Finally whether the shareholders exercise or sell their rights they are the only bene ciaries Reasons for shelf registration include 1 Flexibility in raising money only when necessary without mcurring additional issuance costs 2 As Bhagat Marr and Thompson showed shelf registration is less costly than conventional underwritten issues 3 Issuance of securities is greatly simpli ed Basic empirical regularities in IPOs include 1 underpricing of the offer price 2 bestefforts offerings are generally used for small IPOs and firmcommitment offerings are generally used for large IPOs 3 the underwriter price stabilization of the after market and 4 that issuing costs are higher in negotiated deals than in competitive ones Solutions to Questions and Problems NOTE All end of chapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic a The new market value will be the current shares outstanding times the stock price plus the rights offered times the rights price so New market value 45000090 8000084 47220000 b The number of rights associated with the old shares is the number of shares outstanding divided by the rights offered so Number of rights needed 450000 old shares 80000 new shares 563 rights per new share 398 The new price of the stock will be the new market value of the company divided by the total number of shares outstanding after the rights offer which will be Px 47220000450000 80000 8909 The value of the right Value ofa right 9000 7 8909 091 A rights offering usually costs less it protects the proportionate interests of existing share holders and also protects against underpricing The maximum subscription price is the current stock price or 34 The minimum price is anything greater than 0 The number of new shares will be the amount raised divided by the subscription price so Number of new shares 40000000 30 1333333 shares And the number of rights needed to buy one share will be the current shares outstanding divided by the number of new share offered so Number of rights needed 3400000 shares outstanding 1333333 new shares 255 A shareholder can buy 255 rights on shares for 25534 8670 The shareholder can exercise these rights for 30 at a total cost of 8670 3000 11670 The investor will then have Exrights shares 1 255 Exrights shares 355 The exrights price per share is PX 25534 30355 3287 So the value of a right is Value ofa right 34 7 3287 113 Before the offer a shareholder will have the shares owned at the current market price or Portfolio value 1000 shares34 34000 399 After the rights offer the share price will fall but the shareholder will also hold the rights so Portfolio value 1000 shares32 87 1000 rights113 34000 Using the equation we derived in Problem 2 part c to calculate the price of the stock exrights we can nd the number of shares a shareholder will have exrights which is PK 7 7025 7 N75 50N 1 N 7 4263 The number of new shares is the amount raised divided by the pershare subscription price so Number of new shares 1500000050 300000 And the number of old shares is the number of new shares times the number of shares exrights so Number of old shares 4263300000 1278947 If you receive 1000 shares of each the pro t is Pro t 10008 7 10005 3000 Since you will only receive onehalf of the shares of the oversubscribed issue your pro t will be Expected pro t 5008 7 10005 71000 This is an example of the winner s curse Using X to stand for the required sale proceeds the equation to calculate the total sale proceeds including otation costs is X1 7 08 35000000 X 38043478 required total proceeds from sale So the number of shares offered is the total amount raised divided by the offer price which is Number of shares offered 3 804347831 1227209 This is basically the same as the previous problem except we need to include the 900000 of expenses in the amount the company needs to raise so X1 7 08 35900000 X 39021739 required total proceeds from sale Number of shares offered 3902173931 1258766 We need to calculate the net amount raised and the costs associated with the offer The net amount raised is the number of shares offered times the price received by the company minus the costs associated with the offer so 400 Net amount raised 8000000 shares22 10 7 950000 7 250000 175600000 The company received 175600000 from the stock offering Now we can calculate the direct costs Part of the direct costs are given in the problem but the company also had to pay the underwriters The stock was offered at 24 per share and the company received 2210 per share The difference which is the underwriters spread is also a direct cost The total direct costs were Total direct costs 7 950000 24 7 22108000000 shares 7 16150000 We are given part of the indirect costs in the problem Another indirect cost is the immediate price appreciation The total indirect costs were Total indirect costs 250000 2950 7 248000000 shares 44250000 This makes the total costs Total costs 16150000 44250000 60400000 The otation costs as a percentage of the amount raised is the total cost divided by the amount raised so Flotation cost percentage 60400000 175600000 3440 or 3440 The number of rights needed per new share is Number of rights needed 100000 old shares20000 new shares 5 rights per new share Using PR0 as the rightson price and PS as the subscription price we can express the price per share of the stock eXrights as PX NPR0 PSN 1 a Px 580 806 8000 No change b Px 580 756 7917 Price drops by 083 per share 0 Px 580 656 7750 Price drops by 250 per share In general the new price per share after the offering will be P 7 Current market value Proceeds from offer Old shares New shares The current market value of the company is the number of shares outstanding times the share price or lVIarket value of company 3000040 Market value of company 1200000 401 If the new shares are issued at 40 the share price after the issue will be 1200000800040 30000 8000 P 4000 P If the new shares are issued at 20 the share price after the issue will be 1200000 800020 30000 8000 P 3579 P If the new shares are issued at 10 the share price after the issue will be 1200000 800010 30000 8000 P 3368 P Intermediate a The number of shares outstanding after the stock offer will be the current shares outstanding plus the amount raised divided by the current stock price assuming the stock price doesn t change So Number of shares after offering 8000000 4000000065 8615385 Since the par value per share is 1 the old book value of the shares is the current number of shares outstanding From the previous solution we can see the company will sell 615385 shares and these will have a book value of 65 per share The sum of these two values will give us the total book value of the company If we divide this by the new number of shares outstanding Doing so we find the new book value per share will be New book value per share 800000020 615385658615385 2321 The current EPS for the company is EPSO NTOShareso 115000008000000 shares 144 per share And the current PE is PE0 65144 4522 If the net income increases by 600000 the new EPS will be EP31 NTIsharesl 1210000086l5385 shares 140 per share 402 Assuming the PE remains constant the new share price will be P1 PE0EPS1 4522140 6351 The current markettobook ratio is Current markettobook 65 20 325 Using the new share price and book value per share the new markettobook ratio will be New markettobook 63512321 27357 Accounting dilution has occurred because new shares were issued when the markettobook ratio was less than one market value dilution has occurred because the rm nanced a negative NPV project The cost of the project is given at 40 million The NPV of the project is the new market value of the rm minus the current market value of the rm or NPV 740000000 86153856351 7 800000065 712869565 b For the price to remain unchanged when the PE ratio is constant EPS must remain constant The new net income must be the new number of shares outstanding times the current EPS which gives N11 8615385 sharesl44 per share 12384615 11 The current ROE of the company is ROEO NIOTEO 6300006500000 72600000 1620 or 1620 The new net income will be the ROE times the new total equity or N11 ROE0TE1 16203900000 1100000 807692 The company s current earnings per share are EPSO NTOShares outstandingo 63000045000 shares 1400 The number of shares the company will offer is the cost of the investment divided by the current share price so Number ofnew shares 110000073 15068 The earnings per share after the stock offer will be EP31 80769245000 15068 shares 1345 The current PE ratio is PE0 731400 5214 403 N Assuming the PE remains constant the new stock price will be P1 52141345 7011 The current book value per share and the new book value per share are BVPSO TEOshareso 390000045000 shares 8667 per share BVPSI TElsharesl 3900000 110000060068 shares 8324 per share So the current and new markettobook ratios are Markettobooko 73 8667 08423 Markettobook1 70118324 08423 The NPV of the project is the new market value of the firm minus the current market value of the firm or NPV 1100000 701160068 7 7345000 7173462 Accounting dilution takes place here because the markettobook ratio is less than one Market value dilution has occurred since the fum is investing in a negative NPV project Using the PE ratio to find the necessary EPS after the stock issue we get P1 73 5214EPS1 EPS1 1400 The additional net income level must be the EPS times the new shares outstanding so NI 1415068 shares 210959 And the new ROE is ROE1 2109591100000 1918 or 1918 Next we need to find the NPV of the project The NPV of the project is the new market value of the firm minus the current market value of the rm or NPV 1100000 7360068 7 7345000 0 Accounting dilution still takes place as BVPS still falls from 8667 to 8324 but no market dilution takes place because the firm is investing in a zero NPV project 404 Assume you hold three shares of the company s stock The value of your holdings before you exercise your rights is Value of holdings 363 Value of holdings 189 When you exercise you must remit the three rights you receive for owning three shares and 12 You have increased your equity investment by 12 The value of your holdings after surrendering your rights is New value ofholdings 189 12 New value of holdings 201 After exercise you own four shares of stock Thus the price per share of your stock is Stock price 201 4 Stock price 5025 The value of a right is the difference between the rightson price of the stock and the exrights price of the stock Value of rights Rightson price 7 Exrights price Value ofrights 63 7 5012 Value of rights 1275 The price drop will occur on the exrights date even though the exrights date is neither the expiration date nor the date on which the rights are first exercisable If you purchase the stock before the exrights date you will receive the rights If you purchase the stock on or after the exrights date you will not receive the rights Since rights have value the stockholder receiving the rights must pay for them The stock price drop on the exrights day is similar to the stock price drop on an exdividend day The number of new shares offered through the rights offering is the existing shares divided by the rights per share or New shares 1000000 2 New shares 500000 And the new price per share after the offering will be Current market value Proceeds from offer P Oldshares New shares P 100000027 2000000 1000000 500000 P 1933 405 The subscription price is the amount raised divided by the number of new shares offered or Subscription price 2000000 500000 Subscription price 4 And the value of a right is Value of a right Exrights price 7 Subscription price Rights needed to buy a share of stock Value ofa right 1933 7 4 2 Value ofa right 767 Following the same procedure the number of new shares offered through the rights offering is New shares l000000 4 New shares 250000 And the new price per share after the offering will be P Current market vahe Proceeds from offer Old shares New shares P 100000027 2000000 1000000 250000 P 7 2320 The subscription price is the amount raised divided by the number of number of new shares offered or Subscription price 2000000 250000 Subscription price 8 And the value of a right is Value of a right Exrights price 7 Subscription price Rights needed to buy a share of stock Value ofa right 2320 7 8 4 Value ofa right 380 Since rights issues are constructed so that existing shareholders39 proportionate share will remain unchanged we know that the stockholders wealth should be the same between the two arrangements However a numerical example makes this clearer Assume that an investor holds 4 shares and will exercise under either a or b Prior to exercise the investor39s portfolio value is Current portfolio value Number of shares gtlt Stock price Current portfolio value 427 Current portfolio value 108 406 15 p A 5 After exercise the value of the portfolio will be the new number of shares time the eXrights price less the subscription price paid Under a the investor gets 2 new shares so portfolio value will be New portfolio value 6l933 7 24 New portfolio value 108 Under b the investor gets 1 new share so portfolio value will be New portfolio value 52320 7 l8 New portfolio value 108 So the shareholder39s wealth position is unchanged either by the rights issue itself or the choice of which right s issue the rm chooses The number of new shares is the amount raised divided by the subscription price so Number of new shares 60000000PS And the eXrights number of shares N is equal to N Old shares outstandingNew shares outstanding N 1000000060000000Ps N 01667PS We know the equation for the eXrights stock price is PX NPRO PslNJr 1 We can substitute in the numbers we are given and then substitute the two previous results Doing so and solving for the subscription price we get Px 61N68 PsN 1 61 7 6801667PS PS01667PS 1 617 11333Ps1 01667Ps PS 7 2815 Using PR0 as the rightson price and PS as the subscription price we can express the price per share of the stock eXrights as PX NPRO PslN 1 And the equation for the value of a right is Value of a right PRO 7 PX 407 l on Substituting the eXrights price equation into the equation for the value of a right and rearranging we get Value ofa right PRO 7 NPRO PSN 1 Value of a right N 1PR0 7 NPRO 7 PSN 1 Value ofa right PR0 7PsN 1 The net proceeds to the company on a per share basis is the subscription price times one minus the underwriter spread so Net proceeds to the company 251 7 06 2350 per share So to raise the required funds the company must sell New shares offered 41250002350 175532 The number of rights needed per share is the current number of shares outstanding divided by the new shares offered or Number of rights needed 750000 old shares 175532 new shares Number of rights needed 427 rights per share The eXrights stock price will be PX NPRO PslN1 PX 42745 25527 4121 So the value of a right is Value ofa right 45 7 4121 379 And your proceeds from selling your rights will be Proceeds from selling rights 6000379 2275862 Using the equation for valuing a stock eXrights we nd PX NPRO PslN1 PX 475 405 68 The stock is correctly priced Calculating the value of a right we nd Value of a right PRO 7 PX Value ofa right 75 7 68 7 So the rights are underpriced You can create an immediate profit on the eXrights day if the stock is selling for 68 and the rights are selling for 6 by executing the following transactions Buy 4 rights in the market for 46 24 Use these rights to purchase a new share at the subscription price of 40 Immediately sell this share in the market for 68 creating an instant 4 pro t 408 CHAPTER 21 LEASING Answers to Concepts Review and Critical Thinking Questions 1 Some key differences are l Lease payments are fully taxdeductible but only the interest portion of the loan is 2 The lessee does not own the asset and cannot depreciate it for tax purposes 3 In the event of a default the lessor cannot force bankruptcy and 4 The lessee does not obtain title to the asset at the end of the lease absent some additional arrangement The less profitable one because leasing provides among other things a mechanism for transferring tax bene ts from entities that value them less to entities that value them more Potential problems include 1 Care must be taken in interpreting the IRR a high or low IR is preferred depending on the setup of the analysis and 2 Care must be taken to ensure the IRR under examination is not the implicit interest rate just based on the lease payments a Leasing is a form of secured borrowing It reduces a rm s cost of capital only if it is cheaper than other forms of secured borrowing The reduction of uncertainty is not particularly relevant what matters is the NAL b The statement is not always true For example a lease often requires an advance lease payment or security deposit and may be implicitly secured by other assets of the rm 0 Leasing would probably not disappear since it does reduce the uncertainty about salvage value and the transactions costs of transferring ownership However the use of leasing would be greatly reduced A lease must be disclosed on the balance sheet if one of the following criteria is met I The lease transfers ownership of the asset by the end of the lease In this case the rm essentially owns the asset and will have access to its residual value 2 The lessee can purchase the asset at a price below its fair market value bargain purchase option when the lease ends The rm essentially owns the asset and will have access to most of its residual value 3 The lease term is for 75 or more of the estimated economic life of the asset The firm basically has access to the majority of the bene ts of the asset without any responsibility for the consequences of its disposal 4 The present value of the lease payments is 90 or more of the fair market value of the asset at the start of the lease The rm is essentially purchasing the asset on an installment basis The lease must meet the following IRS standards for the lease payments to be tax deductible I The lease term must be less than 80 of the economic life of the asset If the term is longer the lease is considered to be a conditional sale 2 The lease should not contain a bargain purchase option which the IRS interprets as an equity interest in the asset 3 The lease payment schedule should not provide for very high payments early and very low payments late in the life of the lease This would indicate that the lease is being used simply to avoid taxes 409 p A O p A p A 4 Renewal options should be reasonable and based on the fair market value of the asset at renewal time This indicates that the lease is for legitimate business purposes not tax avoidance As the term implies offbalance sheet nancing involves financing arrangements that are not required to be reported on the rm s balance sheet Such activities if reported at all appear only in the footnotes to the statements Operating leases those that do not meet the criteria in Question 6 provide offbalance sheet nancing For accounting purposes total assets will be lower and some nancial ratios may be arti cially high Financial analysts are generally not fooled by such practices There are no economic consequences since the cash ows of the rm are not affected by how the lease is treated for accounting purposes The lessee may not be able to take advantage of the depreciation tax shield and may not be able to obtain favorable lease arrangements for passing on the tax shield bene ts The lessee might also need the cash ow from the sale to meet immediate needs but will be able to meet the lease obligation cash ows in the future Since the relevant cash ows are all aftertax the aftertax discount rate is appropriate Japan Airlines nancial position was such that the package of leasing and buying probably resulted in the overall best aftertax cost In particular Japan Airlines may not have been in a position to use all of its tax credits and also may not have had the credit strength to borrow and buy the plane without facing a credit downgrade andor substantially higher rates There is the tax motive but beyond this Genesis Lease Limited knows that in the event of a default Japan Airlines would relinquish the plane which would then be released Fungible assets such as planes which can be readily reclaimed and redeployed are good candidates for leasing The plane will be released to Japan Airlines or another air transportation rm used by Genesis Lease Limited or it will simply be sold There is an active market for used aircraft Solutions to Questions and Problems NOTE All end of chapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem m We will calculate cash ows from the depreciation tax shield rst The depreciation tax shield is Depreciation tax shield 4500000435 393750 The aftertax cost of the lease payments will be Aftertax lease payment 1350000l 7 35 877500 410 So the total cash ows from leasing are OCF 393750 877500 1271250 The aftertax cost of debt is Aftertax debt cost 081 7 35 052 Using all of this information we can calculate the NAL as NAL 4500000 7 1271250PVIFA5 2074 1307425 The NAL is positive so you should lease If we assume the lessor has the same cost of debt and the same tax rate the NAL to the lessor is the negative of our company s NAL so NAL 7 1307425 To nd the maximum lease payment that would satisfy both the lessor and the lessee we need to nd the payment that makes the NAL equal to zero Using the NAL equation and solving for the OCF we nd NAL 7 0 7 4500000 7 OCFPVIFA5 20W4 OCF 7 127495424 The OCF for this lease is composed of the depreciation tax shield cash ow as well as the aftertax lease payment Subtracting out the depreciation tax shield cash ow we calculated earlier we nd Aftertax lease payment 127495424 7 393750 88120424 Since this is the aftertax lease payment we can now calculate the breakeven pretax lease payment as Breakeven lease payment 881204241 7 35 135569883 If the tax rate is zero there is no depreciation tax shield foregone Also the aftertax lease payment is the same as the pretax payment and the aftertax cost of debt is the same as the pretax cost So Cost of debt 08 Annual cost of leasing leasing payment 1350000 The NAL to leasing with these assumptions is NAL 4500000 7 1350000PVIFA34 2862877 411 We already calculated the breakeven lease payment for the lessor in Problem 3 The assumptions about the lessor concerning the tax rate have not changed So the lessor breaks even with a payment of 135569883 For the lessee we need to calculate the breakeven lease payment which results in a zero NAL Using the assumptions in Problem 4 we nd NAL 0 4500000 7PMTPVIFA874 PMT 135864362 So the range of lease payments that would satisfy both the lessee and the lessor are Total payment range 135569883 to 135864362 The appropriate depreciation percentages for a 3year MACRS class asset can be found in Chapter 6 The depreciation percentages are 0333 0444 0148 and 0074 The cash ows from leasing are Year 1 450000033335 877500 1401975 Year 2 450000044435 877500 1576800 Year 3 450000014835 877500 1110600 Year 4 450000007435 877500 994050 NAL 4500000 7 14019751052 7 157680010522 7 111060010523 7 99405010524 NAL 72296980 The machine should not be leased However notice that the NAL is higher because of the accelerated tax bene ts due to depreciation It is possible that the accelerated depreciation bene ts could make the NAL positive when compared to straightline depreciation We will calculate cash ows from the depreciation tax shield rst The depreciation tax shield is Depreciation tax shield 435000535 30450 The aftertaX cost of the lease payments will be Afteitax lease payment 10750017 35 69875 So the total cash ows from leasing are OCF 30450 69875 100325 The aftertaX cost of debt is Aftertax debt cost 091 7 35 0585 Using all of this information we can calculate the NAL as NAL 435000 7 85730PVIFA5 85 1066452 The NAL is positive so the company should lease 412 1 Since the lessee has an effective tax rate of zero there is no depreciation tax shield foregone Also the aftertax lease payment is the same as the pretax payment and the aftertax cost of debt is the same as the pretax cost To nd the most the lessee would pay we set the NAL equal to zero and solve for the payment doing so we find the most the lessee will pay is NAL 0 780000 7PMTPVIFA775 PMT 19023474 b We will calculate cash ows from the depreciation tax shield first The depreciation tax shield is Depreciation tax shield 780000535 54600 The aftertax cost of debt is Aftertax debt cost 071 7 35 0455 Using all of this information we can calculate the minimum lease payment for the lessor as NAL 7 0 7 780000 7PMT1 7 35PVIFA4555 54600PVIFA4 55m PMT 7 18973094 0 A lease payment less than 18973094 will give the lessor a negative NAL A payment higher than 19023474 will give the lessee a negative NAL In either case no deal will be struck Therefore these represent the lower and upper bounds of possible lease prices during negotiations Intermediate The pretax cost savings are not relevant to the lease versus buy decision since the firm will de nitely use the equipment and realize the savings regardless of the financing choice made The depreciation tax shield is Depreciation tax shield lost 7000000534 476000 And the aftertax lease payment is Aftertax lease payment 16500001 7 34 1089000 The aftertax cost of debt is Aftertax debt cost 091 7 34 0594 or 594 With these cash ows the NAL is NAL 7000000 7 1089000 7 1089000PVIFA5 944 476000PVIFA5 9475 12394757 The equipment should be leased 413 O p A N To nd the maximum payment we nd where the NAL is equal to zero and solve for the payment Using X to represent the maximum payment NAL 0 7 7000000 7X10594PVIFA5 94 7 476000Pv1FA5 94m x 7 111672956 So the maximum pretax lease payment is Pretax lease payment 1116729561 7 34 169201448 The aftertax residual value of the asset is an opportunity cost to the leasing decision occurring at the end of the project life year 5 Also the residual value is not really a debtlike cash ow s1nce there is uncertainty associated with it at year 0 Nevertheless although a higher discount rate may be appropriate we ll use the a ertax cost of debt to discount the residual value as is common in practice Setting the NAL equal to zero NAL 0 7 7000000 7X10594PVIFA5 94m 7 476000Pv1FA5 94m 7 700000105945 x 7 99937414 So the maximum pretax lease payment is Pretax lease payment 999374141 7 34 151420324 The security deposit is a cash out ow at the beginning of the lease and a cash in ow at the end of the lease when it is returned The NAL with these assumptions is NAL 7000000 7 500000 7 1089000 7 1089000PVIFA5 9474 7 476000PVIFA5 945 500000105945 NAL 7136410 With the security deposit the rm should buy the equipment since the NAL is less than zero We could also solve this problem another way From Problem 9 we know that the NAL without the security deposit is 12394757 so if we nd the present value of the security deposit we can simply add this to 12394757 The present value of the security deposit is PV of security deposit 7500000 500000105945 712531167 So the NAL with the security deposit is NAL 12394757 712531167 71364 10 The lessee is paying taxes so will forego the depreciation tax shield if it leases the equipment The depreciation tax shield for the lessee is Depreciation tax shield 2600000 625 Depreciation tax shield 10833333 414 The aftertax cost of debt for the lessee is Aftertax debt cost 091 7 25 0675 Using all of this information we can calculate the maximum pretax lease payment for the lessee as NAL 0 2600000 7 PMT1 7 25PVIFA6 756 10833333PVIFA6 756 PMT 57724394 For the lessor the depreciation tax shield is Depreciation tax shield 2600000 640 Depreciation tax shield 17333333 The aftertax cost of debt for the lessor is Aftertax debt cost 091 7 40 0540 Using all of this information we can calculate the minimum pretax lease payment for the lessor as NAL 0 2600000 7 PMT1 7 40PVIFA5 4076 17333333PVIFA5 4076 PMT 57580554 1 Since both companies have the same tax rate there is only one lease payment that will result in a zero NAL for each company We will calculate cash ows from the depreciation tax shield rst The depreciation tax shield is Depreciation tax shield 475000334 5383333 The aftertax cost of debt is Aftertax debt cost 101 7 34 0660 Using all of this information we can calculate the lease payment as NAL 0 475000 7PMT1 7 34PVIFA6 6073 5383333PVIFA6 6073 PMT 19067418 4l5 b To generalize the result from part a Let T1 denote the lessor s tax rate Let T2 denote the lessee s tax rate Let P denote the purchase price of the asset Let D equal the annual depreciation expense Let N denote the length of the lease in years Let R equal the pretax cost of debt The value to the lessor is N 7 ValueLm p i Z 1R17T1 And the value to the lessee is 7 N L1T DT ValueLessee 7 P 7 21 2 Since all the values in both equations above are the same except T1 and T2 we can see that the values of the lease to its two parties will be opposite in sign only ile T2 0 Since the lessor s tax bracket is unchanged the zero NAL lease payment is the same as we found in part a The lessee will not realize the depreciation tax shield and the aftertax cost of debt will be the same as the pretax cost of debt So the lessee s maximum lease payment will be NAL 0 475000 PMTPVIFA103 PMT 19100453 Both parties have positive NAL for lease payments between 19067418 and 19100453 14 The decision to buy or lease is made by looking at the incremental cash ows The loan offered by the bank merely helps you to establish the appropriate discount rate Since the deal they are offering is the same as the marketwide rate you can ignore the offer and simply use 9 percent as the pretax discount rate In any capital budgeting project you do not consider the nancing which was to be applied to a speci c project The only exception would be if a speci c and special nancing deal were tied to a speci c project like a lowerthanmarket interest rate loan if you buy a particular car 416 The incremental cash ows from leasing the machine are the lease payments the tax savings on the lease the lost depreciation tax shield and the saved purchase price of the machine The lease payments are due at the beginning of each year so the incremental cash ows are Year 0 Year Year2 Year3 Year4 Lease Leasepayment 1500000 1500000 1500000 1500000 Tax savings on lease 525000 525000 525000 525000 Lost dep tax shield 446250 446250 7446250 7446250 Equipment cost 5100000 4125000 71421250 71421250 71421250 7446250 The aftertax discount rate is Aftertax discount rate 091 7 35 Aftertax discount rate 0585 or 585 So the NAL of leasing is NAL 7 4125000 7 1421500PVIFA5 853 7 446250 105854 NAL 7 74006581 Since the NAL is negative the company should buy the equipment The company is indifferent at the lease payment which makes the NAL of the lease equal to zero The NAL equation of the lease is 0 7 