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# ManagerialFinanceII FRL301

CSU Pomona

GPA 3.91

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This 286 page Class Notes was uploaded by Dominic Erdman on Saturday October 3, 2015. The Class Notes belongs to FRL301 at California State Polytechnic University taught by FarrokhGhazanfari in Fall. Since its upload, it has received 9 views. For similar materials see /class/218188/frl301-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 Fundamentals of Corporate Finance 9th edition Ross Wester eld and Jordan Updated 12202008 CHAPTER I INTRODUCTION TO CORPORATE FINANCE Answers to Concepts Review and Critical Thinking Questions 1 Capital budgeting deciding whether to expand a manufacturing plant capital structure deciding whether to issue new equity and use the proceeds to retire outstanding debt and working capital management modifying the rm s credit collection policy with its customers Disadvantages unlimited liability limited life dif culty in transferring ownership hard to raise capital funds Some advantages simpler less regulation the owners are also the managers sometimes personal tax rates are better than corporate tax rates The primary disadvantage of the corporate form is the double taxation to shareholders of distributed earnings and dividends Some advantages include limited liability ease of transferability ability to raise capital and unlimited life In response to SarbanesOxley small firms have elected to go dark because of the costs of compliance The costs to comply with Sarbox can be several million dollars which can be a large percenng of a small rms pro ts A major cost of going dark is less access to capital Since the rm is no longer publicly traded it can no longer raise money in the public market Although the company will still have access to bank loans and the private equity market the costs associated with raising funds in these markets are usually higher than the costs of raising funds in the public market The treasurer s office and the controller s office are the two primary organizational groups that report directly to the chief nancial officer The controller s office handles cost and nancial accounting tax management and management information systems while the treasurer s office is responsible for cash and credit management capital budgeting and nancial planning Therefore the study of corporate nance is concentrated within the treasury group s functions To maximize the current market value share price of the equity of the rm whether it s publicly traded or not In the corporate form of ownership the shareholders are the owners of the firm 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 A primary market transaction B p n p A p n DJ SOLUTIONS In auction markets like the NYSE brokers and agents meet at a physical location the exchange to match buyers and sellers of assets Dealer markets like NASDAQ consist of dealers operating at dispersed locales who buy and sell assets themselves communicating with other dealers either electronically or literally overthecounter 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 firm has estimated that the cost of improving the safety of one of its products is 30 million However the firm 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 film so that the share price will exceed 35 then they should fight the offer from the outside company If management believes that this bidder or other unidentified 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 firm 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 We would expect agency problems to be less severe in countries with a 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 CHAPTER 1 B3 16 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 CHAPTER 2 FINANCIAL STATEMENTS TAXES AND CASH FLOW Answers to Concepts Review and Critical Thinking Questions 1 Liquidity measures how quickly and easily an asset can be converted to cash without signi cant loss in value It s desirable for rms to have high liquidity so that they have a large factor of safety in meeting shortterm creditor demands However since liquidity also has an opportunity cost associated with itinamely that higher returns can generally be found by investing the cash into productive assetsilow liquidity levels are also desirable to the fum It s up to the rm s financial management staff to nd a reasonable compromise between these opposing needs 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 Historical costs can be objectively and precisely measured whereas market values can be dif cult to estimate and different analysts would come up with different numbers Thus there is a tradeoff between relevance market values and objectivity book values 4 Depreciation is a noncash deduction that re ects adjustments made in asset book values in accordance with the matching principle in nancial accounting Interest expense is a cash outlay but it s a nancing cost not an operating cost 5 Market values can never be negative Imagine a share of stock selling for 4520 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 but it would be fairly ordinary for a start up 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 it 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 CHAPTER 2 B5 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 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 11 Enterprise value is the theoretical takeover price In the event of a takeover an acquirer would have to take on the company39s debt but would pocket its cash Enterprise value differs signi cantly from simple market capitalization in several ways and it may be a more accurate representation of a rm s value In a takeover the value of a rm39s debt would need to be paid by the buyer when taking over a company This enterprise value provides a much more accurate takeover valuation because it includes debt in its value calculation 12 In general it appears that investors prefer companies that have a steady earnings stream If true this encourages companies to manage earnings Under GAAP there are numerous choices for the way a company reports its nancial statements Although not the reason for the choices under GAAP one outcome is the ability of a company to manage earnings which is not an ethical decision Even though earnings and cash ow are often related earnings management should have little effect on cash ow except for tax implications If the market is fooled and prefers steady earnings shareholder wealth can be increased at least temporarily However given the questionable ethics of this practice the company and shareholders will lose value if the practice is discovered 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 1 To nd owner s equity we must construct a balance sheet as follows Balance Sheet CA 5100 CL 4300 NFA 23800 LTD 7400 OE TA 28900 TL amp OE 28 900 We know that total liabilities and owner s equity TL amp OE must equal total assets of 28900 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 28900 7 7400 7 4300 17200 NWC CA7 CL 5100 7 4300 800 B6 SOLUTIONS 2 The income statement for the company is Income Statement Sales 586000 Costs 247000 Depreciation 43000 EBIT 296000 Interest 32000 EBT 264000 Taxes35 92400 Net income i 171600 3 One equation for net income is Net income Dividends Addition to retained earnings Rearranging we get Addition to retained earnings Net income iDividends 171600 773000 98600 4 EPS Net income Shares 171600 85000 202 per share DPS Dividends Shares 73000 85000 086 per share 5 To nd the book value of current assets we use NWC CA 7 CL Rearranging to solve for current assets we get CA NWC CL 380000 1400000 1480000 The market value of current assets and xed assets is given so Book value CA 1480000 Market value CA 1600000 Book value NFA 3 700 000 lVIarket value NFA 4 900 000 Book value assets 5180000 Market value assets 6500000 6 Taxes 01550K 02525K 03425K 039236K 7 100K 75290 7 The average tax rate is the total tax paid divided by net income so Average tax rate 75290 236000 3190 The marginal tax rate is the tax rate on the next 1 of earnings so the marginal tax rate 39 CHAPTER 2 B7 8 To calculate OCF we rst need the income statement Income Statement Sales 27500 Costs 13280 Depreciation 2300 EBIT 11920 Interest 1105 Taxable income 108 15 Taxes 35 3785 Net income 7030 OCF EBIT Depreciation 7 Taxes 11920 2300 7 3785 10435 9 Net capital spending NFAend 7 NFAbe g Depreciation Net capital spending 4200000 7 3400000 385000 Net capital spending 1185000 10 Change in NWC NWCend 7 NWCbe g Change in NWC CAN 7 CLend 7 CAbe g 7 CLbeg Change in NWC 2250 7 1710 7 2100 71380 Change in NWC 540 7 720 7180 11 Cash ow to creditors Interest paid 7 Net new borrowing Cash ow to creditors Interest paid 7 LTDend 7 LTDbeg Cash ow to creditors 170000 7 2900000 7 2600000 Cash ow to creditors 7130000 12 Cash ow to stockholders Dividends paid 7 Net new equity Cash ow to stockholders Dividends paid 7 Commonend APISend 7 Commonbeg APISbeg Cash ow to stockholders 490000 7 815000 5500000 7 740000 5200000 Cash ow to stockholders 115000 Note APIS is the additional paidin surplus 13 Cash ow from assets Cash ow to creditors Cash ow to stockholders 7130000 115000 715000 Cash ow from assets 715000 OCF 7 Change in NWC 7 Net capital spending 715000 OCF 7 785000 7 940000 Operating cash ow 715000 7 85000 940000 Operating cash ow 840000 B8 SOLUTIONS Intermediate 14 To nd the OCF we first calculate net income 9 9 5 1 9 Income Statement Sales 196000 Costs 104000 Other expenses 6800 Depreciation 9100 EBIT 76100 Interest 14800 Taxable income 61300 Taxes 21455 Net income 39845 Dividends 10400 Additions to RE 29445 OCF EBIT Depreciation 7 Taxes 76100 9100 7 21455 63745 CFC Interest 7 Net new LTD 14800 7 77300 22100 Note that the net new longterm debt is negative because the company repaid part of its long term debt CFS Dividends 7 Net new equity 10400 7 5700 4700 We know that CFA CFC CFS so CFA 22100 4700 7 26800 CFA is also equal to OCF 7Net capital spending 7 Change in NWC We already know OCF Net capital spending is equal to Net capital spending Increase in NFA Depreciation 27000 9100 36100 Now we can use CFA OCF 7 Net capital spending 7 Change in NWC 26800 63745 7 36100 7 Change in NWC Solving for the change in NWC gives 845 meaning the company increased its NWC by 845 15 The solution to this question works the income statement backwards Starting at the bottom Net income Dividends Addition to ret earnings 1500 5100 6600 CHAPTER 2 B9 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 6600 1 7 035 10154 Now you can calculate EBIT EBT Interest 10154 4500 14654 The last step is to use EBIT Sales 7 Costs 7 Depreciation 14654 41000 7 19500 7 Depreciation Solving for depreciation we nd that depreciation 6846 The balance sheet for the company looks like this Balance Sheet Cash 195000 Accounts payable 405000 Accounts receivable 137000 Notes payable 160000 Inventory 264000 Current liabilities 565000 Current assets 596000 Longterm debt 1195300 Total liabilities 1760300 Tangible net fixed assets 2800000 Intangible net xed assets 780000 Common stock Accumulated ret earnings 1934000 Total assets 4176000 Total liab amp owners equity 4176000 Total liabilities and owners equity is TL amp OE CL LTD Common stock Retained earnings Solving for this equation for equity gives us Common stock 4176000 7 1934000 7 1760300 481700 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 8400 equity is equal to 1100 and if TA is 6700 equity is equal to 0 We should note here that the book value of shareholders equity can be negative B10 SOLUTIONS 18 1 Taxes Growth 7 01550000 02525000 03413000 7 18170 Taxes Income 7 01550000 02525000 03425000 039235000 0348465000 7 2992000 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 730000 COGS 580000 AampS expenses 105000 Depreciation 135000 EBIT 790000 Interest 75000 Taxable income 7 165000 Taxes 35 a Net income 7165000 b OCF EBIT Depreciation 7 Taxes 790000 135000 7 0 45000 5 1 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 45000 7 0 7 0 45000 Cash ow to stockholders Dividends 7 Net new equity 25000 7 0 25000 Cash ow to creditors Cash ow from assets 7 Cash ow to stockholders Cash ow to creditors 45000 7 25000 20000 Cash ow to creditors Interest 7 Net new LTD Net new LTD Interest 7 Cash ow to creditors 75000 7 20000 55000 a Income Statement Sales 22800 Cost of goods sold 16050 Depreciation 4050 2700 Interest 1830 Taxable income 870 Taxes 34 296 Net income 574 b OCF EBIT Depreciation 7 Taxes 2700 4050 7 296 6454 C 9 s1 9 CHAPTER 2 B11 Change in NWC NWCend 7 NWCbe g CAend CLend CAbeg CLbeg 5930 7 3150 7 4800 7 2700 2780 7 2100 680 Net capital spending NFAend 7 NFAbeg Depreciation 16800 713650 4050 7200 CFA OCF 7 Change in NWC 7 Net capital spending 6454 7 680 7 7200 71426 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 firm invested heavily in both xed assets and net working capital it had to raise a net 1426 in funds from its stockholders and creditors to make these investments Cash ow to creditors Interest 7 Net new LTD 1830 7 0 1830 Cash ow to stockholders Cash ow from assets 7 Cash ow to creditors 71426 7 1830 73256 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 1300 7 73256 4556 The rm had positive earnings in an accounting sense NT gt 0 and had positive cash ow from operations The lm invested 680 in new net working capital and 7200 in new xed assets The rm had to raise 1426 from its stakeholders to support this new investment It accomplished this by raising 4556 in the form of new equity After paying out 1300 of this in the form of dividends to shareholders and 1830 in the form of interest to creditors 1426 was left to meet the firm s cash ow needs for investment Total assets 2008 653 2691 3344 Total liabilities 2008 261 1422 1683 Owners equity 2008 33447 1683 1661 Total assets 2009 707 3240 3947 Total liabilities 2009 293 1512 1805 Owners equity 2009 3947 7 1805 2142 NWC 2008 CA08 7 CL08 653 7 261 392 NWC 2009 CA09 7 CL09 707 7 293 414 Change in NWC NWC09 7 NWC08 414 7392 22 B12 SOLUTIONS 23 e We can calculate net capital spending as Net capital spending Net xed assets 2009 7 Net xed assets 2008 Depreciation Net capital spending 3240 7 2691 738 1287 So the company had a net capital spending cash ow of 1287 We also know that net capital spending is Net capital spending Fixed assets bought 7 Fixed assets sold 1287 1350 7 Fixed assets sold Fixed assets sold 1350 7 1287 63 To calculate the cash ow from assets we must first calculate the operating cash ow The income statement is Income Statement Sales 828000 Costs 386100 Depreciation expense 738 00 EBIT 368100 Interest expense 211 00 EBT 347000 Taxes 35 121550 Net income 2256 50 So the operating cash ow is OCF EBIT Depreciation 7 Taxes 3681 738 7121450 320450 And the cash ow from assets is Cash ow from assets OCF 7 Change in NWC 7 Net capital spending 320450 7 22 71287 189550 d Net new borrowing LTD09 7 LTD08 1512 7 1422 90 Cash ow to creditors Interest 7 Net new LTD 211 7 90 121 Net new borrowing 90 Debt issued 7 Debt retired Debt retired 270 7 90 180 Challenge Net capital spending NFAend 7 NFAbe g Depreciation NFAend 7 NFAbeg Depreciation ADbeg 7 ADbe g NFAend NFAbeg ADend ADbeg NFAend ADend NFAbeg ADbeg FAend FAbeg 24 CHAPTER 2 B13 a 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 b Taxes 01550000 02525000 03425000 039235000 113900 Average tax rate 113900 335000 34 The marginal tax rate on the next dollar of income is 34 percent For corporate taxable income levels of 335000 to 10 million average tax rates are equal to marginal tax rates Taxes 03410000000 0355000000 0383333333 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 0 Taxes 034200000 68000 68000 01550000 02525000 03425000 X100000 X100000 68000 7 22250 X 45750 100000 X 4575 Balance sheet as of Dec 31 2008 Cash 3792 Accounts payable 3984 Accounts receivable 5021 Notes payable 732 Inventory 8927 Current liabilities 4716 Current assets 17740 Longterm debt 12700 Net xed assets 31805 Owners39 equity 32129 Total assets 49545 Total liab amp equity 49545 Balance sheet as of Dec 31 2009 Cash 4041 Accounts payable 4025 Accounts receivable 5892 Notes payable 717 Inventory 9555 Current liabilities 4742 Current assets 19488 Longterm debt 15435 Net xed assets 33921 Owners39 equity 33232 Total assets 53409 Total liab amp equity 53409 B14 SOLUTIONS 2008 Income Statement 2009 Income Statement Sales 723300 Sales 808500 COGS 248700 COGS 294200 Other expenses 59100 Other expenses 51500 Depreciation 103 800 Depreciation 108500 EBIT 311700 EBIT 354300 Interest 48500 Interest 57900 E T 263200 EBT 296400 Taxes 34 89488 Taxes 34 100776 Net income 173712 Net income 195624 Dividends 88200 Dividends 101100 Additions to RE 85512 Additions to RE 94524 OCF EBIT Depreciation 7 Taxes 3543 1085 7 100776 362024 Change in NWC 7 NWCend 7 NWCbeg 7 CA 7 CL end 7 CA 7 CL be 7 19488 7 4742 7 17740 7 4716 7 1722 Net capital spending NFAend 7 NFAbeg Depreciation 339217 31805 1085 3201 Cash ow from assets OCF 7 Change in NWC 7 Net capital spending 362024 7 1722 7 3201 7130276 Cash ow to creditors Interest 7Net new LTD Net new LTD LTD end 7 LTDbe g Cash ow to creditors 579 7 15435 7 12700 72156 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 OEend OEbeg REbeg REend REend REbeg Additions to RE08 Net new equity OEend 7 OEbeg REbe g 7 REbeg Additions to RE08 OEend 7 OEbe g 7 Additions to RE Net new equity 33232 7 32129 7 94524 15776 CFS Dividends 7 Net new equity CFS 1011715776 85324 As a check cash ow from assets is 7130276 CFA Cash ow from creditors Cash ow to stockholders CFA 72156 85324 7130276 CHAPTER 3 WORKING WITH FINANCIAL STATEMENTS Answers to Concepts Review and Critical Thinking Questions 1 9 If inventory is purchased with cash then there is no change in the current ratio If inventory is purchased on credit then there is a decrease in the current ratio if it was initially greater than 10 Reducing accounts payable with cash increases the current ratio if it was initially greater than 10 Reducing shortterm debt with cash increases the current ratio if it was initially greater than 10 As longterm debt approaches maturity the principal repayment and the remaining interest expense become current liabilities Thus if debt is paid off with cash the current ratio increases if it was initially greater than 10 If the debt has not yet become a current liability then paying it off will reduce the current ratio since current liabilities are not affected Reduction of accounts receivables and an increase in cash leaves the current ratio unchanged Inventory sold at cost reduces inventory and raises cash so the current ratio is unchanged Inventory sold for a profit raises cash in excess of the inventory recorded at cost so the current ratio increases 9 5 0039 The rm has increased inventory relative to other current assets therefore assuming current liability levels remain unchanged liquidity has potentially decreased A current ratio of 050 means that the rm has twice as much in current liabilities as it does in current assets the film potentially has poor liquidity If pressed by its shortterm creditors and suppliers for immediate payment the rm might have a difficult time meeting its obligations A current ratio of 150 means the rm has 50 more current assets than it does current liabilities This probably represents an improvement in liquidity shortterm obligations can generally be met com pletely with a safety factor built in A current ratio of 150 however might be excessive Any excess funds sitting in current assets generally earn little or no return These excess funds might be put to better use by investing in productive longterm assets or distributing the funds to shareholders 11 Quick ratio provides a measure of the shortterm liquidity of the lm after removing the effects of inventory generally the least liquid of the rm s current assets b Cash ratio represents the ability of the rm to completely pay off its current liabilities with its most liquid asset cash 0 Total asset turnover measures how much in sales is generated by each dollar of rm assets d Equity multiplier represents the degree of leverage for an equity investor of the rm it measures the dollar worth of firm assets each equity dollar has a claim to 8 Longterm debt ratio measures the percentage of total rm capitalization funded by longterm debt B16 SOLUTIONS 10 5 Times interest earned ratio provides a relative measure of how well the rm s operating earnings can cover current interest obligations Pro t margin is the accounting measure of bottomline pro t per dollar of sales Return on assets is a measure of bottomline pro t per dollar of total assets Return on equity is a measure of bottomline pro t per dollar of equity Priceearnings ratio re ects how much value per share the market places on a dollar of accounting earnings for a firm PW Common size financial statements express all balance sheet accounts as a percentage of total assets and all income statement accounts as a percentage of total sales Using these percentage values rather than nominal dollar values facilitates comparisons between rms of different size or business type Commonbase year nancial statements express each account as a ratio between their current year nominal dollar value and some reference year nominal dollar value Using these ratios allows the total growth trend in the accounts to be measured 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 firm to its peers allows the nancial manager to evaluate whether some aspects of the rm s operations nances or39 investment activities are out of line with the norm thereby providing some guidance on appropriate actions to take to adjust these ratios if appropriate An aspirant group would be a set of rms whose performance the company in question would like to emulate The nancial manager often uses the nancial ratios of aspirant groups as the target ratios for his or her firm some managers are evaluated by how well they match the performance of an identi ed aspirant group Return on equity is probably the most important accounting ratio that measures the