4125000 7PMT1 7 35 7 PMT1735PVIFA5 gym 7 446250 105854 PMT 7 148325212 The different borrowing rates are irrelevant A basic tenant of capital budgeting is that the return of a project depends on the risk of the project Since the lease payments are affected by the riskiness of the lessee the lessee s cost of debt is the appropriate interest rate for the analysis by both companies Since the both companies have the same tax rate there is only one lease payment that will result in a zero NAL for each company We will calculate cash ows from the depreciation tax shield first The depreciation tax shield is Depreciation tax shield 330000334 37400 The aftertaX cost of debt is the lessee s cost of debt which is Aftertax debt cost 0917 34 0594 Using all of this information we can calculate the lease payment as NAL 0 330000 7PMT1 7 34PVIFA5 9473 37400PVIFA5 943 PMT 13018063 417 0 Since the lessor s tax bracket is unchanged the zero NAL lease payment is the same as we found in part b The lessee will not realize the depreciation tax shield and the aftertax cost of debt will be the same as the pretax cost of debt So the lessee s maximum lease payment will be NAL 0 330000 PMTPv1FA93 PMT 13036807 Both parties have positive NAL for lease payments between 13018063 and 13036807 16 The APR of the loan is the lease factor times 2400 so APR 0003422400 821 To calculate the lease payment we rst need the net capitalization cost which is the base capitalized cost plus any other costs minus any down payment or rebates So the net capitalized cost is Net capitalized cost 28000 450 7 2000 Net capitalized cost 26450 The depreciation charge is the net capitalized cost minus the residual value divided by the term of the lease which is Depreciation charge 26450 7 16500 36 Depreciation charge 27639 Next we can calculate the finance charge which is the net capitalized cost plus the residual value times the lease factor or Finance charge 26450 l6500000342 Finance charge 14689 And the taxes on each monthly payment will be Taxes 27639 14689007 Taxes 2963 The monthly lease payment is the sum of the depreciation charge the finance charge and taxes which will be Lease payment 27639 14689 2963 Lease payment 45291 418 17 Challenge With a fouryear loan the annual loan payment will be 4500000 PMTPVIFA874 PMT 135864362 The aftertax loan payment is found by A ertax payment PretaX payment 7 Interest tax shield So we need to nd the interest tax shield To nd this we need a loan amortization table since the interest payment each year is the beginning balance times the loan interest rate of 8 percent The interest tax shield is the interest payment times the tax rate The amortization table for this loan is Beginning Total Interest Principal Ending Year balance payment payment payment balance 1 450000000 135864362 36000000 99864362 350135638 2 350135638 135864362 28010851 107853511 242282127 3 242282127 135864362 19382570 116481792 125800335 4 125800335 135864362 10064027 125800335 000 So the total cash ows each year are A ertax loan payment OCF Total cash ow Year 1 1358643 7 36000035 123264362 7 1271250 73860638 Year 2 1358643 7 2801085135 126060564 7 1271250 71064436 Year 3 1358643 71938257035 129080462 7 1271250 1955462 Year 4 1358643 71006402735 132341953 7 1271250 5216953 So the NAL with the loan payments is NAL 0 7 38606381052 7 106443610522 195546210523 521695310524 NAL 1307425 The NAL is the same because the present value of the aftertax loan payments discounted at the a eitax cost of capital which is the a eitax cost of debt equals 4500000 a The decision to buy or lease is made by looking at the incremental cash ows so we need to nd the cash ows for each alternative The cash ows if the company leases are Cash ows from leasing A ertax cost savings l2000l 7 34 A ertax cost saVings 7920 The tax bene t of the lease is the lease payment times the tax rate so the tax bene t of the lease is Lease tax bene t 2700034 419 Lease tax bene t 9180 We need to remember the lease payments are due at the beginning of the year So if the company leases the cash ows each year will be M M W Year3 Year4 Year5 A ertaX savings 7920 7920 7920 7920 7920 Lease payment 727000 727000 727000 727000 727000 Tax bene t 9180 9180 9180 9180 9180 Net cash ows 717820 79900 79900 79900 79900 7920 The amount the company borrows and the repayment schedule are irrelevant since the company maintains a target debtequity ratio So the cash ows from buying the machine will be Cash ows from purchasing A ertax cost savings 200001 734 A ertax cost savings 13200 And the deprecation tax shield will be Depreciation tax shield 150000 534 Depreciation tax shield 10200 M M W Year3 Year4 Year5 A ertaX savings 13200 13200 13200 13200 13200 Purchase 7150000 Dep tax shield 10200 10200 10200 10200 10200 Net cash ows 7150000 23400 23400 23400 23400 23400 Now we can calculate the incremental cash ows from leasing versus buying by subtracting the net cash ows from buying from the net cash ows from leasing The incremental cash ows from leasing are Year 0 132180 Year 1 733300 Year 2 733300 Lease 7 Buy The aftertax discount rate is A ertax discount rate 1017 34 A ertax discount rate 0660 or 660 So the NAL of leasing is Year 3 733300 NAL 7 132180 7 33300Pv1FA6604 7 15480 10665 NAL 711414 Since the NAL is positive the company should lease the equipment 420 Year 4 733300 Year 5 715480 As long as the company maintains its target debtequity ratio the answer does not depend upon the form of nancing used for the direct purchase A financial lease will displace debt regardless of the form of nancing The amount of displaced debt is the PV of the incremental cash ows from year one through ve PV 33300PVIFA6 60m 15480 106605 PV 12506586 421 CHAPTER 22 OPTIONS AND CORPORATE FINANCE Answers to Concept Questions 1 A call option confers the right without the obligation to buy an asset at a given price on or before a given date A put option confers the right without the obligation to sell an asset at a given price on or before a given date You would buy a call option if you expect the price of the asset to increase You would buy a put option if you expect the price of the asset to decrease A call option has unlimited potential pro t while a put option has limited potential pro t the underlying asset s price cannot be less than zero The buyer of a call option pays money for the right to buy The buyer of a put option pays money for the right to sell The seller of a call option receives money for the obligation to sell The seller of a put option receives money for the obligation to buy 5 1 9 9 An American option can be exercised on any date up to and including the expiration date A European option can only be exercised on the expiration date Since an American option gives its owner the right to exercise on any date up to and including the expiration date it must be worth at least as much as a European option if not more The intrinsic value of a call is MaxS 7 E 0 The intrinsic value of a put is MaxE 7 S 0 The intrinsic value of an option is the value at expiration The call is selling for less than its intrinsic value an arbitrage opportunity exists Buy the call for 10 exercise the call by paying 35 in return for a share of stock and sell the stock for 50 You ve made a riskless 5 pro t The prices of both the call and the put option should increase The higher level of downside risk still results in an option price of zero but the upside potential is greater since there is a higher probability that the asset will nish in the money False The value of a call option depends on the total variance of the underlying asset not just the systematic variance The call option will sell for more since it provides an unlimited profit opportunity while the potential pro t from the put is limited the stock price cannot fall below zero The value of a call option will increase and the value of a put option will decrease 422 p n O p n p A p n DJ p n J p A UI p A 5 The reason they don t show up is that the US government uses cash accounting ie only actual cash in ows and outflows are counted not contingent cash ows From a political perspective they would make the deficit larger so that is another reason not to count them Whether they should be included depends on whether we feel cash accounting is appropriate or not but these contingent liabilities should be measured and reported They currently are not at least not in a systematic fashion Increasing the time to expiration increases the value of an option The reason is that the option gives the holder the right to buy or sell The longer the holder has that right the more time there is for the option to increase or decrease in the case of a put in value For example imagine an outofthe money option that is about to expire Because the option is essentially worthless increasing the time to expiration would obviously increase its value An increase in volatility acts to increase both call and put values because the greater volatility increases the possibility of favorable inthemoney payoffs A put option is insurance since it guarantees the policyholder will be able to sell the asset for a speci c price Consider homeowners insurance If a house burns down it is essent1ally worthless In essence the homeowner is selling the worthless house to the insurance company for the amount of insurance The equityholders of a rm nanced partially with debt can be thought as holding a call option on the assets of the fum with a strike price equal to the debt s face value and a time to expiration equal to the debt s time to maturity If the value of the firm exceeds the face value of the debt when it matures the rm will pay off the debtholders in full leaving the equityholders with the rm s remaining assets However if the value of the firm is less than the face value of debt when it matures the firm must liquidate all of its assets in order to pay off the debtholders and the equityholders receive nothing Consider the following Let VL the value of a firm financed with both debt and equity FVdebt the face value of the rm s outstanding debt at maturity If VL lt FVdebt If VL gt FVdebt Payoff to debtholders VL F Vdebt Payoff to l quot 39 39 39 0 VL 7FVdebt VL VL Notice that the payoff to equityholders is identical to a call option of the form Max0 ST 7 K where the stock price at expiration ST is equal to the value of the firm at the time of the debt s maturity and the strike price K is equal to the face value of outstanding debt Since you have a large number of stock options in the company you have an incentive to accept the second project which will increase the overall risk of the company and reduce the value of the rm s debt However accepting the risky project will increase your wealth as the options are more valuable when the risk of the rm increases Rearranging the putcall parity formula we get S 7 PVE C 7 P Since we know that the stock price and exercise price are the same assuming a positive interest rate the left hand side of the equation must be greater than zero This implies the price of the call must be higher than the price of the put in this situation 423 17 Rearranging the putcall parity formula we get S 7 PVE C 7 P If the call and the put have the same price we know C 7 P 0 This must mean the stock price is equal to the present value of the exercise price so the put is inthemoney 18 A stock can be replicated using a long call to capture the upside gains a short put to re ect the downside losses and a Tbill to re ect the time value component 7 the wait factor Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic 1 a The value of the call is the stock price minus the present value of the exercise price so C0 70 7 601055 1313 The intrinsic value is the amount by which the stock price exceeds the exercise price of the call so the intrinsic value is 10 b The value of the call is the stock price minus the present value of the exercise price so C0 70 7 501055 2261 The intrinsic value is the amount by which the stock price exceeds the exercise price of the call so the intrinsic value is 20 c The value of the put option is 0 since there is no possibility that the put will nish in the money The intrinsic value is also 0 2 a The calls are in the money The intrinsic value of the calls is 3 b The puts are out of the money The intrinsic value of the puts is 0 c The lVIar call and the Oct put are mispriced The call is mispriced because it is selling for less than its intrinsic value If the option expired today the arbitrage strategy would be to buy the call for 280 exercise it and pay 80 for a share of stock and sell the stock for 83 A riskless pro t of 020 results The October put is mispriced because it sells for less than the July put To take advantage of this sell the July put for 390 and buy the October put for 365 for a cash inflow of 025 The exposure of the short position is completely covered by the long position in the October put with a positive cash inflow today 3 a Each contract is for 100 shares so the total cost is Cost 10100 sharescontract760 Cost 7600 424 If the stock price at expiration is 140 the payoff is Payoff 10100140 7110 Payoff 30000 If the stock price at expiration is 125 the payoff is Payoff 10100125 7110 Payoff 15000 Remembering that each contract is for 100 shares of stock the cost is Cost 10100470 Cost 4700 The maximum gain on the put option would occur if the stock price goes to 0 We also need to subtract the initial cost so Maximum gain 10100110 7 4700 Maximum gain 105300 If the stock price at expiration is 104 the position will have a pro t of Pro t 10100110 7 104 7 4700 Profit 1300 At a stock price of 103 the put is in the money As the writer you will make Net loss 4700 710100110 7103 Net loss 72300 At a stock price of 132 the put is out of the money so the writer will make the initial cost Net gain 4700 At the breakeven you would recover the initial cost of 4700 so 4700 10100110 7 ST ST 10530 For terminal stock prices above 10530 the writer of the put option makes a net pro t ignoring transaction costs and the effects of the time value of money The value of the call is the stock price minus the present value of the exercise price so C0 70 760106 C0 1340 425 p A O b Using the equation presented in the text to prevent arbitrage we nd the value of the call is 70 85 7 6585 7 80C0 65106 C0 217 a The value of the call is the stock price minus the present value of the exercise price so C0 60 735105 C0 2667 b Using the equation presented in the text to prevent arbitrage we nd the value of the call is 60 2C0 50105 C0 619 Using putcall parity and solving for the put price we get 47 P 45elt 26lt312gt 380 P 151 Using putcall parity and solving for the call price we get 57 489 606lt 36gtlt5gt C C 296 Using putcall parity and solving for the stock price we get 3 315 85e 048312 612 s 8696 Using putcall parity we can solve for the riskfree rate as follows 4730 265 456M 532 4463 45e R2 gt 09917 67W ln099l7 111e R 2m 00083 711212 Rf 495 Using the BlackScholes option pricing model to nd the price of the call option we nd d1 ln4650 06 5422 x 312 54 x J312 71183 d2 71183 454 x 4312 73883 Nd1 4529 Nd2 3489 426 p A N 03 Putting these values into the BlackScholes model we nd the call price is C 464529 7 50e 6253489 365 Using putcall parity the put price is Put 50e 0625 365 746 690 Using the BlackScholes option pricing model to nd the price of the call option we nd d1 ln9390 04 6222 x 812 62 x M 3706 d2 3706 762 x M 71357 Nd1 6445 Nd2 4460 Putting these values into the BlackScholes model we nd the call price is C 936445 7 906 048124460 2085 Using putcall parity the put price is Put 90e 04812 2085 7 93 1548 The delta of a call option is Nd1 so d1 ln7470 05 5622 x 75 56 x m 4344 Nd1 6680 For a call option the delta is 6680 For a put option the delta is Put delta 6680 7 1 73320 The delta tells us the change in the price of an option for a 1 change in the price of the underlying asset Using the BlackScholes option pricing model with a stock price of 1900000 and an exercise price of 2100000 the price you should receive 1s d1 lnl9000002100000 05 7 2522 x 1212 25 x m 70753 d2 70753 725 x W 73253 Nd1 4700 Nd2 3725 427 4 UI Putting these values into the BlackScholes model we nd the call price is C 19000004700 7 2100000e 51gt3725 14892392 Using the call price we found in the previous problem and putcall parity you would need to pay Put 2100000e 05 148923927 1900000 24650571 You would have to pay 24650571 in order to guarantee the right to sell the land for 2100000 Using the BlackScholes option pricing model to nd the price of the call option we nd d1 1n7480 06 7 5322 x 612 53 gtlt 612 7 0594 d2 0594753 x W 73154 Nd1 5237 Nd2 3762 Putting these values into the BlackScholes model we nd the call price is C 745237 7 80e 65 3762 954 Using putcall parity we nd the put price is Put 80e 65 954 7 74 1318 a The intrinsic value of each option is Call intrinsic value MaXS 7 E 0 0 Put intrinsic value MaXE 7 S 0 6 b Option value consists of time value and intrinsic value so Call option value Intrinsic value Time value 954 0 TV TV 954 Put option value Intrinsic value Time value 1318 6 TV TV 718 0 The time premium theta is more important for a call option than a put option therefore the time premium is in general larger for a call option 428 16 H l The stock price can either increase 15 percent or decrease 15 percent The stock price at expiration will either be Stock price increase 54l 15 6210 Stock price decrease 54l 7 l5 4590 The payoff in either state will be the maximum stock price minus the exercise price or zero which is Payoffifstock price increases Max6210 7 50 0 1210 Payoff if stock price decreases Max4590 7 50 0 0 To get a 15 percent return we can use the following expression to determine the riskneutral probability of a rise in the price of the stock Riskfree rate ProbabilityRiseRetumRse ProbabilityFanXRetumFan 08 ProbabilityRise15 1 7 ProbabilityRise7 15 ProbabilityRse 7667 And the probability of a stock price decrease is ProbabilityFan 1 7 7667 2333 So the risk neutral value of a call option will be Call value 7 7667 x 1210 2333 x 0 1 08 Call value 859 The stock price increase decrease and option payoffs will remain unchanged since the stock price change is the same The new risk neutral probability of a stock price increase is Riskfree rate ProbabilityRiseXRetummse ProbabilityFal1RetumFan 05 ProbabilityRise15 1 7 ProbabilityRise715 ProbabilityRse 6667 And the probability of a stock price decrease is Probabilitde1 1 76667 3333 So the risk neutral value of a call option will be Callvalue 7 6667 x 1210 3333 x 0 1 05 Call value 768 429 1 on p A O N O 22 Intermediate If the exercise price is equal to zero the call price will equal the stock price which is 75 If the standard deviation is zero d1 and d2 go to 8 so Nd1 and Nd2 go to 1 This is the no risk call option formula which is C s 7 Ben C 86 7 80e 05612 798 If the standard deviation is infinite d1 goes to positive in nity so Nd1 goes to 1 and d2 goes to negative in nity so Nd2 goes to 0 In this case the call price is equal to the stock price which is 35 We can use the BlackScholes model to value the equity of a rm Using the asset value of 15800 as the stock price and the face value of debt of 15000 as the exercise price the value of the lm s equity is d ln1580015000 05 3822 x 138 x J1 4583 d2 4583 738 x Ji 0783 Nd1 6766 Nd2 5312 Putting these values into the BlackScholes model we nd the equity value is Equity 158006766715000e 515312 311131 The value of the debt is the firm value minus the value of the equity so D 15800 7311131 1268869 a We can use the BlackScholes model to value the equity of a firm Using the asset value of 17000 as the stock price and the face value of debt of 15000 as the exercise price the value of the rm if it accepts project A is d1 ln1700015000 05 5522 x 1 55 x 11 5935 d2 5935 755 x Ji 0435 Nd1 7236 Nd2 5173 Putting these values into the BlackScholes model we find the equity value is EA 170007236 715000e 0515173 491905 430 DJ The value of the debt is the rm value minus the value of the equity so DA 17000 7491905 1208095 And the value of the rm if it accepts Project B is d1 ln1740015000 05 3422 x 1 34 x J1 7536 d2 7536 7 34 x J1 4136 Nd1 7745 Nd2 6604 Putting these values into the BlackScholes model we find the equity value is EB 174007745 7 15000e 516604 405241 The value of the debt is the rm value minus the value of the equity so DB 17400 7 405241 1334759 9 Although the NPV of project B is higher the equity value with project A is higher While NPV represents the increase in the value of the assets of the rm in this case the increase in the value of the lm s assets resulting from project B is mostly allocated to the debtholders resulting in a smaller increase in the value of the equity Stockholders would therefore prefer project A even though it has a lower NPV 5 1 Yes If the same group of investors have equal stakes in the rm as bondholders and stock holders then totalme value matters and project B should be chosen since it increases the value of the rm to 17400 instead of 17000 9 Stockholders may have an incentive to take on riskier less pro table projects if the rm is leveraged the higher the rm s debt load all else the same the greater is this incentive We can use the BlackScholes model to value the equity of a rm Using the asset value of 27200 as the stock price and the face value of debt of 25000 as the exercise price the value of the lm s equity is d1 ln2720025000 05 5322 x 1 53 gtlt 71 5185 d2 5185 753 x J1 70115 Nd1 6979 Nd2 4954 Putting these values into the BlackScholes model we nd the equity value is 431 4 Equity 272006979 7 2500067051gt4954 720284 The value of the debt is the firm value minus the value of the equity so D 27200 77202 84 1999716 The return on the company s debt is 1999716 2500067R 79989 e R RD iln79989 2233 a The combined value of equity and debt of the two rms is Equity 311131 720284 1031415 Debt 1268869 1999716 3268585 b For the new rm the combined market value of assets is 43000 and the combined face value of debt is 40000 Using BlackScholes to nd the value of equity for the new film we nd d1 ln4300040000 05 2922 x 1 29 x J1 5668 d2 5668 729 x J1 2768 Nd1 7146 Nd2 6090 Putting these values into the BlackScholes model we find the equity value is E 430007146740000e 0516090 755351 The value of the debt is the rm value minus the value of the equity so D 40000 7 755331 3544649 5 1 The change in the value of the rm s equity is Equity value change 755351 7 1031415 7276064 The change in the value of the rm s debt is Debt 3544649 7 3268585 276064 d In a purely nancial merger when the standard deviation of the assets declines the value of the equity declines as well The shareholders will lose exactly the amount the bondholders gain The bondholders gain as a result of the coinsurance effect That is it is less likely that the new company will default on the debt 432 25 a 9 5 1 9 5 Using BlackScholes model to value the equity we get d1 ln2100000025000000 06 3922 x 10 39 x 610 9618 d2 7 9618 7 39 x J10 7 72715 Nd1 8319 Nd2 3930 Putting these values into BlackScholes E 7 210000008319 7 250000006 061 gt3930 7 1207824348 The value of the debt is the rm value minus the value of the equity so D 21000000 7 1207824348 892175652 Using the equation for the PV of a continuously compounded lump sum we get 892175652 7 250000006 R0 35687 7 e Rlquot RD 7110ln35687 1030 Using BlackScholes model to value the equity we get d1 ln2220000025000000 06 3922 x 10 39 x J10 10068 d2 10068 7 39 gtlt 110 72265 Nd1 8430 Nd2 4104 Putting these values into BlackScholes E 7 222000008430 7 55250000006 0610gt4104 7 1308330104 The value of the debt is the rm value minus the value of the equity so D 22200000 7 1308330104 911669896 Using the equation for the PV of a continuously compounded lump sum we get 911669896 7 250000006 R0 36467 7 61 1quot RD 7 71101n36467 7 1009 433 26 a When the rm accepts the new project part of the NPV accrues to bondholders This increases the present value of the bond thus reducing the return on the bond Additionally the new project makes the firm safer in the sense that it increases the value of assets thus increasing the probability the call will end inthemoney and the bondholders will receive their payment In order to solve a problem using the twostate option model we rst need to draw a stock price tree containing both the current stock price and the stock s possible values at the time of the option s expiration Next we can draw a similar tree for the option designating what its value will be at expiration given either of the 2 possible stock price movements Price of stock Call option price with a strike of 85 Today 1 year Today 1 year 98 13 Max0 98 7 85 85 7 70 0 Max0 70 7 85 The stock price today is 85 It will either increase to 98 or decrease to 70 in one year If the stock price rises to 98 the call will be exercised for 85 and a payoff of 13 will be received at expiration If the stock price falls to 70 the option will not be exercised and the payoff at expiration will be zero If the stock price rises its return over the period is 2250 percent 9880 7 1 If the stock price falls its return over the period is 71250 percent 70 80 7 1 We can use the following expression to determine the riskneutral probability of a rise in the price of the stock Riskfree rate ProbabilityRiseRetumRse ProbabilityFanXRetumFan Riskfree rate ProbabilityRiseRetumRise l 7ProbabilityRiseXRetumFan 025 ProbabilityRise02250 17 ProbabilityRise4 1250 ProbabilityRise 4286 or 4286 This means the risk neutral probability of a stock price decrease is ProbabilityFan l 7 ProbabilityRse ProbabilityFan 1 7 4286 ProbabilityFan 5714 or 5714 Using these riskneutral probabilities we can now determine the expected payoff of the call option at expiration The expected payoff at expiration is Expected payoff at expiration 428613 57140 Expected payoff at expiration 557 434 Since this payoff occurs 1 year from now we must discount it back to the value today Since we are using riskneutral probabilities we can use the riskfree rate so PVExpected payoff at expiration 557 1025 PVExpected payoff at expiration 544 Yes there is a way to create a synthetic call option with identical payoffs to the call option described above In order to do this we will need to buy shares of stock and borrow at the risk free rate The number of shares to buy is based on the delta of the option where delta is defined as Delta Swing of option Swing of stock Since the call option will be worth 13 if the stock price rises and 0 if it falls the delta of the option is 13 13 7 0 Since the stock price will either be 98 or 70 at the time of the option s expiration the swing of the stock is 28 98 7 70 With this information the delta of the option is Delta 13 28 Delta 046 Therefore the first step in creating a synthetic call option is to buy 046 of a share of the stock Since the stock is currently trading at 80 per share this will cost 3171 046701 025 In order to determine the amount that we should borrow compare the payoff of the actual call option to the payoff of delta shares at expiration Call Option If the stock price rises to 98 Payoff 13 If the stock price falls to 70 Payoff 0 Delta Shares If the stock price rises to 98 Payoff 04698 4550 If the stock price falls to 80 Payoff 04670 3250 The payoff of his synthetic call position should be identical to the payoff of an actual call option However owning 046 of a share leaves us exactly 3250 above the payoff at expiration regardless of whether the stock price rises or falls In order to reduce the payoff at expiration by 3250 we should borrow the present value of 3250 now In one year the obligation to pay 3250 will reduce the payoffs so that they exactly match those of an actual call option So purchase 046 ofa share of stock and borrow 3171 3250 1025 in order to create a synthetic call option with a strike price of 85 and 1 year until expiration Since the cost of the stock purchase is 3715 to purchase 046 of a share and 3171 is borrowed the total cost of the synthetic call option is Cost of synthetic option 3715 7 3171 Cost of synthetic option 544 435 This is exactly the same price as an actual call option Since an actual call option and a synthetic call option provide identical payoff structures we should not expect to pay more for one than for the other In order to solve a problem using the twostate option model we first draw a stock price tree containing both the current stock price and the stock s possible values at the time of the option s expiration Next we can draw a similar tree for the option designating what its value will be at expiration given either of the 2 possible stock price movements Price of stock Put option price with a strike of 40 Today 6 months Today 6 months 60 0 Max0 40 7 60 30 15 25 Max0 40 7 15 The stock price today is 30 It will either decrease to 15 or increase to 60 in six months If the stock price falls to 15 the put will be exercised and the payoff will be 25 If the stock price rises to 60 the put will not be exercised so the payoff will be zero If the stock price rises its return over the period is 100 6030 7 1 If the stock price falls its return over the period is 750 1530 71 Use the following expression to determine the riskneutral probability of a rise in the price of the stock Riskfree rate ProbabilityRiseXReturnmse ProbabilityFal1ReturnFan Riskfree rate ProbabilityRiseReturnRse l 7ProbabilityRiseXReturnFan The riskfree rate over the next six months must be used in the order to match the timing of the expected stock price change Since the riskfree rate per annum is 8 percent the riskfree rate over the next six months is 392 percent 108 2 71 so 0392 ProbabilityRiseXl 1 7 ProbabilityRise750 ProbabilityRise 3595 or 3595 Which means the riskneutral probability of a decrease in the stock price is Probabilitde1 l 7 ProbabilityRse ProbabilityFan 1 73595 ProbabilityFan 6405 or 6405 Using these riskneutral probabilities we can determine the expected payoff to put option at expiration as Expected payoff at expiration 35950 640525 Expected payoff at expiration 1601 436 Since this payoff occurs 6 months from now we must discount it at the riskfree rate in order to nd its present value which is PVExpected payoff at expiration 1601 108 2 PVExpected payoff at expiration 1541 Yes there is a way to create a synthetic put option with identical payoffs to the put option described above In order to do this we need to short shares of the stock and lend at the risk free rate The number of shares that should be shorted sell is based on the delta of the option where delta is de ned as Delta Swing of option Swing of stock Since the put option will be worth 0 if the stock price rises and 25 if it falls the swing of the call option is 725 0 7 25 Since the stock price will either be 60 or 15 at the time of the option s expiration the swing of the stock is 45 60 7 15 Given this information the delta of the put option is Delta Swing of option Swing of stock Delta 725 45 Delta 4156 Therefore the rst step in creating a synthetic put option is to short 056 of a share of stock Since the stock is currently trading at 30 per share the amount received will be 1667 056 X 30 as a result of the short sale In order to determine the amount to lend compare the payoff of the actual put option to the payoff of delta shares at expiration Put option If the stock price rises to 60 Payoff 0 If the stock price falls to 15 Payoff 25 Delta shares If the stock price rises to 60 Payoff 415660 73333 If the stock price falls to 15 Payoff 45615 7833 The payoff of the synthetic put position should be identical to the payoff of an actual put option However shorting 056 of a share leaves us exactly 3333 below the payoff at expiration whether the stock price rises or falls In order to increase the payoff at expiration by 3333 we should lend the present value of 3333 now In six months we will receive 3333 which will increase the payoffs so that they exactly match those of an actual put option So the amount to lend is Amount to lend 7 3333 108 2 Amount to lend 3208 Since the short sale results in a positive cash ow of 1667 and we will lend 3208 the total cost of the synthetic put option is Cost of synthetic put 3208 7 1667 Cost of synthetic put 1541 437 This is exactly the same price as an actual put option Since an actual put option and a synthetic put option provide identical payoff structures we should not expect to pay more for one than for the other The company would be interested in purchasing a call option on the price of gold with a strike price of 875 per ounce and 3 months until expiration This option will compensate the company for any increases in the price of gold above the strike price and places a cap on the amount the rm must pay for gold at 875 per ounce In order to solve a problem using the twostate option model rst draw a price tree containing both the current price of the underlying asset and the underlying asset s possible values at the time of the option s expiration Next draw a similar tree for the option designating what its value will be at expiration given either of the 2 possible stock price movements Price of gold Call option price with a strike of 875 Today 3 months Today 3 months 975 100 Max0 975 7 875 815 7 740 0 Max0 740 7 875 The price of gold is 815 per ounce today If the price rises to 975 the company will exercise its call option for 875 and receive a payoff of 100 at expiration If the price of gold falls to 740 the company will not exercise its call option and the lm will receive no payoff at expiration If the price of gold rises its return over the period is 1963 percent 975 815 7 1 If the price of gold falls its return over the period is 7920 percent 740 815 71 Use the following expression to determine the riskneutral probability of a rise in the price of gold Riskfree rate ProbabilityRiseXRetummse ProbabilityFal1RetumFan Riskfree rate ProbabilityRiseRetumRse 7 1 ProbabilityRiseRetumFan The riskfree rate over the next three months must be used in the order to match the timing of the expected price change Since the riskfree rate per annum is 650 percent the riskfree rate over the next three months is 159 percent 10650 4 7 1 so 0159 ProbabilityRiseX 1963 1 7ProbabilityRise70920 ProbabilityRise 3742 or 3742 And the riskneutral probability of a price decline is ProbabilityFan 1 7 ProbabilityRse ProbabilityFan 1 73742 ProbabilityFall 6258 or 6258 Using these riskneutral probabilities we can determine the expected payoff to of the call option at expiration which Will be 438 Expected payoff at expiration 3742100 62580 Expected payoff at expiration 3742 Since this payoff occurs 3 months from now it must be discounted at the riskfree rate in order to find its present value Doing so we nd PVExpected payoff at expiration 3742 10650 4 PVExpected payoff at expiration 3683 Therefore given the information about gold s price movements over the next three months a European call option with a strike price of 875 and three months until expiration is worth 3683 today Yes there is a way to create a synthetic call option with identical payoffs to the call option described above In order to do this the company will need to buy gold and borrow at the risk free rate The amount of gold to buy is based on the delta of the option where delta is de ned as Delta Swing of option Swing of price of gold Since the call option will be worth 100 if the price of gold rises and 0 if it falls the swing of the call option is 100 100 7 0 Since the price of gold will either be 975 or 740 at the time of the option s expiration the swing of the price of gold is 235 975 7 740 Given this information the delta of the call option is Delta Swing of option Swing of price of gold Delta 100 235 Delta 043 Therefore the rst step in creating a synthetic call option is to buy 043 of an ounce of gold Since gold currently sells for 815 per ounce the company will pay 34681 043 gtlt 815 to purchase 043 of an ounce of gold In order to determine the amount that should be borrowed compare the payoff of the actual call option to the payoff of delta shares at expiration Call Option If the price of gold rises to 975 Payoff 100 If the price of gold falls to 740 Payoff 0 Delta Shares If the price of gold rises to 975 Payoff 043975 41489 If the price of gold falls to 740 Payoff 043740 31489 The payoff of this synthetic call position should be identical to the payoff of an actual call option However buying 043 of a share leaves us exactly 31489 above the payoff at expiration whether the price of gold rises or falls In order to decrease the company s payoff at expiration by 31489 it should borrow the present value of 31489 now In three months the company must pay 31489 which will decrease its payoffs so that they exactly match those of an actual call option So the amount to borrow today is Amount to borrow today 31489 10650 4 439 Amount to borrow today 30997 d Since the company pays 34681 in order to purchase gold and borrows 30997 the total cost of the synthetic call option is 3683 34681 7 30997 This is exactly the same price for an actual call option Since an actual call option and a synthetic call option provide identical payoff structures the company should not expect to pay more for one than for the other 29 To construct the collar the investor must purchase the stock sell a call option with a high strike price and buy a put option with a low strike price So to find the cost of the collar we need to nd the price of the call option and the price of the put option We can use BlackScholes to find the price of the call option which will be Price 0fcall option with 11 0 strike price d1 1n85110 07 5022 x 612 50 x 612 7 74535 d2 74535 750 x m 7 708070 Nd1 3251 Nd2 2098 Putting these values into the BlackScholes model we find the call price is C 853251 7 110 76122098 535 Now we can use BlackScholes and putcall parity to nd the price of the put option with a strike price of 65 Doing so we nd Price ofput option with 65 strike price d1 1n8565 07 5022 x 612 50 x 612 7 10345 d2 710345 750 x M 06810 Nd1 8496 Nd2 7521 Putting these values into the BlackScholes model we find the call price is C 858496765e 076127521 2501 Rearranging the putcall parity equation we get P C 7 S Xe Rt P 825017 85 65e 07612 P 277 440 5 So the investor will buy the stock sell the call option and buy the put option so the total cost is Total cost of collar 85 7 535 277 Total cost of collar 8243 Challenge a b 5 1 9 Using the equation for the PV of a continuously compounded lump sum we get PV 40000 x 6052 3619350 Using BlackScholes model to value the equity we get d1 lnl900040000 05 6022 x 2 60 gtlt 12 73352 d2 733527 60 X J2 711837 Nd1 3687 Nd2 1183 Putting these values into BlackScholes E 1900036877 40000e 0521183 272575 And using putcall parity the price of the put option is Put 40000e 05 272575 7 19000 1991925 The value of a risky bond is the value of a riskfree bond minus the value of a put option on the rm s equity so Value ofrisky bond 3619350 7 1991925 1627425 Using the equation for the PV of a continuously compounded lump sum to nd the return on debt we get 1627425 400006 40686 6 RD 712ln40686 4496 or 4496 The value of the debt with ve years to maturity at the riskfree rate is PV 40000 x 6055 3115203 441 Using BlackScholes model to value the equity we get d1 ln1900040000 05 6022 x 5 60 x J5 3023 d2 3023 760 x J5 710394 Nd1 6188 Nd2 1493 Putting these values into BlackScholes E 1900061887 40000e 55gt1493 710526 And using putcall parity the price of the put option is Put 40000e 55 710526 7 19000 1925729 The value of a risky bond is the value of a riskfree bond minus the value of a put option on the rm s equity so Value ofrisky bond 3115203 7 1925729 1189474 Using the equation for the PV of a continuously compounded lump sum to nd the return on debt we get Return on debt 1189474 400006 29737 erR5 RD 715ln29737 2426 The value of the debt declines because of the time value of money ie it will be longer until shareholders receive their payment However the required return on the debt declines Under the current situation it is not likely the company will have the assets to pay off bondholders Under the new plan where the company operates for ve more years the probability of increasing the value of assets to meet or exceed the face value of debt is higher than if the company only operates for two more years Using the equation for the PV of a continuously compounded lump sum we get PV 50000 x 6065 3704091 442 b Using BlackScholes model to value the equity we get 5 1 d1 ln4600050000 06 5022 x 5 50 x J 7528 d2 7528 750 x J5 73653 Nd1 7742 Nd2 3575 Putting these values into BlackScholes E 4600077427 50000e 65gt3575 2237293 And using putcall parity the price of the put option is Put 850000606 2237293 7 46000 1341384 The value of a risky bond is the value of a riskfree bond minus the value of a put option on the rm s equity so Value of risky bond 3704091 7 1341384 2362707 Using the equation for the PV of a continuously compounded lump sum to nd the return on debt we get Return on debt 2362707 8500006 4725 e M RD 715ln4725 1499 Using the equation for the PV of a continuously compounded lump sum we get PV 850000 x e706 3704091 Using BlackScholes model to value the equity we get d1 ln4600050000 06 6022 x 5 60 x 5 8323 d2 8323 760 gtlt J5 75094 Nd1 7974 Nd2 3052 Putting these values into BlackScholes E 460007974750000e 0653052 2537250 443 5 And using putcall parity the price of the put option is Put 7 50000e 06 2537250 7 46000 7 1641341 The value of a risky bond is the value of a riskfree bond minus the value of a put option on the rm s equity so Value ofrisky bond 3704091 7 1641341 2062750 Using the equation for the PV of a continuously compounded lump sum to nd the return on debt we get Return on debt 2062750 7 500006 125 7 4 RD 7 715ln41255 7 1771 The value of the debt declines Since the standard deviation of the company s assets increases the value of the put option on the face value of the bond increases which decreases the bond s current value From 0 and d bondholders lose 2062750 7 2362707 7299957 From 0 and d stockholders gain 2537250 7 2237293 299957 This is an agency problem for bondholders Management acting to increase shareholder wealth in this manner will reduce bondholder wealth by the exact amount by which shareholder wealth is increased Since the equityholders of a rm nanced partially with debt can be thought of as holding a call option on the assets of the rm with a strike price equal to the debt s face value and a time to expiration equal to the debt s time to maturity the value of the company s equity equals a call option with a strike price of 320 million and 1 year until expiration In order to value this option using the twostate option model first draw a tree containing both the current value of the rm and the rm s possible values at the time of the option s expiration Next draw a similar tree for the option designating what its value will be at expiration given either of the 2 possible changes in the rm s value The value of the company today is 300 million It will either increase to 380 million or decrease to 210 million in one year as a result of its new project If the film s value increases to 380 million the equityholders will exercise their call option and they will receive a payoff of 60 million at expiration However if the film s value decreases to 210 million the equityholders will not exercise their call option and they will receive no payoff at expiration 444 Equityholders call option price with a strike of 320 Value of company in millions in millions Today 1 year Today 1 year 380 60 Max0 380 7 320 300 210 0 Max0 210 7 320 If the project is successful and the company s value rises the percentage increase in value over the period is 2667 percent 380 300 7 1 If the project is unsuccessful and the company s value falls the percentage decrease in value over the period is 730 percent 210 300 71 We can determine the riskneutral probability of an increase in the value of the company as Riskfree rate ProbabilityRiseRetumRse ProbabilityFal1ReturnFan Riskfree rate ProbabilityRiseRetumRse 1 7 ProbabilityRiseReturnFan 007 ProbabilityRise2667 1 7ProbabilityRise730 ProbabilityRise 6529 or 6529 And the riskneutral probability of a decline in the company value is ProbabilityFan 1 7 ProbabilityRse ProbabilityFan 1 76529 ProbabilityFan 3471 or 3471 Using these riskneutral probabilities we can determine the expected payoff to the equityholders call option at expiration which will be Expected payoff at expiration 652960000000 34710 Expected payoff at expiration 3917647059 Since this payoff occurs 1 year from now we must discount it at the riskfree rate in order to nd its present value So PVExpected payoff at expiration 3917647059 107 PVExpected payoff at expiration 3661352391 Therefore the current value of the company s equity is 3661352391 The current value of the company is equal to the value of its equity plus the value of its debt In order to find the value of company s debt subtract the value of the company s equity from the total value of the company VL Debt Equity 300000000 7 Debt 3661352391 Debt 7 26338647609 445 To nd the price per share we can divide the total value of the equity by the number of shares outstanding So the price per share is Price per share Total equity value Shares outstanding Price per share 366135239l 500000 Price per share 7323 The market value of the rm s debt is 26338647609 The present value of the same face amount of riskless debt is 29906542056 320000000 107 The rm s debt is worth less than the present value of riskless debt since there is a risk that it will not be repaid in full In other words the market value of the debt takes into account the risk of default The value of riskless debt is 29906542056 Since there is a chance that the company might not repay its debtholders in full the debt is worth less than 29906542056 The value of Strudler today is 300 million It will either increase to 445 million or decrease to 185 million in one year as a result of the new project If the rm s value increases to 445 million the equityholders will exercise their call option and they will receive a payoff of 125 million at expiration However if the firm s value decreases to 185 million the equityholders will not exercise their call option and they will receive no payoff at expiration Equityholders call option price with a strike of 320 Value of company in millions in millions Today 1 year Today 1 year 445 125 Max0 445 7 320 300 185 0 Max0 185 7320 If the project is successful and the company s value rises the increase in the value of the company over the period is 4833 percent 445 300 7 1 If the project is unsuccessful and the company s value falls decrease in the value of the company over the period is 73833 percent 185 300 71 We can use the following expression to determine the riskneutral probability of an increase in the value of the company Riskfree rate ProbabilityRiseRetumRse ProbabilityFal1ReturnFan Riskfree rate ProbabilityRiseRetulese l ProbabilityRiseReturnFan 007 ProbabilityRise4833 1 7ProbabilityRise73833 ProbabilityRise 5231 or 5231 percent So the riskneutral probability of a decrease in the company value is ProbabilityFan l 7 ProbabilityRse ProbabilityFan 175231 ProbabilityFan 4769 or 4769 446 33 a 9 Using these riskneutral probabilities we can determine the expected payoff to the equityholders call option at expiration which is Expected payoff at expiration 5231125000000 47690 Expected payoff at expiration 6538461538 Since this payoff occurs 1 year from now we must discount it at the riskfree rate in order to nd its present value So PVExpected payoff at expiration 653 8461538 107 PVExpected payoff at expiration 6110711718 Therefore the current value of the rm s equity is 611071 17 18 The current value of the company is equal to the value of its equity plus the value of its debt In order to nd the value of the company s debt we can subtract the value of the company s equity from the total value of the company which yields VL Debt Equity 300000000 Debt 6110711718 Debt 238892882 82 The riskier project increases the value of the company s equity and decreases the value of the company s debt If the company takes on the riskier project the company is less likely to be able to pay off its bondholders Since the risk of default increases if the new project is undertaken the value of the company s debt decreases Bondholders would prefer the company to undertake the more conservative project Going back to the chapter on dividends the price of the stock will decline by the amount of the dividend less any tax effects Therefore we would expect the price of the stock to drop when a dividend is paid reducing the upside potential of the call by the amount of the dividend The price of a call option will decrease when the dividend yield increases Using the BlackScholes model with dividends we get d1 ln106100 05 7 02 5022 x 5 50 x B 3840 d2 3840 750 gtlt J3 0305 Nd1 6495 Nd2 5121 C 106 02564957 100e 0555121 1821 447 34 a 03 UI DJ 5 DJ 1 DJ on Going back to the chapter on dividends the price of the stock will decline by the amount of the dividend less any tax effects Therefore we would expect the price of the stock to drop when a dividend is paid The price of put option will increase when the dividend yield increases b Using putcall parity to nd the price of the put option we get 106602 P7 100605 1821 P7 1080 Nd1 is the probability that Z is less than or equal to Nd1 so 1 7 Nd1 is the probability that Z is greater than Nd1 Because of the symmetry of the normal distribution this is the same thing as the probability that Z is less than N11 So Nd11 N1 From putcall parity P Exe39R C7S Substituting the BlackScholes call option formula for C and using the result in the previous question produces the put option formula E x e39R C75 7 E x a S xNd1 7E x a xNd2 is S XNd1 7 l E X e39R gtltl 7Nd2 E X e39R XN12 7S X N11 U TJ TJ TJ Based on BlackScholes the call option is worth 50 The reason is that present value of the exercise price is zero so the second term disappears Also all is in nite so Nd1 is equal to one The problem is that the call option is European with an infinite expiration so why would you pay anything for it since you can never exercise it The paradox can be resolved by examining the price of the stock Remember that the call option formula only applies to a nondividend paying stock If the stock will never pay a dividend it and a call option to buy it at any price must be worthless The delta of the call option is Nd1 and the delta of the put option is Nd1 7 1 Since you are selling a put option the delta of the portfolio is Nd1 7 Nd1 7 1 This leaves the overall delta of your position as 1 This position will change dollar for dollar in value with the underlying asset This position replicates the dollar action on the underlying asset 448 CHAPTER 23 OPTIONS AND CORPORATE FINANCE EXTENSIONS AND APPLI CA T I ONS Answers to Concepts Review and Critical Thinking Questions 1 One of the purposes to give stock options to CEOs instead of cash is to tie the performance of the rm s stock with the compensation of the CEO In this way the CEO has an incentive to increase shareholder value 2 Most businesses have the option to abandon under bad conditions and the option to expand under good conditions 3 Virtually all projects have embedded options which are ignored in NPV calculations and likely leads to undervaluation 4 As the volatility increases the value of an option increases As the volatility of coal and oil increases the option to burn either increases However if the prices of coal and oil are highly correlated the value of the option would decline If coal and oil prices both increase at the same time the option to switch becomes less valuable since the company will likely save less money 5 The advantage is that the value of the land may increase if you wait Additionally if you wait the best use of the land other than sale may become more valuable 6 The company has an option to abandon the mine temporarily which is an American put If the option is exercised which the company is doing by not operating the mine it has an option to reopen the mine when it is profitable which is an American call Of course if the company does reopen the mine it has another option to abandon the mine again which is an American put 7 Your colleague is correct but the fact that an increased volatility increases the value of an option is an important part of option valuation All else the same a call option on a venture that has a higher volatility will be worth more since the upside potential is greater Even though the downside is also greater with an option the downside is irrelevant since the option will not be exercised and will expire worthless no matter how low the asset falls With a put option the reverse is true in that the option becomes more valuable the further the asset falls and if the asset increases in value the option is allowed to expire 8 Real option analysis is not a technique that can be applied in isolation The value of the asset in real option analysis is calculated using traditional cash ow techniques and then real options are applied to the resulting cash ows 9 Insurance is a put option Consider your homeowner s insurance If your house were to burn down you would receive the value of the policy from your insurer In essence you are selling your burned house putting to the insurance company for the value of the policy the strike price 449 p n O In a market with competitors you must realize that the competitors have real options as well The decisions made by these competitors may often change the payoffs for your company s options For example the first entrant into a market can often be rewarded with a larger market share because the name can become synonymous with the product think of Qtips and Kleenex Thus the option to become the first entrant can be valuable However we must also consider that it may be better to be a later entrant in the market Either way we must realize that the competitors actions will affect our options as well Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic 1 a The inputs to the BlackScholes model are the current price of the underlying asset S the strike price of the option K the time to expiration of the option in fractions of a year t the variance 62 of the underlying asset and the continuouslycompounded riskfree interest rate R Since these options were granted atthemoney the strike price of each option is equal to the current value of one share or 55 We can use BlackScholes to solve for the option price Doing so we nd d1 lnSK R 522t 52012 d1 ln5555 06 4522 x 5 45 x J 8013 d2 8013 745 x J5 72050 Find Nd1 and Nd2 the area under the normal curve from negative in nity to dl and negative infinity to d2 respectively Doing so Nd1 N08013 07885 Nd2 M02050 04188 Now we can find the value of each option which will be C SNd1 iKe R39Nd2 C 5507885 7 55e 06504188 2630 Since the option grant is for 30000 options the value of the grant is Grant value 300002630 Grant value 78912334 450 b Because he is riskneutral you should recommend the alternative with the highest net present value Since the expected value of the stock option package is worth more than 750000 he would prefer to be compensated with the options rather than with the immediate bonus 0 If he is riskaverse he may or may not prefer the stock option package to the immediate bonus Even though the stock option package has a higher net present value he may not prefer it because it is undiversi ed The fact that he cannot sell his options prematurely makes it much more risky than the immediate bonus Therefore we cannot say which alternative he would prefer The total compensation package consists of an annual salary in addition to 15000 atthemoney stock options First we will nd the present value of the salary payments Since the payments occur at the end of the year the payments can be valued as a threeyear annuity which will be PVSalary 375000Pv1FA3 PVSalary 94923550 Next we can use the BlackScholes model to determine the value of the stock options Doing so we n d1 lnSK R 522t 62012 d1 1113434 05 7422 x 3 74 x J5 7579 d2 7579774 x 75238 Find Ndl and Ndz the area under the normal curve from negative in nity to dl and negative infmity to d2 respectively Doing so Ndl N07579 07757 Ndz N45238 03002 Now we can find the value of each option which will be C SNd1 7 Ke R39Nmz C 3407757 7 34e 05303002 C 1759 Since the option grant is for 15000 options the value of the grant is Grant value 150001759 Grant value 26385243 The total value of the contract is the sum of the present value of the salary plus the option value or Contract value 94923550 26385243 Contract value 121308793 451 Since the contract is to sell up to 5 million gallons it is a call option so we need to value the contract accordingly Using the binomial mode we will nd the value of u and d which are SmH u e u 846V123 u 12586 d lu d 1 12586 d 07945 This implies the percentage increase if gasoline increases will be 26 percent and the percentage decrease if prices fall will be 21 percent So the price in three months with an up or down move will e PUlo 17412586 PUlo 7 219 PM 7 17407945 PDown 7 138 The option is worthless if the price decreases If the price increases the value of the option per gallon is Value with price increase 2 19 7 205 Value with price increase 014 Next we need to nd the risk neutral probability of a price increase or decrease which will be 06 l23 026Probability of rise 42ll 7 Probability ofrise Probability ofrise 04751 And the probability of a price decrease is Probability of decrease l 7 04751 Probability of decrease 05249 The contract will not be exercised if gasoline prices fall so the value of the contract with a price decrease is zero So the value per gallon of the call option contract will be C 047510 14 7 052490 1 7 00612 3 0066 This means the value of the entire contract is Value of contract 00665000000 Value of contract 32755379 452 When solving a question dealing with real options begin by identifying the optionlike features of the situation First since the company will exercise its option to build if the value of an office building rises the right to build the office building is similar to a call option Second an office building would be worth 185 million today This amount can be viewed as the current price of the underlying asset S Third it will cost 20 million to construct such an office building This amount can be viewed as the strike price of a call option K since it is the amount that the rm must pay in order to exercise its right to erect an office building Finally since the firm s right to build on the land lasts only 1 year the time to expiration t of the real option is one year We can use the two state model to value the option to build on the land First we need to find the return of the land if the value rises or falls The return will be RRise 22400000 7 18500000 18500000 RRise 2108 or 2108 RFall 17500000 7 18500000 18500000 RFall 70541 or 7541 Now we can find the riskneutral probability of a rise in the value of the building as Value of building millions Value of real call option with a strike of 20 millions Today 1 year Today 1 year 224 24 Max0 224 7 20 185 175 0 Max0 175 7 20 Riskfree rate ProbabilityRiseRetumRse ProbabilityFanReturnFan Riskfree rate ProbabilityRiseRetumRse l ProbabilityRiseReturnFan 0048 ProbabilityRise02108 1 iProbabilityRise70541 ProbabilityRse 03853 So a probability of a fall is Probabilitde1 l iProbabilityRise ProbabilityFan 1 703853 Probabilitde1 06147 Using these riskneutral probabilities we can determine the expected payoff of the real option at expiration Expected payoff at expiration 38532400000 61470 Expected payoff at expiration 92473469 453 Since this payoff will occur 1 year from now it must be discounted at the riskfree rate in order to nd its present value which is PV 92473469 1048 PV 88238043 Therefore the right to build an of ce building over the next year is worth 88238043 today Since the offer to purchase the land is less than the value of the real option to build the company should not accept the offer When solving a question dealing with real options begin by identifying the optionlike features of the situation First since the company will only choose to drill and excavate if the price of oil rises the right to drill on the land can be viewed as a call option Second since the land contains 375000 barrels of oil and the current price of oil is 58 per barrel the current price of the underlying asset S to be used in the BlackScholes model is Stock price 37500058 Stock price 21750000 Third since the company will not drill unless the price of oil in one year will compensate its excavation costs these costs can be viewed as the real option s strike price K Finally since the winner of the auction has the right to drill for oil in one year the real option can be viewed as having a time to expiration t of one year Using the BlackScholes model to determine the value of the option we nd d1 lnSK R 522t 62012 d1 ln2l75000035000000 04 5022 x 1 50 X J1 76215 d2 76215 750 x Ji 711215 Find Nd1 and Nd2 the area under the normal curve from negative in nity to dl and negative infmity to d2 respectively Doing so Nd1 M06215 02671 Nd2 NH 1215 01310 Now we can find the value of call option which will be C SNd1 7 Ke R39Nd2 C 2175000002671735000000e 4101310 C 140371165 This is the maximum bid the company should be willing to make at auction 454 Intermediate When solving a question dealing with real options begin by identifying the optionlike