bottomline performance of the firm with respect to the equity shareholders The Du Pont identity emphasizes the role of a rm s pro tability asset utilization efficiency and nancial leverage in achieving an ROE gure For example a firm with ROE of 20 would seem to be doing well but this gure may be misleading if it were marginally pro table low pro t margin and highly levered high equity multiplier If the rm s margins were to erode slightly the ROE would be heavily impacted The booktobill ratio is intended to measure whether demand is growing or falling It is closely followed because it is a barometer for the entire hightech industry where levels of revenues and earnings have been relatively volatile 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 a For an electric utility such as Con Ed expressing costs on a per kilowatt hour basis would be a way to compare costs with other utilities of different sizes For a retailer such as Sears expressing sales on a per square foot basis would be useful in comparing revenue production against other retailers c For an airline such as Southwest expressing costs on a per passenger mile basis allows for comparisons with other airlines by examining how much it costs to y one passenger one mile 9 CHAPTER 3 B17 d For an online service provider such as AOL using a per call basis for costs would allow for comparisons with smaller services Aper subscriber basis would also make sense e For a hospital such as Holy Cross revenues and costs expressed on a per bed basis would be useful f For a college textbook publisher such as McGrawHillIrwin the leading publisher of nance textbooks for the college market the obvious standardization would be per book sold 11 Reporting the sale of Treasury securities as cash ow from operations is an accounting trick and as such should constitute a possible red ag about the companies accounting practices For most companies the gain from a sale of securities should be placed in the nancing section Including the sale of securities in the cash ow from operations would be acceptable for a nancial company such as an investment or commercial bank p A N Increasing the payables period increases the cash ow from operations This could be bene cial for the company as it may be a cheap form of nancing but it is basically a one time change The payables period cannot be increased inde nitely as it will negatively affect the company s credit rating if the payables period becomes too long 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 1 Using the formula for NWC we get NWC CA 7 CL CA CL NWC 3720 1370 5090 So the current ratio is Current ratio CA CL 50903720 137 times And the quick ratio is Quick ratio CA7 Inventory CL 5090 7 1950 3720 084 times 2 We need to nd net income first So Pro t margin Net income Sales Net income SalesPro t margin Net income 29000000008 2320000 ROA Net income TA 2320000 17500000 1326 or 1326 B18 SOLUTIONS To nd ROE we need to nd total equity TL amp OE TD TE TE TL amp OE 7 TD TE 17500000 7 6300000 11200000 ROE Net income TE 2320000 11200000 2071 or 2071 3 Receivables turnover Sales Receivables Receivables turnover 3943709 431287 914 times Days sales in receivables 365 days Receivables turnover 365 914 3992 days The average collection period for an outstanding accounts receivable balance was 3992 days 4 Inventory turnover COGS Inventory Inventory turnover 4105612 407534 1007 times Days sales in inventory 365 days Inventory turnover 365 1007 3623 days On average a unit of inventory sat on the shelf 3623 days before it was sold 5 Total debt ratio 063 TD TA Substituting total debt plus total equity for total assets we get 063 TD TD TE Solving this equation yields 063TE 037TD Debtequity ratio TD TE 063 037 170 Equity multiplier l DE 270 6 Net income Addition to RE Dividends 430000 175000 605000 Earnings per share NI Shares 605000 210000 288 per share Dividends per share Dividends Shares 175000 210000 083 per share Book value per share TE Shares 5300000 210000 2524 per share Markettobook ratio Share price BVPS 63 2524 250 times PE ratio Share price EPS 63 288 2187 times Sales per share Sales Shares 4500000 210000 2143 PS ratio Share price Sales per share 63 2143 294 times gt1 9 gt0 p n H p A N CHAPTER 3 B19 ROE 7 PMTATEM ROE 7 055115280 7 1771 or 1771 This question gives all of the necessary ratios for the DuPont Identity except the equity multiplier so using the DuPont Identity ROE 7 PMTATEM ROE 7 1827 7 068195EM EM 1827068195 138 DE 7 EM71 138 717 038 Decrease in inventory is a source of cash Decrease in accounts payable is a use of cash Increase in notes payable is a source of cash Increase in accounts receivable is a use of cash Changes in cash sources 7uses 375 7 190 210 7 105 290 Cash increased by 290 Payables turnover COGS Accounts payable Payables turnover 28384 6105 465 times Days sales in payables 365 days Payables turnover Days sales in payables 365 465 7851 days The company left its bills to suppliers outstanding for 7851 days on average A large value for this ratio could imply that either 1 the company is having liquidity problems making it dif cult to pay off its shortterm obligations or 2 that the company has successfully negotiated lenient credit terms from its suppliers New investment in xed assets is found by Net investment in FA NFAend 7 NFAbeg Depreciation Net investment in FA 835 148 983 The company bought 983 in new xed assets this is a use of cash The equity multiplier is EM1DE EM 1 065 165 One formula to calculate return on equity is ROE 7 ROAE ROE 7 085165 7 1403 or 1403 B20 SOLUTIONS ROE can also be calculated as ROE NT TE So net income is NI ROETE NI 1403540000 75735 13 through 15 2008 13 2009 13 14 15 Assets Current assets Cash 8436 286 10157 313 12040 10961 Accounts receivable 21530 729 23406 721 10871 09897 Inventory 38760 1312 42650 1314 11004 10017 Total 68726 2326 76213 2348 11089 10095 Fixed assets Net plant and equipment 226706 76 74 2483 06 7652 1095 3 09971 Total assets 295432 100 324519 100 10985 10000 Liabilities and Owners Equity Current liabilities Accounts payable 43050 14 57 46821 1443 10876 09901 Notes payable 183 84 622 173 82 536 0945 5 08608 Total 61434 2079 64203 1978 10451 09514 Longterm debt 25000 846 32000 9 86 12800 11653 Owners39 equity Common stock and paidin surplus 40000 1354 40000 1233 10000 09104 Accumulated retained earnings 168998 5720 188316 5803 11143 10144 Total 208998 7074 228316 7036 10924 09945 Total liabilities and owners39 equity 295432 100 324519 100 10985 10000 The commonsize balance sheet answers are found by dividing each category by total assets For example the cash percentage for 2008 is 8436 295432 0286 or 286 This means that cash is 286 of total assets CHAPTER 3 B21 The commonbase year answers for Question 14 are found by dividing each category value for 2009 by the same category value for 2008 For example the cash commonbase year number is found by 10157 8436 12040 This means the cash balance in 2009 is 12040 times as large as the cash balance in 2008 The commonsize commonbase year answers for Question 15 are found by dividing the common size percentage for 2009 by the commonsize percentage for 2008 For example the cash calculation is found by 313 286 10961 This tells us that cash as a percentage of assets increased by 961 16 2008 SourcesUses 2008 Assets Current assets Cash 8436 1721 U 10157 Accounts receivable 21530 1876 U 23406 Inventory 38760 3890 U 42650 Total 68726 7487 U 76213 Fixed assets Net plant and equipment 226706 21600 U 248306 Total assets 295432 29087 U 324519 Liabilities and Owners Equity Current liabilities Accounts payable 43050 3771 S 46821 Notes payable 183 84 71002 U 17382 Total 61434 2769 3 64203 Longterm debt 25000 7000 3 32000 Owners39 equity Common stock and paidin surplus 40000 0 40000 Accumulated retained earnings 168998 19318 S 188316 Total 208998 19318 3 228316 Total liabilities and owners equity 295432 29087 S 324519 The rm used 29087 in cash to acquire new assets It raised this amount of cash by increasing liabilities and owners equity by 29087 In particular the needed funds were raised by internal nancing on a net basis out of the additions to retained earnings an increase in current liabilities and by an issue of longterm debt B22 SOLUTIONS 17 Current assets Current liabilities 68726 61434 112 times 76213 64203 119 times a Cun39ent ratio Current ratio 2008 Current ratio 2009 b Quick ratio Quick ratio 2008 Quick ratio 2009 Current assets 7 Inventory Current liabilities 67726 7 38760 61434 049 times 76213 7 42650 64203 052 times 0 Cash ratio Cash Current liabilities Cash ratio 2008 8436 61434 014 times Cash ratio 2009 10157 64203 016 times d NWC ratio NWC Total assets NWC ratio 2008 68726 7 61434 295432 247 NWC ratio 2009 76213 7 64203 324519 370 e Debtequity ratio Total debt Total equity Debtequity ratio 2008 61434 25000 208998 041 times Debtequity ratio 2009 64206 32000 228316 042 times Equity multiplier 1 DE Equity multiplier 2008 1 041 141 Equity multiplier 2009 1 042 142 f Total debt ratio Total assets 7 Total equity Total assets Total debt ratio 2008 295432 7 208998 295432 029 Total debt ratio 2009 324519 7 228316 324519 030 Longterm debt ratio Longterm debt Longterm debt Total equity Longterm debt ratio 2008 25000 25000 208998 011 Longterm debt ratio 2009 32000 32000 228316 012 Intermediate This is a multistep problem involving several ratios The ratios given are all part of the DuPont Identity The only DuPont Identity ratio not given is the pro t margin If we know the pro t margin we can find the net income since sales are given So we begin with the DuPont Identity ROE 015 Pl1TATEl1 Pl1S TA1 DE Solving the DuPont Identity for profit margin we get PM 7 ROETA 1 DES PM 0153105 1 14 5726 7 0339 Now that we have the pro t margin we can use this number and the given sales gure to solve for net income PM 0339NIS NI 03395726 19406 19 CHAPTER 3 B23 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 0087 N1 Sales 218000 Sales Sales 2505747 Credit sales are 70 percent of total sales so Credit sales 2515747070 1754023 Now we can nd receivables turnover by Receivables turnover Credit sales Accounts receivable 1754023 132850 1320 times Days sales in receivables 365 days Receivables turnover 365 1320 2765 days The solution to this problem requires a number of steps First remember that CA NFA TA So if we find 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 125875 109375 To nd the total assets we must rst nd the total debt and equity from the information given So we find the sales using the pro t margin PM NT Sales N1 PMSales 0955870 54910 We now use the net income figure as an input into ROE to find the total equity ROE NT TE TENIROE 54910 185 296811 Next we need to nd the longterm debt The longterm debt ratio is Longterm debt ratio 045 LTD LTD TE Inverting both sides gives 1045 LTD TE LTD 1 TE LTD Substituting the total equity into the equation and solving for longterm debt gives the following 2222 1 296811 LTD LTD 296811 1222 242845 B24 SOLUTIONS N H N N N 43 Now we can nd the total debt of the company TD CL LTD 875 242845 330345 And with the total debt we can nd the TDampE which is equal to TA TA TD TE 330345 296811 627156 And finally we are ready to solve the balance sheet identity as NFA TA7 CA 627156 7109375 517781 Child Pro t margin NT S 300 50 06 or 6 Store Pro t margin NT S 22500000 750000000 03 or 3 The advertisement is referring to the store s pro t margin but a more appropriate earnings measure for the firm s owners is the return on equity ROE NT TENITA7TD ROE 22500000 420000000 7 280000000 1607 or 1607 The solution requires substituting two ratios into a third ratio Rearranging D TA FirmA D TA 35 TA7E TA 35 TA TA 7E TA 7 35 FirmB D TA 30 TA7ETA 30 TA TA 7E TA 7 30 17ETA35 17ETA30 ETA65 ETA30 E 65TA E 70 TA Rearranging ROA we nd NITA12 NITA11 NI 12TA NI 11TA Since ROE NT E we can substitute the above equations into the ROE formula which yields ROE 7 12TA 65TA 7 12 65 1846 ROE 7 11TA 70 TA 7 11 70 71571 This problem requires you to work backward through the income statement First recognize that Net income 1 7 tEBT Plugging in the numbers given and solving for EBT we get EBT 13168 1 7 034 1995152 Now we can add interest to EBT to get EBIT as follows EBIT EBT Interest paid 1995152 3605 2355652 CHAPTER 3 B25 To get EBITD earnings before interest taxes and depreciation the numerator in the cash coverage ratio add depreciation to EBIT EBITD EBIT Depreciation 2355652 23 82 2593852 Now simply plug the numbers into the cash coverage ratio and calculate Cash coverage ratio EBITD Interest 2593852 3605 720 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 is given we start with the current ratio Current ratio 140 CA CL CA 365000 CA 511000 Using the quick ratio we solve for inventory Quick ratio 085 CA7Inventory CL 511000 7 Inventory 365000 Inventory CA7 Quick ratio gtlt CL Inventory 511000 7 085 X 365000 Inventory 200750 Inventory turnover 582 COGS Inventory COGS 200750 COGS 7 1164350 PM NI S 7 13482000 138793 700971 or 7971 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 nancial 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 rms andor divisions across national economic borders The net income in dollars is NI PM gtlt Sales NI 40971274213000 726636355 Short term solvency ratios Current ratio Current assets Current liabilities Current ratio 2008 56260 38963 144 times Current ratio 2009 60550 43235 140 times Quick ratio Current assets 7Inventory Current liabilities Quick ratio 2008 56260 7 23084 38963 085 times Quick ratio 2009 60550 7 24650 43235 083 times Cash ratio Cash Current liabilities Cash ratio 2008 21860 38963 056 times Cash ratio 2009 22050 43235 051 times B26 SOLUTIONS Asset utilization ratios Total asset turnover Sales Total assets Total asset turnover 305830 321075 095 times Inventory turnover Cost of goods sold Inventory Inventory turnover 210935 24650 856 times Receivables turnover Sales Accounts receivable Receivables turnover 305830 13850 2208 times Longterm solvency ratios Total debt ratio Total assets 7 Total equity Total assets Total debt ratio 2008 290328 7 176365 290328 039 Total debt ratio 2009 321075 7 192840 321075 040 Debtequity ratio Total debt Total equity Debtequity ratio 2008 38963 75000 176365 065 Debtequity ratio 2009 43235 85000 192840 066 Equity multiplier 1 DE Equity multiplier 2008 1 065 165 Equity multiplier 2009 1 066 166 Times interest earned EBIT Interest Times interest earned 68045 11930 570 times Cash coverage ratio EBIT Depreciation Interest Cash coverage ratio 68045 26850 11930 795 times Profitability ratios Pro t margin Net income Sales Pro t margin 36475 305830 01193 or 1193 Return on assets Net income Total assets Return on assets 36475 321075 01136 or 1136 Return on equity Net income Total equity Return on equity 36475 192840 01891 or 1891 27 The DuPont identity is ROE Pl1TATEM ROE 7 01193095166 7 01891 or 1891 29 Earnings per share Earnings per share Dividends per share Dividends per share Book value per share Book value per share SMOLIRA GOLF CORP Statement of Cash Flows For 2009 Cash beginning ofthe year Operating activities Net income Plus Depreciation Increase in accounts payable Increase in other current liabilities Less Increase in accounts receivable Increase in inventory Net cash from operating activities Investment activities Fixed asset acquisition Net cash from investment activities Financing activities Increase in notes payable Dividends paid Increase in longterm debt Net cash from nancing activities Net increase in cash Cash end ofyear Net income Shares Shares price Earnings per share 43 146 2947 times Dividends Shares Total equity Shares 36475 25000 146 per share 20000 25000 080 per share 21860 36475 26850 3530 1742 2534 1566 64497 53307 53307 1000 20000 10000 11000 190 22050 192840 25000 shares 771 per share CHAPTER 3 B27 B28 SOLUTIONS Markettobook ratio Markettobook ratio Share price Book value per share 43771 557 times PEG ratio PEG ratio PE ratio Growth rate 2947 9 327 times First we will nd the market value of the company s equity which is Market value of equity Shares X Share price Market value of equity 2500043 1075000 The total book value of the company s debt is Total debt Current liabilities Longterm debt Total debt 43235 85000 128235 Now we can calculate Tobin s Q which is Tobin s Q Market value of equity Book value of debt Book value of assets Tobin s Q 1075000 128235 321075 Tobin s Q 375 Using the book value of debt implicitly assumes that the book value of debt is equal to the market value of debt This will be discussed in more detail in later chapters but this assumption is generally true Using the book value of assets assumes that the assets can be replaced at the current value on the balance sheet There are several reasons this assumption could be awed First in ation during the life of the assets can cause the book value of the assets to understate the market value of the assets Since assets are recorded at cost when purchased in ation means that it is more expensive to replace the assets Second improvements in technology could mean that the assets could be replaced with more productive and possibly cheaper assets If this is true the book value can overstate the market value of the assets Finally the book value of assets may not accurately represent the market value of the assets because of depreciation Depreciation is done according to some schedule generally straightline or MACRS Thus the book value and market value can often diverge CHAPTER 4 LONGTERM FINANCIAL PLANNING AND GROWTH Answers to Concepts Review and Critical Thinking Questions 1 5 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 Ifthe 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 profit margin increases if it changes its dividend policy by increasing the retention ratio or its total asset turnover increases The internal growth rate is greater than 15 because at a 15 growth rate the negative EFN indicates that there is excess internal nancing Ifthe intemal growth rate is greater than 15 then the sustainable growth rate is certame greater than 15 because there is additional debt nancing used in that case assuming the rm is not 100 equityfmanced As the retention ratio is increased the rm has more internal sources of funding so the EFN will decline Conversely as the retention ratio is decreased the EFN will rise If the firm pays out all its earnings in the form of dividends then the rm has no internal sources of funding ignoring the effects of accounts payable the intemal growth rate is zero in this case and the EFN will rise to the change in total assets The sustainable growth rate is greater than 20 because at a 20 growth rate the negative EFN indicates that there is excess nancing still available Ifthe rm is 100 equity financed then the sustainable and internal growth rates are equal and the internal growth rate would be greater than 20 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 If the retention ratio is increased the rm will have more intemal funding sources available and it will have to take on more debt to keep the debt equity 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 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 B30 SOLUTIONS 9 Apparently not In hindsight the rm may have underestimated costs and also underestimated the extra demand from the lower price 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 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 up front 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 final answer for each problem is found without rounding during any step in the problem 1 Basic It is important to remember that equity will not increase by the same percentage as the other assets If every other item on the income statement and balance sheet increases by 15 percent the pro forma income statement and balance sheet will look like this Pro forma income statement Pro forma balance sheet Sales 26450 Assets 18170 Debt 5980 Costs 19205 Equity 12190 Net income g 7245 Total 18170 Total 9 18170 In order for the balance sheet to balance equity must be Equity Total liabilities and equity 7 Debt Equity 18170 7 5980 Equity 12190 Equity increased by Equity increase 12190 7 10600 Equity increase 1590 CHAPTER 4 B31 Net income is 7245 but equity only increased by 1590 therefore a dividend of Dividend 7 7245 7 1590 Dividend 5655 must have been paid Dividends paid is the plug variable Here we are given the dividend amount so dividends paid is not a plug variable If the company pays out onehalf of its net income as dividends the pro forma income statement and balance sheet will look like this Pro forma income statement Pro forma balance sheet Sales 2645000 Assets 1817000 Debt 598000 Costs 1920500 Equity 1422250 Net income 724500 Total 1817000 Total 19422 50 Dividends 362250 Add toRE 362250 Note that the balance sheet does not balance This is due to EFN The EFN for this company is EFN Total assets 7 Total liabilities and equity EFN 18170 71942250 EFN 7125250 An increase of sales to 7424 is an increase of Sales increase 7424 7 6300 6300 Sales increase 18 or 18 Assuming costs and assets increase proportionally the pro forma nancial statements will look like this Pro forma income statement Pro forma balance sheet Sales 7434 Assets 21594 Debt 12400 Costs 4 590 Equity 874 Net income 2844 Total 21594 Total 21144 If no dividends are paid the equity account will increase by the net income so Equity 5900 2844 Equity 8744 So the EFN is EFN Total assets 7 Total liabilities and equity EFN 21594 7 21144 450 B32 SOLUTIONS 4 An increase of sales to 21840 is an increase of Sales increase 21840 7 19500 19500 Sales increase 12 or 12 Assuming costs and assets increase proportionally the pro forma nancial statements will look like this Pro forma income statement Pro forma balance sheet Sales 21840 Assets 109760 Debt 52500 Costs 16800 Equity 79208 EBIT 5040 Total 109760 Total 99456 Taxes 40 2 016 Net income 3024 The payout ratio is constant so the dividends paid this year is the payout ratio from last year times net income or Dividends 1400 27003024 Dividends 1568 The addition to