features of the situation First since Sardano will only choose to manufacture the steel rods if the price of steel falls the lease which gives the rm the ability to manufacture steel can be viewed as a put option Second since the rm will receive a xed amount of money if it chooses to manufacture the rods Amount received 45000 steel rods24 7 16 Amount received 360000 The amount received can be viewed as the put option s strike price K Third since the project requires Sardano to purchase 400 tons of steel and the current price of steel is 630 per ton the current price of the underlying asset S to be used in the BlackScholes formula is Stock price 400 tons630 per ton Stock price 252000 Finally since Sardano must decide whether to purchase the steel or not in six months the rm s real option to manufacture steel rods can be viewed as having a time to expiration t of six months In order to calculate the value of this real put option we can use the BlackScholes model to determine the value of an otherwise identical call option then infer the value of the put using putcall parity Using the BlackScholes model to determine the value of the option we nd d1 lnSK R 522t 62012 d1 ln252000360000 045 4522 x 612 45 x l612 708911 d2 418911745 x 4612 712093 Find Nd1 and Nd2 the area under the normal curve from negative in nity to dl and negative in nity to d2 respectively Doing so Nd1 N48911 01864 Nd2 N712093 01133 Now we can find the value of call option which will be C SNd1 iKe R39Nd2 C 2520000 1864 7 360000e 04561201133 C 711089 Now we can use putcall parity to find the price of the put option which is C P S 7 Ke Rt 711089 P 252000 7 360000e 045612 P 10710133 This is the most the company should be willing to pay for the lease 455 In one year the company will abandon the technology if the demand is low since the value of abandonment is higher than the value of continuing operations Since the company is selling the technology in this case the option is a put option The value of the put option in one year if demand is low will be Value of put with low demand 8200000 7 7000000 Value of put with low demand 1200000 Of course if demand is high the company will not sell the technology so the put will expire worthless We can value the put with the binomial model In one year the percentage gain on the project if the demand is high will be Percentage increase with high demand 13400000 7 11600000 11600000 Percentage increase with high demand 1552 or 1552 And the percentage decrease in the value of the technology with low demand is Percentage decrease with high demand 7000000 7 11600000 11600000 Percentage decrease with high demand 73966 or 73966 Now we can find the riskneutral probability of a rise in the value of the technology as Riskfree rate ProbabilityRiseRetumRise ProbabilityFal1RetumFan Riskfree rate ProbabilityRiseRetumRise 1 7ProbabilityRiseRetumFan 006 ProbabilityRiseXO 1552 1 7 ProbabilityRise73966 ProbabilityRSe 08275 So a probability of a fall is Probabilitde1 1 7ProbabilityRse ProbabilityFan 17 08275 Probabilitde1 01725 Using these riskneutral probabilities we can determine the expected payoff of the real option at expiration With high demand the option is worthless since the technology will not be sold and the value of the technology with low demand is the 12 million we calculated previously So the value of the option to abandon is Value of option to abandon 82750 17251200000 1 06 Value of option to abandon 19528302 456 Using the binomial mode we will find the value of u and d which are SMH ue u 12239 d lu d112239 d08170 This implies the percentage increase if the stock price increases will be 22 percent and the percentage decrease if the stock price falls will be 18 percent The monthly interest rate is Monthly interest rate 00512 Monthly interest rate 00042 Next we need to nd the risk neutral probability of a price increase or decrease which will be 00042 022Probability of rise 70181 iProbability of rise Probability ofrise 04599 And the probability of a price decrease is Probability of decrease 1 7 04599 Probability of decrease 05401 The following gure shows the stock price and put price for each possible move over the next two months Stockprice D 8689 Put price 0 Stock price B 7099 Put price 3 77 Stock priceA 5800 Stock price E 5 800 Put price 1105 Put price 700 Stock price C 4739 Put price 1734 Stockprice F 3872 Putprice 2628 457 The stock price at node A is the current stock price The stock price at node B is from an up move which means Stock price B 58l2239 Stock price B 7099 And the stock price at node D is two up moves or Stock price D 5 812239 12239 Stock price D 8689 The stock price at node C is from a down move or Stock price C 5808l70 Stock price C 4739 And the stock price at node F is two down moves or Stock price F 5 80817008170 Stock price F 3872 Finally the stock price at node E is from an up move followed by a down move or a down move followed by an up move Since the binomial tree recombines both calculations yield the same result which is Stock price E 581223908170 580817012239 Stock price E 5800 Now we can value the put option at the expiration nodes namely D E and F The value of the put option at these nodes is the maximum of the strike price minus the stock price or zero So Put value D Max65 7 8689 80 Put value D 0 Put value E Max65 7 58 0 Put value E 7 Put value F lIax65 7 3872 0 Put value F 2628 The value of the put at node B is the present value of the expected value We find the expected value by using the value of the put at nodes D and E since those are the only two possible stock prices after node B So the value of the put at node B is Put value B 45990 54017 10042 Put value B 377 458 O Similarly the value of the put at node C is the present value of the expected value of the put at nodes E and F since those are the only two possible stock prices after node C So the value of the put at node C is Put value C 45997 54012628 10042 Put value C 1734 Using the put values at nodes B and C we can now find the value of the put today which is Put value A 4599377 54011734 10042 Put value A 1105 Challenge Since the exercise style is now American the option can be exercised prior to expiration At node B we would not want to exercise the put option since it would be out of the money at that stock price However if the stock price falls next month the value of the put option if exercised is Value ifexercised 65 7 4739 Value ifexercised 1761 This is greater then the present value of waiting one month so the option will be exercised early in one month if the stock price falls This is the value of the put option at node C Using this put value we can now find the value of the put today which is Put value A 4599377 54011761 10042 Put value A 1120 This is slightly higher than the value of the same option with a European exercise style An American option must be worth at least as much as a European option and can be worth more Remember an option always has value until it is exercised The option to exercise early in an American option is an option itself therefore it can often has some value Using the binomial mode we will find the value of u and d which are SMH ue 830 12 u 12363 d lu d112363 d08089 This implies the percentage increase is if the stock price increases will be 24 percent and the percentage decrease if the stock price falls will be 19 percent The six month interest rate is Six month interest rate 0062 Six month interest rate 003 459 Next we need to nd the risk neutral probability of a price increase or decrease which will be 003 024Probability of rise 41191 7 Probability of rise Probability ofrise 05173 And the probability of a price decrease is Probability of decrease 1 7 05173 Probability of decrease 04827 The following gure shows the stock price and call price for each possible move over each of the six month steps Stock priceA Call price Value prepayment 61 8155 55 Value postpayment B 611655 55 Call price 12267307 50000000 6161619 Value prepayment 40442895 Value postpayment C 39792895 Call price 0 Value D Call price Value E Call price Value F Call price Value G Call price 76423258 24423258 49474242 0 49196398 0 32186797 0 First we need to nd the building value at every step along the binomial tree The building value at node A is the current building value The building value at node B is from an up move which means I Building value B 5000000012363 Building value B 61815555 460 At node B the accrued rent payment will be made so the value of the building after the payment will be reduced by the amount of the payment which means the building value at node B is Building value B after payment 61815555 7 650000 Building value B after payment 61165555 To nd the building value at node D we multiply the afterpayment building value at node B by the up move or Building value D 6116555512363 Building value D 76423258 To nd the building value at node E we multiply the afterpayment building value at node B by the down move or Building value E 6116555508089 Building value E 49474242 The building value at node C is from a down move which means the building value will be Building value E 5000000008089 Building value E 40442895 At node C the accrued rent payment will be made so the value of the building after the payment will be reduced by the amount of the payment which means the building value at node C is Building value C after payment 40442895 7 650000 Building value C after payment 39792895 To nd the building value at node F we multiply the afterpayment building value at node C by the down move or Building value F 3979289512363 Building value F 49196398 Finally the building value at node G is from a down move from node C so the building value is Building value G 3979289508089 Building value G 32186797 Note that because of the accrued rent payment in six months the binomial tree does not recombine during the next step This occurs whenever a xed payment is made during a binomial tree For example when using a binomial tree for a stock option a xed dividend payment will mean that the tree does not recombine With the expiration values we can value the call option at the expiration nodes namely D E F and G The value of the call option at these nodes is the maximum of the building value minus the strike price or zero We do not need to account for the value of the building after the accrued rent payments in this case since if the option is exercised you will receive the rent payment So Call value D Max76423258 7 52000000 0 461 Call value D 24423258 Call value E Max49474242 7 52000000 0 Call value E 0 Call value F Max49196398 7 52000000 0 Call value F 0 Call value G Max32186797 7 52000000 0 Call value G 0 The value of the call at node B is the present value of the expected value We find the expected value by using the value of the call at nodes D and E since those are the only two possible building values after node B So the value of the call at node B is Call value B 517324423258 4827 0 103 Call value B 12267307 Note that you would not want to exercise the option early at node B The value of the option at node B if exercised is the value of the building including the accrued rent payment minus the strike price or Option value at node B ifexercised 61165555 7 52000000 Option value at node B if exercised 9165555 Since this is less than the value of the option if it left alive the option will not be exercised With a call option unless a large cash payment dividend is made it is generally not valuable to exercise the call option early The reason is that the potential gain is unlimited In contrast the potential gain on a put option is limited by the strike price so it may be valuable to exercise an American put option early if it is deep in the money We can value the call at node C which will be the present value of the expected value of the call at nodes F and G since those are the only two possible building values after node C Since neither node has a value greater than zero obviously the value of the option at node C will also be zero Now we need to find the value of the option today which is Call value A 517312267307 48270 104 Call value A 6161619 462 CHAPTER 24 WARRANTS AND CONVERTIBLES Answers to Concepts Review and Critical Thinking Questions 1 A warrant is issued by the company and when a warrant is exercised the number of shares increases A call option is a contract between investors and does not affect the number of shares of e mu 1 If the stock price is less than the exercise price of the warrant at expiration the warrant is worthless Prior to expiration however the warrant will have value as long as there is some probability that the stock price will rise above the exercise price in the time remaining until expiration Therefore if the stock price is below the exercise price of the warrant the lower bound on the price of the warrant is zero b If the stock price is above the exercise price of the warrant the warrant must be worth at least the difference between these two prices If warrants were selling for less than the difference between the current stock price and the exercise price an investor could earn an arbitrage pro t ie an immediate cash in ow by purchasing warrants exercising them immediately and selling the stock 0 If the warrant is selling for more than the stock it would be cheaper to purchase the stock than to purchase the warrant which gives its owner the right to buy the stock Therefore an upper bound on the price of any warrant is the rm s current stock price An increase in the stock price volatility increases the bond price If the stock price becomes more volatile the conversion option on the stock becomes more valuable The two components of the value of a convertible bond are the straight bond value and the option value An increase in interest rates decreases the straight value component of the convertible bond Conversely an increase in interest rates increases the value of the equity call option Generally the effect on the straight bond value will be much greater so we would expect the bond value to fall although not as much as the decrease in a comparable straight bond When warrants are exercised the number of shares outstanding increases This results in the value of the firm being spread out over a larger number of shares often leading to a decrease in value of each individual share The decrease in the pershare price of a company s stock due to a greater number of shares outstanding is known as dilution In an ef cient capital market the difference between the market value of a convertible bond and the value of straight bond is the fair price investors pay for the call option that the convertible or the warrant provides 463 CHAPTER 24 B 464 There are three potential reasons 1 To match cash ows that is they issue securities whose cash ows match those of the rm 2 To bypass assessing the risk of the company risk synergy For example the risk of company startups is hard to evaluate 3 To reduce agency costs associated with raising money by providing a package that reduces bondholderstockholder con icts Because the holder of the convertible has the option to wait and perhaps do better than what is implied by current stock prices Theoretically conversion should be forced as soon as the conversion value reaches the call price because other conversion policies will reduce shareholder value If conversion is forced when conversion values are above the call price bondholders will be allowed to exchange less valuable bonds for more valuable common stock In the opposite situation shareholders are giving bondholders the excess value No the market price of the warrant will not equal zero Since there is a chance that the market price of the stock will rise above the 31 per share exercise price before expiration the warrant still has some value Its market price will be greater than zero As a practical matter warrants that are far out ofthemoney may sell at 0 due to transaction costs Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic The conversion price is the par value divided by the conversion ratio or Conversion price 1000 184 Conversion price 5435 The conversion ratio is the par value divided by the conversion price or Conversion ratio 1000 7026 Conversion ratio 1423 First we need to find the conversion price which is the par value divided by the conversion ratio or Conversion price 1000 1280 Conversion price 7813 The conversion premium is the necessary increase in stock price to make the bond convertible So the conversion premium is Conversion premium 7813 7 6118 61 18 Conversion premium 02770 or 2770 464 CHAPTER 24 B 465 a The conversion ratio is de ned as the number of shares that will be issued upon conversion Since each bond is convertible into 1850 shares of Hannon s common stock the conversion ratio of the convertible bonds is 1850 b The conversion price is de ned as the face amount of a convertible bond that the holder must surrender in order to receive a single share Since the conversion ratio indicates that each bond is convertible into 1850 shares the conversion price is Conversion price 1000 1850 Conversion price 5405 c The conversion premium is defined as the percentage difference between the conversion price of the convertible bonds and the current stock price So the conversion premium is Conversion premium 5405 7 3820 3820 Conversion premium 04150 or 4150 d The conversion value is defined as the amount that each convertible bond would be worth if it were immediately converted into common stock So the conversion value is Conversion value 38201850 Conversion value 70670 8 If the stock price increases by 2 the new conversion value will be Conversion value 40201850 Conversion value 74370 The total exercise price of each warrant is shares each warrant can purchase times the exercise price which in this case will be Exercise price 341 Exercise price 123 Since the shares of stock are selling at 47 the value of three shares is Value of shares 347 Value of shares 141 Therefore the warrant effectively gives its owner the right to buy 141 worth of stock for 123 It follows that the minimum value of the warrant is the difference between these numbers or lVIinimum warrant value 141 7 123 lVIinimum warrant value 18 If the warrant were selling for less than 18 an investor could earn an arbitrage pro t by purchasing the warrant exerc1s1ng it immediately and selling the stock Here the warrant holder pays less than 18 while receiving the 18 difference between the price of three shares and the exercise price 465 CHAPTER 24 B 466 Since a convertible bond gives its holder the right to a fixed payment plus the right to convert it must be worth at least as much as its straight value Therefore if the market value of a convertible bond is less than its straight value there is an opportunity to make an arbitrage pro t by purchasing the bond and holding it until expiration In Scenario A the market value of the convertible bond is 1000 Since this amount is greater than the convertible s straight value 900 Scenario A is feasible In Scenario B the market value of the convertible bond is 900 Since this amount is less than the convertible s straight value 950 Scenario B is not feasible 1 Using the conversion price we can determine the conversion ratio which is Conversion ratio 1000 34 Conversion ratio 29 1 So each bond can be exchanged for 2941 shares of stock This means the conversion price of the bond is Conversion price 294129 Conversion price 85294 Therefore the minimum price the bond should sell for is 85294 Since the bond price is higher than this price the bond is selling at the straight value plus a premium for the conversion feature A convertible bond gives its owner the right to convert his bond into a xed number of shares The market price of a convertible bond includes a premium over the value of immediate conversion that accounts for the possibility of increases in the price of the rm s stock before the maturity of the bond If the stock price rises a convertible bondholder will convert and receive valuable shares of equity If the stock price decreases the convertible bondholder holds the bond and retains his right to a xed interest and principal payments You can convert or tender the bond ie surrender the bond in exchange for the call price If you convert you get stock worth 2150 gtlt 52 1118 If you tender you get 1100 110 percent of par It s a nobrainer convert 1 Since the stock price is currently below the exercise price of the warrant the lower bound on the price of the warrant is zero If there is only a small probability that the rm s stock price will rise above the exercise price of the warrant the warrant has little value An upper bound on the price of the warrant is 51 the current price of the common stock One would never pay more than 51 to receive the right to purchase a share of the company s stock if the rm s stock were only worth 51 If the stock is trading for 58 per share the lower bound on the price of the warrant is 3 the difference between the current stock price and the warrant s exercise price If warrants were selling for less than this amount an investor could earn an arbitrage pro t by purchasing warrants exercising them immediately and selling the stock As always the upper bound on the price of a warrant is the current stock price In this case one would never pay more than 58 for the right to buy a single share of stock when he could purchase a share outright for 58 466 CHAPTER 24 B 467 Intermediate The minimum convertible bond value is the greater of the conversion price or the straight bond price To find the conversion price of the bond we need to determine the conversion ratio which is Conversion ratio 1000 130 Conversion ratio 769 So each bond can be exchanged for 769 shares of stock This means the conversion price of the bond is Conversion price 76926 Conversion price 200 And the straight bond value is P 301711 05560 055 10001 1 0556 P 56375 So the minimum price of the bond is 56375 If the stock price were growing by 13 percent per year forever each share of its stock would be worth approximately 26113 after 1 years Since each bond is convertible into 769 shares the conversion value of the bond equals 26769113 after tyears In order to calculate the number of years that it will take for the conversion value to equal 1100 set up the following equation 26769113 1100 I 1395 years The percentage of the company stock currently owned by the CEO is Percentage of stock 750000 5000000 Percentage of stock 1500 or 1500 The conversion price indicates that for every 34 of face value of convertible bonds outstanding the company will be obligated to issue a new share upon conversion So the new number of shares the company must issue will be New shares issued 30000000 34 New shares issued 88235294 So the new number of shares of company stock outstanding will be New total shares 5000000 88235294 New total shares 588235294 467 CHAPTER 24 B 468 After the conversion the percentage of company stock owned by the CEO will be New percentage of stock 750000 588235294 New percentage of stock 1275 or 1275 Before the warrant was issued the film s assets were worth Value of assets 9 oZ of platinum850 per 02 Value of assets 7650 So the price per share is Price per share 7650 8 Price per share 95625 When the warrant was issued the firm received 850 increasing the total value of the film s assets to 8500 7650 850 If the 8 shares of common stock were the only outstanding claims on the firm s assets each share would be worth 106250 8500 8 shares However since the warrant gives warrant holder a claim on the rm s assets worth 850 the value of the rm s assets available to stockholders is only 7650 8500 7 850 Since there are 8 shares outstanding the value per share remains at 95625 7650 8 shares after the warrant issue Note that the fum uses the warrant price of 850 to purchase one more ounce of platinum If the price of platinum is 975 per ounce the total value of the rm s assets is 9750 10 oZ of platinum gtlt 975 per oz If the warrant is not exercised the value of the rm s assets would remain at 9750 and there would be 8 shares of common stock outstanding so the stock price would be 121875 If the warrant is exercised the firm would receive the warrant s 1000 strike price and issue one share of stock The total value of the rm s assets would increase to 10750 9750 1000 Since there would now be 9 shares outstanding and no warrants the price per share would be 119444 10750 9 shares Since the 121875 value of the share that the warrant holder will receive is greater than the 1000 exercise price of the warrant investors will expect the warrant to be exercised The rm s stock price will re ect this information and will be priced at 119444 per share on the warrant s expiration date 13 The value of the company s assets is the combined value of the stock and the warrants So the value of the company s assets before the warrants are exercised is Company value l500000025 l0000007 Company value 382000000 When the warrants are exercised the value of the company will increase by the number of warrants times the exercise price or Value increase l000000l9 Value increase 19000000 So the new value of the company is New company value 3 82000000 19000000 468 4 UI CHAPTER 24 B 469 New company value 401000000 This means the new stock price is New stock price 401000000 16000000 New stock price 2506 Note that since the warrants were exercised when the price per warrant 7 was above the exercise value of each warrant 6 25 7 19 the stockholders gain and the warrant holders lose Challenge The straight bond value today is Straight bond value 868Pv1FA9ms 100010925 Straight bond value 78390 And the conversion value of the bond today is Conversion value 35501000150 Conversion value 23667 We expect the bond to be called when the conversion value increases to 1250 so we need to nd the number of periods it will take for the current conversion value to reach the expected value at which the bond will be converted Doing so we nd 23667112t 1250 t 1469 years The bond will be called in 1469 years The bond value is the present value of the expected cash ows The cash ows will be the annual coupon payments plus the conversion price The present value of these cash ows is Bond value 68PVIFA1469 12501091469 89503 The value of a single warrant W equals w w x Ca11S V K KW where the number of shares of common stock outstanding the number of warrants outstanding a call option on an underlying asset worth S with a strike price K the rm s value net of debt the strike price of each warrant w Call S K V Kw Therefore the value of a single warrant W equals W w X CallS V K KW 6000000 6000000 750000 x CallS 105000000 6000000 K 20 8889 x CallS 81750 K 20 469 5 CHAPTER 24 B 470 In order to value the call option use the BlackScholes formula Solving for d1 and d2 we nd d1 lnSK R 12550 62012 d1ln175020 007 10151 015x112 d1 00296 d2 d1 7 52012 d2 00296 7 015 x 1 2 d2 03577 Next we need to nd Nd1 and Nd2 the area under the normal curve from negative in nity to dl and negative infmity to d2 respectively Nd1 N00296 05118 Nd2 N43577 03603 According to the BlackScholes formula the price of a European call option C on a nondividend paying common stock is C SNd1 7 Ke R39Nd2 C 175005118 7 20e 071 03603 C 224 Therefore the price of a single warrant W equals W 8889 X CallS 1750 K 20 w 8889224 w 199 To calculate the number of warrants that the company should issue in order to pay off 18 million in six months we can use the BlackScholes model to find the price of a single warrant then divide this amount into the present value of 18 million to nd the number of warrants to be issued So the value of the liability today is PV of liability 180000008 6612 PV of liability 1746801960 The company must raise this amount from the warrant issue The value of the company s assets will increase by the amount of the warrant issue after the issue but this increase in value from the warrant issue is exactly offset by the bond issue Since the cash in ow from the warrants offsets the lm s debt the value of the warrants will be exactly the same as if the cash from the warrants were used to immediately pay off the debt We can use the market value of the company s assets to nd the current stock price which is Stock price 210000000 3200000 Stock price 6563 470 CHAPTER 24 B 471 The value of a single warrant W equals w w x CallS K w 3200000 3200000 w x Call6563 75 Since the rm must raise 1746801960 as a result ofthe warrant issue we know W gtlt W must equal 1746801960 Therefore it can be stated that 1746801960 wW 1746801960 w3200000 3200000 w x Call6563 75 Using the BlackScholes formula to value the warrant which is a call option we nd d1 lnSK R 1262t 62012 d1 1n6563 75 06 125026 12 502 x 6 12 2 d1 01161 d2 d1 7 62012 d2 411617502 x 6 12 2 d2 04696 Next we need to nd Nd1 and Nd2 the area under the normal curve from negative in nity to dl and negative infmity to d2 respectively Nd1 M01161 04538 Nd2 M04696 03193 According to the BlackScholes formula the price of a European call option C on a nondividend paying common stock is C SNd1 7 Ke R39Nd2 C 656304538775e 00661203193 C 654 Using this value in the equation above we nd the number of warrants the company must sell is 1746801960 W3200000 3200000 wl X Call6563 75 1746801960 W 3200000 3200000 W x 654 W 16156877 471 CHAPTER 25 DERIVATIVES AND HEDGING RISK Answers to Concepts Review and Critical Thinking Questions 1 Since the firm is selling futures it wants to be able to deliver the lumber therefore it is a supplier Since a decline in lumber prices would reduce the income of a lumber supplier it has hedged its price risk by selling lumber futures Losses in the spot market due to a fall in lumber prices are offset by gains on the short position in lumber futures Buying call options gives the firm the right to purchase pork bellies therefore it must be a consumer of pork bellies While a rise in pork belly prices is bad for the consumer this risk is offset by the gain on the call options if pork belly prices actually decline the consumer enjoys lower costs while the call option expires worthless Forward contracts are usually designed by the parties involved for their speci c needs and are rarely sold in the secondary market so forwards are somewhat customized nancial contracts All gains and losses on the forward position are settled at the maturity date Futures contracts are standardized to facilitate liquidity and to allow them to be traded on organized futures exchanges Gains and losses on futures are markedtomarket daily Default risk is greatly reduced with futures since the exchange acts as an intermediary between the two parties guaranteeing performance Default risk is also reduced because the daily settlement procedure keeps large loss positions from accumulating You might prefer to use