retained earnings is Addition to retained earnings 3024 7 1568 Addition to retained earnings 1456 And the new equity balance is Equity 7 45500 1456 Equity 46956 So the EFN is EFN Total assets 7 Total liabilities and equity EFN 109760 7 99456 EFN 10304 5 Assuming costs and assets increase proportionally the pro forma nancial statements will look like this Pro forma income statement Pro forma balance sheet Sales 483000 CA 414000 CL 214500 Costs 379500 FA 908500 LTD 365000 Taxable income 103500 Equit 615986 Taxes 34 35190 TA 1322500 Total DampE 1222486 Net income 68310 CHAPTER 4 B33 The payout ratio is 40 percent so dividends will be Dividends 7 040683 10 Dividends 27324 The addition to retained earnings is Addition to retained earnings 68310 7 27324 Addition to retained earnings 40986 So the EFN is EFN Total assets 7 Total liabilities and equity EFN 13225 71222486 EFN 100014 To calculate the internal growth rate we rst need to calculate the ROA which is ROA NI TA ROA 7 2262 39150 ROA 7 0578 or 578 The plowback ratio b is one minus the payout ratio so b1730 b70 Now we can use the internal growth rate equation to get Internal growth rate ROA X b 1 7 ROA X b Internal growth rate 0057870 1 7 0057870 Internal growth rate 0421 or 421 To calculate the sustainable growth rate we rst need to calculate the ROE which is ROE NI TE ROE 7 2262 21650 ROE 7 1045 or 1045 The plowback ratio b is one minus the payout ratio so b1730 b70 Now we can use the sustainable growth rate equation to get Sustainable grth rate ROE X b 1 7 ROE X b Sustainable grth rate 0104570 l 7 0104570 Sustainable grth rate 0789 or 789 B34 SOLUTIONS 8 The maximum percentage sales increase is the sustainable growth rate To calculate the sustainable growth rate we first need to calculate the ROE which is ROE NI TE ROE 8910 56000 ROE 1591 or 1591 The plowback ratio b is one minus the payout ratio so b1730 b70 Now we can use the sustainable growth rate equation to get Sustainable grth rate ROE X b l 7 ROE X b Sustainable grth rate 159l70 17159l70 Sustainable grth rate 1253 or 1253 So the maximum dollar increase in sales is Maximum increase in sales 420001253 Maximum increase in sales 526403 9 Assuming costs vary with sales and a 20 percent increase in sales the pro forma income statement will look like this HEIR JORDAN CORPORATION Pro Forrna Income Statement Sales 4560000 Costs 2208000 Taxable income 2352000 Taxes 34 799680 Net income 1552320 The payout ratio is constant so the dividends paid this year is the payout ratio from last year times net income or Dividends 7 5200129361552320 Dividends 624000 And the addition to retained earnings will be Addition to retained earnings 1552320 7 6240 Addition to retained earnings 928320 CHAPTER 4 B3 5 10 Below is the balance sheet with the percentage of sales for each account on the balance sheet Notes payable total current liabilities longterm debt and all equity accounts do not vary directly with sales HEIR JORDAN CORPORATION Balance Sheet 55 55 Assets Liabilities and Owners Equity Current assets Current liabilities Cash 3050 803 Accounts payable 1300 342 Accounts receivable 6900 1816 Notes payable 6800 na Inventory 7600 2000 Total 8100 na Total 17550 4618 Longterm debt 25000 na Fixed assets Owners equity Net plant and Common stock and equipment 34500 9079 paidin surplus 15000 n a Retained earnings 3950 n a To a 18950 na Total liabilities and owners Total assets 52050 13697 equity 52050 na Assuming costs vary with sales and a 15 percent increase in sales the pro forma income statement will look like this HEIR JORDAN CORPORATION Pro Forma Income Statement Sales 4370000 Costs 2116000 Taxable income 2254000 Taxes 34 7 66360 Net income 1487640 The payout ratio is constant so the dividends paid this year is the payout ratio from last year times net income or Dividends 5200129361487640 Dividends 598000 And the addition to retained earnings will be Addition to retained earnings 1487640 7 5980 Addition to retained earnings 889640 The new accumulated retained earnings on the pro forma balance sheet will be New accumulated retained earnings 3950 889640 New accumulated retained earnings 1284640 B36 SOLUTIONS The pro forma balance sheet will look like this HEIR JORDAN CORPORATION Pro Forma Balance Sheet p A N p A DJ Assets Liabilities and Owners Equity Current assets Current liabilities Cash 350750 Accounts payable 149500 Accounts receivable 793500 Notes payable 680000 Inventory 874000 Total 829500 Total 20 1 82 50 Longterm debt 2500000 Fixed assets Net plant and Owners equity equipment 3967500 Common stock and paidin surplus 1500000 Retained earnings 1284640 otal 2784640 Total liabilities and owners Total assets 5985750 equity 6114140 So the EFN is EFN Total assets 7 Total liabilities and equity EFN 5985750 7 6114140 EFN 7128390 We need to calculate the retention ratio to calculate the intemal growth rate The retention ratio is b1720 b80 Now we can use the intemal growth rate equation to get Internal growth rate ROA X b 1 7 ROA gtlt b Internal growth rate 0880 170880 Internal growth rate 0684 or 684 We need to calculate the retention ratio to calculate the sustainable growth rate The retention ratio is b1725 b75 Now we can use the sustainable growth rate equation to get Sustainable grth rate ROE X b 1 7 ROE X b Sustainable grth rate 1575 171575 Sustainable grth rate 1268 or 1268 14 CHAPTER 4 B37 We rst must calculate the ROE to calculate the sustainable growth rate To do this we must realize two other relationships The total asset turnover is the inverse of the capital intensity ratio and the equity multiplier is 1 DE Using these relationships we get ROE PMTATEM ROE 7 0821751 40 ROE 1531 or 1531 The plowback ratio is one minus the dividend payout ratio so b 7 17 12000 43000 b 7 7209 Now we can use the sustainable growth rate equation to get Sustainable grth rate ROE X b 1 7 ROE X b 1 Sustainable grth rate 153l7209 l 7 153l7209 Sustainable grth rate 1240 or 1240 We must rst calculate the ROE using the DuPont ratio to calculate the sustainable growth rate The 1S ROE PMTATEM ROE 078250180 ROE 3510 or 3510 The plowback ratio is one minus the dividend payout ratio so b1760 b40 Now we can use the sustainable growth rate equation to get Sustainable growth rate ROE X b l 7 ROE X b Sustainable growth rate 351040 17351040 Sustainable growth rate 1633 or 1633 Intermediate To determine full capacity sales we divide the current sales by the capacity the company is currently using so Full capacity sales 550000 95 Full capacity sales 578947 The maximum sales growth is the full capacity sales divided by the current sales so Maximum sales growth 578947 550000 7 1 Maximum sales growth 0526 or 526 B3 8 SOLUTIONS 17 gt0 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 440000 578947 Fixed assets Full capacity sales 76 Next we calculate the total dollar amount of xed assets needed at the new sales figure Total xed assets 76630000 Total xed assets 478800 The new xed assets necessary is the total xed assets at the new sales figure minus the current level of xed assts New fixed assets 478800 7 440000 New fixed assets 38800 We have all the variables to calculate ROE using the DuPont identity except the pro t margin If we nd ROE we can solve the DuPont identity for pro t margin We can calculate ROE 39om the sustainable growth rate equation For this equation we need the retention ratio so b1730 b70 Using the sustainable growth rate equation and solving for ROE we get Sustainable grth rate ROE X b 1 7ROE X b 12 ROE70 17 ROE70 ROE 1531 or 1531 Now we can use the DuPont identity to find the pro t margin as ROE PMTATEM 1531 PM1 0751 120 PM 1531 1 075220 PM 7 0522 or 522 We have all the variables to calculate ROE using the DuPont identity except the equity multiplier Remember that the equity multiplier is one plus the debtequity ratio If we nd ROE we can solve the DuPont identity for equity multiplier then the debtequity ratio We can calculate ROE from the sustainable growth rate equation For this equation we need the retention ratio so b1730 b70 Using the sustainable growth rate equation and solving for ROE we get Sustainable grth rate ROE X b l 7 ROE X b 115 7 ROE70 1 7ROE70 ROE 1473 or 1473 9 H CHAPTER 4 B39 Now we can use the DuPont identity to find the equity multiplier as ROE 7 PMTATEM 1473 7 0621 60EM EM 7 147360 062 EM 7 143 So the DE ratio is DE EM 7 1 DE 143 7 l DE 043 We are given the profit margin Remember that ROA PMTAT We can calculate the ROA from the internal growth rate formula and then use the ROA in this equation to find the total asset tumover The retention ratio is b1725 b75 Using the intemal growth rate equation to find the ROA we get Internal growth rate ROA X b l 7 ROA X b 07 ROA75 1 7 ROA75 ROA 0872 or 872 Plugging ROA and PM into the equation we began with and solving for TAT we get ROA 7 PMTAT 0872 7 05PM TAT 7 0872 05 TAT 7 174 times We should begin by calculating the DE ratio We calculate the DE ratio as follows Total debt ratio 65 TD TA Inverting both sides we get 1 65 TA TD Next we need to recognize that TATD1TETD Substituting this into the previous equation we get 1651TETD B40 SOLUTIONS N N Subtract 1 one from both sides and inverting again we get DE116571 DE 186 With the DE ratio we can calculate the EM and solve for ROE using the DuPont identity ROE Pl1TATEl1 ROE 0481251 186 ROE 1714 or 1714 Now we can calculate the retention ratio as b1730 b70 Finally putting all the numbers we have calculated into the sustainable growth rate equation we get Sustainable grth rate ROE X b l 7 ROE X b Sustainable grth rate 171470 1 7 171470 Sustainable grth rate 1364 or 1364 To calculate the sustainable growth rate we rst must calculate the retention ratio and ROE The retention ratlo 1s b 7 17 9300 17500 b 7 4686 And the ROE is ROE 17500 58000 ROE 3017 or 3017 So the sustainable growth rate is Sustainable grth rate ROE X b 1 7 ROE X b Sustainable grth rate 30174686 1 7 30174686 Sustainable growth rate 1647 or 1647 Ifthe company grows at the sustainable growth rate the new level of total assets is New TA 1164786000 58000 16771084 To find the new level of debt in the company s balance sheet we take the percentage of debt in the capital structure times the new level of total assets The additional borrowing will be the new level of debt minus the current level of debt So New TD D D ETA New TD 7 86000 86000 5800016771084 New TD 7 10016064 CHAPTER 4 B41 And the additional borrowing will be Additional borrowing 10016004 7 86000 Additional borrowing 1416064 The growth rate that can be supported with no outside nancing is the internal growth rate To calculate the internal growth rate we rst need the ROA which is ROA 17500 86000 58000 ROA 1215 or 1215 This means the internal growth rate is Internal growth rate ROA X b 1 7 ROA X b Internal growth rate 121546861712154686 Internal growth rate 0604 or 604 Since the company issued no new equity shareholders equity increased by retained earnings Retained earnings for the year were Retained earnings NI 7 Dividends Retained earnings 19000 7 2500 Retained earnings 16500 So the equity at the end of the year was Ending equity 135000 16500 Ending equity 151500 The ROE based on the end of period equity is ROE 19000 151500 ROE 1254 or 1254 The plowback ratio is Plowback ratio Addition to retained earningsNT Plowback ratio 16500 19000 Plowback ratio 8684 or 8684 Using the equation presented in the text for the sustainable growth rate we get Sustainable grth rate ROE X b 1 7 ROE X b Sustainable growth rate 12548684 1 7 12548684 Sustainable growth rate 1222 or 1222 The ROE based on the beginning of period equity is ROE 16500 135000 ROE 1407 or 1407 B42 SOLUTIONS Using the shortened equation for the sustainable growth rate and the beginning of period ROE we get Sustainable growth rate ROE X b Sustainable grth rate 1407 X 8684 Sustainable grth rate 1222 or 1222 Using the shortened equation for the sustainable grth rate and the end of period ROE we get Sustainable grth rate ROE X b Sustainable grth rate 1254 X 8684 Sustainable grth rate 1089 or 1089 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 The ROA using end ofperiod assets is ROA 19000 250000 ROA 0760 or 760 The beginning of period assets had to have been the ending assets minus the addition to retained earnings so Beginning assets Ending assets 7 Addition to retained earnings Beginning assets 250000 7 16500 Beginning assets 233500 And the ROA using beginning ofperiod assets is ROA 19000 233500 ROA 0814 or 814 Using the intemal growth rate equation presented in the text we get Internal growth rate ROA X b 1 7 ROA X b Internal growth rate 08148684 17 08148684 Internal growth rate 0707 or 707 Using the formula ROA X b and end of period assets Internal growth rate 0760 X 8684 Internal growth rate 0660 or 660 Using the formula ROA X b and beginning of period assets Internal growth rate 0814 X 8684 Internal growth rate 0707 or 707 CHAPTER 4 B43 25 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 Taxes35 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 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 B44 SOLUTIONS N I So the EFN is EFN Total assets 7 Total liabilities and equity EFN 679080 7 675232 EFN 3848 First we need to calculate full capacity sales which is Full capacity sales 929000 80 Full capacity sales 1161250 The capital intensity ratio at full capacity sales is Capital intensity ratio Fixed assets Full capacity sales Capital intensity ratio 413000 1161250 Capital intensity ratio 35565 The xed assets required at full capacity sales is the capital intensity ratio times the projected sales eve Total xed assets 355651161250 396480 So EFN is EFN 7 183480 396480 7 613806 7 7895272 Note that this solution assumes that xed assets are decreased sold so the company has a 100 percent xed asset utilization Ifwe assume xed assets are not sold the answer becomes EFN 183480 413000 7 613806 7166154 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 CHAPTER 4 B45 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 68000 17000 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 44400 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 Tota 418632 Total liabilities and owners Total assets g 697080 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 7697080 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 B46 SOLUTIONS 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 44400 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 43697080 291600 Equity needed 57697080 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 Assets MOOSE TOURS INC Pro Forma Balance Sheet CHAPTER 4 B47 Liabilities and Owners Equity Current assets Cash Accounts receivable Inventory Total Fixed assets Net plant and equipment Total assets Challenge Sales Costs Other expenses EBIT Interest Taxable income Taxes 35 Net income Dividends Add to RE 8 30360 44400 104280 183480 495600 8 697080 Current liabilities Accounts payable Notes payable Total Longterm debt Owners equity Common stock and paidin surplus Retained earnings otal 81600 17000 98600 193 000 140000 247 480 mm Total liabilities and owners equity The pro forma income statements for all three growth rates will be MOOSE TOURS INC Pro Forma Income Statement 15 Sales 20 Sales Growth Growth 1068350 1114800 831450 867600 21850 22800 215050 224400 14000 14000 201050 210400 70368 73640 130683 136760 39205 41028 91478 95732 g 6970 80 25 Sales Growth 1161250 903750 23750 233750 14000 219750 76913 142838 42851 99986 We will calculate the EFN for the 15 percent growth rate first 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 B48 SOLUTIONS 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 15 Sales Growth MOOSE TOURS INC Pro Forma Balance Sheet Assets Liabilities and Owners Equity Current assets Current liabilities Cash 29095 Accounts payable 78200 Accounts receivable 46805 Notes payable 17000 Inventory 9993 5 Total 95200 Total 17583 5 Longterm debt 158000 Fixed assets Net plant and Owners equity equipment 474950 Common stock and paidin surplus 140000 Retained earnings 2743 78 Tota 414378 Total liabilities and owners Total assets 650785 equity 667578 So the EFN is EFN Total assets 7 Total liabilities and equity EFN 650785 7 667578 EFN 716793 At a 20 percent growth rate and assuming the payout ratio is constant the dividends paid will be Dividends 3373 5 1 12450 136760 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 CHAPTER 4 B49 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 3373 511245014283 8 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 B50 SOLUTIONS 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 5 16250 Common stock and paidin surplus 140000 Retained earnings 282886 otal 422886 Total liabilities and owners Total assets g 707375 equity g 682886 So the EFN is EFN Total assets 7 Total liabilities and equity EFN 707375 7 682886 EFN 24889 The pro forma income statements for all three growth rates will be MOOSE TOURS INC Pro Forma 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 238450 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 30810102700135948 Dividends 40784 And the addition to retained earnings will be CHAPTER 4 B51 Addition to retained earnings 135948 7 40784 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 7556427l4l 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 58047 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 g 735670 equity g 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 30810102700154993 Dividends 46498 B52 SOLUTIONS 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 CHAPTER 4 B53 Sales growth rate 35 and debtequity ratio 75255 MOOSE TOURS INC Pro Forma Balance Sheet DJ Assets Liabilities and Owners Equity Current assets Current liabilities Cash 3415 5 Accounts payable 91800 Accounts receivable 54945 Notes payable 17000 Inventory 1 173 15 Total 108800 Total 206415 Longterm debt 215848 Fixed assets Net plant and Owners equity equipment 5 57550 Common stock and paidin surplus 140000 Retained earnings 2913 95 o a 431395 Total liabilities and owners Total assets g 763965 equity g 756043 So the excess debt raised is Excess debt 756043 7 763965 Excess debt 77922 At a 35 percent growth rate the firm will need funds in the amount of 7922 in addition to the external debt already raised So the EFN will be EFN 57848 7922 EFN 7 65770 We must need the ROE to calculate the sustainable growth rate The ROE is ROE P11TATEM ROE 06711351 030 ROE 0645 or 645 Now we can use the sustainable growth rate equation to nd the retention ratio as Sustainable grth rate ROE X b 1 7ROE X b Sustainable grth rate 12 0645b 1 7 0645b b 166 This implies the payout ratio is Payout ratio 1 7 b Payout ratio 1 7 166 Payout ratio 7066 B54 SOLUTIONS 43 N This is a negative dividend payout ratio of 66 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 l or total earnings retention The maximum sustainable growth rate for this company is Maximum sustainable growth rate ROE X b l 7 ROE X b Maximum sustainable growth rate 0645l l 706451 Maximum sustainable growth rate 0690 or 690 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 NI gtlt bl g 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 7 PMSbl g EFN 7 Ag 7 PMSb 7 PMSbg EFN 7 7PMSb 7 A 7 PMSb g We start with the EFN equation we derived in Problem 31 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 R0A7NIA 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 A7PMSb A Internal growth rate bROA l 7 bROA 43 CHAPTER 4 B5 5 To derive the sustainable growth rate we must realize that to maintain a constant DE ratio with no external equity nancing EFN must equal the addition to retained earnings times the DE ratio EFN 7 D EPMSb1 g EFN 7 Ag 7 PMSb1 g Solving for g and then dividing numerator and denominator by A Sustainable grth rate PMSbl DE A 7 PMSbl DE Sustainable grth rate ROA1 DE b l 7 ROA1 DE b Sustainable grth 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 grth rate ROEE X b l 7 ROEE X b Sustainable grth rate NITEE X b l 7NITEE X b We multiply this equation by TEE TEE Sustainable grth rate NI TEE X b l 7 NI TEE X b X TEE TEE Sustainable grth rate NI X b TEE 7NI X b Recognize that the numerator is equal to beginning of period equity that is TEE 7NI 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 grth 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 B56 SOLUTIONS We multiply this equation by TAE TAE Internal growth rate NI TAE X b l 7 NT TAE X b X TAE TAE Internal growth rate NI X b TAE 7N1 X b Recognize that the numerator is equal to beginning of period assets that is TAE 7N1 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 The internal growth rate equation is Internal growth rate ROAB X b CHAPTER 5 INTRODUCTION TO VALUATION THE TIME VALUE OF MONEY Answers to Concepts Review and Critical Thinking Questions 1 The four parts are the present value PV the future value FV the discount rate r and the life of the investment I Compounding refers to the growth of a dollar amount through time via reinvestment of interest earned It is also the process of determining the future value of an investment Discounting is the process of determining the value today of an amount to be received in the future Future values grow assuming a positive rate of return present values shrink The future value rises assuming it s positive the present value falls It would appear to be both deceptive and unethical to run such an ad without a disclaimer or explanation It s a re ection of the time value of money TMCC gets to use the 24099 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 s nancial position could deteriorate Either event would tend to depress the security s price B5 8 SOLUTIONS 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 1 The simple interest per year is 5000 X 08 400 So after 10 years you will have 400 X 10 4000 in interest The total balance will be 5000 4000 9000 With compound interest we use the future value formula FV PV1 r39 FV 500010810 1079462 The difference is 1079462 7 9000 179462 2 To nd the FV ofa lump sum we use FV PV1 r FV 225011011 641951 FV 87521087 1499939 FV 76355117 68776417 FV 1837961078 31579575 3 To nd