forwards instead of futures if your hedging needs were different from the standard contract size and maturity dates offered by the futures contract The firm is hurt by declining oil prices so it should sell oil futures contracts The rm may not be able to create a perfect hedge because the quantity of oil it needs to hedge doesn t match the standard contract size on crude oil futures or perhaps the exact settlement date the company requires isn t available on these futures Also the fum may produce a different grade of crude oil than that speci ed for delivery in the futures contract The firm is directly exposed to uctuations in the price of natural gas since it is a natural gas user In addition the firm is indirectly exposed to uctuations in the price of oil If oil becomes less expensive relative to natural gas its competitors will enjoy a cost advantage relative to the firm Buying the call options is a form of insurance policy for the film If cotton prices rise the firm is protected by the call while if prices actually decline they can just allow the call to expire worthless However options hedges are costly because of the initial premium that must be paid The futures contract can be entered into at no initial cost with the disadvantage that the firm is locking in one price for cotton it can t pro t from cotton price declines 472 p A O p A p A 3quot The put option on the bond gives the owner the right to sell the bond at the option s strike price If bond prices decline the owner of the put option pro ts However since bond prices and interest rates move in opposite directions if the put owner pro ts from a decline in bond prices he would also profit from a rise in interest rates Hence a call option on interest rates is conceptually the same thing as a put option on bond prices The company would like to lock in the current low rates or at least be protected from a rise in rates allowing for the possibility of bene t if rates actually fall The former hedge could be implemented by selling bond futures the latter could be implemented by buying put options on bond prices or buying call options on interest rates A swap contract is an agreement between parties to exchange assets over several time intervals in the future The swap contract is usually an exchange of cash ows but not necessarily so Since a forward contract is also an agreement between parties to exchange assets in the future but at a single point in time a swap can be viewed as a series of forward contracts with different settlement dates The rm participating in the swap agreement is exposed to the default risk of the dealer in that the dealer may not make the cash ow payments called for in the contract The dealer faces the same risk from the contracting party but can more easily hedge its default risk by entering into an offsetting swap agreement with another party The firm will borrow at a xed rate of interest receive xed rate payments from the dealer as part of the swap agreement and make oating rate payments back to the dealer the net position of the firm is that it has effectively borrowed at oating rates Transaction exposure is the shortterm exposure due to uncertain prices in the near future Economic exposure is the longterm exposure due to changes in overall economic conditions There are a variety of instruments available to hedge transaction exposure but very few longterm hedging instruments exist It is much more dif cult to hedge against economic exposure since fundamental changes in the business generally must be made to offset longrun changes in the economic environment The risk is that the dollar will strengthen relative to the yen since the xed yen payments in the future will be worth fewer dollars Since this implies a decline in the exchange rate the film should sell yen futures The way the interest rate is quoted will affect the calculation of which currency is strengthening a Buy oil and natural gas futures contracts since these are probably your primary resource costs If it is a coal red plant a crosshedge might be implemented by selling natural gas futures since coal and natural gas prices are somewhat negatively related in the market coal and natural gas are somewhat substitutable b Buy sugar and cocoa futures since these are probably your primary commodity inputs 0 Sell corn futures since a record harvest implies low corn prices d Buy silver and platinum futures since these are primary commodity inputs required in the manufacture of photographic lm 8 Sell natural gas futures since excess supply in the market implies low prices f Assuming the bank doesn t resell its mortgage portfolio in the secondary market buy bond futures g Sell stock index futures using an index most closely associated with the stocks in your fund such as the SampP 100 or the Major Market Index for large bluechip stocks 473 14 p A UI p A 5 h Buy Swiss franc futures since the risk is that the dollar will weaken relative to the franc over the next six months which implies a rise in the SFr exchange rate i Sell euro futures since the risk is that the dollar will strengthen relative to the Euro over the next three months which implies a decline in the exchange rate Sysco must have felt that the combination of xed plus swap would result in an overall better rate In other words the variable rate available via a swap may have been more attractive than the rate available from issuing a oatingrate bond He is a little na39139ve about the capabilities of hedging While hedging can signi cantly reduce the risk of changes in foreign exchange markets it cannot completely eliminate it Basis risk is the primary reason that hedging cannot reduce 100 of any film s exposure to price uctuations Basis risk arises when the price movements of the hedging instrument do not perfectly match the price movements of the asset being hedged Kevin will be hurt if the yen loses value relative to the dollar over the next eight months Depreciation in the yen relative to the dollar results in a decrease in the exchange rate Since Kevin is hurt by a decrease in the exchange rate he should take on a short position in yen per dollar futures contracts to hedge his risk Solutions to Questions and Problems NOTE All end of chapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic The initial price is 2649 per metric ton and each contract is for 10 metric tons so the initial contract value is Initial contract value 2649 per ton 10 tons per contract 26490 And the final contract value is Final contract value 2431 per tonlO tons per contract 24 10 You will have a loss on this futures position of Loss on futures contract 26490 7 24310 2180 474 The price quote is 1351 per ounce and each contract is for 5000 ounces so the initial contract value is Initial contract value 1351 per oz5000 oz per contract 67550 At a nal price of 1397 per ounce the value of the position is Final contract value 1397 per oz5000 oz per contract 69850 Since this is a short position there is a net loss of 69850 7 67550 2300 per contract Since you sold ve contracts the net loss is Net loss 52300 11500 At a nal price of 1263 per ounce the value of the position is Final contract value 1263 per oz5000 oz per contract 63150 Since this is a short position there is a net gain of 67550 7 63150 4400 Since you sold ve contracts the net gain is Net gain 54400 22000 With a short position you make a pro t when the price falls and incur a loss when the price rises The call options give the manager the right to purchase oil futures contracts at a futures price of 35 per barrel The manager will exercise the option if the price rises above 35 Selling put options obligates the manager to buy oil futures contracts at a futures price of 35 per barrel The put holder will exercise the option if the price falls below 35 The payoffs per barrel are Oil futures price 30 32 35 38 40 Value of call option position 0 0 0 3 5 Value of put option position 75 73 0 0 0 Total value 75 73 0 3 5 The payoff pro le is identical to that of a forward contract with a 35 strike price 475 When you purchase the contracts the initial value is Initial value 1010095l Initial value 95l000 At the end of the rst day the value of your account is Day 1 account value 10100943 Day 1 account value 943000 So your cash ow is Day 1 cash ow 943000 7 95 1000 Day 1 cash ow 78000 The day 2 account value is Day 2 account value 10100946 Day 2 account value 946000 So your cash ow is Day 2 cash ow 946000 7 943000 Day 2 cash ow 3000 The day 3 account value is Day 3 account value 10100953 Day 3 account value 953000 So your cash ow is Day 3 cash ow 953000 7 946000 Day 3 cash ow 7000 The day 4 account value is Day 4 account value 10100957 Day 4 account value 957000 So your cash ow is Day 4 cash ow 957000 7 953000 Day 4 cash ow 4000 You total pro t for the transaction is Pro t 957000 7 951000 Profit 7 6000 476 When you purchase the contracts your cash out ow is Cash out ow 2542000141 Cash out ow 1480500 At the end of the rst day the value of your account is Day 1 account value 2542000137 Day 1 account value 1438500 Remember on a short position you gain when the price declines and lose when the price increases So your cash ow is Day 1 cash ow 1480500 7 1438500 Day 1 cash ow 42000 The day 2 account value is Day 2 account value 2542000142 Day 2 account value 1491000 So your cash ow is Day 2 cash ow 1438500 71491000 Day 2 cash ow 752500 The day 3 account value is Day 3 account value 2542000145 Day 3 account value 1522500 So your cash ow is Day 3 cash ow 1491000 71522500 Day 3 cash ow 731500 The day 4 account value is Day 4 account value 2542000151 Day 4 account value 1585500 So your cash ow is Day 4 cash ow 1522500 71585500 Day 4 cash ow 763000 You total pro t for the transaction is Pro t 1480500 71585500 Profit 7 7105000 477 The duration of a bond is the average time to payment of the bond s cash ows weighted by the ratio of the present value of each payment to the price of the bond Since the bond is selling at par the market interest rate must equal 8 percent the annual coupon rate on the bond The price of a bond selling at par is equal to its face value Therefore the price of this bond is 1000 The relative value of each payment is the present value of the payment divided by the price of the bond The contribution of each payment to the duration of the bond is the relative value of the payment multiplied by the amount of time in years until the payment occurs So the duration of the bond is M PV of payment Relative value Pavment weight 1 7407 007407 007407 2 6859 006859 013717 3 85734 085734 257202 Price of bond 1000 Duration 278326 The duration of a bond is the average time to payment of the bond s cash ows weighted by the ratio of the present value of each payment to the price of the bond Since the bond is selling at par the market interest rate must equal 8 percent the annual coupon rate on the bond The price of a bond selling at par is equal to its face value Therefore the price of this bond is 1000 The relative value of each payment is the present value of the payment divided by the price of the bond The contribution of each payment to the duration of the bond is the relative value of the payment multiplied by the amount of time in years until the payment occurs So the duration of the bond is M PV of payment Relative value Pavment weight 1 7407 007407 007407 2 6859 006859 013717 3 6351 006351 019052 4 79383 079383 317533 Price of bond 1000 Duration 357710 The duration of a portfolio of assets or liabilities is the weighted average of the duration of the portfolio s individual items weighted by their relative market values a The total market value of assets in millions is Market value of assets 31 630 390 98 346 lVIarket value of assets 1495 478 So the market value weight of each asset is Federal funds deposits 31 1495 0021 Accounts receivable 630 1495 0421 Shortterm loans 390 1495 0261 Longterm loans 98 1495 0066 Mortgages 346 1495 0231 Since the duration of a group of assets is the weighted average of the durations of each individual asset in the group the duration of assets is Duration of assets 00210 0421020 0261065 0066525 02311285 Duration of assets 357 years The total market value of liabilities in millions is Market value of liabilities 585 310 305 lVIarket value of liabilities 1200 Note that equity is not included in this calculation since it is not a liability So the market value weight of each asset is Checking and savings deposits 585 1200 0488 Certi cates of deposit 310 1200 0258 Longterm financing 305 1200 0254 Since the duration of a group of liabilities is the weighted average of the durations of each individual asset in the group the duration of liabilities is Duration of liabilities 04880 0258160 02549 80 Duration of liabilities 290 years Since the duration of assets does not equal the duration of its liabilities the bank is not immune from interest rate risk Intermediate You re concerned about a rise in corn prices so you would buy May contracts Since each contract is for 5000 bushels the number of contracts you would need to buy is Number of contracts to buy 1400005000 28 By doing so you re effectively locking in the settle price in May 2009 of 376 per bushel of corn or Total price for 140000 bushels 283765000 526400 479 b If the price of corn at expiration is 392 per bushel the value of you futures position is Value of futures position 392 per bu5000 bu per contract28 contracts 548800 Ignoring any transaction costs your gain on the futures position will be Gain 548800 7 526400 22400 While the price of the corn your rm needs has become 22400 more expensive since February your pro t from the futures position has netted out this higher cost 10 a XYZ has a comparative advantage relative to ABC in borrowing at xed interest rates while ABC has a comparative advantage relative to XYZ in borrowing at oating interest rates Since the spread between ABC and XYZ s xed rate costs is only 1 while their differential is 2 in oating rate markets there is an opportunity for a 3 total gain by entering into a xed for oating rate swap agreement b If the swap dealer must capture 2 of the available gain there is 1 left for ABC and XYZ Any division of that gain is feasible in an actual swap deal the divisions would probably be negotiated by the dealer One possible combination is 5 for ABC and 5 for XYZ A 105 l L 100 ABC r Dealer XYZ 39I l LIBOR1 LIBOR 25 Ll39BOR l Debt Market 10 Debt Market 11 The duration of a liability is the average time to payment of the cash ows required to retire the liability weighted by the ratio of the present value of each payment to the present value of all payments related to the liability In order to determine the duration of a liability first calculate the present value of all the payments required to retire it Since the cost is 30000 at the beginning of each year for four years we can find the present value of each payment using the PV equation PV FV 1 Rt So the PV each year of college is Year 1 PV 30000 1093 2316550 Year 2 PV 30000 1094 2125276 Year 3 PV 30000 1095 1949794 Year 4 PV 30000 1096 1788802 480 So the total PV of the college cost is PV of college 2316550 2125276 1949794 1788802 PV of college 8180422 Now we can set up the following table to calculate the liability s duration The relative value of each payment is the present value of the payment divided by the present value of the entire liability The contribution of each payment to the duration of the entire liability is the relative value of the payment multiplied by the amount of time in years until the payment occurs M PV of payment Relative value Payment weight 7 2316550 028318 084955 8 2125276 025980 103920 9 1949794 023835 119174 10 1788802 021867 131201 PV of college 8180422 Duration 439250 p A N The duration of a bond is the average time to payment of the bond s cash ows weighted by the ratio of the present value of each payment to the price of the bond We need to nd the present value of the bond s payments at the market rate The relative value of each payment is the present value of the payment divided by the price of the bond The contribution of each payment to the duration of the bond is the relative value of the payment multiplied by the amount of time in years until the payment occurs Since this bond has semiannual coupons the years will include halfyears So the duration of the bond is M PV of payment Relative value Payment weight 05 2871 003034 001517 10 2747 002903 002903 15 2629 002778 004168 20 86372 091284 182568 Price of bond 94619 Duration 191156 13 Let R equal the interest rate change between the initiation of the contract and the delivery of the asset Cash flaws from Strategy 1 39 Today I Year Purchase silver is 0 0 Borrow S0 7S01 R Total cash ow 0 7S01 R Cash aws from Strategy 2 Today I Year Purchase silver 0 7F Total cash ow 0 7F 481 Notice that each strategy results in the ownership of silver in one year for no cash out ow today Since the payoffs from both the strategies are identical the two strategies must cost the same in order to preclude arbitrage The forward price F of a contract on an asset with no carrying costs or convenience value equals the current spot price of the asset S0 multiplied by 1 plus the appropriate interest rate change between the initiation of the contract and the delivery date of the asset a The forward price of an asset with no carrying costs or convenience value is Forward price S01 R Since you will receive the bond s face value of 1000 in 11 years and the 11 year spot interest rate is currently 9 percent the current price of the bond is Current bond price l000 10911 Current bond price 38753 Since the forward contract defers delivery of the bond for one year the appropriate interest rate to use in the forward pricing equation is the oneyear spot interest rate of 6 percent Forward price 38753l06 Forward price 41078 If both the 1year and 11year spot interest rates unexpectedly shift downward by 2 percent the appropriate interest rates to use when pricing the bond is 7 percent and the appropriate interest rate to use in the forward pricing equation is 4 percent Given these changes the new price of the bond will be New bond price 1000 10711 New bond price 47509 And the new forward price of the contract is Forward price 47509l04 Forward price 49410 The forward price of an asset with no carrying costs or convenience value is Forward price S0l R Since you will receive the bond s face value of 1000 in 18 months we can nd the price of the bond today which will be Current bond price 1000 1047332 Current bond price 93303 482 Since the forward contract defers delivery of the bond for six months the appropriate interest rate to use in the forward pricing equation is the six month EAR so the forward price will be Forward price 933031036112 Forward price 94972 b It is important to remember that 100 basis points equals 1 percent and one basis point equals 001 Therefore if all rates increase by 30 basis points each rate increases by 0003 So the new price of the bond today will be New bond price 1000 1 0473 00332 New bond price 92903 Since the forward contract defers delivery of the bond for six months the appropriate interest rate to use in the forward pricing equation is the six month EAR increased by the interest rate change So the new forward price will be Forward price 929030 0361 003 2 Forward price 94702 Challenge 16 The nancial engineer can replicate the payoffs of owning a put option by selling a forward contract and buying a call For example suppose the forward contract has a settle price of 50 and the exercise price of the call is also 50 The payoffs below show that the position is the same as owning a put with an exercise price of 50 Price of coal 40 45 50 55 60 Value of call option position 0 0 0 5 10 Value of forward position 10 5 0 75 710 Total value 10 5 0 0 0 Value of put position 10 5 0 0 0 The payoffs for the combined position are exactly the same as those of owning a put This means that in general the relationship between puts calls and forwards must be such that the cost of the two strategies will be the same or an arbitrage opportunity exists In general given any two of the instruments the third can be synthesized 483 CHAPTER 26 SHORTTERM FINANCE AND PLANNING Answers to Concepts Review and Critical Thinking Questions 1 These are rms with relatively long inventory periods andor relatively long receivables periods Thus such rms tend to keep inventory on hand and they allow customers to purchase on credit and take a relatively long time to pay These are rms that have a relatively long time between the time that purchased inventory is paid for and the time that inventory is sold and payment received Thus these are rms that have relatively short payables periods andor relatively long receivable cycles 1 Use The cash balance declined by 200 to pay the dividend b Source The cash balance increased by 500 assuming the goods bought on payables credit were sold for cash 0 Use The cash balance declined by 900 to pay for the xed assets d Use The cash balance declined by 625 to pay for the higher level of inventory 8 Use The cash balance declined by 1200 to pay for the redemption of debt Carrying costs will decrease because they are not holding goods in inventory Shortage costs will probably increase depending on how close the suppliers are and how well they can estimate need The operating cycle will decrease because the inventory period is decreased Since the cash cycle equals the operating cycle minus the accounts payable period it is not possible for the cash cycle to be longer than the operating cycle if the accounts payable is positive Moreover it is unlikely that the accounts payable period would ever be negative since that implies the rm pays its bills before they are incurred Shortage costs are those costs incurred by a lm when its investment in current assets is low There are two basic types of shortage costs 1 Trading or order costs Order costs are the costs of placing an order for more cash or more inventory 2 Costs related to safety reserves These costs include lost sales lost customer goodwill and disruption of production schedules A longterm growth trend in sales will require some permanent investment in current assets Thus in the real world net working capital is not zero Also the variation across time for assets means that net working capital is unlikely to be zero at any point in time This is a liquidity reason It lengthened its payables period thereby shortening its cash cycle 484 9 Their receivables period increased thereby increasing their operating and cash cycles p n O It is sometimes argued that large firms take advantage of smaller firms by threatening to take their business elsewhere However considering a move to another supplier to get better terms is the nature of competitive free enterprise p A p A They would like to The payables period is a subject of much negotiation and it is one aspect of the price a firm pays its suppliers A firm will generally negotiate the best possible combination of payables period and price Typically suppliers provide strong financial incentives for rapid payment This issue is discussed in detail in a later chapter on credit policy p A N BlueSky will need less financing because it is essentially borrowing more from its suppliers Among other things BlueSky will likely need less shortterm borrowing from other sources so it will save on interest expense Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic 1 a No change A dividend paid for by the sale of debt will not change cash since the cash raised from the debt offer goes immediately to shareholders b No change The real estate is paid for by the cash raised from the debt so this will not change the cash balance c No change Inventory and accounts payable will increase but neither will impact the cash account d Decrease The shortterm bank loan is repaid with cash which will reduce the cash balance e Decrease The payment of taxes is a cash transaction f Decrease The preferred stock will be repurchased with cash g No change Accounts receivable will increase but cash will not increase until the sales are paid off h Decrease The interest is paid with cash which will reduce the cash balance i Increase When payments for previous sales or accounts receivable are paid off the cash balance increases since the payment must be made in cash j Decrease The accounts payable are reduced through cash payments to suppliers 485 k Decrease Here the dividend payments are made with cash which is generally the case This is different from part a where debt was raised to make the dividend payment I No change The shortterm note will not change the cash balance m Decrease The utility bills must be paid in cash n Decrease A cash payment will reduce cash 0 Increase If marketable securities are sold the company will receive cash from the sale The total liabilities and equity of the company are the net book worth or market value of equity plus current liabilities and longterm debt so Total liabilities and equity 10380 1450 7500 Total liabilities and equity 19330 This is also equal to the total assets of the company Since total assets are the sum of all assets and cash is an asset the cash account must be equal to total assets minus all other assets so Cash 19330 715190 7 2105 Cash 2035 We have NWC other than cash so the total NWC is NWC 2105 2035 NWC 4140 We can find total current assets by using the NWC equation NWC is equal to NWC CA 7 CL 4140 CA7 1450 CA 5590 1 Increase If receivables go up the time to collect the receivables would increase which increases the operating cycle b Increase If credit repayment times are increased customers will take longer to pay their bills which will lead to an increase in the operating cycle 0 Decrease If the inventory turnover increases the inventory period decreases d No change The accounts payable period is part of the cash cycle not the operating cycle 486 Decrease If the receivables turnover increases the receivables period decreases No change Payments to suppliers affects the accounts payable period which is part of the cash cycle not the operating cycle Increase Increase If the terms of the cash discount are made less favorable to customers the accounts receivable period will lengthen This will increase both the cash cycle and the operating cycle Increase No change This will shorten the accounts payable period which will increase the cash cycle It will have no effect on the operating cycle since the accounts payable period is not part of the operating cycle Decrease Decrease If more customers pay in cash the accounts receivable period will decrease This will decrease both the cash cycle and the operating cycle Decrease Decrease Assume the accounts payable period and inventory period do not change Fewer raw materials purchased will reduce the inventory period which will decrease both the cash cycle and the operating cycle Decrease No change If more raw materials are purchased on credit the accounts payable period will tend to increase which would decrease the cash cycle We should say that this may not be the case The accounts payable period is a decision made by the company s management The company could increase the accounts payable account and still make the payments in the same number of days This would leave the accounts payable period unchanged which would leave the cash cycle unchanged The change in credit purchases made on credit will not affect the inventory period or the accounts payable period so the operating cycle will not change Increase Increase If more goods are produced for inventory the inventory period will increase This will increase both the cash cycle and operating cycle A 45day collection period implies all receivables outstanding from the previous quarter are collected in the current quarter and 90 7 4590 12 of current sales are collected So Q1 Q2 Q3 Q4 Beginning receivables 27500 35000 31500 40500 Sales 70000 63000 81000 93000 Cash quot quot 762500 766500 772000 787000 Ending receivables 35000 31500 40500 46500 487 b A 60day collection period implies all receivables outstanding from the previous quarter are collected in the current quarter and 906090 13 of current sales are collected So Q1 Q2 Q3 Q4 Beginning receivables 27500 46667 42000 54000 Sales 70000 63000 81000 93000 Cash quot quot 750833 767667 769000 785000 Ending receivables 46667 42000 54000 62000 c A 30day collection period implies all receivables outstanding from the previous quarter are collected in the current quarter and 903090 23 of current sales are collected So Q1 Q2 Q3 Q4 Beginning receivables 27500 23333 21000 27000 Sales 70000 63000 81000 93000 Cash quot quot 774167 765333 775000 789000 Ending receivables 233 33 21000 27000 31000 The operating cycle is the inventory period plus the receivables period The inventory turnover and inventory period are Inventory turnover COGS Average inventory Inventory turnover 10581715382 161472 Inventory turnover 67124 times Inventory period 365 daysInventory turnover Inventory period 365 days67124 Inventory period 5438 days And the receivables turnover and receivables period are Receivables turnover Credit sales Average receivables Receivables turnover 14362512169 126822 Receivables turnover 1156 times Receivables period 365 daysReceivables turnover Receivables period 365 days115589 Receivables period 315774 days So the operating cycle is Operating cycle 5438 days 3158 days Operating cycle 8595 days 488 The cash cycle is the operating cycle minus the payables period The payables turnover and payables period are Payables turnover COGSAverage payables Payables turnover 1058 l713408 141082 Payables turnover 76913 times Payables period 365 daysPayables turnover Payables period 365 days76913 Payables period 4746 days So the cash cycle is Cash cycle 8595 days 7 4746 days Cash cycle 3850 days The firm is receiving cash on average 3850 days after it pays its bills 1 The payables period is zero since the company pays immediately Sales in the year following this one are projected to be 15 greater in each quarter Therefore Q1 sales for the next year will be 830 115 95450 The payment in each period is 30 percent of next period s sales so Q 1 Q2 Q 3 Q4 Payment of accounts 223 50 271 50 29400 28635 b Since the payables period is 90 days the payment in each period is 30 percent of the current period sales so Q 1 Q2 Q 3 Q4 Payment of accounts 24900 22350 27150 29400 0 Since the payables period is 60 days the payment in each period is 23 of last quarter s orders plus 13 of this quarter s orders or Quarterly payments 2330 times current sales l330 next period sales Q1 Q2 Q3 Q4 Payment of accounts 24050 239 50 27900 29145 489 Since the payables period is 60 days the payables in each period will be Payables each period 23 of last quarter s orders 13 of this quarter