the PV ofa lump sum we use PVFV1r PV 15451 1076 1029565 PV 51557 1137 2191485 PV 886073 114 4351690 PV 550164 10918 11663132 CHAPTER 5 B59 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 r FVPvfkl FV 297 2401 r2 r 297 240 2 7 1 1124 FV 1080 3601 r1 r 1080360110 71 1161 FV 185382 390001 r r 185382 l 39000115 71 1095 FV 531618 382611 r r 531618 38261 30 71 917 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 560109 t ln1284 560 In 109 963 years FV 4341 810110 t ln4341 810 In 110 1761 years FV 364518 18400117 t ln364518 18400 In 117 1902 years FV 173439 21500115 t ln173439 21500 111 115 1494 years 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 r FVPWW 1 r 290000 55000 18 7 1 0968 or 968 B60 SOLUTIONS 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 I lnFVPVln1 r The length of time to double your money is FV 2 1007 1 ln 2 In 107 1024 years The length of time to quadruple your money is FV 4 1007 I ln 4 In 107 2049 years 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 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 r FVPvfk 1 r 314600 200300 71 0666 or 666 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 lnFVPV1n1 r t In 170000 40000 In 1053 2802 years To find the PV ofa lump sum we use PV FV 1 r PV 650000000 107420 15589340013 H N 43 A UI CHAPTER 5 B61 To nd the PV ofa lump sum we use PvFV1rt PV 1000000 11080 48819 To nd the FV ofa lump sum we use FV PV1 r2 Fv 5010451 5 508371 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 r FVPV1 71 r 1260000 150 112 71 0840 or 840 To nd the FV of the first prize we use FV PV1 r2 Fv 12600001084033 1805640994 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 71 r 43125 1 71 0990 or 990 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 r FVPV 71 r 10311500 812377500 71 7 446 Notice that the interest rate is negative This occurs when the FV is less than the PV B62 SOLUTIONS 1 l Intermediate 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 r FVPV1 71 a PV 100000 1 r30 24099 r 100000 24099 0 7 0486 or 486 13 PV 38260 1 r 7 24099 r 38260 24099 12 71 0393 or 393 0 PV 100000 1 r18 38260 r 100000 38260 18 71 0548 or 548 To nd the PV ofa lump sum we use PvFV1rt PV 170000 1129 6130370 To nd the FV ofa lump sum we use FV PV1 r2 FV 4000111 5 43812097 FV 400011135 15429940 Better start early We need to nd the FV of a lump sum However the money will only be invested for six years so the number of periods is six FV PV1 r2 FV 2000010846 3244933 CHAPTER 5 B63 20 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 EV formula that is FV PV1 r Solving for t we get tlnFVPVln1 r tln7500010000ln111 1931 So the money must be invested for 1931 years However you will not receive the money for another two years From now you ll wait 2 years 1931 years 2131 years Calculator Solutions 1 Enter 10 8 5000 PMT Solve for 1079462 1079462 7 9000 179462 2 Enter 11 10 2250 PMT F Solve for 6419 51 Enter 7 8 8752 PMT Solve for 1499939 Enter 14 17 763 5 5 PV PMT F V Solve for 68776417 Enter 8 7 183796 PV PMT Solve for 31579575 3 Enter 7 1545 1 6 Solve for 1029565 B64 SOLUTIONS Enter Solve for Enter Solve for Enter Solve for 4 Enter Solve for Enter Solve for Enter Solve for Enter Solve for 5 Enter Solve for Enter Solve for Enter Solve for 7 23 18 2 10 15 30 963 1761 N 1902 13 14 9 IY 1124 IY 1161 IY 1095 917 9 10 17 2191485 FY 4351690 11663132 240 P v I 1g 4 360 P v I 3 9000 P 139 I 1 1 e6 as u 1 LN 11 co m 1 on a o m 1 co co N 3 8261 5 60 PV 810 18400 Pquot H 4 PMT PMT PMT PMT PMT PMT PMT PMT PMT 51557 3 mo mm w 4 4b a 0 l J W i297 1 as r o oo o a L r 1 4g 4 4 as uquot m J 1 H 4 1 as m Ox 4 kl co 5 Enter Solve for 6 Enter Solve for 7 Enter Solve for Enter Solve for 8 Enter Solve for 9 Enter Solve for 10 Enter Solve for 1 1 Enter Solve for 12 Enter Solve for 1494 18 1024 N 2049 7 N 2802 15 IY 968 7 7 IY 666 530 450 21500 55000 1 1 200300 40000 15589340013 48819 50 CHAPTER 5 B6 5 1 e6 1 Ln 4 Ln 0 a 1 R o 30 o o o i2 1 0 O O a 1 e6 113 4 1 ea l o o o o 5 H 4 650000000 H I O O O O O O a a H 4 ea kl m 4 O 00 Ln 1 a B66 SOLUTIONS 13 Enter Solve for Enter Solve for 14 Enter Solve for 15 Enter Solve for 16 1 Enter Solve for 16 b Enter Solve for 16 0 Enter Solve for 17 Enter Solve for 18 Enter Solve for Enter Solve for 112 30 12 18 9 840 840 IY 990 I 4146 IY 4 86 393 IY 548 12 11 11 5150 1260000 1 112377500 524099 124099 i38260 6130370 4000 4000 1260000 1805640494 i 43125 10311500 100000 FV39 38260 100000 FV 170000 43812097 15429940 CHAPTER 5 B67 19 Enter 6 840 20000 PMT Solve for 3 244933 20 Enter 11 i10000 75000 Solve for 1931 From now you ll wait 2 1931 2131 years CHAPTER 6 DISCOUNTED CASH FLOW VALUATION Answers to Concepts Review and Critical Thinking Questions 1 The four pieces are the present value PV the periodic cash ow C the discount rate r and the number of payments or the life of the annuity 1 Assuming positive cash ows both the present and the future values will rise Assuming positive cash ows the present value will fall and the future value will rise It s deceptive but very common The basic concept of time value of money is that a dollar today is not worth the same as a dollar tomorrow The deception is particularly irritating given that such lotteries are usually government sponsored If the total money is xed you want as much as possible as soon as possible The team or more accurately the team owner wants just the opposite 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 The subsidy is the present value on the day the loan is made of the interest that would have accrued up until the time it actually begins to accrue The problem is that the subsidy makes it easier to repay the loan not obtain it However ability to repay the loan depends on future employment not current need For example consider a student who is currently needy but is preparing for a career in a highpaying area such as corporate nance Should this student receive the subsidy How about a student who is currently not needy but is preparing for a relatively low paying job such as becoming a college professor CHAPTER 6 B69 10 In general viatical settlements are ethical In the case of a viatical settlement it is simply an exchange of cash today for payment in the future although the payment depends on the death of the seller The purchaser of the life insurance policy is bearing the risk that the insured individual will live longer than expected Although viatical settlements are ethical they may not be the best choice for an individual In a Business Week article October 31 2005 options were examined for a 72 year old male with a life expectancy of 8 years and a 1 million dollar life insurance policy with an annual premium of 37000 The four options were 1 Cash the policy today for 100000 2 Sell the policy in a viatical settlement for 275000 3 Reduce the death benefit to 375000 which would keep the policy in force for 12 years without premium payments 4 Stop paying premiums and don t reduce the death bene t This will run the cash value of the policy to zero in 5 years but the viatical settlement would be worth 475000 at that time If he died within 5 years the bene ciaries would receive 1 million Ultimately the decision rests on the individual on what they perceive as best for themselves The values that will affect the value of the viatical settlement are the discount rate the face value of the policy and the health of the individual selling the policy 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 1 To solve this problem we must find the PV of each cash ow and add them To nd the PV of a lump sum we use PVFV1r PV10 950 110 10401102 11301103 10751104 330637 PV18 950 118 1040118Z 11301183 1075 1184 279422 PV24 950 124 1040 1242 1130 1243 1075 1244 248988 2 To find the PVA we use the equation PVA C1711 r r At a 5 percent interest rate X5 PVA 6000lt1711059 05 4264693 Y5 PVA 80001711056 05 4060554 B70 SOLUTIONS And at a 15 percent interest rate X15 PVA 60001 711159 15 2862950 Y15 PVA 8000171115615 3027586 Notice that the PV of cash ow X has a greater PV at a 5 percent interest rate but a lower PV at a 15 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 the higher interest rate these bigger cash ows early are more important since the cost of waiting the interest rate is so much greater 3 To solve this problem we must find the FV of each cash ow and add them To nd the FV of a lump sum we use Fv 7 PV1 r FV8 7 9401083 10901082 1340108 1405 7 530771 FV11 7 9401113 10901112 1340111 1405 7 552096 FV24 7 9401243 10901242 1340124 1405 7 653481 Notice we are nding the value at Year 4 the cash ow at Year 4 is simply added to the FV of the other cash ows In other words we do not need to compound this cash ow 4 To find the PVA we use the equation PVA C1711 r r PVA15 yrs PVA 5300lt17110715 07 4827194 PVA40 yrs PVA 53001711074 07 7065806 PVA75 yrs PVA 5300lt17110775 07 7524070 To find the PV of a perpetuity we use the equation PV C r PV 5300 07 7571429 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 75 year annuity and the present value of the perpetuity imply that the value today of all perpetuity payments beyond 75 years is only 47359 9 CHAPTER 6 B71 Here we have the PVA the length of the annuity and the interest rate We want to calculate the annuity payment Using the PVA equation PVA C1711 r2 r PVA 7 34000 7 C1 7 11076515 0765 We can now solve this equation for the annuity payment Doing so we get C 34000 874548 388772 To find the PVA we use the equation PVA7 C1711 y r PVA7 7300017110858 085 7 41166036 Here we need to nd the FVA The equation to nd the FVA is FVA C1 r 7 1 r FVA for 20 years 4000111220 7 1 112 26278116 FVA for 40 years 4000111240 7 1 112 245907263 Notice that because of exponential growth doubling the number of periods does not merely double the FVA Here we have the FVA the length of the annuity and the interest rate We want to calculate the annuity payment Using the FVA equation FVA 7 C1 r 7 1 r 90000 7 C106810 7 1 068 We can now solve this equation for the annuity payment Doing so we get C 90000 1368662 657577 Here we have the PVA the length of the annuity and the interest rate We want to calculate the annuity payment Using the PVA equation PVA C1711 a r 50000 7 C17110757075 We can now solve this equation for the annuity payment Doing so we get C 50000 529660 944002 This cash ow is a perpetuity To find the PV of a perpetuity we use the equation PV C r PV 25000 072 34722222 B72 SOLUTIONS 11 p A N Here we need 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 7 C r 375000 7 25000 r We can now solve for the interest rate as follows r 25000 375000 0667 or 667 For discrete compounding to nd the EAR we use the equation EAR 1 APRm39quot 71 EAR 1 08 44 71 0824 or 824 EAR 1 16 1212 71 1723 or 1723 EAR 1 12 365365 71 1275 or 1275 To find the EAR with continuous compounding we use the equation EAReq71 EARe15 71 1618 or 1618 Here we are given the EAR and need to nd the APR Using the equation for discrete compounding EAR 1APRmm 71 We can now solve for the APR Doing so we get APR 7 m1 EAR 71 EAR 0860 1 APR 22 71 APR 21086012 71 0842 or 842 EAR 1980 1 APR 1212 71 APR 1211980112 71 1820 or 1820 EAR 0940 1 APR 5252 71 APR 5210940152 71 0899 or 899 Solving the continuous compounding EAR equation EAR eq7 1 We get APR ln1 EAR APRln1 1650 APR 1527 or 1527 14 p n Ch p A l CHAPTER 6 B73 For discrete compounding to nd the EAR we use the equation EAR 1 APRm39quot 71 So for each bank the EAR is First National EAR 1 1420 1212 7 1 1516 or 1516 First United EAR 1 1450 22 71 1503 or 1503 Notice that the higher APR does not necessarily mean the higher EAR The number of compounding periods within a year will also affect the EAR The reported rate is the APR so we need to convert the EAR to an APR as follows EAR1APRm quot71 APR m1 EAR 71 APR 3651161365 7 1 1485 or 1485 This is deceptive because the borrower is actually paying annualized interest of 16 per year not the 1485 reported on the loan contract For this problem we simply need to nd the FV of a lump sum using the equation FV PV1 r It is important to note that compounding occurs semiannually To account for this we will diVide the interest rate by two the number of compounding periods in a year and multiply the number of periods by two Doing so we get FV 210010842 850593 For this problem we simply need to nd the FV of a lump sum using the equation FV PV1 r It is important to note that compounding occurs daily To account for this we will diVide the interest rate by 365 the number of days in a year ignoring leap year and multiply the number of periods by 365 Doing so we get FV in 5 years 45001 0933655365 716364 FV in 10 years 45001 0933651 365 1140394 FV in 20 years 45001 0933652 365 2889997 B74 SOLUTIONS 18 N G N p A For this problem we simply need to nd the PV of a lump sum using the equation PVFVl r It is important to note that compounding occurs daily To account for this we will diVide the interest rate by 365 the number of days in a year ignoring leap year and multiply the number of periods by 365 Doing so we get PV 58000 1 10365 2880471 The APR is simply the interest rate per period times the number of periods in a year In this case the interest rate is 30 percent per month and there are 12 months in a year so we get APR 1230 360 To nd the EAR we use the EAR formula EAR 1 APRm39quot 71 EAR 1 3012 71 222981 Notice that we didn t need to diVide the APR by the number of compounding periods per year We do this diVision to get the interest rate per period but in this problem we are already given the interest rate per period We rst need to nd the annuity payment We have the PVA the length of the annuity and the interest rate Using the PVA equation PVA C1711 a r 68500 C1711069126 06912 Solving for the payment we get C 68500 50622252 135315 To nd the EAR we use the EAR equation EAR 1 APRmm71 EAR 1 069 1212 7 1 0712 or7l2 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 18000 5001711013 013 N 43 A CHAPTER 6 B75 Now we solve for r 11013 17 18000500013 11013 0532 1013 10532 18797 1 ln 18797 ln 1013 4886 months Here we are trying to find 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 find 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 EAR1APRm quot71 EAR 1 333352 71 31391651569 Here we need to nd the interest rate that equates the perpetuity cash ows with the PV of the cash ows Using the PV of a perpetu1ty equatlon PV C r 95000 1800 r We can now solve for the interest rate as follows r 1800 95000 0189 or 189 per month The interest rate is 189 per month To find the APR we multiply this rate by the number of months in a year so APR 12189 2274 And using the equation to find an EAR EAR1APRm quot71 EAR 1 018912 71 2526 This problem requires us to nd the FVA The equation to find the FVA is FVA C1 r 7 1 r FVA 3001101236 711012 67814638 B76 SOLUTIONS 25 N CA N l N D In the previous problem the cash ows are monthly and the compounding period is monthly This assumption still holds Since the cash ows are annual we need to use the EAR to calculate the future value of annual cash ows It is important to remember that you have to make sure the compounding periods of the interest rate is the same as the timing of the cash ows In this case we have annual cash ows so we need the EAR since it is the true annual interest rate you will earn So nding the EAR EAR 1APRmm71 EAR 1 101212 7 1 1047 or 1047 Using the FVA equation we get FVA C1 r 7 1 r FVA 3600110473 7 1 1047 64762345 The cash ows are simply an annuity with four payments per year for four years or 16 payments We can use the PVA equation PVA C1711 r2 r PVA 23001711006516 0065 3484371 The cash ows are annual and the compounding period is quarterly so we need to calculate the EAR to make the interest rate comparable with the timing of the cash ows Using the equation for the EAR we get EAR1APRmm71 EAR 1 114 7 1 1146 or 1146 And now we use the EAR to find the PV of each cash ow as a lump sum and add them together PV 725 11146 980111462 1360111464 232036 Here the cash ows are annual and the given interest rate is annual so we can use the interest rate given We simply nd the PV of each cash ow and add them together PV 1650 10845 4200 108453 2430108454 657086 Intermediate 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 07 10 7 First Complex Bank pays compound interest so the interest paid by this bank will be the FV factor of 1 or 1 r10 CHAPTER 6 B77 Setting the two equal we get 0710 1 r10 71 r 17 10 7 1 0545 or 545 Here we need to convert an EAR into interest rates for different compounding periods Using the equation for the EAR we get EAR1APRmm71 EAR 171r271 rl17mil 0817 or 817per six months EAR 17 1 r47139 r 11714 71 0400 or 400 per quarter EAR171r1271 r11711271 0132 or 132per month Notice that the effective six month rate is not twice the effective quarterly rate because of the effect of compounding Here we need to nd the EV 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 5000 1 015126 503762 This is the balance in six months The FV in another six months will be FV 503762118126 550835 The problem asks for the interest accrued so to nd the interest we subtract the beginning balance from the EV The interest accrued is Interest 550835 7 500000 50835 We need to nd the annuity payment in retirement Our retirement savings ends and the retirement withdrawals begin so the PV of the retirement withdrawals will be the EV 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 7001111236 7 1 1112 196316382 Bond account FVA 3001 0612 3607 1 0612 30135451 So the total amount saved at retirement is 196316382 30135451 226451833 Solving for the withdrawal amount in retirement using the PVA equation gives us PVA 226451833 C171109123 0912 C 226451833 1191616 19003763 withdrawal per month B78 SOLUTIONS 33 We need to nd the EV of a lump sum in one year and two years It is important that we use the number of months in compounding since interest is compounded monthly in this case So FV in one year 11011712 115 FV in two years 11011724 132 There is also another common alternative solution We could nd the EAR and use the number of years as our compounding periods So we will nd the EAR first EAR 1 011712 7 1 1498 or 1498 Using the EAR and the number of years to nd the EV we get FV in one year 1114981 115 FV in two years 1114982 132 Either method is correct and acceptable We have simply made sure that the interest compounding period is the same as the number of periods we use to calculate the EV Here we are nding the annuity payment necessary to achieve the same FV The interest rate given is a 12 percent APR with monthly deposits We must make sure to use the number of months in the equation So using the EVA equation Starting today FVA C1121248 711212 C 1000000 1176477 8500 Starting in 10 years FVA C1121236 711212 C 1000000 349496 28613 Starting in 20 years FVA C11212 7 11212 C 1000000 989255 101086 Notice that a deposit for half the length of time ie 20 years versus 40 years does not mean that the annuity payment is doubled In this example by reducing the savings period by onehalf the deposit necessary to achieve the same ending value is about twelve times as large Since we are looking to quadruple our money the PV and FV are irrelevant as long as the FV is three times as large as the PV The number of periods is four the number of quarters per year So FV 3 11 no r 3161 or 3161 36 CHAPTER 6 B79 Since we have an APR compounded monthly and an annual payment we must first convert the interest rate to an EAR so that the compounding period is the same as the cash ows EAR 1 10 1212 71104713 or 104713 PVA1 95000 171 11047132 104713 16383909 PVAZ 45000 7000017111047132 104713 16572354 You would choose the second option since it has a higher PV We can use the present value of a growing perpetuity equation to nd the value of your deposits today Doing so we nd PV C 1rrg 1rrg X 1 t g1 r PV 1000000108 705 7108 705 x 1 051 08 PV 1901656318 Since your salary grows at 4 percent per year your salary next year will be Next year s salary 50000 1 04 Next year s salary 52000 This means your deposit next year will be Next year s deposit 5200005 Next year s deposit 2600 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 nd the value of your deposits today Doing so we nd PV C 1rrg 1rrg X 1 t g1 t r PV 26001117047111704 x 104111quot PV 3439945 Now we can nd the future value of this lump sum in 40 years We nd FV PV1 r FV 343664511140 FV 223599431 This is the value of your savings in 40 years B80 SOLUTIONS 39 The relationship between the PVA and the interest rate is 40 4 H 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 9000 per year for 10 years at the various interest rates given are PVA10 900017111015 10 6845472 PVA5 900017110515 05 9341692 PVA15 900017111515 15 5262633 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 20000 3401 0612 71 0612 Solving for t we get 1005 1 200003400612 1 ln 1294118 ln 1005 5169 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 73000 14501 7 1 1 r6 r To find 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 lowers the PVA and decreasing the interest rate increases the PVA Using a spreadsheet we nd r 0594 The APR is the periodic interest rate times the number of periods in the year so APR 120594 713 42 A A CHAPTER 6 B81 The amount of principal paid on the loan is the PV of the monthly payments you make So the present value of the 1150 monthly payments is PVA 11501 7 1 1 06351236 063512 18481742 The monthly payments of 1150 will amount to a principal payment of 18481742 The amount of principal you will still owe is 240000 718481742 5518258 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 55182581 063512360 36893654 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 1700 110 154545 PV onear 3 CF 2100 1103 157776 PV onear 4 CF 2800 1104 191244 So the PV ofthe missing CF is 6550 7154545 7157776 7191244 151435 The question asks for the value of the cash ow in Year 2 so we must find the future value of this amount The value of the missing CF is 151435110Z 7 183236 To solve this problem we simply need to find the PV of each lump sum and add them together It is important to note that the first cash ow of 1 million occurs today so we do