s orders Payables each period 2375 times current sales l375 next period sales Q1 Q2 Q3 Q4 Payment of accounts 67750 76750 70000 67250 Wages taxes other expenses 16600 21000 19400 17200 Longterm nancing expenses 7300 7300 7300 7300 Total 91650 105050 96700 91750 a The November sales must have been the total uncollected sales minus the uncollected sales from December divided by the collection rate two months after the sale so November sales 79800 7 572000l5 November sales 15066667 b The December sales are the uncollected sales from December divided by the collection rate of the previous months sales so December sales 57200035 December sales 16342857 0 The collections each month for this company are Collections l5Sales from 2 months ago 20Last months sales 65 Current sales January collections l5l5066667 20l6342857 65173000 January collections 16773571 February collections 1516342857 20173000 65184000 February collections 17871429 March collections 15173000 20184000 65205000 March collections 19600000 490 10 The sales collections each month will be Sales collections 35current month sales 60previous month sales Given this collection the cash budget will be April May June Beginning cash balance 448000 398160 508544 Cash receipts Cash collections from credit sales 414400 586560 625440 Total cash available 862400 984720 1133984 Cash disbursements Purchases 249600 235200 280800 Wages taxes and expenses 63600 77136 80480 Interest 18240 18240 18240 Equipment purchases 132800 145600 0 Total cash quot 39 464240 476176 379520 Ending cash balance 398160 508544 754464 1 1 Item SourceUse Amount Cash Source 2 150 Accounts receivable Use 74780 Inventories Use 75560 Property plant and equipment Use 7l7765 Accounts payable Source 2080 Accrued expenses Use 7745 Longterm debt Source 10000 Common stock Source 5000 Accumulated retained earnings Source 3 170 Intermediate p n N First we need to calculate the sales from the last quarter of the previous year Since 50 percent of the sales were collected in that quarter the sales gure must have been Sales last quarter of pervious year 72000000 17 50 Sales last quarter of pervious year 144000000 Now we can estimate the sales growth each quarter and calculate the net sales including the seasonal adjustments The sales gures for each quarter will be Quarter 1 Quarter 2 Quarter 4 Quarter 4 Sales basic trend 150000000 180000000 216000000 259200000 Seasonal adjustment 0 716000000 78000000 21000000 Sales projection 150000000 164000000 208000000 280200000 491 Since 50 percent of sales are collected in the quarter the sales are made and 45 percent of sales are collected in the quarter after the sales are made the cash budget is Quarter 1 Quarter 2 Quarter 4 Quarter 4 Collected within quarter 75000000 82000000 104000000 140100000 Collection from previous quarter 64800000 67500000 73800000 93600000 Cash collections from sales 139800000 149500000 177800000 233700000 13 a A 45day collection period means sales collections each quarter are Collections 12 current sales 12 old sales A 36day payables period means payables each quarter are Payables 35 current orders 25 old orders So the cash in ows and disbursements each quarter are Q1 Q2 Q3 Q4 Beginning receivables 6800 10500 9000 122 50 Sales 21000 18000 24500 28000 Collection of accounts 17300 19500 21250 26250 Ending receivables 10500 9000 12250 14000 Payment of accounts 8640 98 55 11970 11520 Wages taxes and expenses 6300 5400 7350 8400 Capital expenditures 8000 Interest amp dividends 1200 1200 1200 1200 Total cash disbursements 16140 244 55 20520 21120 Total cash collections 17300 19500 21250 26250 Total cash quot 39 16140 24455 20520 21120 Net cash in ow 1160 749 55 730 5130 492 The company s cash budget will be WILDCAT INC Cash Budget in millions Q1 Q2 Q3 Q4 Beginning cash balance 6400 7560 2605 3335 Net cash in ow 1160 4955 730 5130 Ending cash balance 7560 2605 3335 8465 Minimum cash balance 73000 73000 73000 73000 Cumulative surplus de cit 4560 7395 335 5465 With a 30 million minimum cash balance the shortterm nancial plan will be WILDCAT INC ShortTerm Financial Plan in millions Q1 Q2 Q3 Q4 Beginning cash balance 3000 3000 3000 3000 Net cash in ow 1160 4955 730 5130 New shortterm investments 71228 0 489 75140 Income on shortterm investments 068 093 0 010 Shortterm investments sold 0 4628 0 0 New shortterm borrowing 0 234 0 0 Interest on shortterm borrowing 0 0 4107 0 Shortterm borrowing repaid 0 0 7234 0 Ending cash balance 3000 3000 3000 3000 Minimum cash balance 73000 73000 73000 73000 Cumulative surplus deficit 0 0 0 0 Beginning shortterm investments 3400 4628 0 0 Ending shortterm investments 4628 0 489 5150 Beginning shortterm debt 0 0 234 0 Ending shortterm debt 0 234 0 0 Below you will nd the interest paid or received for each quarter Q1 excess funds at start of quarter of 34 invested for 1 quarter earns 0234 068 income Q2 excess funds of 4628 invested for 1 quarter earns 024628 093 in income Q3 shortage funds of 234 borrowed for 1 quarter costs 03234 007 in interest Q4 excess funds of 4 89 invested for 1 quarter earns 024 89 010 in income Net cash cost 068 093 7 007 010 163 493 a With a minimum cash balance of 50 million the shortterm nancial plan will be WILDCAT INC ShortTerm Financial Plan in millions Q1 Q2 Q3 Q4 Beginning cash balance 5000 5000 5000 5000 Net cash in ow 1160 41955 730 5130 New shortterm investments 71188 0 0 73426 Income on shortterm investments 028 052 0 0 Shortterm investments sold 0 2588 0 0 New shortterm borrowing 0 2315 0 0 Interest on shortterm borrowing 0 0 4169 4150 Shortterm borrowing repaid 0 0 61 71655 Ending cash balance 5000 5000 5000 5000 Minimum cash balance 75000 75000 75000 75000 Cumulative surplus deficit 0 0 0 0 Beginning shortterm investments 1400 2588 0 0 Ending shortterm investments 2588 0 0 3426 Beginning shortterm debt 0 0 2315 1655 Ending shortterm debt 0 2315 1655 0 Below you will nd the interest paid or received for each quarter Ql excess funds at start of quarter of 14 invested for 1 quarter earns 02l4 028 income Q2 excess funds of 2588 invested for 1 quarter earns 022588 052 in income Q3 shortage of funds of 2315 borrowed for 1 quarter costs 0323 l5 069 in interest Q4 shortage of funds of 1655 borrowed for 1 quarter costs 03l655 050 in interest Net cash cost 028 052 7 069 7 050 7039 494 b And with a minimum cash balance of 10 million the shortterm financial plan will be WILDCAT INC ShortTerm Financial Plan in millions Q1 Q2 Q3 Q4 Beginning cash balance 1000 1000 1000 1000 Net cash in ow 1160 41955 730 5130 New shortterm investments 71268 0 7767 75145 Income on shortterm investments 108 133 037 015 Shortterm investments sold 0 4822 0 0 New shortterm borrowing 0 0 0 0 Interest on shortterm borrowing 0 0 0 0 Shortterm borrowing repaid 0 0 0 0 Ending cash balance 1000 1000 1000 1000 Minimum cash balance 71000 71000 71000 71000 Cumulative surplus deficit 0 0 0 0 Beginning shortterm investments 5400 6668 1846 2613 Ending shortterm investments 6668 1846 2613 7848 Beginning shortterm debt 0 0 0 0 Ending shortterm debt 0 0 0 0 Below you will find the interest paid or received for each quarter Q1 excess funds at start of quarter of 54 invested for 1 quarter earns 0254 108 income Q2 excess funds of 6668 invested for 1 quarter earns 026668 133 in income Q3 excess funds of 1846 invested for 1 quarter earns 021846 037 in income Q4 excess funds of 2613 invested for 1 quarter earns 022613 052 in income Net cash cost 108 133 037 052 331 Since cash has an opportunity cost the rm can boost its profit if it keeps its minimum cash balance low and invests the cash instead However the tradeoff is that in the event of unforeseen circumstances the fum may not be able to meet its shortrun obligations if enough cash is not available 495 The current assets of Cleveland Compressor are nanced largely by retained earnings From 2009 to 2010 total current assets grew by 7212 Only 2126 of this increase was financed by the growth of current liabilities Pnew York Pneumatic s current assets are largely nanced by current liabilities Bank loans are the most important of these current liabilities They grew 3077 to finance an increase in current assets of 8333 Cleveland Compressor holds the larger investment in current assets It has current assets of 92616 while Pnew York Pneumatic has 78434 in current assets The main reason for the difference is the larger sales of Cleveland Compressor Cleveland Compressor is more likely to incur shortage costs because the ratio of current assets to sales is 057 That ratio for Pnew York Pneumatic is 086 Similarly Pnew York Pneumatic is incurring more carrying costs for the same reason a higher ratio of current assets to sales 496 CHAPTER 2 7 CASH MANAGEMENT Answers to Concepts Review and Critical Thinking Questions 1 Yes Once a rm has more cash than it needs for operations and planned expenditures the excess cash has an opportunity cost It could be invested by shareholders in potentially more pro table ways Question 10 discusses another reason If it has too much cash it can simply pay a dividend or more likely in the current financial environment buy back stock It can also reduce debt If it has insuf cient cash then it must either borrow sell stock or improve pro tability Probably not Creditors would probably want substantially more Cash management is associated more with the collection and disbursement of cash Liquidity management is broader and concerns the optimal level of liquid assets needed by a rm Thus for example a company s stockpiling of cash is liquidity management whereas evaluating a lockbox system is cash management Such instruments go by a variety of names but the key feature is that the dividend adjusts keeping the price relatively stable This price stability along with the dividend tax exemption makes so called adjustable rate preferred stock very attractive relative to interestbearing instruments Net disbursement oat is more desirable because the bank thinks the rm has more money than it actually does and the firm is therefore receiving interest on funds it has already spent The rm has a net disbursement oat of 500000 If this is an ongoing situation the rm may be tempted to write checks for more than it actually has in its account 1 About the only disadvantage to holding Tbills are the generally lower yields compared to alternative money market investments b Some ordinary preferred stock issues pose both credit and price risks that are not consistent with most shortterm cash management plans 0 The primary disadvantage of NCDs is the normally large transactions sizes which may not be feasible for the shortterm investment plans of many smaller to mediumsized corporations d The primary disadvantages of the commercial paper market are the higher default risk characteristics of the security and the lack of an active secondary market which may excessively restrict the exibility of corporations to meet their liquidity adjustment needs 8 The primary disadvantages of RANs is that some possess nontrivial levels of default risk and also corporations are somewhat restricted in the type and amount of these taxexempts that they can hold in their portfolios 497 p n O p A p A p A N f The primary disadvantage of the repo market is the generally very short maturities available The concern is that excess cash on hand can lead to poorly thoughtout management decisions The thought is that keeping cash levels relatively low forces management to pay careful attention to cash ow and capital spending A potential advantage is that the quicker payment often means a better price The disadvantage is that doing so increases the rm s cash cycle This is really a capital structure decision If the rm has an optimal capital structure paying off debt moves it to an underleveraged position However a combination of debt reduction and stock buy backs could be structured to leave capital structure unchanged It is unethical because you have essentially tricked the grocery store into making you an interestfree loan and the grocery store 1s harmed because it could have earned interest on the money instead of loaning it to you Solutions to Questions and Problems NOTE All end of chapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic The average daily oat is the average amount of checks received per day times the average number of days delay divided by the number of days in a month Assuming 30 days in a month the average daily oat is Average daily oat 415600030 Average daily oat 20800 a The disbursement oat is the average monthly checks written times the average number of days for the checks to clear so Disbursement oat 4l4000 Disbursement oat 56000 The collection oat is the average monthly checks received times the average number of days for the checks to clear so Collection oat 2726000 Collection oat 752000 The net oat is the disbursement oat plus the collection oat so Net oat 56000 7 52000 Net oat 4000 498 The new collection oat will be Collection oat l726000 Collection oat 726000 And the new net oat will be Net oat 56000 726000 Net oat 30000 The collection oat is the average daily checks received times the average number of days for the checks to clear so Collection oat 3l9000 Collection oat 57000 The rm should pay no more than the amount of the oat or 57000 to eliminate the oat The maximum daily charge the rm should be willing to pay is the collection oat times the daily interest rate so Maximum daily charge 5700000019 Maximum daily charge 1083 Total oat 417000 56000 Total oat 98000 The average daily oat is the total oat divided by the number of days in a month Assuming 30 days in a month the average daily oat is Average daily oat 9800030 Average daily oat 326667 The average daily receipts are the average daily checks received divided by the number of days in a month Assuming a 30 day month Average daily receipts 17000 6000 30 Average daily receipts 76667 The weighted average delay is the sum of the days to clear a check times the amount of the check divided by the average daily receipts so Weighted average delay 4l700023000 5600023000 Weighted average delay 426 days 499 The average daily collections are the number of checks received times the average value of a check so Average daily collections 1088500 Average daily collections 918000 The present value of the lockbox service is the average daily receipts times the number of days the collection is reduced so PV 2 day reduction9l8000 PV 1836000 The daily cost is a perpetuity The present value of the cost is the daily cost divided by the daily interest rate So PV of cost 22500016 PV of cost 1406250 The firm should take the lockbox service The NPV of the lockbox is the cost plus the present value of the reduction in collection time so NPV 71406250 1836000 NPV 429750 The annual savings excluding the cost would be the future value of the savings minus the costs so Annual savings 1836000100016365 7 1836000 Annual savings 11040605 And the annual cost would be the future value of the daily cost which is an annuity so Annual cost 225FVIFA3657 016 Annual cost 8456346 So the annual net savings would be Annual net savings 11040605 7 8456346 Annual net savings 2584259 1 The average daily oat is the sum of the percentage each check amount is of the total checks received times the number of checks received times the amount of the check times the number of days until the check clears divided by the number of days in a month Assuming a 30 day month we get Average daily oat 605300552 40530080330 Average daily oat 28620 On average there is 28620 that is uncollected and not available to the film 500 The total collections are the sum of the percentage of each check amount received times the total checks received times the amount of the check so Total collections 60530055 40530080 Total collections 344500 The weighted average delay is the sum of the average number of days a check of a speci c amount is delayed times the percentage that check amount makes up of the total checks received so Weighted average delay 260530055344500 340530080 344500 Weighted average delay 249 days The average daily oat is the weighted average delay times the average checks received per day Assuming a 30 day month we get Average daily oat 24934450030 days Average daily oat 28620 The most the rm should pay is the total amount of the average oat or 28620 The average daily interest rate is 107 1 R365 R 01854 per day The daily cost of oat is the average daily oat times the daily interest rate so Daily cost of the oat 286200001854 Daily cost of the oat 531 The most the rm should pay is still the average daily oat Under the reduced collection time assumption we get New average daily oat 1534450030 New average daily oat 17225 The present value of adopting the system is the number of days collections are reduced times the average daily collections so PV 33851105 PV 1276275 The NPV of adopting the system is the present value of the savings minus the cost of adopting the system The cost of adopting the system is the present value of the fee per transaction times the number of transactions This is a perpetuity so NPV 1276275 7 0503850002 NPV 313775 501 The net cash ows is the present value of the average daily collections times the daily interest rate minus the transaction cost per day so Net cash ow per day l2762750002 7 050385 Net cash ow per day 6276 The net cash ow per check is the net cash ow per day divided by the number of checks received per day or Net cash ow per check 6276385 Net cash ow per check 016 Alternatively we could nd the net cash ow per check as the number of days the system reduces collection time times the average check amount times the daily interest rate minus the transaction cost per check Doing so we con rm our previous answer as Net cash ow per check 3l1050002 7 050 Net cash ow per check 016 per check The reduction in cash balance from adopting the lockbox is the number of days the system reduces collection time times the average daily collections so Cash balance reduction 3l45000 Cash balance reduction 43 5000 The dollar return that can be earned is the average daily interest rate times the cash balance reduction The average daily interest rate is AVerage daily rate 7 109 365 1 Average daily rate 0236 per day The daily dollar return that can be earned from the reduction in days to clear the checks is Daily dollar return 435000000236 Daily dollar return 10272 If the company takes the lockboX it will receive three payments early with the rst payment occurring today We can use the daily interest rate from part b so the savings are Savings 7 145000 145000PVIFA 023W Savings 43489732 If the lockbox payments occur at the end of the month we need the effective monthly interest rate which is Monthly interest rate 109112 7 1 Monthly interest rate 07207 502 O p A Assuming the lockbox payments occur at the end of the month the lockbox payments which are a perpetuity will be PV CR 43489732 C 007207 C 313435 It could also be assumed that the lockboX payments occur at the beginning of the month If so we would need to use the PV of a perpetuity due which is PV C C R Solving for C cPv gtltR1R C 43489732 X 007207 1 007207 C 311202 The interest that the company could earn will be the amount of the checks times the number of days it will delay payment times the number of weeks that checks will be disbursed times the daily interest rate so Interest 93000752200015 Interest 253890 The bene t of the new arrangement is the 4 million in accelerated collections since the new system will speed up collections by one day The cost is the new compensating balance but the company will recover the existing compensating balance so NPV 4000000 7 500000 7 400000 NPV 3900000 The company should proceed with the new system The savings are the NPV times the annual interest rate so Net savings 390000005 Net savings 195000 Intermediate To nd the NPV of taking the lockboX we rst need to calculate the present value of the savings The present value of the savings will be the reduction in collection time times the average daily collections so PV 2750980 PV 1470000 And the daily interest rate is Daily interest rate 10701365 7 1 Daily interest rate 00019 or 019 per day 503 N The transaction costs are a perpetuity The cost per day is the cost per transaction times the number of transactions per day so the NPV of taking the lockboX is NPV 7 1470000 7 03575000019 NPV 7 5401517 Without the fee the lockbox system should be accepted To calculate the NPV of the lockboX with the annual fee we can simply use the NPV of the lockbox without the annual fee and subtract the addition cost The annual fee is a perpetuity so with the fee the NPV of taking the lockbox is NPV 5401517 7 500007 NPV 71741340 With the fee the lockboX system should not be accepted To nd the minimum number of payments per day needed to make the lockbox system feasible is the number of checks that makes the NPV of the decision equal to zero The average daily interest rate is Daily interest rate 1051365 7 1 Daily interest rate 0134 per day The present value of the savings is the average payment amount times the days the collection period is reduced times the number of customers The costs are the transaction fee and the annual fee Both are perpetuities The total transaction costs are the transaction costs per check times the number of checks The equation for the NPV of the project where N is the number of checks transacted per day is NPV 0 53001N 7 010N000134 7 2000005 400000 5300N 7 74805N 455195N 400000 N 8787 m 88 customers per day 504 APPENDIX 2 7A a Decrease This will lower the trading costs which will cause a decrease in the target cash balance b Decrease This will increase the holding cost which will cause a decrease in the target cash balance 0 Increase This will increase the amount of cash that the fum has to hold in noninterest bearing accounts so they will have to raise the target cash balance to meet this requirement d Decrease If the credit rating improves then the firm can borrow more easily allowing it to lower the target cash balance and borrow if a cash shortfall occurs 8 Increase If the cost of borrowing increases the firm will need to hold more cash to protect against cash shortfalls as its borrowing costs become more prohibitive f Either This depends somewhat on what the fees apply to but if direct fees are established then the compensating balance may be lowered thus lowering the target cash balance If on the other hand fees are charged on the number of transactions then the rm may wish to hold a higher cash balance so they are not transferring money into the account as often The target cash balance using the BAT model is Cquot 2T x Fm 2 cquot 28500250612 C 266145 The initial balance should be 266145 and whenever the balance drops to 0 another 266145 should be transferred in The holding cost is the average daily cash balance times the interest rate so Holding cost 130005 Holding cost 6500 The trading costs are the total cash needed times the replenishing costs divided by the average daily balance times two so Trading cost 43000813002 Trading cost 13231 The total cost is the sum of the holding cost and the trading cost so Total cost 6500 13231 Total cost 19731 505 The target cash balance using the BAT model is Cquot 2T x F 12 cquot 243000805 2 Cquot 370945 They should increase their average daily cash balance to New average cash balance 3709452 New average cash balance 185472 This would minimize the costs The new total cost would be New total cost 18457205 4300082185472 New total cost 18547 a The opportunity costs are the amount transferred times the interest rate divided by two so Opportunity cost 1500052 Opportunity cost 3750 The trading costs are the total cash balance times the trading cost per transaction divided by the amount transferred so Trading cost 16000251500 Trading cost 26667 The rm keeps too little in cash because the trading costs are much higher than the opportunity costs b The target cash balance using the BAT model is Cquot 2T x mm Cquot 2160002505 2 Cquot 4000 The total cash needed is the cash shortage per month times twelve months so Total cash 12140000 Total cash 1680000 The target cash balance using the BAT model is Cquot 2T x F m 2 Cquot 21680000500057 2 Cquot 17167902 506 The company should invest Invest 690000 7 17167902 Invest 51832098 of its current cash holdings in marketable securities to bring the cash balance down to the optimal level Over the rest of the year sell securities Sell securities l680000 17167902 Sell securities 979 m 10 times The lower limit is the minimum balance allowed in the account and the upper limit is the maximum balance allowed in the account When the account balance drops to the lower limit Securities sold 80000 7 43000 Securities sold 37000 in marketable securities will be sold and the proceeds deposited in the account This moves the account balance back to the target cash level When the account balance rises to the upper limit then Securities purchased 125000 7 80000 Securities purchased 45000 of marketable securities will be purchased This expenditure brings the cash level back down to the target balance of 80000 The target cash balance using the MillerOrr model is Cquot L 34 XF x czR13 cquot 1500 3440702000211 Cquot 238790 The upper limit is Uquot 3 x Cquot 7 2 x L Uquot 3238790 7 21500 Uquot 416371 507 O When the balance in the cash account drops to 1500 the rm sells Sell 238790 71500 Sell 88790 of marketable securities The proceeds from the sale are used to replenish the account back to the optimal target level of Cl Conversely when the upper limit is reached the rm buys Buy 4163717 238790 Buy 177581 of marketable securities This expenditure lowers the cash level back down to the optimal level of 238790 As variance increases the upper limit and the spread will increase while the lower limit remains unchanged The lower limit does not change because it is an exogenous variable set by management As the variance increases however the amount of uncertainty increases When this happens the target cash balance and therefore the upper limit and the spread will need to be higher If the variance drops to zero then the lower limit the target balance and the upper limit will all be the same The average daily interest rate is Daily rate 1071365 7 1 Daily rate 000185 or 0185 per day The target cash balance using the MillerOrr model is CL34xeczR 3 Cl 160000 34300890000000185 3 Cquot 17026047 The upper limit is Uquot 3 x Cquot 7 2 x L Uquot 317026047 72160000 Uquot 19078141 Using the BAT model and solving for R we get Cquot 2T x mm 2700 22800010R 2 R 2280001027002 R 0768 or 768 508 CHAPTER 28 CREDIT AND IN V ENTORY MANAGEMENT Answers to Concepts Review and Critical Thinking Questions 1 9amp999 A sight draft is a commercial draft that is payable immediately A time draft is a commercial draft that does not require immediate payment A bankers acceptance is when a bank guarantees the future payment of a commercial draft A promissory note is an IOU that the customer signs A trade acceptance is when the buyer accepts the commercial draft and promises to pay it in the future Trade credit is usually granted on open account The invoice is the credit instrument Credit costs cost of debt probability of default and the cash discount Nocredit costs lost sales The sum of these are the carrying costs 1 2 3 MA NQMrkSNNt Character determines if a customer is willing to pay his or her debts Capacity determines if a customer is able to pay debts out of operating cash ow Capital determines the customer s nancial reserves in case problems occur with operating cash ow Collateral assets that can be liquidated to pay off the loan in case of default Conditions customer s ability to weather an economic downturn and whether such a ownturn is likely Perishability and collateral value Consumer deman Cost pro tability and standardization Credit risk The size of the account Competition Customer type If the credit period exceeds a customer s operating cycle then the rm is nancing the receivables and other aspects of the customer s business that go beyond the purchase of the selling film s merchandise 1 B A is likely to sell for cash only unless the product really works If it does then they might grant longer credit periods to entice buyers b A Landlords have significantly greater collateral and that collateral is not mobile 0 A Since A s customers turn over inventory less frequently they have a longer inventory period and thus will most likely have a longer credit period as well d B Since A s merchandise is perishable and B s is not B will probably have a longer credit period 509 p n O e A Rugs are fairly standardized and they are transportable while carpets are custom t and are not particularly transportable The three main categories of inventory are raw material initial inputs to the rm s production process workinprogress partially completed products and nished goods products ready for sale From the rm s perspective the demand for nished goods is independent from the demand for the other types of inventory The demand for raw material and workinprogress is derived from or dependent on the firm s needs for these inventory types in order to achieve the desired levels of nished goods IIT systems reduce inventory amounts Assuming no adverse effects on sales inventory turnover will increase Since assets will decrease total asset turnover will also increase Recalling the DuPont equation an increase in total asset turnover all else being equal has a positive effect on ROE Carrying costs should be equal to order costs Since the carrying costs are low relative to the order costs the firm should increase the inventory level Since the price of components can decline quickly Dell does not have inventory which is purchased and then declines quickly in value before it is sold If this happens the inventory may be sold at a loss While this approach is valuable it is difficult to implement For example Dell manufacturing plants will often have areas set aside that are for the suppliers When parts are needed it is a matter of going across the oor to get new parts In fact most computer manufacturers are trying to implement similar inventory systems Solutions to Questions and Problems NOTE All end of chapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic a There are 30 days until account is overdue Ifyou take the full period you must remit Remittance 400125 Remittance 50000 b There is a 1 percent discount offered with a 10 day discount period If you take the discount you will only have to remit Remittance l 7 0150000 Remittance 49500 c The implicit interest is the difference between the two remittance amounts or Implicit interest 50000 7 49500 Implicit interest 500 510 2 The number of days credit offered is Days credit 30 7 10 Days credit 20 days The receivables turnover is