not need to discount that cash ow The PV of the lottery winnings is PV 1000000 1500000109 20000001092 25000001093 3000000109 35000001095 40000001096 45000001097 50000001098 55000001099 60000001091 PV 7 2281287340 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 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 0802900000 2320000 B82 SOLUTIONS 4 4 4 Ch l on Using the PVA equation PVA 2320000 150001 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 financial calculator using a spreadsheet or by trial and error Ifyou 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 0560 The APR is the monthly interest rate times the number of months in the year so APR 120560 672 And the EAR is EAR 1 0056012 71 0693 or 693 The pro t the rm 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 1650001134 10119759 And the film s profit is Pro t 10119759 7 9400000 719759 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 94000 7 165000 1 r4 r 7 165000 94000 7 1 7 1510 or 1510 We want to nd the value of the cash ows today so we will nd the PV of the annuity and then bring the lump sum PV back to today The annuity has 18 payments so the PV of the annuity is PVA 7 400017111018 10 7 3280565 Since this is an ordinary annuity equation this is the PV one period before the rst payment so it is the PV at t 7 To nd the value today we nd the PV of this lump sum The value today is PV 3280565 1107 1683448 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 rst The PV of these cash ows is PVA2 7 1500 1 7 1 1 071296 0712 7 11002135 49 5 5 G H CHAPTER 6 B83 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 11002135 1 111284 5112033 Now we need to nd the PV of the annuity for the rst seven years The value of these cash ows today is PVAI 150017111112841112 8760436 The value of the cash ows today is the sum of these two cash ows so PV 5112033 8760436 13872468 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 1200 1 08512180 7108512 43414362 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 43414362 New PV 43414362120 13076155 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 3500 062 5645161 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 5645161 10627 3705141 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 25000 2416671 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 nd r 2361 per month B84 SOLUTIONS 5 N So the APR is APR 122361 2833 And the EAR is EAR 10236112 71 3231 or 3231 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 10 l2 00833 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 1008336 7 1 0511 or 511 We can now use this rate to nd the PV of the annuity The PV of the annuity is PVA year 8 7000171105111 0511 5377672 Note this is the value one period six months before the first payment so it is the value at year 8 So the value at the various times the questions asked for uses this value 8 years from now PV year 5 5377672 105116 3988833 Note 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 EAR 1 008312 71 1047 or 1047 So we can nd the PV at year 5 using the following method as well PV year 5 5377672 110473 3988833 The value of the annuity at the other times in the problem is PV year 3 5377672 1051110 7 3268488 PV year 3 5377672 110475 7 3268488 PV year 0 5377672 1051116 7 2424367 PV year 0 5377672 110478 7 2424367 a If the payments are in the form of an ordinary annuity the present value will be PVA C1711 r r PVA7 100001711 11511 PVA7 3695897 CHAPTER 6 B85 If the payments are an annuity due the present value will be PVAdue 1 r PVA PVAdue 1 113695897 PVAdue 4102446 b We can nd the future value of the ordinary annuity as FVAC1r 7 lr FVA 100001115 7 1111 FVA 7 6227801 Ifthe payments are an annuity due the future value will be FVAdue 1 r FVA FVAdue 1 116227801 FVAdue 6912860 0 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 54 We need to use the PVA due equation that is 5 UI PVAdue 1 r PVA Using this equation PVAdue 68000 1 078512 X C17110785126 078512 6755806 C1 7 1 1 0785126 078512 C 136499 Notice when we nd the payment for the PVA due we simply discount the PV of the annuity due back one period We then use this value as the PV of an ordinary annuity The payment for a loan repaid with equal payments is the annuity payment with the loan value as the PV of the annuity So the loan payment will be PVA 42000 C 1 7 1 1 085 08 C 1051917 The interest payment is the beginning balance times the interest rate for the period and the principal payment is the total payment minus the interest payment The ending balance is the beginning balance minus the principal payment The ending balance for a period is the beginning balance for the next period The amortization table for an equal payment is B86 SOLUTIONS 56 5 l Beginning Total Interest Principal Ending M Balance Payment Payment Payment Balance 1 4200000 1051917 336000 715917 3484083 2 3484083 1051917 278727 773190 2710892 3 2710892 1051917 216871 835046 1875847 4 1875847 1051917 150068 901849 973997 5 973997 1051917 77920 973997 000 In the third year 216871 of interest is paid Total interest over life ofthe loan 3360 278727 216871 150068 77920 Total interest over life of the loan 1059586 This amortization table calls for equal principal payments of 8400 per year The interest payment is the beginning balance times the interest rate for the period and the total payment is the principal payment plus the interest payment The ending balance for a period is the beginning balance for the next period The amortization table for an equal principal reduction is Beginning Total Interest Principal Ending Year Balance Payment Payment Payment Balance 1 4200000 1176000 336000 840000 3360000 2 3360000 1108800 268800 840000 2520000 3 2520000 1041600 201600 840000 1680000 4 1680000 974400 134400 840000 840000 5 840000 907200 67200 840000 000 In the third year 2016 of interest is paid Total interest over life ofthe loan 3360 2688 2016 1344 672 10080 Notice that the total payments for the equal principal reduction loan are lower This is because more principal is repaid early in the loan which reduces the total interest expense over the life of the loan Challenge 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 find the APR based on monthly compounding and then diVide by 12 So the preretirement APR is EAR 10 1 APR 1212 71 APR 12110112 7 1 0957 or 957 And the postretirement APR is EAR 07 1 APR 1212 71 APR 12107 12 71 0678 or 678 CHAPTER 6 B87 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 0678121225067812 288549645 PV 7 900000 1 067812300 7 16582426 So at retirement he needs 288549645 16582426 305132071 He will be saving 2500 per month for the next 10 years until he purchases the cabin The value of his savings after 10 years will be FVA 2500 1 095712 1 7 1 095712 49965964 After he purchases the cabin the amount he will have left is 49965964 7 380000 11965964 He still has 20 years until retirement When he is ready to retire this amount will have grown to Fv 7 119659641095712122 80501023 So when he is ready to retire based on his current savings he will be short 3051320717 80501023 224631048 This amount is the FV of the monthly savings he must make between years 10 and 30 So nding the annuity payment using the FVA equation we find his monthly savings will need to be FVA 7 224631048 7 C 1 104812 2 7 1 104812 C 7 312744 To answer this question we should nd 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 99 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 99 4501 7 1 1 0712 3 0712 1467291 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 23000 1 0712 7 1865482 The PV of the decision to purchase is 32000 71865482 1334518 B88 SOLUTIONS 5 D In this case it is cheaper to buy the car than leasing it since the PV of the purchase 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 32000 7 PV of resale price 1467291 PV of resale price 1732709 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 17327091 0712 7 2136301 To nd the quarterly salary for the player we rst need to nd 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 055365365 71 565 The PV of the current contract offer is the sum of the PV of the cash ows So the PV is PV 7000000 450000010565 5000000105652 6000000105653 6800000105654 7900000105655 8800000105656 PV 3861048257 The player wants the contract increased in value by 1400000 so the PV of the new contract will be PV 3861048257 1400000 4001048257 The player has also requested a signing bonus payable today in the amount of 9 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 4001048257 7 9000000 3101048257 To find the quarterly payments rst realize that the interest rate we need is the effective quarterly rate Using the daily interest rate we can find 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 05536591 25 7 1 01384 or 1384 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 3101048257 7 C171101384 01384 C 7 152746376 CHAPTER 6 B89 60 To nd the APR and EAR we need to use the actual cash ows of the loan In other words the interest rate 6 H quoted in the problem is only relevant to determine the total interest under the terms given The cash ows of the loan are the 25000 you must repay in one year and the 21250 you borrow today The interest rate of the loan is 25000 212501 r r 25000 21250 7 1 1765 or 1765 Because of the discount you only get the use of 21250 and the interest you pay on that amount is 1765 not 15 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 12108 12 71 0772 or 772 To nd the value today of the back pay from two years ago we will nd the EV of the annuity and then nd the FV of the lump sum Doing so gives us FVA 4700012 1 07721212 7 1 077212 4869939 Fv 4869939108 5259534 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 5000012 1 07721212 71077212 5180786 Next we nd the value today of the five year s future salary PVA 5500012171 1 077212 lt5077212 22753914 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 5259534 5180786 22753914 7 100000 20000 45194234 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 B90 SOLUTIONS 62 Again to find the interest rate of a loan we need to look at the cash ows of the loan Since this loan is in 6 A the form of a lump sum the amount you will repay is the FV of the principal amount which will be Loan repayment amount 10000108 10800 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 10800 7 97001 r r 10800 9700 7 1134 or 1134 This is the same question as before with different values So Loan repayment amount 10000111 11100 Amount received 100001 7 02 9800 11100 7 98001 r r 7 11100 9800 7 1 7 1327 or 1327 The effective rate is not affected by the loan amount since it drops out when solving for r 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 2300 application fee you will need to borrow 242300 to have 240000 after deducting the fee Solving for the payment under these circumstances we get PVA 7 242300 7 C 17 1100566736 005667 where 005667 7 06812 C 7 157961 We can now use this amount in the PVA equation with the original amount we wished to borrow 240000 Solving for r we nd PVA 240000 1579611711 r36 r Solving for r with a spreadsheet on a nancial calculator or by trial and error gives r 05745 per month APR 1205745 689 EAR 7 1 00574512 717 712 6 UI 66 CHAPTER 6 B91 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 1 0681212 71 702 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 4115 per month for three years so the interest rate on the loan is PVA 1000 411517 1 1 r36 r Solving for r with a spreadsheet on a nancial calculator or by trial and error gives r 230 per month APR 12230 2761 EAR 1 023012 71 3139 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 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 7 1050001711072 07 7 111237150 This amount is the same for all three parts of this question Q 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 111237150 7 C10730 7 1 07 C7 1177601 9 Here we need to nd a lump sum savings amount Using the FV for a lump sum equation we get FV 7 111237150 7 PV1073 PV 7 14612904 B92 SOLUTIONS c 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 of trust fund deposit 15000010710 29507270 So the amount your friend still needs at retirement is FV 111237150 7 29507270 81729880 Using the FVA equation and solving for the payment we get 81729880 C10730 7107 C 865225 This is the total annual contribution but your friend s employer will contribute 1500 per year so your friend must contribute Friend39s contribution 865225 7 1500 715225 67 We will calculate the number of periods necessary to repay the balance with no fee first We simply need to use the PVA equation and solve for the number of payments Without fee and annual rate 1980 PVA 10000 20017 110165 0165 where 0165 19812 Solving for t we get 110165 17 100002000165 110165 175 t In 1175 ln 10165 1 10650 months Without fee and annual rate 620 PVA 10000 2001711005167 005167 where 005167 06212 Solving for t we get 11005167 17 10000200005 167 11005167 7417 t In 17417 1n 1005167 1 5799 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 CHAPTER 6 B93 With fee and annual rate 620 PVA 10200 200 1711005167 005167 where 005167 08212 Solving for t we get 11005167 1710200200005167 11005167 7365 t In 17365 ln 1005167 1 5935 months 68 We need to nd the FV of the premiums to compare with the cash payment promised at age 65 We have to nd the value of the premiums at year 6 rst since the interest rate changes at that time So Fv1 9001125 158611 sz 900112 141617 FV3 10001123 140493 FV4 10001122 125440 FV5 11001121 123200 Value at year six 158611 141617 140493 125440 123200 1100 Value at year six 799360 Finding the FV of this lump sum at the child s 65 h birthday FV 79936010859 74945256 The policy is not worth buying the future value of the deposits is 74945256 but the policy contract will pay off 500000 The premiums are worth 24945256 more than the policy payoff Note we could also compare the PV of the two cash ows The PV of the premiums is PV 900112 9001122 10001123 1000112quot 11001125 11001126 PV 404981 And the value today ofthe 500000 at age 65 is PV 50000010859 533296 PV 5332961126 270184 The premiums still have the higher cash ow At time zero the difference is 134797 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 134797 the difference in the cash ows at time zero for six years at a 12 percent interest rate and then for 59 years at an 8 percent interest rate How much will it B94 SOLUTIONS 6 D be worth Without doing calculations you know it will be worth 24945256 the difference in the cash ows at time 65 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 7 750000 7 C171 1 0811236 08112 C7 555561 Now at time 8 we need to nd the PV of the payments which have not been made The balloon payment will be PVA 555561171108112 m08112 PVA 7 68370032 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 rquot r And the FV of the savings is FVA 90001 r6 7 1 r Setting these two equations equal to each other we get 200001 7 1 1 r4 r 9000 1 r67 1 r Reducing the equation gives us 1 r6 7110001 r4 29000 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 nd r 807 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 280001711 r20 rl N 43 CHAPTER 6 B95 Setting them equal and solving for r we get 20000 r 28000 1 e 1 1 r20 r 20000 28000 1e 1 1 r20 7143 20 1 1 r r 0646 or 646 The cash ows in this problem occur every two years so we need to find 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 103653652 7 1 2214 or 2214 We can use this interest rate to find the PV of the perpetuity Doing so we nd PV 15000 2214 6776007 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 flows are two years apart we have found the value of the perpetuity one period two years before the first payment which is one year ago We need to compound this value for one year to nd the value today The value of the cash ows today is PV 67760071 10365365 7488544 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 nd the value of the perpetuity two years from today So the value of these cash ows today is PV 6776007 l 2214 5547878 To solve for the PVA due A L F C L 1r 1 r2 1r PVAdue C7ML lr 1rquot1 PVAdue 1r L l C 2 C t 1r 1 r 1r PVAdue 1 r PVA And the FVA due is FVACC1 rC1 r2 C1rH FVAdue C1rC1r2 C1rt FVAdue 1 rC C1 r C1 0H FVAdue 1rFVA B96 SOLUTIONS 74 We need to nd the lump sum payment into the retirement account The present value of the desired amount at retirement is PV FVl r PV 20000001 11quot0 PV 3076882 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 C11g1 r l reg 3076882 C1 71 031 7 11quot0 117 03 C 259156 This is the amount you need to save next year So the percentage of your salary is Percentage of salary 259156 40000 Percentage of salary 0648 or 648 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 a The APR is the interest rate per week times 52 weeks in a year so APR 527 364 EAR 1 0752 71 327253 or 327353 0 In a discount loan the amount you receive is lowered by the discount and you repay the full principal With a 7 percent discount you would receive 930 for every 10 in principal so the weekly interest rate would be 10 9301 r r 10 930 7 1 0753 or 753 Note the dollar amount we use is irrelevant In other words we could use 093 and 1 93 and 100 or any other combination and we would get the same interest rate Now we can nd the APR and the EAR APR 52753 39140 EAR 1 7 075352 7 1 425398 or 425398 CHAPTER 6 B97 c Using the cash ows from the loan we have the PVA and the annuity payments and need to find the interest rate so PVA 6892 251 7 1 1 rquot r Using a spreadsheet trial and error or a nancial calculator we nd r 1675 per week APR 521675 87099 EAR 1167552 7 1 31417472 or 31417472 76 To answer this we need to 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 and a perpetuity of CzR C3R C4R The present value of this perpetuity is PV CR R CR2 So the present value equation of a perpetuity that increases by C each period is PV CR CR2 B98 SOLUTIONS 77 l 9 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 as FV PV1R39 2 11Rt Solving for t we nd 1112 tln 1 R t 1112 1111 R Since R is expressed as a percentage in this case we can write the expression as t ln2 ln1 R100 To simplify the equation we can make use of a Taylor Series expansion 1111 R R7R22 R33 7 Since R is small we can truncate the series after the first term ln1 R R Combine this with the solution for the doubling expression t 1112 R100 t 1001112 R t 693147 R This is the exact approximate expression Since 693147 is not easily divisible and we are only concerned with an approximation 72 is substituted 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 18Rt 2 8R1 Rt 1112 Rt 693147 t 693147 R Since we are using interest rates while the equation uses decimal form to make the equation correct with percentages we can multiply by 100 t 693147R Calculator Solutions 1 2 Enter Solve for Enter Solve for Enter Solve for Enter Solve for 3 Enter Solve for Enter Solve for Enter Solve for 330637 9 6 9 5 3 2 1 279422 5 4264693 5 PV 40605 54 15 PV 28629 50 15 PV 30275 86 8 940 8 1090 8 1340 248988 6000 8000 6000 8000 FV 118413 127138 144720 1405 530771 FV39 118413 FV39 127138 144720 CHAPTER 6 B99 B100 SOLUTIONS Enter Solve for Enter Solve for Enter Solve for 3 2 1 FV 128557 134299 148740 1405 552096 Enter Solve for Enter Solve for Enter Solve for 2 1 11 940 11 1090 11 1340 24 940 24 1090 24 1340 FV 179223 167598 166160 1405 653481 4 Enter Solve for Enter Solve for Enter Solve for 15 40 75 7 7 7 PV 4827194 7065806 PV 7524070 5300 5300 5300 128557 FV39 134299 148740 FV39 179223 167598 FV39 166160 5 Enter Solve for 6 Enter Solve for 7 Enter Solve for Enter Solve for 8 Enter Solve for 9 Enter Solve for 12 Enter Solve for Enter Solve for Enter Solve for 13 Enter Solve for 15 8 20 7 8 16 12 824 765 85 112 112 68 75 824 1723 1275 86 34000 41166036 70000 4 12 365 2 PMT 388772 73000 4000 4000 657577 PMT 944002 CHAPTER 6 B 101 FV FV 26278116 245907263 90000 B102 SOLUTIONS Enter Solve for Enter Solve for 14 Enter Solve for Enter Solve for 15 Enter Solve for 16 Enter Solve for 17 Enter Solve for Enter Solve for Enter Solve for 18 Enter Solve for 1820 899 142 145 1485 17 X 2 5x365 N 10 x 365 20 x 365 78365 N 198 940 1516 EFF 1503 16 84 93 365 93 365 93 365 10 365 12 52 12 2 365 2100 4500 4500 4500 2880471 850593 716364 1140394 2889997 58000 19 Enter Solve for 20 Enter Solve for Enter Solve for 2 1 Enter Solve for 22 Enter Solve for 23 Enter Solve for 24 Enter Solve for 25 Enter Solve for Enter Solve for 26 Enter Solve for 360 60 69 N 4886 173333 2274 30x12 N 1000 30 EFF 2229 8 1 69 12 712 13 EFF 31391651569 2526 10 12 1047 1047 12 68500 12 18000 7 52 12 PV 12 3484371 135315 i500 300 3600 2300 CHAPTER 6 B 103 FV 67814638 64762345 B104 SOLUTIONS 27 Enter 1100 NOM Solve for 1146 F04 I 1146 NPV CPT 232036 28 I 845 NPV CPT 657086 30 Enter 17 NOM Solve for 1633 1633 2 817 Enter 17 NOM Solve for 1601 1601 4 400 Enter 17 NOM Solve for 1580 158012 132 2 4 12 31 Enter 6 150 12 Solve for Enter 6 18 12 Solve for 550835 7 5000 50835 32 Stock account Enter 360 11 12 N Solve for Bond account Enter 360 6 12 Solve for CHAPTER 6 B 105 5000 503762 503762 550835 700 PV 196316382 300 Savings at retirement 196316382 30135451 226451833 Enter 300 9 12 N Solve for 33 Enter 12 117 Solve for Enter 24 117 Solve for 34 Enter 480 12 12 N IY Solve for Enter 360 12 12 Solve for 30135451 226451833 PMT FV39 1900376 1 115 1 132 1000000 8500 1000000 28613 B106 SOLUTIONS Enter Solve for 35 Enter Solve for 36 Enter Solve for Enter Solve for 39 Enter Solve for Enter Solve for Enter Solve for 40 Enter Solve for 240 N 123 1000 2 CFO C01 F01 I 1047 NPV CPT 16572394 15 5169 12 12 IY 3161 1047 1047 45000 75000 2 15 6 12 PV i1 12 PV 16383909 6845472 9342692 FY 5262633 PMT 101086 95000 9000 9000 9000 ir3 40 1000000 3 20000 41 Enter 60 Solve for 0594 0594 x 12 713 42 Enter 360 635 12 Solve for 240000 718481742 5518258 Enter 360 635 12 N Solve for 43 503565 PV ofmjssing CF 6550 7 503565 151435 Value of missing CF Enter 2 10 Solve for 73000 PV 18481742 5518258 151435 11450 PMT 1150 CHAPTER 6 B 107 FV 36893654 183236 B108 SOLUTIONS 44 1000000 1500000 1 2500000 1 2800000 1 3000000 1 3500000 1 4000000 1 4500000 5000000 5500000 1 6000000 I 9 NPV CPT 22812873 45 Enter 360 802900000 i15000 PMT Solve for 0560 APR 0560 x 12 672 Enter 672 12 NOM Solve for 693 46 Enter 4 13 PMT Solve for 10119759 Pro t 10119759 7 94000 719759 Enter 4 ir94000 PMT Solve for 1510 165000 165000 47 Enter 18 10 4000 PMT Solve for 3280565 Enter 7 10 3280565 PV39 PMT Solve for 1683448 48 Enter 84 7 12 1500 quot PMT Solve for 8760436 Enter 96 11 12 1500 N PMT Solve for 11002135 Enter 84 7 12 11002135 PV39 PMT FV39 Solve for 5112033 8760436 5112033 13872468 49 Enter 15 X 12 8512 1200 N IY PMT FV39 Solve for 43414362 FV 43414362 PV 80815PV 43414362120 13076155 50 PV t 14 3500 0062 5645161 Enter 7 62 5645161 PV39 PMT Solve for 3705141 51 Enter 12 25000 r241667 IY PMT F V Solve for 2361 APR 2361 x 12 2833 Enter 2833 12 NOM Solve for 3231 CHAPTER 6 B 109 B110 SOLUTIONS 52 Monthly rate 10 12 0083 semiannual rate 100836 71 511 Enter 10 Solve for Enter 6 Solve for Enter 10 Solve for Enter 16 Solve for 53 a Enter 5 Solve for 2quot 1 BGN 2quot 1 SET Enter 5 Solve for b Enter 5 Solve for 2quot 1 BGN 2quot 1 SET Enter 5 Solve for 511 511 511 511 IY 11 11 11 11 5377672 FY 3988833 326 84 88 2424367 3695897 4102446 PV 7000 i10000 i10000 PMT i10000 i10000 PMT 5377672 FY 5377672 5377672 6227801 FY 6912860 CHAPTER 6 B 111 54 2 BGN 2quot 1 SET Enter 60 785 12 68000 N PMT FV Solve for 136499 57 Preretirement APR Enter 10 12 NOM Solve for 957 Postretirement APR Enter 7 12 NOM Solve for 678 At retirement he needs Enter 300 678 12 20000 900000 N PMT FV Solve for 305132071 In 10 years his savings will be worth Enter 120 772 12 2500 39 PMT Solve for 49965964 After purchasing the cabin he will have 49965964 7 380000 11965964 Each month between years 10 and 30 he needs to save Enter 240 957 12 11965964 305132071i N Solve for 312744 58 PV of purchase Enter 6 7 12 23000 Solve for 1865482 32000 71865482 1334518 B112 SOLUTIONS PV of lease Enter 36 7 12 450 FV Solve for 1457399 1457391 99 1467291 Buy the car You would be indifferent when the PV of the two cash ows are equal The present value of the purchase decision must be 1467291 Since the difference in the two cash ows is 32000 7 1467291 1732709 this must be the present value of the future resale price of the car The break even resale price of the car is Enter 36 7 12 1732709 PMT Solve for 2136301 59 Enter 550 365 NOM Solve for 565 7000000 4500000 1 5000000 1 6000000 1 6800000 1 7900000 1 8800000 1 I 565 NPV CPT 3861048257 New contract value 3861048257 1400000 4001048257 PV of payments 4001048257 7 9000000 3101048257 Effective quarterly rate 1 0553659125 71 01384 or 1384 Enter 24 1384 3101048257 PMT F V Solve for 152746376 IY 1765 8 EFF 772 12 H 8 Y I 772 12 H 772 12 H 21250 PV 4869939 FY PV PV 227539 14 47000 12 PMT 50000 12 PMT 55000 12 PMT CHAPTER 6 B 1 13 i25000 FV 4869939 FY 5259534 FY 5 180786 FV Award 5259534 5180786 22753914 100000 20000 45194234 60 Enter 1 Solve for 61 Enter N OM Solve for 772 Enter 12 Solve for Enter 1 Solve for Enter 12 Solve for Enter 60 Solve for 62 Enter 1 Solve for 63 Enter 1 Solve for IY 1134 IY 1429 9700 9800 i10800 i11200 64 Refundable fee With the 2300 application fee you will need to borrow 242300 to have 240000 after deducting the fee Solve for the payment under these circumstances Enter 30 gtlt 12 N Solve for 680 12 242300 PV39 PMT 157961 FV B114 SOLUTIONS Enter 30 x 12 240000 i157961 PMT FV Solve for 05745 APR 05745 x 12 689 Enter 6 89 12 NOM Solve for 712 Without refundable fee APR 680 Enter 680 12 NOM Solve for 702 65 Enter 36 1000 i4115 PMT FV Solve for 230 APR 230 x 12 2761 Enter 2761 12 NOM Solve for 3139 66 What she needs at age 65 Enter 20 7 105000 PMT FV Solve for 111237150 a Enter 30 7 111237150 PV PMT FV Solve for 1177601 b Enter 30 7 111237150 PMT Solve for 14612904 0 Enter 10 7 150000 PMT Solve for 29507270 CHAPTER 6 B l 15 At 65 she is short 111237150 7 29507250 81729880 Enter 30 7 i81729880 PMT Solve for 865225 Her employer will contribute 1500 per year so she must contribute 865225 71500 715225 per year 67 Without fee Enter 19 8 12 10000 r200 N PMT Solve for 10650 Enter 68 12 10000 i200 Solve for 5799 With fee Enter 68 12 10200 i200 Solve for 5935 68 Value at Year 6 Enter 5 12 900 PMT Solve for 158611 Enter 4 12 900 PMT Solve for 141617 Enter 3 12 1000 PMT FV Solve for 140493 Enter 2 12 1000 Solve for 125440 B116 SOLUTIONS Enter 1 12 1100 Solve for 1232 So at Year 5 the value is 158611 141617 140493 125440 1232 1100 799360 At Year 65 the value is Enter 59 8 799360 PMT Solve for 74945256 The policy is not worth buying the future value of the deposits is 74945256 but the policy contract will pay off 500000 69 Enter 30 x 12 81 12 750000 39 PV PMT FV39 Solve for 555561 Enter 22 x 12 81 12 555561 PMT Solve for 68370032 70 9000 9000 20000 75 1 APR 7 x 52 364 Enter 364 52 NOM Solve for 327253 b Enter 1 930 ir1000 Solve for 753 CHAPTER 6 B 1 17 APR 753 x 52 39140 Enter 39140 52 Solve for 425398 0 Enter 4 6892 i25 Solve for 1675 APR 1675 x 52 87099 Enter 87099 52 Solve for 31417472 CHAPTER 7 INTEREST RATES AND BOND VALUATION Answers to Concepts Review and Critical Thinking Questions 1 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 Ifthe bid price were higher than the ask price 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 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 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 fixed 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 provide the needed nominal dollar amounts Pension funds for example often must plan for pension payments many years in the future If those payments are xed 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 difficult 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 p n 43 CHAPTER 7 B 1 19 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 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 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 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 yield to maturity is the required rate of return on a bond expressed as a nominal annual interest rate For noncallable bonds the yield to maturity and required rate of return are interchangeable terms Unlike YTM and required return the coupon rate is not a return used as the interest rate in bond cash ow valuation but is a xed percentage of par over the life of the bond used to set the coupon payment amount For the example given the coupon rate on the bond is still 10 percent and the YTM is 8 percent Price and yield move in opposite directions if interest rates rise the price of the bond will fall This is because the xed coupon payments determined by the xed coupon rate are not as valuable when interest rates riseihence the price of the bond decreases B120 SOLUTIONS 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 1 000 We will use this par value in all problems unless a ali erent par value is explicitly stated 3 The price of any bond is the PV of the interest payment plus the PV of the par value Notice this problem assumes an annual coupon The price of the bond will be P 751 7 11 087510 0875 10001 1 08751 91889 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 1 1 r which stands for Eresent yalue lnterest Eactor PVIFAR1711 rt r which stands for Eresent yalue Interest 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 remainder of the solutions key Here we need to nd the YTM of a bond The equation for the bond price is P 934 90PV1FAR79 1000PV1FR79 Notice the equation cannot be solved directly for R Using a spreadsheet a financial calculator or trial and error we find R YTM 1015 If you are using trial and error to nd the YTM of the bond you might be wondering how to pick an interest rate to start the process First we know the YTM has to be higher than the coupon rate since the bond is a discount bond That still leaves a lot of interest rates to check One way to get a starting point is to use the following equation which will give you an approximation of the YTM Approximate YTM Annual interest payment Price difference from par Years to maturity Price Par value 2 Solving for this problem we get Approximate YTM 90 64 9 934 1000 2 1004 This is not the exact YTM but it is close and it will give you a place to start CHAPTER 7 B 121 Here we need to nd the coupon rate of the bond All we need to do is to set up the bond pricing equation and solve for the coupon payment as follows P 1045 CPVIFA7 5 1000PVIF75735 Solving for the coupon payment we get C 8054 The coupon payment is the coupon rate times par value Using this relationship we get Coupon rate 8054 1000 0805 or 805 To nd the price of this bond we need to realize that the maturity of the bond is 10 years The bond was issued one year ago with 11 years to maturity so there are 10 years left on the bond Also the coupons are semiannual so we need to use the semiannual interest rate and the number of semiannual periods The price of the bond is P 3450PVIFA3 mm 1000PV1F3 720 96510 Here we are finding the YTM of a semiannual coupon bond The bond price equation is P 1050 42PVIFAR20 1000PVIFR720 Since we cannot solve the equation directly for R using a spreadsheet a nancial calculator or trial and error we nd R 3837 Since the coupon payments are semiannual this is the semiannual interest rate The YTM is the APR of the bond so YTM 2 X 3837 767 Here we need to nd the coupon rate of the bond All we need to do is to set up the bond pricing equation and solve for the coupon payment as follows P 924 CPVIFA3 4mg 1000PVIF3 4mg Solving for the coupon payment we get C 29 84 Since this is the semiannual payment the annual coupon payment is 2 X 2984 5968 And the coupon rate is the annual coupon payment divided by par value so Coupon rate 5968 1000 Coupon rate 0597 or 597 B122 SOLUTIONS H H p A N p A DJ The approximate relationship between nominal interest rates R real interest rates r and in ation h is R r h Approximate r 07 7 038 032 or 320 The Fisher equation which shows the exact relationship between nominal interest rates real interest rates and in ation is 1R1r1h 1 07 7 1 rl 038 Exactr 7 1 07 1 038 7 1 7 0308 or 308 The Fisher equation which shows the exact relationship between nominal interest rates real interest rates and in atlon 1s 1 R1 r1h R 7 1 0471 03 7 1 7 0784 or 784 The Fisher equation which shows the exact relationship between nominal interest rates real interest rates and in atlon 1s 1 R 1r1h h 7 1 14 1 09 7 1 7 0459 or 459 The Fisher equation which shows the exact relationship between nominal interest rates real interest rates and in ation is 1 13 1 t r1 h r 1 114 1048 7 1 7 0630 or 630 This is a bond since the maturity is greater than 10 years The coupon rate located in the first column of the quote is 6125 The bid price is Bid price 12007 120 732 12021875 gtlt 1000 12021875 The previous day s ask price is found by Previous day s asked price Today s asked price 7 Change 120 832 7 532 120 332 The previous day s price in dollars was Previous day s dollar price l20406 gtlt 1000 120406 14 CHAPTER 7 B 123 This is a premium bond because it sells for more than 100 of face value The current yield is Current yield Annual coupon payment Price 75 13515625 5978 The YTM is located under the Asked Yield column so the YTM is 447 The bidask spread is the difference between the bid price and the ask price so BidAsk spread 13506 713505 132 Intermediate Here we are nding the YTM of semiannual coupon bonds for various maturity lengths The bond price equation is P aPVIFAmg 1000PVIFR X P0 80Pv1FA613 1000Pv1F613 117705 P1 80Pv1FA6u 1000Pv1F612 116768 P3 80Pv1FA10 1000Pv1F610 114720 P8 80Pv1FA5 1000PVIF675 108425 P12 80PVIFA51 1000Pv1F61 101887 P13 1000 Y P0 60Pv1FA813 1000Pv1F813 84192 P1 60PVIFA8712 1000PVIF8712 84928 P3 60PVIFA8710 1000PVIF8710 86580 P8 60PVIFA875 1000PVIF875 92015 P12 60PVIFA871 1000Pv1F81 98148 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 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 B124 SOLUTIONS 16 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 9 percent If the YTM suddenly rises to 11 percent P5 45PVIFA5 5 1000PVIF5 56 95004 PDave 45PVIFA5 540 1000PVIF5 540 83954 The percentage change in price is calculated as Percentage change in price New price 7 Original price Original price APSam 95004 7 1000 1000 7 500 APDaveOO 83954 7 1000 1000 71605 If the YTM suddenly falls to 7 percent P5quot 45PVIFA3576 1000PVIF3575 105329 PDave 45PVIFA35740 1000PVIF35740 121355 APSam 105329710001000 533 APDaVeoO 121355 71000 1000 2136 All else the same the longer the maturity of a bond the greater is its price sensitivity to changes in interest rates 17 Initially at a YTM of 8 percent the prices of the two bonds are P 20PVIFA4718 1000PVIF4718 74681 PK 60PVIFA4718 1000PVIF4718 125319 If the YTM rises from 8 percent to 10 percent P 20PVIFA5718 1000PVIF5718 64931 PK 60PVIFA5718 1000PVIF5718 111690 The percentage change in price is calculated as Percentage change in price New price 7 Original price Original price APJ 649317 74681 74681 71306 APK 111690 7125319125319 71088 on 0 CHAPTER 7 B 125 If the YTM declines from 8 percent to 6 percent P 20Pv1FA318 1000Pv1F318 86246 PK 60Pv1FA318 1000Pv1F318 141261 APJ 86246 7 74681 74681 1549 APK 1412617125319125319 1272 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 1068 46PVIFAR718 1000PVIFR718 Using a spreadsheet nancial calculator or trial and error we find R 406 This is the semiannual interest rate so the YTM is YTM 2 x 406 812 The current yield is Current yield Annual coupon payment Price 92 1068 0861 or 861 The effective annual yield is the same as the EAR so using the EAR equation from the previous chapter Effective annual yield 1 004062 7 1 0829 or 829 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 930 40PV1FAR40 1000PVIFR740 Using a spreadsheet nancial calculator or trial and error we find R 4373 This is the semiannual interest rate so the YTM is YTM 2 x 4373 875 B126 SOLUTIONS 20 N p A N N 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 742 X 26 1233 And we calculate the clean price as Clean price Dirty price iAccrued interest 968 7 1233 95567 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 682 X 46 2267 And we calculate the dirty price as Dirty price Clean price Accrued interest 1073 2267 109567 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 0755 80P0 P0 800755 105960 Now that we have the price of the bond the bond price equation is P 105960 801711072 072 10001072 We can solve this equation for t as follows 1059601072 1111111072 7 111111 1000 11111 51511072 21570 1072 I log 21570 log 1072 1106 m 11 years The bond has 11 years to maturity CHAPTER 7 B 127 23 The bond has 14 years to maturity so the bond price equation is P 108960 36PV1FAR728 1000PVIFR728 Using a spreadsheet nancial calculator or trial and error we nd R 3116 This is the semiannual interest rate so the YTM is YTM 2 x 3116 623 The current yield is the annual coupon payment divided by the bond price so Current yield 572 108960 0661 or 661 24 a b 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 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 insuf cient 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 Current yield is de ned 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 25 The price of a zero coupon bond is the PV of the par so a b P0 1000104550 11071 In one year the bond will have 24 years to maturity so the price will be P1 10001045 8 12090 B128 SOLUTIONS 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 Previous IRS regulations required a straightline calculation of interest The total interest received by the bondholder is Total interest 1000 711071 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 The company will prefer straightline methods when allowed because the valuable interest deductions occur earlier in the life of the bond 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 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 300001040 32400000 The repayment of the zero coupon bond will be the par value times the number of bonds issued so Zeroes repayment 3155891000 315588822 27 CHAPTER 7 B 129 c The total coupon payment for the coupon bonds will be the number bonds times the coupon payment For the cash ow of the coupon bonds we need to account for the tax deductibility of the interest payments To do this we will multiply the total coupon payment times one minus the tax rate So Coupon bonds 3000080 1735 1560000 cash out ow Note that this is cash out ow since the company is making the interest payment For the zero coupon bonds the rst year interest payment is the difference in the price of the zero at the end of the year and the beginning of the year The price of the zeroes in one year will be P1 100010458 10282 The year 1 interest deduction per bond will be this price minus the price at the beginning of the year which we found in part b so Year 1 interest deduction per bond 10282 7 9506 776 The total cash ow for the zeroes will be the interest deduction for the year times the number of zeroes sold times the tax rate The cash ow for the zeroes in year 1 will be Cash ows for zeroes in Year 1 31558977635 85680000 Notice the cash ow for the zeroes is a cash in ow This is because of the tax deductibility of the imputed interest expense That is the company gets to write off the interest expense for the year even though the company did not have a cash ow for the interest expense This reduces the company s tax liability which is a cash in ow During the life of the bond the zero generates cash in ows to the rm in the form of the interest tax shield of debt We should note an important point here If you nd the PV of the cash ows from the coupon bond and the zero coupon bond they will be the same This is because of the much larger repayment amount for the zeroes We found the maturity of a bond in Problem 22 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 22 the number of periods can be any positive number We first need to find the real interest rate on the savings Using the Fisher equation the real interest rate is 1R1r1h 1 111 rl 038 r0694 or 694 B130 SOLUTIONS N 0 Now we can use the future value of an annuity equation to nd the annual deposit Doing so we nd FVA C1 at 1 r 1500000 C10694quot 7 1 0694 C 7 763776 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 120PVIFA775 1000PVIF775 111669 P1 120PVIFA774 1000PVIF74 109719 Current yield l20 111669 1075 or 1075 The capital gains yield is Capital gains yield New price 7 Original price Original price Capital gains yield 109719 7 111169 111669 70175 or 7175 The current price of Bond D and the price of Bond D in one year is D P0 60PVIFA775 1000PVIF775 88331 P1 60PVIFA774 1000PVIF734 90281 Current yield 60 88381 0679 or 679 Capital gains yield 902817 88331 88331 0221 or 221 All else held constant premium bonds pay high current income while haVing price depreciation as maturity nears discount bonds do not pay high current income but have price appreciation as maturity nears For either bond the total return is still 9 but this return is distributed differently between current income and capital gains 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 1060 70PVIFAR710 1000PVIF Mylo Using a spreadsheet nancial calculator or trial and error we find R YTM 618 H N CHAPTER 7 B 131 b To find 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 70PVIFA5 188 1000PVIF5 1mg 111692 To calculate the HPY we need to nd the interest rate that equates the price we paid for the bond with the cash ows we received The cash ows we received were 70 each year for two years and the price of the bond when we sold it The equation to nd our HPY is P0 1060 70PVIFAR2 111692PVIFR72 Solving for R we get R HPY 917 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 1100PV1FA3516PVIF3512 f 1400PV1FA3 512PVIF3528 f 20000PV1F3 540 PM 1901878 Notice that for the coupon payments of 1400 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 20000Pv113540 505145 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 rst noncallable bond which means our investment in Bond 3 the other noncallable bond will be 17 X The equation is C2 11 X C317X 825 650X1217X 825 650X 12 712x X 068181 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 068181P1 031819P3 P2 068181 106375 03181913496875 P2 1154730 B132 SOLUTIONS U U 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 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 Abetter 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 Ifthe YTC were calculated on this bond it would be positive To nd the present value we need to nd the real weekly interest rate To find the real return we need to use the effective annual rates in the Fisher equation So we nd the real EAR is 1R1r1h 10841r1037 r 0453 or 453 Now to nd the weekly interest rate we need to nd the APR Using the equation for discrete compounding EAR 1 APRm39quot 71 We can solve for the APR Doing so we get APR m1 EAR 71 APR 5210453152 71 APR 0443 or 443 So the weekly interest rate is Weekly rate APR 52 Weekly rate 0443 52 Weekly rate 0009 or 009 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 present value of the cost of the roses is PVA C1711 a r PVA 51 7 11 00093 52 0009 PVA 431213 CHAPTER 7 B 133 35 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 1131 r1 h 1 11 1 r1 04 r 0673 or 673 APR m1 EAR 71 APR 121 0673 12 7 1 APR 0653 or 653 Monthly rate APR 12 Monthly rate 0653 12 Monthly rate 0054 or 054 Bond account 1 1 31 r1 h 1 07 1 r1 04 r 0288 or 288 APR m1 EAR 71 APR 121 0288 12 7 1 APR 0285 or 285 Monthly rate APR 12 Monthly rate 0285 12 Monthly rate 0024 or 024 Now we can find the future value of the retirement account in real terms The future value of each account will be Stock account FVA C 1 r 1 r FVA 9001 005436 710054 FVA 100170405 Bond account FVA C 1 r 1 r FVA 4501 00243607 1 0024 FVA 25547517 The total future value of the retirement account will be the sum of the two accounts or Account value 100170405 25547517 Account value 125717922 B134 SOLUTIONS 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 1 R1 r1 11 1 09 1 r1 04 r 0481 or 481 APR m1 EAR 71 APR 121 0481 12 7 1 APR 0470 or 470 Monthly rate APR 12 Monthly rate 0470 12 Monthly rate 0039 or 039 