Receivables turnover 365 Average collection period Receivables turnover 36536 Receivables turnover 10139 times And the average receivables are Average receivables SalesReceivables period Average receivables 47000000 10139 Average receivables 4635616 1 The average collection period is the percentage of accounts taking the discount times the discount period plus the percentage of accounts not taking the discount times the days until full payment is required so Average collection period 65 10 days 3530 days Average collection period 17 days b And the average daily balance is Average balance 130017001712365 Average balance 123517808 The daily sales are Daily sales 19400 7 Daily sales 277143 Since the average collection period is 34 days the average accounts receivable is Average accounts receivable 27714334 Average accounts receivable 9422857 The interest rate for the term of the discount is Interest rate 0 1 99 Interest rate 0101 or 101 And the interest is for 35 7 10 25 days 511 So using the EAR equation the effective annual interest rate is EAR 1 Periodic ratem 7 1 EAR 10101 25 71 EAR 1580 or 1580 a The periodic interest rate is Interest rate 0298 Interest rate 0204 or 204 And the EAR is EAR 10202036525 7 1 EAR 3431 or 3431 b The EAR is EAR 101013655 71 EAR 0761 or 761 c The EAR is EAR 10101 20 71 EAR 2013 or 2013 The receivables turnover is Receivables turnover 365 Average collection period Receivables turnover 36539 Receivables turnover 93590 times And the annual credit sales are Annual credit sales Receivables turnover gtlt Average daily receivables Annual credit sales 9359047500 Annual credit sales 44455128 The total sales of the firm are equal to the total credit sales since all sales are on credit so Total credit sales 5600425 Total credit sales 2380000 The average collection period is the percentage of accounts taking the discount times the discount period plus the percentage of accounts not taking the discount times the days until full payment is required so Average collection period 60 10 4040 Average collection period 22 days 512 The receivables turnover is 365 divided by the average collection period so Receivables turnover 36522 Receivables turnover 16591 times And the average receivables are the credit sales divided by the receivables turnover so Average receivables 23 80000 16591 Average receivables 14345205 If the rm increases the cash discount more people will pay sooner thus lowering the average collection period If the ACP declines the receivables turnover increases which will lead to a decrease in the average receivables The average collection period is the net credit terms plus the days overdue so Average collection period 30 8 Average collection period 38 days The receivables turnover is 365 divided by the average collection period so Receivables turnover 36538 Receivables turnover 96053 times And the average receivables are the credit sales divided by the receivables turnover so Average receivables 8400000 96053 Average receivables 87452055 1 The cash outlay for the credit decision is the variable cost of the engine If this is a onetime order the cash in ow is the present value of the sales price of the engine times one minus the default probability So the NPV per unit is NPV 71600000 1700518700001029 NPV 20821186 per unit The company should ll the order b To nd the breakeven probability of default 7 we simply use the NPV equation from part a set it equal to zero and solve for 7 Doing so we get NPV 0 481600000 17 n18700001029 739 1196 or 1196 We would not accept the order if the default probability was higher than 1196 percent 513 10 1 p A c If the customer will become a repeat customer the cash in ow changes The cash in ow is now one minus the default probability times the sales price minus the variable cost We need to use the sales price minus the variable cost since we will have to build another engine for the customer in one period Additionally this cash in ow is now a perpetuity so the NPV under these assumptions is NPV 781600000 1 7 0051870000 7 1600000029 NPV 766379310 per unit The company should fill the order The breakeven default probability under these assumptions is NPV 0 7 71600000 7 17 n1870000 71600000029 739 7 8281 or 8281 We would not accept the order if the default probability was higher than 8281 percent This default probability is much higher than in part b because the customer may become a repeat customer d It is assumed that if a person has paid his or her bills in the past they will pay their bills in the future This implies that if someone doesn t default when credit is rst granted then they will be a good customer far into the future and the possible gains from the future business outweigh the possible losses from granting credit the rst time The cost of switching is any lost sales from the existing policy plus the incremental variable costs under the new policy so Cost of switching 7201305 4951380 7 1305 Cost of switching 976725 The benefit of switching is any increase in the sales price minus the variable costs per unit times the incremental units sold so Bene t of switching 720 74951380 7 1305 Bene t of switching 16875 The bene t of switching is a perpetuity so the NPV of the decision to switch is NPV 7976275 16875015 NPV 14827500 The rm will have to bear the cost of sales for one month before they receive any revenue from credit sales which is why the initial cost is for one month Receivables will grow over the one month credit period and will then remain stable with payments and new sales offsetting one another The carrying costs are the average inventory times the cost of carrying an individual unit so Carrying costs 250029 11250 514 N 03 The order costs are the number of orders times the cost of an order so Order costs 521700 88400 The economic order quantity is EOQ 2T x FCC 2 EOQ 252250017009 2 EOQ 700793 The rm s policy is not optimal since the carrying costs and the order costs are not equal The company should increase the order size and decrease the number of orders The carrying costs are the average inventory times the cost of carrying an individual unit so Carrying costs 30024l 6150 The order costs are the number of orders times the cost of an order so Restocking costs 5295 4940 The economic order quantity is EOQ 2T x FCC12 EOQ 2523009541 2 EOQ 26887 The number of orders per year will be the total units sold per year divided by the EOQ so Number of orders per year 5230026887 Number of orders per year 5802 The lm s policy is not optimal since the carrying costs and the order costs are not equal The company should decrease the order size and increase the number of orders Intermediate The total carrying costs are Carrying costs Q2 x CC where CC is the carrying cost per unit The restocking costs are Restocking costs F x TQ Setting these equations equal to each other and solving for Q we nd CC x Q2 F x TQ Q22xeTCC Q2F x TCC1ZEOQ 515 14 p n UI The cash ow from either policy is Cash ow P 7 VQ So the cash ows from the old policy are Cash ow from old policy 91 7 473850 Cash ow from old policy 169400 And the cash ow from the new policy would be Cash ow from new policy 94 7 473940 Cash ow from new policy 185180 So the incremental cash ow would be Incremental cash ow 185180 7 169400 Incremental cash ow 15780 The incremental cash ow is a perpetuity The cost of initiating the new policy is Cost of new policy 7PQ VQ39 7 Q So the NPV of the decision to change credit policies is NPV 7 7913850 473940 7 3850 15780025 NPV 7 276620 The cash ow from the old policy is Cash ow from old policy 290 7 2301105 Cash ow from old policy 66300 And the cash ow from the new policy will be Cash ow from new policy 295 7 2341125 Cash ow from new policy 68625 The incremental cash ow which is a perpetuity is the difference between the old policy cash ows and the new policy cash ows so Incremental cash ow 68625 7 66300 Incremental cash ow 2325 516 GK 1 The cost of switching credit policies is Cost ofnew policy 7PQ Qv39 7v v39Q39 7 Q In this cost equation we need to account for the increased variable cost for all units produced This includes the units we already sell plus the increased variable costs for the incremental units So the NPV of switching credit policies is NPV 7 72901105 1105234 7 230 2341 125 7110523250095 NPV 7 78481316 If the cost of subscribing to the credit agency is less than the savings from collection of the bad debts the company should subscribe The cost of the subscription is Cost of the subscription 450 5500 Cost of the subscription 2950 And the savings from having no bad debts will be Savings from not selling to bad credit risks 490500004 Savings from not selling to bad credit risks 9800 So the company s net savings will be Net savings 9800 7 2950 Net savings 6850 The company should subscribe to the credit agency Challenge The cost of switching credit policies is Cost of new policy 7PQ Qv39 7 v v39Q39 7 Q And the cash ow from switching which is a perpetuity is Cash ow from new policy Q39P39 7 v 7 QP 7 v To nd the breakeven quantity sold for switching credit policies we set the NPV equal to zero and solve for Q39 Doing so we nd NPV 7 0 7 7913850 47Q39 7 3850 Q3994 7 47 7 385091 7 47025 0 7 78350350 7 47Q39 180950 1880Q39 7 6776000 1833Q39 7 6945400 Q39 7 378909 517 18 p A O N O We can use the equation for the NPV we constructed in Problem 17 Using the sales gure of 4100 units and solving for P39 we get NPV 7 0 7 H913850 7 474100 7 3850 P39 7 474100 7 91 7 473850025 0 7 7350350 7 11750 164000P39 7 7708000 7 6776000 164000P39 7 14846100 P39 7 9053 From Problem 15 the incremental cash ow from the new credit policy will be Incremental cash ow Q39P39 7v39 7 QP 7v And the cost of the new policy is Cost of new policy 7PQ Qv39 7 V v39Q39 7 Q Setting the NPV equal to zero and solving for P39 we get NPV 0 72901105 234 7 2301105 2341 125 711051125P397 234 7 1105290 7 2300095 0 7320450 4420 4680 11842105P39 7 2771052632 7 697894737 11842105P39 3501902368 P39 29572 Since the company sells 700 suits per week and there are 52 weeks per year the total number of suits sold is Total suits sold 700 X 52 36400 And the EOQ is 500 suits so the number of orders per year is Orders per year 36400 500 7280 To determine the day when the next order is placed we need to determine when the last order was placed Since the suits arrived on Monday and there is a 3 day delay from the time the order was placed until the suits arrive the last order was placed Friday Since there are ve days between the orders the next order will be placed on Wednesday Alternatively we could consider that the store sells 100 suits per day 700 per week 7 days This implies that the store will be at the safety stock of 100 suits on Saturday when it opens Since the suits must arrive before the store opens on Saturday they should be ordered 3 days prior to account for the delivery time which again means the suits should be ordered in Wednesday 518 APPENDIX 28A The cash ow from the old policy is the quantity sold times the price so Cash ow from old policy 40000510 Cash ow from old policy 20400000 The cash ow from the new policy is the quantity sold times the new price all times one minus the default rate so Cash ow from new policy 40000537l 7 03 Cash ow from new policy 20835600 The incremental cash ow is the difference in the two cash ows so Incremental cash ow 20835600 7 20400000 Incremental cash ow 435600 The cash ows from the new policy are a perpetuity The cost is the old cash ow so the NPV of the decision to switch is NPV 20400000 435600025 NPV 72976000 1 The old price as a percentage of the new price is 909184 98 So the discount is Discount 17 98 02 or 2 The credit terms will be Credit terms 2 15 net 30 b We are unable to determine for certain since no information is given concerning the percentage of customers who will take the discount However the maximum receivables would occur if all customers took the credit so Receivables 330090 Receivables 297000 at a maximum 0 Since the quantity sold does not change variable cost is the same under either plan 519 No because d 7 71 02 7 11 d 7 TI 709 or 79 Therefore the NPV will be negative The NPV is NPV 7 7330090 3300918402 7 1101 NPV 7 73023592 The breakeven credit price is P1 rl 71 9010189 P 7 10213 This implies that the breakeven discount is Breakeven discount 1 7 90102 l3 Breakeven discount 1188 or 1188 The NPV at this discount rate is NPV 7330090 3300102 l3 1188 7 1101 NPV m 0 The cost of the credit policy switch is the quantity sold times the variable cost The cash in ow is the price times the quantity sold times one minus the default rate This is a onetime lump sum so we need to discount this value one period Doing so we find the NPV is NPV 7 715760172151140102 NPV 7 201176 The order should be taken since the NPV is positive To find the breakeven default rate 7 we just need to set the NPV equal to zero and solve for the breakeven default rate Doing so we get NPV 7 0 7 71576017n151140102 739 7 3200 or 3200 Effectively the cash discount is Cash discount 1140 7 l090ll40 Cash discount 0439 or 439 Since the discount rate is less than the default rate credit should not be granted The rm would be better off taking the 1090 upfront than taking an 80 chance of making 1140 520 The cash discount is Cash discount 75 7 7l75 Cash discount 0533 or 533 The default probability is one minus the probability of payment or Default probability l 7 90 Default probability 10 Since the default probability is greater than the cash discount credit should not be granted the NPV of doing so is negative Due to the increase in both quantity sold and credit price when credit is granted an additional incremental cost is incurred of Additional cost 620033 7 32 6900 7 620033 Additional cost 29300 The breakeven price under these assumptions is NPV 0 7829300 7 620071 69001 7 10P39 7 3317 6200717 32100753 7 1 NPV 734100 7 440200 273940311339 7 1004447808 7 1066646816 2118524624 27394031P39 P39 7732 The credit report is an additional cost so we have to include it in our analysis The NPV when using the credit reports is NPV 620032 7 90690033 7 620071 7 6900150 690009075 7 33 7 150 7 620071732100753 71 NPV 198400 7 204930 7 440200 7 10350 38445773 NPV 787462227 The reports should not be purchased and credit should not be granted 521 We can express the old cash ow as Old cash ow P 7 VQ And the new cash ow will be New cash ow P 7Vl 7 0LQ39 OLQ39 l 771P39 7V So the incremental cash ow is Incremental cash ow 7P 7VQ P 7Vl 7 0LQ39 OLQ39 l 7 7P39 7 V Incremental cash ow P 7VQ39 7 Q OLQ39 l 7 7P39 7 P Thus 7 397 7 39 PVQ39QaQ 17rP39P NPV P VQ Q ocPQ R J 522 CHAPTER 29 MERGERS AND ACQUISITIONS Answers to Concepts Review and Critical Thinking Questions 1 In the purchase method assets are recorded at market value and goodwill is created to account for the excess of the purchase price over this recorded value In the pooling of interests method the balance sheets of the two firms are simply combined no goodwill is created The choice of accounting method has no direct impact on the cash ows of the firms EPS will probably be lower under the purchase method because reported income is usually lower due to the required amortization of the goodwill created in the purchase 1 False Although the reasoning seems correct in general the new firms do not have monopoly power This is especially true since many countries have laws limiting mergers when it would create a monopoly b True When managers act in their own interest acquisitions are an important control device for shareholders It appears that some acquisitions and takeovers are the consequence of underlying con icts between managers and shareholders 0 False Even if markets are efficient the presence of synergy will make the value of the combined firm different from the sum of the values of the separate rms Incremental cash ows provide the positive NPV of the transaction d False In an efficient market traders will value takeovers based on fundamental factors regardless of the time horizon Recall that the evidence as a whole suggests ef ciency in the markets Mergers should be no different 8 False The tax effect of an acquisition depends on whether the merger is taxable or nontaxable In a taxable merger there are two opposing factors to consider the capital gains effect and the writeup effect The net effect is the sum of these two effects f True Because of the coinsurance effect wealth might be transferred from the stockholders to the bondholders Acquisition analysis usually disregards this effect and considers only the total value Diversi cation doesn t create value in and of itself because diversi cation reduces unsystematic not systematic risk As discussed in the chapter on options there is a more subtle issue as well Reducing unsystematic risk bene ts bondholders by making default less likely However if a merger is done purely to diversify ie no operating synergy then the NPV of the merger is zero If the NPV is zero and the bondholders are better off then stockholders must be worse off A rm might choose to split up because the newer smaller firms may be better able to focus on their particular markets Thus reverse synergy is a possibility An added advantage is that performance evaluation becomes much easier once the split is made because the new firm s nancial results and stock prices are no longer cornmingled 523 p A O It depends on how they are used If they are used to protect management then they are not good for stockholders If they are used by management to negotiate the best possible terms of a merger then they are good for stockholders One of the primary advantages of a taxable merger is the writeup in the basis of the target lm s assets while one of the primary disadvantages is the capital gains tax that is payable The situation is the reverse for a taxfree merger The basic determinant of tax status is whether or not the old stockholders will continue to participate in the new company which is usually determined by whether they get any shares in the bidding rm An LBO is usually taxable because the acquiring group pays off the current stockholders in full usually in cash Economies of scale occur when average cost declines as output levels increase A merger in this particular case might make sense because Eastern and Western may need less total capital investment to handle the peak power needs thereby reducing average generation costs Among the defensive tactics often employed by management are seeking white knights threatening to sell the crown jewels appealing to regulatory agencies and the courts if possible and targeted share repurchases Frequently antitakeover charter amendments are available as well such as poison pills poison puts golden l 39 locku n l ity amendments but these require shareholder approval so they can t be immediately used if time is short While target rm shareholders may bene t from management actively fighting acquisition bids in that it encourages higher bidding and may solicit bids from other parties as well there is also the danger that such defensive tactics will discourage potential bidders from seeking the rm in the rst place which harms the shareholders In a cash offer it almost surely does not make sense In a stock offer management may feel that one suitor is a better longrun investment than the other but this is only valid if the market is not efficient In general the highest offer is the best one Various reasons include 1 Anticipated gains may be smaller than thought 2 Bidding firms are typically much larger so any gains are spread thinly across shares 3 Management may not be acting in the shareholders best interest with many acquisitions 4 Competition in the market for takeovers may force prices for target rms up to the zero NPV level and 5 Market participants may have already discounted the gains from the merger before it is announced Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem Basic For the merger to make economic sense the acquirer must feel the acquisition will increase value by at least the amount of the premium over the market value so Minimum economic value 620000000 7 585000000 35000000 524 1 Since neither company has any debt using the pooling method the asset value of the combined rm must equal the value of the equity so Assets Equity 2600021 200009 726000 b With the purchase method the assets of the combined rm will be the book value of Firm X the acquiring company plus the market value of Firm Y the target company so Assets from X 260002l 546000 book value Assets from Y 20000l9 380000 market value The purchase price of Firm Y is the number of shares outstanding times the sum of the current stock price per share plus the premium per share so Purchase price on 20000l9 5 480000 The goodwill created will be Goodwill 480000 7 380000 100000 And the total asset of the combined company will be Total assets XY Total equity XY 546000 380000 100000 1026000 In the pooling method all accounts of both companies are added together to total the accounts in the new company so the postmerger balance sheet will be Jurion Ca postmerger Current assets 10600 Current liabilities 6400 Fixed assets 30100 Longterm debt 9700 Equity 24600 Total 40700 Total 40700 Since the acquisition is funded by longterm debt the postmerger balance sheet will have longterm debt equal to the original longterm debt of Jurion s balance sheet plus the new longterm debt issue so Postmerger longterm debt 8500 17000 25500 Goodwill will be created since the acquisition price is greater than the market value The goodwill amount is equal to the purchase price minus the market value of assets Generally the market value of current assets is equal to the book value so Goodwill created 17000 7l2000 market value FA 7 2600 market value CA 2400 Current liabilities and equity will remain the same as the premerger balance sheet of the acquiring rm Current assets will be the sum of the two rm s premerger balance sheet accounts and the xed assets will be the sum of the premerger fixed assets of the acquirer and the market value of xed assets of the target rm The postmerger balance sheet will be 525 Juriori Ca postmerger Current assets 10600 Current liabilities 4500 Fixed assets 35000 Longterm debt 25500 Goodwill 2400 Equity 18000 Total 48000 Total 48000 In the pooling method all accounts of both companies are added together to total the accounts in the new company so the postmerger balance sheet will be Silver Enterprises postmerger Current assets 5600 Current liabilities 3800 Other assets 1350 Longterm debt 1800 Net xed assets 11800 Equity 13150 Total 18750 Total 18750 Since the acquisition is funded by longterm debt the postmerger balance sheet will have longterm debt equal to the original longterm debt of Silver s balance sheet plus the new longterm debt issue so Postmerger longterm debt 1800 9100 10900 Goodwill will be created since the acquisition price is greater than the market value The goodwill amount is equal to the purchase price minus the market value of assets Since the market value of xed assets of the target firm is equal to the book value and the book value of all other assets is equal to market value we can subtract the total assets from the purchase price so Goodwill created 9100 7 5650 market value TA 3450 Current liabilities and equity will remain the same as the premerger balance sheet of the acquiring rm Current assets and other assets will be the sum of the two rm s premerger balance sheet accounts and the xed assets will be the sum of the premerger xed assets of the acquirer and the market value of xed assets of the target rm Note in this case the market value and the book value of xed assets are the same The postmerger balance sheet will be Silver Enterprises postmerger Current assets 5600 Current liabilities 2600 Other assets 1350 Longterm debt 10900 Net xed assets 11800 Equity 8700 Goodwill 3450 Total 22200 Total 22200 526 The cash cost is the amount of cash offered so the cash cost is 70 million To calculate the cost of the stock offer we rst need to calculate the value of the target to the acquirer The value of the target rm to the acquiring rm will be the market value of the target plus the PV of the incremental cash ows generated by the target firm The cash ows are a perpetuity so Vi 65000000 160000012 78833333 The cost of the stock offer is the percentage of the acquiring rm given up times the sum of the market value of the acquiring rm and the value of the target rm to the acquiring rm So the equity cost will be Equity cost 4098000000 78833333 70533333 The NPV of each offer is the value of the target rm to the acquiring rm minus the cost of acquisition so NPV cash 78333333 7 70000000 8333333 NPV stock 78333333 7 70533333 7800000 Since the NPV is greater with the cash offer the acquisition should be in cash The EPS of the combined company will be the sum of the earnings of both companies divided by the shares in the combined company Since the stock offer is one share of the acquiring rm for three shares of the target rm new shares in the acquiring rm will increase by onethird of the number of shares of the target company So the new EPS will be EPS 450000 675000180000 1390000 5357 The market price of Stultz will remain unchanged if it is a zero NPV acquisition Using the PE ratio we nd the current market price of Stultz stock which is P 21675000180000 7875 If the acquisition has a zero NPV the stock price should remain unchanged Therefore the new PE will be PE 78755357 1470 The value of Flannery to Stultz must be the market value of the company since the NPV of the acquisition is zero Therefore the value is vquot 450000525 2362500 527 The cost of the acquisition is the number of shares offered times the share price so the cost is Cost 13900007875 2362500 So the NPV of the acquisition is NPV 0 vquot AV 7 Cost 2362500 AV 7 2362500 AV 0 Although there is no economic value to the takeover it is possible that Stultz is motivated to purchase Flannery for other than financial reasons The decision hinges upon the risk of surviving That is consider the wealth transfer from bondholders to stockholders when risky projects are undertaken Highrisk projects will reduce the expected value of the bondholders claims on the rm The telecommunications business is riskier than the utilities business If the total value of the firm does not change the increase in risk should favor the stockholder Hence management should approve this transaction If the total value of the rm drops because of the transaction and the wealth effect is lower than the reduction in total value management should reject the project a The NPV of the merger is the market value of the target firm plus the value of the synergy minus the acquisition costs so NPV 140026 5500 7 140029 1300 Since the NPV goes directly to stockholders the share price of the merged rm will be the market value of the acquiring firm plus the NPV of the acquisition divided by the number of shares outstanding so Share price 290039 13002900 3945 The merger premium is the premium per share times the number of shares of the target firm outstanding so the merger premium is Merger premium l40029 7 26 4200 The number of new shares will be the number of shares of the target times the exchange ratio so New shares created 14003 5 840 new shares The value of the merged rm will be the market value of the acquirer plus the market value of the target plus the synergy bene ts so vBT 290039 140026 5500 155000 528 The price per share of the merged rm will be the value of the merged rm divided by the total shares of the new firm which is P 1550002900 840 4144 e The NPV of the acquisition using a share exchange is the market value of the target rm plus synergy bene ts minus the cost The cost is the value per share of the merged rm times the number of shares offered to the target rm shareholders so NPV 140026 5500 7 8404144 708717 Intermediate 11 The cash offer is better for the target rm shareholders since they receive 29 per share In the share offer the target rm s shareholders will receive Equity offer value 3 529 1560 per share From Problem 10 we know the value of the merged rm s assets will be 155000 The number of shares in the new rm will be Shares in new lm 2900 1400x that is the number of shares outstanding in the bidding form plus the number of shares outstanding in the target rm times the exchange ratio This means the post merger share price will be P 1550002900 1400x To make the target rm s shareholders indifferent they must receive the same wealth so 1400xP 140029 This equation shows that the new offer is the shares outstanding in the target company times the exchange ratio times the new stock price The value under the cash offer is the shares outstanding times the cash offer price Solving this equation for P we find P 29 x Combining the two equations we find 1550002900 1400x 29 x x 07351 There is a simpler solution that requires an economic understanding of the merger terms If the target rm s shareholders are indifferent the bidding rm s shareholders are indifferent as well That is the offer is a zero sum game Using the new stock price produced by the cash deal we nd Exchange ratio 263945 7351 529 12 The cost of the acquisition is p A DJ 14 Cost 25022 7 5500 Since the stock price of the acquiring rm is 50 the rm will have to give up Shares offered 550050 110 shares a NPV NPV NPV NPV a The EPS of the merged rm will be the combined EPS of the existing rms divided by the new shares outstanding so EPS 1600 700600 110 324 The PE of the acquiring rm is Original PE 501600600 1875 times Assuming the PE ratio does not change the new stock price will be New P 3241875 6074 Ifthe market correctly analyzes the earnings the stock price will remain unchanged since this is a zero NPV acquisition so New PE 50324 1543 times The new share price will be the combined market value of the two existing companies divided by the number of shares outstanding in the merged company So P 60050 25020600 110 4930 And the PE ratio of the merged company will be PE 4930324 1522 times At the proposed bid price this is a negative NPV acquisition for A since the share price declines They should revise their bid downward until the NPV is zero Beginning with the fact that the NPV of a merger is the value of the target minus the cost we get V 7 Cost AV VB 7 Cost AV 7 Cost 7 VB AV 7Merger premium The synergy will be the present value of the incremental cash ows of the proposed purchase Since the cash ows are perpetual the synergy value is Synergy value 500000 08 530 Synergy value 6250000 The value of FlashinthePan to FlybyNight is the synergy plus the current market value of F lashinthePan which is Value 6250000 10000000 Value 16250000 The value of the cash option is the amount of cash paid or 13 million The value of the stock 39 quot39 VI in the merged company times the value of the l is the