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 a r 125717922 C1711 0039300 0039 C 713482 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 r FV 7134821 043 5 FV 6169029 Calculator Solutions 3 Enter 10 875 75 1000 Solve for 91889 4 Enter 9 i934 90 1000 N IY Solve for 1015 5 Enter 13 75 i1045 1000 Solve for 8054 Coupon rate 8054 1000 805 CHAPTER 7 B 13 5 6 Enter 20 370 3450 1000 Solve for 96510 7 Enter 20 i1050 42 1000 PV Solve for 3837 3837 x 2 767 8 Enter 29 340 i924 1000 Solve for 2984 29 84 10002 597 15 Bond X Po Enter 13 6 80 1000 Solve for 117705 P1 Enter 12 6 80 1000 Solve for 116768 P3 Enter 10 6 80 1000 Solve for 114720 Pa Enter 5 6 80 1000 PV39 Solve for 108425 P12 Enter 6 80 1000 1 Solve for 101887 B136 SOLUTIONS Bond Y Po Enter 13 8 60 1000 Solve for 84192 P1 Enter 12 8 60 1000 Solve for 84928 P3 Enter 10 8 60 1000 Solve for 86580 Pa Enter 5 8 60 1000 Solve for 92015 P12 Enter 1 8 60 1000 Solve for 98148 16 If both bonds sell at par the initial YTM on both bonds is the coupon rate 9 percent If the YTM suddenly rises to 11 percent PSam Enter 6 55 45 1000 PMT Solve for 95004 APSam 95004 71000 1000 7 500 PDave Enter 40 55 45 1000 PMT Solve for 83954 APDM 83954 71000 1000 71605 If the YTM suddenly falls to 7 percent PSam Enter 6 35 45 1000 PMT Solve for 105329 APsm 7 105329 710001000 7 533 CHAPTER 7 B 137 PDave Enter 40 35 45 1000 PMT Solve for 1121355 APDM 121355 710001000 2136 All else the same the longer the maturity of a bond the greater is its price sensitivity to changes in interest rates 17 Initially at a YTM of 8 percent the prices of the two bonds are PI Enter 18 4 20 1000 PMT Solve for 746 81 PK Enter 18 4 60 1000 PMT Solve for 125319 If the YTM rises from 8 percent to 10 percent P1 Enter 18 5 20 1000 PMT Solve for 64931 APJ 649317 74681 74681 71306 PK Enter 18 5 60 1000 PMT Solve for 111690 APK 111690 7 125319 125319 71088 If the YTM declines from 8 percent to 6 percent PI Enter 18 3 20 1000 PMT V Solve for 86246 APJ 86246 7 74681 74681 1549 PK Enter 18 3 60 1000 PMT Solve for 1 41261 APK 1412617125319125319 7 1272 All else the same the lower the coupon rate on a bond the greater is its price sensitivity to changes in interest rates B138 SOLUTIONS 18 Enter 18 i1068 46 1000 Solve for 406 406 x 2 812 Enter 812 2 Solve for 829 19 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 i930 35 1000 PMT Solve for 4373 4373 x 2 875 22 Current yield 0755 90P0 P0 900755 105960 Enter 72 i105960 80 1000 Solve for 1106 1106 or m 11 years 23 Enter 28 i108960 36 1000 Solve for 3116 3116 X 2 623 25 a P0 Enter 50 45 1000 Solve for 11071 b P1 Enter 48 45 1000 Solve for year 1 interest deduction 12090 7 11071 1019 P19 Enter 1 45 1000 Solve for 915 73 year 25 interest deduction 1000 7 91573 8427 CHAPTER 7 B 139 c Total interest 1000 711071 88929 Annual interest deduction 88929 25 3557 d The company will prefer straightline method when allowed because the valuable interest deductions occur earlier in the life of the bond 26 a The coupon bonds have an 8 coupon rate which matches the 8 required return so they will sell at par of bonds 300000001000 30000 For the zeroes Enter 60 4 1000 Solve or 95 06 f 300000009506 7 315589 will be issued b Coupon bonds repayment 300001080 32400000 Zeroes repayment 3155891000 315588822 c Coupon bonds 30000801 735 1560000 cash out ow Zeroes Enter 5 8 4 1000 Solve for 102 82 year 1 interest deduction 10282 7 9506 776 31558977635 85680000 cash in ow During the life of the bond the zero generates cash in ows to the rm in the form of the interest tax shield of debt 29 Bond P Po Enter 5 7 120 1000 Solve for 111669 P1 Enter 4 7 120 1000 Solve for 109719 Current yield 120 111669 1075 Capital gains yield 109719 7111669111669 7175 Bond D Po Enter 7 1000 5 60 Solve for 8 8331 B140 SOLUTIONS P1 Enter 4 7 60 1000 Solve for 902 81 Current yield 60 88331 679 Capital gains yield 902817 88331 88331 221 All else held constant premium bonds pay high current income while haVing price depreciation as maturity nears discount bonds do not pay high current income but have price appreciation as maturity nears For either bond the total return is still 9 but this return is distributed differently between current income and capital gains 30 a Enter 10 i1060 70 1000 F V Solve for 618 This is the rate of return you expect to earn on your investment when you purchase the bond b Enter 8 518 70 1000 Solve for 111692 The HPY is Enter 2 i 1060 70 111692 Solve for 917 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 31 PM I 35 NPV CPT 1901878 PN Enter 35 20000 40 Solve for 505145 CHAPTER 8 STOCK VALUATION Answers to Concepts Review and Critical Thinking Questions 1 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 pro table 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 finite 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 rm 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 fixed 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 Ifthe 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 In a corporate election you can buy votes by buying shares so money can be used to in uence or even determine the outcome Many would argue the same is true in political elections but in principle at least no one has more than one vote 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 Investors buy such stock because they want it recognizing that the shares have no voting power Presumably investors pay a little less for such shares than they would otherwise B142 SOLUTIONS p n p A Presumably the current stock value re ects the risk timing and magnitude of all future cash flows both shortterm and longterm If this is correct then the statement is false If this assumption is violated the twostage dividend growth model is not valid In other words the price calculated will not be correct Depending on the stock it may be more reasonable to assume that the dividends fall from the high growth rate to the low perpetual growth rate over a period of years rather than in one year 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 The constant dividend growth model is P2D2X1gR g So the price of the stock today is P0 D0 1 g Rig 195 106 11 7 06 4134 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 gR7g DO 1 g4 Rig 195 106411706 4924 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 D0 1 g16 R 7g 195 10616 11 7 06 9907 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 P01g3 41341 063 4924 And the stock price in 15 years will be P15 P01 g 413410615 9907 We need to find the required return of the stock Using the constant growth model we can solve the equation for R Doing so we nd R D1 P0 g 210 4800 05 0938 or 938 CHAPTER 8 B 143 The dividend yield is the dividend next year divided by the current price so the dividend yield is Dividend yield D1 P0 210 4800 0438 or 438 The capital gains yield or percentage increase in the stock price is the same as the dividend growth rate so Capital gains yield 5 Using the constant growth model we find the price of the stock today is P0 D1 R 7g 304 11 7 038 4222 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 063 052 1150 or 1150 We know the stock has a required return of 11 percent and the dividend and capital gains yield are equal so Dividend yield 1211 055 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 D105547 259 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 259 D01 055 D0 259 1055 245 The price of any financial instrument is the PV of the future cash ows The future dividends of this stock are an annuity for 11 years so the price of the stock is the PVA which will be P0 975PVIFA10711 6333 The price 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 most preferred stock pays a xed dividend so the growth rate is zero Using this equation we find the price per share of the preferred stock is R DP0 550108 0509 or 509 B144 SOLUTIONS 9 9 H We can use the constant dividend growth model which is P2D2X1gRg So the price of each company s stock today is 235 08 7 05 7833 235 11 7 05 3917 235 147 05 2611 Red stock price Yellow stock price Blue stock price As the required return increases the stock price decreases This is a function of the time value of money A higher discount rate decreases the present value of cash ows It is also important to note that relatively small changes in the required return can have a dramatic impact on the stock price Intermediate This stock has a constant growth rate of dividends but the required return changes twice To nd the value of the stock today we will begin by nding 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 gR7g Do 1 g7R7g 350 105710705 9850 Now we can nd the price of the stock in Year 3 We need to nd the price here since the required r39etum 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 350105quot 112 35010551122 35010561123 98501123 P3 8081 Finally we can nd 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 350105 114 3501052 1142 3501053 1143 8081 1143 PO 6347 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 constant dividend grth formula is P D2 X 1 g Rig 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 rst payment So the price of the stock in Year 9 will be P9 D10 R7g 1000147 05 11111 p A N CHAPTER 8 B 145 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 11111 1149 3417 The price of a stock is the PV of the future dividends This stock is paying four dividends so the price of the stock is the PV of these dividends using the required return The price of the stock is P0 10111 141112 181113 221114 261115 6345 With supemorrnal dividends we find the price of the stock when the dividends level off at a constant growth rate and then find the PV of the future stock price plus the PV of all dividends during the supernormal growth period The stock begins constant growth in Year 4 so we can nd the price of the stock in Year 4 at the beginning of the constant dividend growth as R D4 1 g Rig 200105 127 05 3000 The price of the stock today is the PV of the rst four dividends plus the PV of the Year 3 stock price So the price of the stock today will be P0 1100111 8001112 5001113 2001114 30001114 4009 With supemorrnal 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 futures stock price plus the PV of all dividends during the supernormal 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 Ds 1 tgReg Do 1 g13 1 g2Reg P3 1801303106 13 7 06 P3 5988 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 180130113 1801302 1132 18013031133 59881133 P0 4870 We could also use the twostage dividend growth model for this problem which is Po 1301g1Rg1llt11 g11 RT1 g11tRTDo1g1Rg1l P0 18013013 730171301133130113318010613 7 06 P0 4870 Here we need to find the dividend next year for a stock experiencing supemorrnal 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 1253 B146 SOLUTIONS And the dividend in Year 4 will be the dividend in Year 3 times one plus the growth rate or D D0 1253 115 The stock begins constant growth in Year 4 so we can find 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 gReg Now we can substitute the previous dividend in Year 4 into this equation as follows P4 Do 1 g13 1 g2 1 g3Rg P4 D0 1253 115 108 13 7 08 4852D0 When we solve this equation we nd that the stock price in Year 4 is 4852 times as large as the dividend today Now we need to find 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 D0125113 D0125Z1132 D012531133 D012531151134 4852D01134 We can factor out Do in the equation and combine the last two terms Doing so we get P0 76 D0125113 12521132 12531133 1253115 4852 1134 Reducing the equation even further by solving all of the terms in the braces we get 76 3479D0 D0 76 3479 Do 218 This is the dividend today so the projected dividend for the next year will be D1 218025 D1 273 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 Do 1 gReg P0 10461 7 04 115 7 704 PO 6478 We are given the stock price the dividend growth rate and the required return and are asked to find the dividend Using the constant dividend growth model we get P064D0 1gR7g N 9 0 CHAPTER 8 B 147 Solving this equation for the dividend gives us Do 6410 7 045 1045 D0 337 The price of a share of preferred stock is the dividend payment divided by the required return We know the dividend payment in Year 20 so we can nd the price of the stock in Year 19 one year before the first dividend payment Doing so we get P19 2000 064 P19 31250 The price of the stock today is the PV of the stock price in the future so the price today will be P0 31250 1064 P0 9615 The annual dividend paid to stockholders is 148 and the dividend yield is 21 percent Using the equation for the dividend yield Dividend yield Dividend Stock price We can plug the numbers in and solve for the stock price 021 148 P0 P0 148021 7048 The Net Chg of the stock shows the stock decreased by 023 on this day so the closing stock price yesterday was Yesterday s closing price 7048 023 7071 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 19 Stock price EPS 7048 EPS EPS 7048 19 371 We know that EPS is just the total net income divided by the number of shares outstanding so EPS NT Shares 371 NT 25000000 N1 37125000000 92731830 We can use the twostage dividend growth model for this problem which is Po Do1g1Rg111 g11 RlTgt1 g11RTDo1 g2Rg2 P0 12512813 728171281138 128113812510613 7 06 P0 6955 B148 SOLUTIONS 21 23 We can use the twostage dividend growth model for this problem which is Po Do1g1Rrg1l11 g11 RTt1 g11RTDo1t g2Rg2 P0 174125 12 7 251 712511211 1251121117410612 706 P0 14214 Challenge We are asked to nd the dividend yield and capital gains yield for each of the stocks All of the stocks have a 15 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 D01 g R 7g 450110197 10 5500 Dividend yield DlPo 4501105500 09 or 9 Capital gains yield 19 7 09 10 or 10 P0 D01 g R 7g 45019 70 2368 Dividend yield DlPo 4502368 19 or 19 Capital gains yield 19 719 0 P0 D01 g R 7g 450170519 05 1781 Dividend yield DlPo 4500951781 24 or 24 Capital gains yield 19 7 24 705 or 75 Z P2 D21 g R 7g D01 g121 g2R7g2 4501202112197 12 10368 P0 450 120 119 450 1202 1192 10368 1192 8233 Dividend yield DlPo 45012096 10 066 or 66 Capital gains yield 19 7 066 124 or 124 In all cases the required return is 19 but the return is distributed differently between current income and capital gains High growth 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 1 Using the constant growth model the price of the stock paying annual dividends will be P0 7 D01 g R 7g 7 320106127 06 7 5653 24 CHAPTER 8 B 149 b If the company pays quarterly dividends instead of annual dividends the quarterly dividend will be onefourth of annual dividend or Quarterly dividend 320l06 4 0848 To nd 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 nd 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 11225 7 l 0287 The effective annual dividend will be the EVA of the quarterly dividend payments at the effective quarterly required return In this case the effective annual dividend will be Effective D1 0848FVIFA2 8774 354 Now we can use the constant growth model to nd the current stock price as P0 354127 06 5902 Note that we cannot simply nd the quarterly effective required return and growth rate to find the value of the stock This would assume the dividends increased each quarter not each year Here we have a stock with supemormal growth but the dividend growth changes every year for the first four years We can nd 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 be P3 245120115110105 11 7 05 6508 The price of the stock today will be the PV of the first three dividends plus the PV of the stock price in Year 3 so P0 245120111 2451201151112 2451201151101113 65081113 P0 5570 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 2451201 R 2451201151 R2 2451201151101 R3 245120115110105R 7051 R3 6382 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 1024 B150 SOLUTIONS 26 Even though the question concerns a stock with a constant growth rate we need to begin with the equation for twostage growth given in the chapter which is P01901g111g1 B Rg1 1R 1R We can expand the equation see Problem 27 for more detail to the following P0Do1g111g1 1g1 D1g2 Rgl 1R 1R R g2 Since the grth rate is constant g1 g2 so P0D01g1 1g 1g D1g Rg lR lR Rg Since we want the first I dividends to constitute onehalf of the stock price we can set the two terms on the right hand side of the equation equal to each other which gives us D0lg 11 gj 1 g D1 g Rg 1R 1R Rg Do1g Rg Since appears on both sides of the equation we can eliminate this which leaves 171gt1gt 1R 1R Solving this equation we get 2 t 1 1g 1 g 1R 1 R t 1 2 L g 1R 1 2 12 g 1R 39 CHAPTER 8 B 151 1g tln ln05 1n05 h1 1 g 1 R This expression will tell you the number of dividends that constitute onehalf of the current stock price To nd the value of the stock with twostage dividend growth consider that the present value of the first I dividends is the present value of a growing annuity Additionally to nd the price of the stock we need to add the present value of the stock price at time I So the stock price today is P0 PV of t dividends PVPt Using g1 to represent the rst growth rate and substituting the equation for the present value of a growing annuity we get 1g1 E l P0 D1 PVP R g1 Since the dividend in one year will increase at g1 we can rewrite the expression as 1g1 E J P0D01g1 PvPt R39gl Now we can rewrite the equation again as 1R PVPt R 39 g1 To nd the price of the stock at time t we can use the constant dividend growth model or The dividend at t I will have grown at g1 for tperiods and at g for one period so Dt1 Do1glt182 B152 SOLUTIONS N on So we can rewrite the equation as RDnangu R39gz Next we can nd value today of the future stock price as D1g1 1 g x 1 PVP 72 R g2 1 R which can be written as 1 2 D1 PVP71 x R 2 39 2 Substituting this into the stock price equation we get P DmampL1amp 1amp R g1 1 R 1 R In this equation the first term on the right hand side is the present value of the first I dividends and the second term is the present value of the stock price when constant dividend growth forever begins X D1 g2 R39gz To nd the expression when the growth rate for the first stage is exactly equal to the required return consider we can nd the present value of the dividends in the rst stage as D0 1 g1 1 1 R1 Do1 gal 112 D01g13 113 PV Since g1 is equal to R each of the terns reduces to PVD0D0D0 PVtXD0 So the expression for the price of a stock when the rst growth rate is exactly equal to the required return is t PttgtltD0 D0 X1g1 X1g2 R 7 g 2 CHAPTER 9 NET PRESENT VALUE AND OTHER INVESTMENT CRITERIA Answers to Concepts Review and Critical Thinking Questions 1 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 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 IfNPV 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 payback before this cutoff and reject projects that take longer to payback b The worst problem associated with payback period is that it ignores the time value of money In addition the selection of a hurdle point for payback period is an arbitrary exercise that lacks any steadfast rule or method The payback period is biased towards shortterm projects it fully ignores any cash ows that occur after the cutoff point 6 Despite its shortcomings payback is often used because 1 the analysis is straightforward and simple and 2 accounting numbers and estimates are readily available Materiality considerations often warrant a payback analysis as sufficient maintenance projects are another example where the detailed analysis of other methods is often not needed Since payback is biased towards liquidity it may be a useful and appropriate analysis method for shortterm projects where cash management is most important a The discounted payback is calculated the same as is regular payback with the exception that each cash ow in the series is first converted to its present value Thus discounted payback provides a measure of fmancialeconomic breakeven because of this discounting just as regular payback provides a measure of accounting breakeven because it does not discount the cash ows Given some predetermined cutoff for the discounted payback period the decision rule is to accept projects whose discounted cash ows payback before this cutoff period and to reject all other projects B154 SOLUTIONS b The primary disadvantage to using the discounted payback method is that it ignores all cash ows that occur after the cutoff date thus biasing this criterion towards shortterm projects As a result the method may reject projects that in fact have positive NPVs or it may accept projects with large future cash outlays resulting in negative NPVs In addition the selection of a cutoff point is again an arbitrary exercise 0 Discounted payback is an improvement on regular payback because it takes into account the time value of money For conventional cash ows and strictly positive discount rates the discounted payback will always be greater than the regular payback period 5 a The average accounting return is interpreted as an average measure of the accounting performance of a project over time computed as some average pro t measure attributable to the project divided by some average balance sheet value for the project This text computes AAR as average net income with respect to average total book value Given some predetermined cutoff for AAR the decision rule is to accept projects with an AAR in excess of the target measure and reject all other projects b AAR is not a measure of cash ows and market value but a measure of financial statement accounts that often bear little resemblance to the relevant value of a project In addition the selection of a cutoff is arbitrary and the time value of money is ignored For a nancial manager both the reliance on accounting numbers rather than relevant market data and the exclusion of time value of money considerations are troubling Despite these problems AAR continues to be used in practice because 1 the accounting information is usually available 2 analysts often use accounting ratios to analyze firm performance and 3 managerial compensation is often tied to the attainment of certain target accounting