l of merged company so Stock acquisition value 3016250000 26000000 Stock acquisition value 12675000 The NPV is the value of the acquisition minus the cost so the NPV of each alternative is NPV of cash offer 16250000 7 13000000 NPV of cash offer 3250000 NPV of stock offer 16250000 7 12675000 NPV of stock offer 3575000 The acquirer should make the stock offer since its NPV is greater The number of shares after the acquisition will be the current number of shares outstanding for the acquiring rm plus the number of new shares created for the acquisition which is Number of shares after acquisition 30000000 12000000 Number of shares after acquisition 42000000 And the share price will be the value of the combined company divided by the shares outstanding which will be New stock price 720000000 42000000 New stock price 17 14 Let 0 equal the fraction of ownership for the target shareholders in the new rm We can set the percentage of ownership in the new rm equal to the value of the cash offer so oc 720000000 205000000 a 2847 or 2847 So the shareholders of the target rm would be equally as well off if they received 2847 percent of the stock in the new company as if they received the cash offer The ownership percentage of the target rm shareholders in the new rm can be expressed as Ownership New shares issued New shares issued Current shares of acquiring rm 2847 New shares issued New shares issued 30000000 New shares issued 11941748 531 To nd the exchange ratio we divide the new shares issued to the shareholders of the target rm by the existing number of shares in the target rm so Exchange ratio New shares Existing shares in targetme Exchange ratio 11941748 20000000 Exchange ratio 5971 An exchange ratio of 5971 shares of the merged company for each share of the target company owned would make the value of the stock offer equivalent to the value of the cash offer The value of each company is the sum of the probability of each state of the economy times the value of the company in that state of the economy so ValueBemley 70280000 30100000 ValueBemley 226000 ValueRolls 70250000 3070000 ValueRons 196000 The value of each company s equity is sum of the probability of each state of the economy times the value of the equity in that state of the economy The value of equity in each state of the economy is the maximum of total company value minus the value of debt or zero Since Rolls is an all equity company the value of its equity is simply the total value of the rm or 196000 The value of Bentley s equity in a boom is 155000 280000 company value minus 125000 debt value and the value of Bentley s equity in a recession is zero since the value of its debt is greater than the value of the company in that state of the economy So the value of Bentley s equity is EquityBemley 70155000 300 EquityBemley 108500 The value of Bentley s debt in a boom is the full face value of 125000 In a recession the value of the company s debt is 100000 since the value of the debt cannot exceed the value of the company So the value of Bentley s debt today is Destemley 70125000 30100000 Destemley 117500 Note this is also the value of the company minus the value of the equity or Destentley 226000 7 108500 Destemley 117500 532 The combined value of the companies the combined equity value and combined debt value is Combined value 226000 196000 Combined value 422000 Combined equity value 108500 196000 Combined equity value 304500 Combined debtvalue 117500 To find the value of the merged company we need to nd the value of the merged company in each state of the economy which is Boom merged value 280000 250000 Boom merged value 530000 Recession merged value 100000 70000 Recession merged value 170000 So the value of the merged company today is Merged company value 70530000 30l70000 Merged company value 422000 Since the merged company will still have 125000 in debt the value of the equity in a boom is 405000 and the value of equity in a recession is 45000 So the value of the merged company s equity is Merged equity value 70405000 3045000 Merged equity value 297000 The merged company will have a value greater than the face value of debt in both states of the economy so the value of the company s debt is 125000 There is a wealth transfer in this case The combined equity value before the merger was 304500 but the value of the equity in the merged company is only 297000 a loss of 7500 for stockholders The value of the debt in the combined companies was only 117500 but the value of debt in the merged company is 125000 since there is no chance of default The bondholders gained 7500 exactly the amount the stockholders lost If the value of Bentley s debt before the merger is less than the lowest firm value there is no coinsurance effect Since there is no possibility of default before the merger bondholders do not gain after the merger 533 Challenge To nd the value of the target to the acquirer we need to nd the share price with the new growth rate We begin by nding the required return for shareholders of the target rm The earnings per share of the target are EPSp 640000500000 128 per share The price per share is Pp 10128 1280 And the dividends per share are DPSp 380000500000 076 The current required return for Palmer shareholders which incorporates the risk of the company is RE 0761041280 04 1018 The price per share of Palmer with the new growth rate is Pp 0761061018 7 06 1930 The value of the target firm to the acquiring rm is the number of shares outstanding times the price per share under the new growth rate assumptions so v 5000001930 964790419 The gain to the acquiring rm will be the value of the target rm to the acquiring rm minus the market value of the target so Gain 964790419 7 5000001280 324790419 The NPV of the acquisition is the value of the target rm to the acquiring rm minus the cost of the acquisition so NPV 964790419 7 50000013 314790419 The most the acquiring rm should be willing to pay per share is the offer price per share plus the NPV per share so Maximum bid price 13 314790419500000 1930 Notice that this is the same value we calculated earlier in part a as the value of the target to the acquirer 534 The price of the stock in the merged rm would be the market value of the acquiring rm plus the value of the target to the acquirer divided by the number of shares in the merged rm so PFp 40600000 9647904191000000 150000 4369 The NPV of the stock offer is the value of the target to the acquirer minus the value offered to the target shareholders The value offered to the target shareholders is the stock price of the merged rm times the number of shares offered so NPV 964790419 7 1500004369 309382973 Yes the acquisition should go forward and Plant should offer cash since the NPV is higher Using the new growth rate in the dividend growth model along with the dividend and required return we calculated earlier the price of the target under these assumptions is Pp 076105 1018 7 05 1542 And the value of the target rm to the acquiring rm is V 5000001542 771014493 The gain to the acquiring rm will be Gain 771014493 7 5000001542 131014493 The NPV of the cash offer is now NPV cash 771014493 7 50000013 121014493 And the new price per share of the merged rm will be PFp 40600000 7710144931000000 150000 4201 And the NPV of the stock offer under the new assumption will be NPV stock 771014493 7 1500004201 140882168 Even with the lower projected growth rate the stock offer still has a positive NPV However the NPV of the stock offer is now higher Plant should purchase Palmer with a stock offer of 150000 shares 535 To nd the distribution of joint values we rst must nd the joint probabilities To do this we need to nd the joint probabilities for each possible combination of weather in the two towns The weather conditions are independent therefore the joint probabilities are the products of the individual probabilities Possible states Joint probability RainRain 1 1 01 RainWarm 14 04 RainHot 15 05 WarmRain 4 1 04 WarmWarm 4 4 16 WarmHot 4 5 20 HotRain 5 1 05 HotWarm 54 20 HotHot 5 5 25 Next note that the revenue when rainy is the same regardless of which town So since the state quotRainWarmquot has the same outcome revenue as quotWarmRainquot their probabilities can be added The same is true of quotRainHotquot quotHotRainquot and quotWarmHotquot quotHotWarmquot Thus the joint probabilities are Possible states Joint probabilitv RainRain 0 1 RainWarm 08 RainHot 10 WarmWarm 16 WarmHot 40 HotHot 25 Finally the joint values are the sums of the values of the two companies for the particular state Possible states Joint value RainRain 200000 200000 400000 RainWarm 200000 350000 550000 RainHot 200000 800000 1000000 WarmWarm 350000 350000 700000 WarmHot 350000 800000 1150000 HotHot 800000 800000 1600000 536 Recall that if a rm cannot service its debt the bondholders receive the value of the assets Thus the value of the debt is reduced to the value of the company if the face value of the debt is greater than the value of the company If the value of the company is greater than the value of the debt the value of the debt is its face value Here the value of the common stock is always the residual value of the firm over the value of the debt So the value of the debt and the value of the stock in each state is Possible states Joint Prob Joint Value Debt Value Stock Value RainRain 400000 400000 0 RainWarm 08 550000 550000 0 RainHot 10 1000000 700000 300000 WarmWarm 16 700000 700000 0 WarmHot 40 1150000 700000 450000 HotHot 25 1600000 700000 900000 The bondholders are better off if the value of the debt after the merger is greater than the value of the debt before the merger The value of the debt is the smaller of the debt value or the company value So the value of the debt of each individual company before the merger in each state is Possible states Probabilitv Debt Value Ram 10 200000 Warm 40 350000 Hot 50 350000 Individual debt value l200000 4350000 5350000 Individual debt value 335000 This means the total value of the debt for both companies premerger must be Total debt value premerger 2335000 Total debt value premerger 670000 To get the expected debt value postmerger we can use the joint probabilities for each possible state and the debt values corresponding to each state we found in part c Using this information to find the value of the debt in the postmerger rm we get Total debt value postmerger 01400000 08550000 10700000 16700000 40700000 25700000 Total debt value postmerger 685000 The bondholders are better off by 15000 Since we have already shown that the total value of the combined company is the same as the sum of the value of the individual companies the implication is that the stockholders are worse off by 15000 537 CHAPTER 30 FINANCIAL DISTRESS Answers to Concepts Review and Critical Thinking Questions 1 Financial distress is often linked to insolvency Stockbased insolvency occurs when a rm has a negative net worth Flowbased insolvency occurs when operating cash ow is insuf cient to meet current obligations 2 Financial distress frequently can serve as a rm s early warning sign for trouble Thus it can be bene cial since it may bring about new organizational forms and new operating strategies 3 A prepackaged bankruptcy is where the rm and most creditors agree to a private reorganization before bankruptcy takes place After the private agreement the rm les for formal bankruptcy The biggest advantage is that a prepackaged bankruptcy is usually cheaper and faster than a traditional bankruptcy 4 Just because a rm is experiencing nancial distress doesn t necessarily imply the firm is worth more dead than alive 5 Liquidation occurs when the assets of a rm are sold and payments are made to creditors usually based upon the APR Reorganization is the restructuring of the rm39s nances 6 The absolute priority rule is the priority rule of the distribution of the proceeds of the liquidation It begins with the rst claim to the last in the order administrative expenses unsecured claims after a ling of involuntary bankruptcy petition wages employee bene t plans consumer claims taxes secured and unsecured loans preferred stocks and common stocks 7 Bankruptcy allows rms to issue new debt that is senior to all previously incurred debt This new debt is called DIP debtor in possession debt If DIP loans were not senior to all other debt a rm in bankruptcy would be unable to obtain financing necessary to continue operations while in bankruptcy since the lender would be unlikely to make the loan 8 One answer is that the right to le for bankruptcy is a valuable asset and the financial manager acts in shareholders best interest by managing this asset in ways that maximize its value To the extent that a bankruptcy ling prevents a race to the courthouse steps it would seem to be a reasonable use of the process 9 As in the previous question it could be argued that using bankruptcy laws as a sword may simply be the best use of the asset Creditors are aware at the time a loan is made of the possibility of bankruptcy and the interest charged incorporates it If the only way a firm can continue to operate is to reduce labor costs it may be a bene t to everyone including employees 538 p n O There are four possible reasons why rms may choose legal bankruptcy over private workout I It may be less expensive although legal bankruptcy is usually more expensive 2 Equity investors can use legal bankruptcy to hold out 3 A complicated capital structure makes private workouts more difficult 4 Con icts of interest between creditors equity investors and management can make private workouts impossible Solutions to Questions and Problems NOTE All end of chapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem 1 Basic Under the absolute priority rule APR claims are paid out in full to the extent there are assets In this case assets are 28500 so you should propose the following Distribution of Original claim value Trade credit 4800 4800 Secured mortgage notes 8000 8000 Senior debentures 10000 10000 Junior debentures 15000 5700 0 0 Equity There are many possible reorganization plans so we will make an assumption that the mortgage bonds are fully recognized as senior debentures the senior debentures will receive junior debentures in the value of 65 cents on the dollar and the junior debentures will receive any remaining value as equity With these assumptions the reorganization plan will look like this Original claim quot 39 claim Mortgage bonds 19000 Senior debenture l9000 Senior debentures 9500 Junior debenture 6175 Junior debentures 7500 Equity l825 Since we are given shares outstanding and a share price the company must be publicly traded First we need to calculate the market value of equity which is lVIarket value of equity 5000l8 90000 We also need the book value of debt Since we have the value of total assets and the book value of equity the book value of debt must be the difference between these two gures or Book value of debt Total assets iBook value of equity Book value of debt 42000 7 19000 Book value of debt 23000 539 Now we can calculate the Zscore for a publicly traded company which is Zscore 33EBITTotal assets 12NWCTotal assets 10SalesTotal assets 6Market value of equityBook value of equity 14Accumulated retained eamings Total assets Zscore 33650042000 l2310042000 6100042000 69000019000 141350042000 Zscore 5344 Since this company is private we must use the Zscore for private companies and non manufacturers which is Zscore 656NWCTotal assets 326Accumulated retained eamingsTotal assets l05EBITTotal assets 672Bookvalue of equityTotal liabilities Zscore 656680075000 3261900075000 105830075000 6722600049000 Zscore 5103 540 CHAPTER 31 INTERNATIONAL CORPORATE FINANCE Answers to Concepts Review and Critical Thinking Questions 1 a The dollar is selling at a premium because it is more expensive in the forward market than in the spot market SFr 153 versus SFr 150 b The franc is expected to depreciate relative to the dollar because it will take more francs to buy one dollar in the future than it does today 0 In ation in Switzerland is higher than in the United States as are nominal interest rates The exchange rate will increase as it will take progressively more pesos to purchase a dollar This is the relative PPP relationship a The Australian dollar is expected to weaken relative to the dollar because it will take more Al in the future to buy one dollar than it does today b The in ation rate in Australia is higher 0 Nominal interest rates in Australia are higher relative real rates in the two countries are the same A Yankee bond is most accurately described by d No For example if a country s currency strengthens imports become cheaper good but its exports become more expensive for others to buy bad The reverse is true for currency depreciation Additional advantages include being closer to the final consumer and thereby saving on transportation signi cantly lower wages and less exposure to exchange rate risk Disadvantages include political risk and costs of supervising distant operations One key thing to remember is that dividend payments are made in the home currency More generally it may be that the owners of the multinational are primarily domestic and are ultimately concerned about their wealth denominated in their home currency because unlike a multinational they are not intemationally diversi ed a False If prices are rising faster in Great Britain it will take more pounds to buy the same amount of goods that one dollar can buy39 the pound will depreciate relative to the dollar b False The forward market would already re ect the projected deterioration of the euro relative to the dollar Only if you feel that there might be additional unanticipated weakening of the euro that isn t re ected in forward rates today will the forward hedge protect you against additional declines 541 p n O p n p A p A N p n DJ p A J p n UI 0 True The market would only be correct on average while you would be correct all the time a American exporters their situation in general improves because a sale of the exported goods for a xed number of euros will be worth more dollars American importers their situation in general worsens because the purchase of the imported goods for a xed number of euros will cost more in dollars b American exporters they would generally be better off if the British government s intentions result in a strengthened pound American importers they would generally be worse off if the pound strengthens 0 American exporters they would generally be much worse off because an extreme case of scal expansion like this one will make American goods prohibitively expensive to buy or else Brazilian sales if xed in reais would become worth an unacceptably low number of dollars American importers they would generally be much better off because Brazilian goods will become much cheaper to purchase in dollars IRP is the most likely to hold because it presents the easiest and least costly means to exploit any arbitrage opportunities Relative PPP is least likely to hold since it depends on the absence of market imperfections and frictions in order to hold strictly It all depends on whether the forward market expects the same appreciation over the period and whether the expectation is accurate Assuming that the expectation is correct and that other traders do not have the same information there will be value to hedging the currency exposure One possible reason investment in the foreign subsidiary might be preferred is if this investment provides direct diversi cation that shareholders could not attain by investing on their own Another reason could be if the political climate in the foreign country was more stable than in the home country Increased political risk can also be a reason you might prefer the home subsidiary investment Indonesia can serve as a great example of political risk If it cannot be diversi ed away investing in this type of foreign country will increase the systematic risk As a result it will raise the cost of the capital and could actually decrease the NPV of the investment Yes the firm should undertake the foreign investment If after taking into consideration all risks a project in a foreign country has a positive NPV the firm should undertake it Note that in practice the stated assumption that the adjustment to the discount rate has taken into consideration all political and diversi cation issues is a huge task But once that has been addressed the net present value principle holds for foreign operations just as for domestic If the foreign currency depreciates the US parent will experience an exchange rate loss when the foreign cash ow is remitted to the US This problem could be overcome by selling forward contracts Another way of overcoming this problem would be to borrow in the country where the project is located False If the nancial markets are perfectly competitive the difference between the Eurodollar rate and the US rate will be due to differences in risk and government regulation Therefore speculating in those markets will not be bene cial 542 16 The difference between a Eurobond and a foreign bond is that the foreign bond is denominated in the currency of the country of origin of the issuing company Eurobonds are more popular than foreign bonds because of 39 quot quotEquot F 39 39 39 are 39 securities Solutions to Questions and Problems NOTE All endofchapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problem m 1 Using the quotes from the table we get a 100 069251 6925 b 14441 c 5M1441 7220217 d Singapore dollar e Mexican peso f P110302l1441 1 P159281 This is a cross rate g The most valuable is the Kuwait dinar The least valuable is the Vietnam dong 2 a You would prefer 100 since 1005528 1 5528 b You would still prefer 100 Using the exchange rate and the SF exchange rate to find the amount of Swiss francs 100 will buy we get 10018091 1SF 9188 SF 1968981 c Using the quotes in the book to nd the SF cross rate we nd SF 1809ll19188 1 SF 19690 1 The SF exchange rate is the inverse of the SF exchange rate so 1SF 19690 05079SF1 543 F180 10252 per The yen is selling at a premium because it is more expensive in the forward market than in the spot market 00095886 versus 00097542 F90 05535 The pound is selling at a premium because it is less expensive in the forward market than in the spot market 18066 versus l809l The value of the dollar will fall relative to the yen since it takes more dollars to buy one yen in the future than it does today The value of the dollar will rise relative to the pound because it will take less dollars to buy one pound in the future than it does today The US dollar since one Canadian dollar will buy Can1Can118l 08475 The cost in US dollars is Can2l9Can1l8l 186 Among the reasons that absolute PPP doesn t hold are tariffs and other barriers to trade transactions costs taxes and different tastes The US dollar is selling at a discount because it is less expensive in the forward market than in the spot market Canl 18 versus Canl 13 The Canadian dollar is expected to appreciate in value relative to the dollar because it takes fewer Canadian dollars to buy one US dollar in the future than it does today Interest rates in the United States are probably higher than they are in Canada The cross rate in terms is 110l165 1 l815 1 The yen is quoted high low relative to the pound Take out a loan for 1 and buy 06061 Use the 06061 to purchase yen at the crossrate which will give you 183 06061 ll0909 Use the pounds to buy back dollars and repay the loan The cost to repay the loan will be 1109091 110 10083 You arbitrage profit is 00083 per dollar used 544 We can rearrange the interest rate parity condition to answer this question The equation we will use is RFC FT SW30 RUS Using this relationship we find Great Britain RFC 05564 7 05528 05528 026 325 Japan RFC 10252 7 10429 10429 026 090 Switzerland RFC SFr 10803 7 SFr 10884SFr 10884 026 186 If we invest in the US for the next three months we will have 30000000100253 3022556297 If we invest in Great Britain we must exchange the dollars today for pounds and exchange the pounds for dollars in three months After making these transactions the dollar amount we would have in three months would be 30000000 0541100413 0531 3094354355 The company should invest in Great Britian Using the relative purchasing power parity equation Ft SO X 1hFC hUslt We nd Z451 Z4271 hFc 7 hUS3 hFc 7hUS 7 Z451z427 3 7 1 hFc 7hUS 0184 In ation in Poland is expected to exceed that in the US by 184 annually over this period The pro t will be the quantity sold times the sales price minus the cost of production The production cost is in Singapore dollars so we must convert this to US dollars Doing so we nd that if the exchange rates stay the same the pro t will be Pro t 7 30000125 7 S16850S143181 Pro t 7 21947898 If the exchange rate rises we must adjust the cost by the increased exchange rate so Pro t 7 30000125 7 S1685011S143181 Pro t 7 54043543 545 If the exchange rate falls we must adjust the cost by the decreased exchange rate so Pro t 30000125 7 S1685009S143181 Pro t 78172801 14 To calculate the breakeven change in the exchange rate we need to nd the exchange rate that make the cost in Singapore dollars equal to the selling price in US dollars so 125 S16850ST ST Sl3481 ST 70585 or 7585 decline a If IRP holds then F180 Kr 6841 07 7 04 2 F180 Kr 69418 Since given F180 is Kr696 an arbitrage opportunity exists the forward premium is too high Borrow Krl today at 7 interest Agree to a 180day forward contract at Kr 696 Convert the loan proceeds into dollars Kr 1 1Kr 684 014620 Invest these dollars at 4 ending up with 014905 Convert the dollars back into krone as 014905Kr 6961 Kr 103742 Repay the Kr 1 loan ending with a pro t of Kr103742 7Kr103393 Kr 000349 b To find the forward rate that eliminates arbitrage we use the interest rate parity condition so F180 Kr 68410770412 F180 Kr 69418 The international Fisher effect states that the real interest rate across countries is equal We can rearrange the international Fisher effect as follows to answer this question RUS hUs RFC hFC hFC RFC hUs RUs a hAUS 04 041 7 025 hAUS 056 or 56 b hCAN 06 0417025 hCAN 076 or 76 c hTAI 09 0417025 hTAI 106 or 106 546 13 p A J a The yen is expected to get stronger since it will take fewer yen to buy one dollar in the future than it does today b hUs thAp m 11632 7 11815 11815 hUS 711W 7 0155 or 7155 1701554 71 70605 or 405 The approximate in ation differential between the US and Japan is 7605 annually We need to nd the change in the exchange rate over time so we need to use the relative purchasing power parity relationship F1 s0 x11hFc 7 hug Using this relationship we find the exchange rate in one year should be F1 209105770351 F1 HUF 21360 The exchange rate in two years should be F2 209105770352 F2 HUF 21830 And the exchange rate in ve years should be F5 209105770355 F5 HUF 23302 Intermediate First we need to forecast the future spot rate for each of the next three years From interest rate and purchasing power parity the expected exchange rate 1s HST 1 RUs 1 RFC1T 80 So ES1 10480 104101 122 12282 Esz 10480 104102 812 12365 ES310480104103 122e 12448 547 Now we can use these future spot rates to nd the dollar cash ows The dollar cash ow each year will be Year 0 cash ow 18000000122 e2196000000 Year 1 cash ow 360000012282 442153314 Year 2 cash ow 410000012365 506949610 Year 3 cash ow 5100000 1220000012448 2153463887 And the NPV of the project will be NPV 721960000 442153314113 SE5069496101132 21534638871133 NPV 84760521 a Implicitly it is assumed that interest rates won t change over the life of the project but the exchange rate is projected to decline because the Euroswiss rate is lower than the Eurodollar rate b We can use relative purchasing power parity to calculate the dollar cash ows at each time The equation is EST SFr 1721 07 e 08T EST 17299T So the cash ows each year in US dollar terms will be 1 E M US 0 725000000 17200 7l453488372 1 7200000 17028 422832981 2 7200000 16858 427104021 3 7200000 16689 431418203 4 7200000 16522 435775963 5 7200000 16357 440177740 And the NPV is NPV 71453488372 422832981113 4271040211132 SE4314182031133 fl 4357759631134 fl 4401777401135 NPV 60360061 0 Rearranging the relative purchasing power parity equation to find the required return in Swiss francs we get Rsrr 11310770871 RSFr 1187 So the NPV in Swiss francs is NPV iSFr 25000000 SFr 7200000PVIFA118M5 548 17 NPV SFr 103819305 Converting the NPV to dollars at the spot rate we get the NPV in US dollars as NPV SFr 1038193051SFr 172 NPV 60360061 a To construct the balance sheet in dollars we need to convert the account balances to dollars At the current exchange rate we get Assets solaris 23000 X solaris 120 1916667 Debt solaris 9000 gtlt solaris 120 750000 Equity solaris 14000 X solaris 120 1916667 b In one year if the exchange rate is solaris 140 the accounts will be Assets solaris 23000 gtlt solaris 140 1642857 Debt solaris 9000 gtlt solaris 140 642857 Equity solaris 14000 X 5 solaris 140 1000000 b If the exchange rate is solaris 112 the accounts will be Assets solaris 23000 gtlt solaris 112 2053571 Debt solaris 9000 gtlt solaris 112 803571 Equity solaris 14000 gtlt solaris 112 1250000 Challenge First we need to construct the end of year balance sheet in solaris Since the company has retained earnings the equity account will increase which necessarily implies the assets will also increase by the same amount So the balance sheet at the end of the year in solaris will be Balance Sheet solaris Liabilities 900000 Equity 1525000 Assets 2425000 Total liabilities amp equity 2425000 Now we need to convert the balance sheet accounts to dollars which gives us Assets solaris 24250 X solaris 124 1955645 Debt solaris 9000 gtlt solaris 124 725806 Equity solaris 15250 gtlt solaris 124 1229839 a The domestic Fisher effect is 1RUS 1 rUS1 hUS 1 rUs 1 RUS1 hUS This relationship must hold for any country that is 1rFC1RFC1hFC 549 The international Fisher effect states that real rates are equal across countries so 1 rUs 1 RUS1 hUS1RFc1 hFc1 rFC The exact form of unbiased interest rate parity is ES F s0 1 RFc1 RUS39 The exact form for relative PPP is ES SO 1 hFc1 115t For the home currency approach we calculate the expected currency spot rate at time t as ES 05107105t 051019t We then convert the euro cash ows using this equation at every time and nd the present value Doing so we nd NPV 7 2M 05 09lI1019 0511 09M10 192 05112 09M10193 051113 NPV 31623072 For the foreign currency approach we rst nd the return in the euros as RFC 11010710571 0121 Next we nd the NPV in euros as NPV 7 2M 09l11121 09l111212 09l111213 15811536 And nally we convert the euros to dollars at the current exchange rate which is NPV 15811536 05l 31623072 550

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