ratio goals 6 a NPV is simply the present value of a project s cash ows NPV speci cally measures after considering the time value of money the net increase or decrease in firm wealth due to the project The decision rule is to accept projects that have a positive NPV and reject projects with a negative b 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 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 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 firm will increase by 2500 if the project is accepted 7 a The IRR is the discount rate that causes the NPV of a series of cash ows to be exactly zero IRR can thus be interpreted as a nancial breakeven rate of return at the IRR the net value of the project is zero The IRR decision rule is to accept projects with IRRs greater than the discount rate and to reject projects with IRS less than the discount rate b IR is the interest 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 ambiguously ranks some mutually exclusive projects However for standalone projects with conventional cash ows IRR and NPV are interchangeable techniques C IR is frequently used because it is easier for many nancial managers and analysts to rate performance in relative terms such as 12 than in absolute terms such as 46000 IRR may be a preferred method to NPV in situations where an appropriate discount rate is unknown are uncertain in this situation IRR would provide more information about the project than would H H 3quot CHAPTER 9 B 15 5 a 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 profitability index decision rule is to accept projects with a PI greater than one and to reject projects with a PI less than one b PI NPV costcost l 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 buck of each particular project For a project with future cash ows that are an annuity Payback I C And the IRR is 0 71 C IRR 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 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 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 bene ts 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 definitely be used by the US government and would go a long way toward balancing the budget B156 SOLUTIONS 13 The MIRR 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 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 ows 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 C 0 c 758100 1 C IR 10 2 110 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 Ifyou 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 CHAPTER 9 B 157 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 1 To calculate the payback period we need to find the time that the project has recovered its initial investment After three years the project has created 1600 1900 2300 5800 in cash ows The project still needs to create another 6400 7 5800 600 in cash ows During the fourth year the cash ows from the project will be 1400 So the payback period will be 3 years plus what we still need to make divided by what we will make during the fourth year The payback period is Payback 3 600 1400 343 years 2 To calculate the payback period we need to find the time that the project has recovered its initial investment The cash flows in this problem are an annuity so the calculation is simpler If the initial cost is 2400 the payback period is Payback 3 105 765 314 years There is a shortcut to calculate the future cash ows are an annuity Just divide the initial cost by the annual cash ow For the 2400 cost the payback period is Payback 2400 765 314 years For an initial cost of 3600 the payback period is Payback 3600 765 471 years The payback period for an initial cost of 6500 is a little trickier Notice that the total cash inflows after eight years will be Total cash in ows 8765 6120 If the initial cost is 6500 the project never pays back Notice that if you use the shortcut for annuity cash ows you get Payback 6500 765 850 years This answer does not make sense since the cash ows stop after eight years so again we must conclude the payback period is never B158 SOLUTIONS 3 Project Ahas cash ows of 19000 in Year 1 so the cash ows are short by 21000 of recapturing the initial investment so the payback for Project A is Payback l 21000 25000 184 years Project B has cash ows of Cash ows 7 14000 17000 24000 7 55000 during this rst three years The cash ows are still short by 5000 of recapturing the initial investment so the payback for Project B is B Payback 3 5000 270000 3019 years Using the payback criterion and a cutoff of 3 years accept project A and reject project B 4 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 4200l14 368421 Value today of Year 2 cash ow 5300ll42 407818 Value today of Year 3 cash ow 61001l43 411733 Value today of Year 4 cash ow 7400l144 438139 To nd the discounted payback we use these values to nd the payback period The discounted rst year cash ow is 368421 so the discounted payback for a 7000 initial cost is Discounted payback 1 7000 7 36842l4078l8 181 years For an initial cost of 10000 the discounted payback is Discounted payback 2 10000 7 3684217 4078184ll733 254 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 13000 the discounted payback is Discounted payback 3 13000 73684217 407818 7 411733 438139 326 years 5 R 0 3 2100 4300 349 years discounted payback regular payback 349 years R 5 4300105 43001052 43001053 7 1170997 4300105quot 7 353762 discounted payback 3 15000 7 ll70997 353762 393 years CHAPTER 9 B 159 R 19 4300PV1FA195 1466204 The project never pays back Our de nition of AAR is the average net income divided by the average book value The average net income for this project is Average net income 1938200 2201600 1876000 1329500 4 1836325 And the average book value is Average book value 15000000 0 2 7500000 So the AARfor this project is AAR Average net income Average book value 1836325 7500000 2448 or 2448 The IRR 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 0 7 34000 160001IRR 18000171RR2 15000171RR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 2097 Since the IR is greater than the required return we would accept the project The NPV of a project is the PV of the out ows minus the PV of the in ows The equation for the NPV of this project at an 11 percent required return is NPV 7 34000 16000111 180001112 150001113 599149 At an 11 percent required return the NPV is positive so we would accept the project The equation for the NPV of the project at a 30 percent required return is NPV 7 34000 16000130 180001302 150001303 7421393 At a 30 percent required return the NPV is negative so we would reject the project The NPV of a project is the PV of the outflows minus the PV of the in ows Since the cash inflows are an annuity the equation for the NPV of this project at an 8 percent required return is NPV 7 7138000 28500PVIFA89 4003631 At an 8 percent required return the NPV is positive so we would accept the project B160 SOLUTIONS The equation for the NPV of the project at a 20 percent required return is NPV 7138000 28500PVIFA20 9 72311745 At a 20 percent required return the NPV is negative so we would reject the project We would be indifferent to the project if the required return was equal to the IRR of the project since at that required return the NPV is zero The IRR of the project is 0 7138000 28500PVIFAIRR79 IRR 1459 The IRR 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 0 7 19500 98001IRR 1030011RR2 86001IRR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 2264 The NPV of a project is the PV of the outflows minus the PV of the in ows At a zero discount rate and only at a zero discount rate the cash ows can be added together across time So the NPV of the project at a zero percent required return is NPV 719500 9800 10300 8600 9200 The NPV at a 10 percent required return is NPV 719500 980011 10300112 8600113 438279 The NPV at a 20 percent required return is NPV 7l9500 980012 10300122 8600123 79630 And the NPV at a 30 percent required return is NPV 719500 980013 10300132 8600133 7195244 Notice that as the required return increases the NPV of the project decreases This will always be true for projects with conventional cash ows Conventional cash ows are negative at the beginning of the project and positive throughout the rest of the project CHAPTER 9 B 161 The IRR is the interest rate that makes the NPV of the project equal to zero The equation for the IRR of Project A is 0 743000 230001IRR 1790011RR2 124001IRR3 94001IRR4 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 2044 The equation for the IRR of Project B is 0 743000 70001IRR 1380011RR2 240001IRR3 260001IRR4 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 1884 Examining the IRS of the projects we see that the IRRA is greater than the IRRB so IRR decision rule implies accepting project A This may not be a correct decision however because the IRR criterion has a ranking problem for mutually exclusive projects To see if the IRR decision rule is correct or not we need to evaluate the project NPVs The NPV of Project A is vaA 43000 23000111 179001112 124001113 9400111 NPvA 750761 And the NPV of Project B is vaB 43000 7000111 138001112 240001113 26000111 NPvB 918229 The NPVB is greater than the NPVA so we should accept Project B To nd the crossover rate we subtract the cash ows from one project from the cash ows of the other project Here we will subtract the cash ows for Project B from the cash ows of Project A Once we nd these differential cash ows we nd the IRR The equation for the crossover rate is Crossover rate 0 160001R 41001R2 7 116001R3 7 166001R4 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that R 1530 At discount rates above 1530 choose project A for discount rates below 1530 choose project B indifferent between A and B at a discount rate of 1530 B162 SOLUTIONS 13 The IRR is the interest rate that makes the NPV of the project equal to zero The equation to calculate the IRR of Project X is 0 715000 81501IRR 5050171RR2 680011RR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 1657 For Project Y the equation to nd the IR is 0 715000 770017IRR 5150171RR2 7250171RR3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 1645 To nd the crossover rate we subtract the cash ows from one project from the cash ows of the other project and nd the IRR of the differential cash ows We will subtract the cash ows from Project Y from the cash ows from Project X It is irrelevant which cash ows we subtract from the other Subtracting the cash ows the equation to calculate the IRR for these differential cash ows is Crossover rate 0 4501R 7 1001R2 7 4501R3 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that R 1173 The table below shows the NPV of each project for different required returns Notice that Project Y always has a higher NPV for discount rates below 1173 percent and always has a lower NPV for discount rates above 1173 percent E NPvX NPvX 0 500000 510000 5 321650 326736 10 169159 170323 15 37659 35678 20 75876620 75881134 25 7176640 7183200 a The equation for the NPV of the project is NPV 745000000 78000000117 14000000112 1348214286 The NPV is greater than 0 so we would accept the project CHAPTER 9 B 163 b The equation for the IRR of the project is 0 745000000 780000001IRR 7 140000001IRR2 From Descartes rule of signs we know there are potentially two IRRs since the cash ows change signs twice From trial and error the two IRRs are IRR 5300 77967 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 15 The pro tability index is defined as the PV of the cash in ows diVided by the PV of the cash out ows 16 The equation for the pro tability index at a required return of 10 percent is PI 730011 6900112 5700113 14000 1187 The equation for the pro tability index at a required return of 15 percent is PI 7300115 69001152 57001153 14000 1094 The equation for the pro tability index at a required return of 22 percent is PI 7300122 69001222 57001223 14000 0983 We would accept the project if the required return were 10 percent or 15 percent since the PI is greater than one We would reject the project if the required return were 22 percent since the PI is less than one a The pro tability index is the PV of the future cash ows diVided by the initial investment The cash ows for both projects are an annuity so P1I 27000PVIFA1073 53000 1267 PI11 9100PV1FA1073 16000 1414 The pro tability index decision rule implies that we accept project 11 since P111 is greater than the P11 b The NPV of each project is NPVI 53000 27000PV1FA103 1414500 NPVH 716000 9100PV1FA1073 663035 The NPV decision rule implies accepting Project 1 since the NPVI is greater than the NPVH B164 SOLUTIONS Using the pro tability index to compare mutually exclusive projects can be ambiguous when the magnitude of the cash ows for the two projects are of different scale In this problem projectI is roughly 3 times as large as project 11 and produces a larger NPV yet the pro tability index criterion implies that project 11 is more acceptable The payback period for each project is A 3 180000390000 346 years B 2 900018000 250 years The payback criterion implies accepting project B because it pays back sooner than project A The discounted payback for each project is A 20000115 500001152 500001153 8807430 390000115quot 22298377 Discounted payback 3 390000 7 880743022298377 395 years B 19000115 120001152 180001153 3743076 10500115 600341 Discounted payback 3 40000 7 374307660034l 343 years The discounted payback criterion implies accepting project B because it pays back sooner than A The NPV for each project is A NPV 300000 20000115 500001152 500001153 390000115quot NPV 1105807 B NPV 40000 19000115 120001152 180001153 10500115 NPV 343416 NPV criterion implies we accept project A because project A has a higher NPV than project B The IR for each project is A 300000 200001IRR 500001IRR2 500001IRR3 3900001IRR4 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 1620 CHAPTER 9 B 165 B 40000 1900011RR 120001IRR2 1800011RR3 105001IRR Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 1950 IRR decision rule implies we accept project B because IRR for B is greater than IRR for A e The pro tability index for each project is A PI 20000115 500001152 500001153 3900001154 300000 1037 B PI 19000115 120001152 180001153 105001154 40000 1086 Pro tability index criterion implies accept project B because its PI is greater than project A s f In this instance the NPV criteria implies that you should accept project A while pro tability index payback period discounted payback and IRR imply that you should accept project B The final decision should be based on the NPV since it does not have the ranking problem associated with the other capital budgeting techniques Therefore you should accept project A 18 At a zero discount rate and only at a zero discount rate the cash ows can be added together across time So the NPV of the project at a zero percent required return is NPV 7684680 263279 294060 227604 174356 274619 If the required return is in nite future cash ows have no value Even if the cash ow in one year is 1 trillion at an infinite rate of interest the value of this cash ow today is zero So if the future cash ows have no value today the NPV of the project is simply the cash ow today so at an in nite interest rate NPV 7684680 The interest rate that makes the NPV of a project equal to zero is the IRR The equation for the IRR of this project is 0 7684680 2632791IRR 29406011RR2 2276041IRR3 17435611RR4 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 1623 B166 SOLUTIONS 19 The MIRR for the project with all three approaches is Discounting approach In the discounting approach we nd the value of all cash out ows to time 0 while any cash in ows remain at the time at which they occur So the discounting the cash out ows to time 0 we nd Time 0 cash ow 7l6000 7 5100 1105 Time 0 cash ow 7l916670 So the MIRR using the discounting approach is 0 7l916670 61001MIRR 78001l11RR2 84001lHRR3 65001lIIRR4 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that MIRR 1818 Reinvestment approach In the reinvestment approach we nd the future value of all cash except the initial cash ow at the end of the project So reinvesting the cash ows to time 5 we nd Time 5 cash ow 6100110 78001103 84001102 6500110 7 5100 Time 5 cash ow 3152681 So the MIRR using the discounting approach is 0 4816000 31526811MIRR5 3152681 16000 1MIRR5 MIRR 3152681 16000 71 MIRR 1453 or 1453 Combination approach In the combination approach we nd the value of all cash out ows at time 0 and the value of all cash in ows at the end of the project So the value of the cash ows is Time 0 cash ow 7l6000 7 5100 1105 Time 0 cash ow 7l9l6670 Time 5 cash ow 6100110 78001103 84001102 6500110 Time 5 cash ow 3662681 So the IVIIRR using the discounting approach is 0 71916670 36626811MIRR5 3662681 1916670 1MIRR5 MIRR 3662681 1916670 71 MIRR 1383 or 1383 CHAPTER 9 B 167 Intermediate 20 With different discounting and reinvestment rates we need to make sure to use the appropriate interest rate The MIRR for the project with all three approaches is Discounting approach In the discounting approach we nd the value of all cash outflows to time 0 at the discount rate while any cash in ows remain at the time at which they occur So the discounting the cash out ows to time 0 we find Time 0 cash ow 7l6000 7 5100 1115 Time 0 cash ow 7l902660 So the MIRR using the discounting approach is 0 7 5 1902660 61001MIRR 78001lIIRR2 84001lHRR3 65001IIIRR4 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that MIRR 1855 Reinvestment approach In the reinvestment approach we nd the future value of all cash except the initial cash ow at the end of the project using the reinvestment rate So the reinvesting the cash ows to time 5 we nd Time 5 cash ow 6100108 78001083 84001082 6500108 7 5100 Time 5 cash ow 2984250 So the MIRR using the discounting approach is 0 4816000 29842501MIRR5 2984250 16000 1MIRR5 MIRR 2984250 16000 7 1 MIRR 1328 or 1328 Combination approach In the combination approach we nd the value of all cash out ows at time 0 using the discount rate and the value of all cash in ows at the end of the project using the reinvestment rate So the value of the cash ows is Time 0 cash ow 7l6000 7 5100 1115 Time 0 cash ow 7l902660 Time 5 cash ow 6100108 78001083 84001082 6500108 Time 5 cash ow 3494250 B168 SOLUTIONS N p A So the MIRR using the discounting approach is 0 71902660 34942501MIRR5 3494250 1902660 1MIRR5 MIRR 3494250 1902660 7 1 MIRR 1293 or 1293 Since the NPV index has the cost subtracted in the numerator NPV index PI 7 l a To have a payback equal to the project s life given C is a constant cash ow for N years C IN b To have a positive NPV I lt C PVIFAMY N Thus C gt I PVIFARW N 0 Bene ts C PVIFAW N 2 X costs 21 C 2I PVIFAW N Challenge Given the seven year payback the worst case is that the payback occurs at the end of the seventh year Thus the worstcase NPV 7724000 7240001127 7396499 17 The best case has infmite cash ows beyond the payback point Thus the bestcase NPV is infmite The equation for the IRR of the project is 0 71512 85861 IR 7 182101IRR2 171001IRR3 7 60001 1RR 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 if that NPV is greater than zero for required returns between 25 and 3333 or between 4286 and 6667 1 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 If you remember back to the chapter on stock valuation we presented a formula for valuing a stock with constant growth in dividends This formula is actually the formula for a growing perpetuity so we can use it here The PV of the future cash ows from the project is PV of cash in ows CJR 7g PV of cash in ows 8500013 7 06 121428571 CHAPTER 9 B 169 NPV is the PV of the out ows minus the PV of the in ows so the NPV is NPV ofthe project 71400000 121428571 718571429 The NPV is negative so we would reject the project b 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 71400000 85000 13 7g Solving for g we get g 0693 or 693 26 The IR of the project is 5 8000 3400011RR 450001IRR2 Using a spreadsheet nancial calculator or trial and error to nd the root of the equation we nd that IRR 2214 At an interest rate of 12 percent the NPV is NPV 58000 7 34000112 7 450001122 NPV 78823087 At an interest rate of zero percent we can add cash ows so the NPV is NPV 58000 7 34000 7 45000 NPV 782100000 And at an interest rate of 24 percent the NPV is NPV 58000 7 34000124 7 450001242 NPV 7 131426 The cash ows for the project are unconventional Since the initial cash ow is positive and the remaining cash ows are negative the decision rule for IRR in invalid in this case The NPV profile is upward sloping indicating that the project is more valuable when the interest rate increases B170 SOLUTIONS 27 The IRR 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 IRR2 Even though it appears there are two IRRs a spreadsheet financial 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 ibiVbz 74110 211 X In this case the equation is 7 726000 i 726000 7 42000013000 X 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 IRR First we need to find the future value of the cash ows for the one year in which they are blocked by the government So reinvesting each cash in ow for one year we find Year 2 cash ow 205000l04 213200 Year 3 cash ow 265000l04 275600 Year 4 cash ow 346000l04 359840 Year 5 cash ow 220000l04 228800 So the NPV ofthe project is NPV 7450000 2132001112 2756001113 359840111 2288001115 NPV 7262633 And the IRR of the project is 0 7450000 2132001 1RR2 2756001 IRR3 3598401IRR4 2288001 IRR5 Using a spreadsheet nancial calculator or trial and error to find the root of the equation we nd that IRR 1089 CHAPTER 9 B 171 While this may look like a MIRR calculation it is not an 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 Calculator Solutions 7 734000 16000 1 18000 1 15000 1 I 11 1 30 NPV CPT NPV CPT 599149 7421393 CFo 138000 CFo 138000 CFo 7138000 C01 28500 C01 28500 C01 28500 F01 9 F01 9 F01 9 I 8 1 20 IRR CPT NPV CPT NPV CPT 1459 4003631 72311745 B172 SOLUTIONS 10 IR CPT 2264 11 719500 9800 1 10300 8600 1 I 0 I 10 NPV CPT NPV CPT 9200 438279 719500 9800 1 10300 1 8600 1 I 20 I 30 NPV CPT NPV CPT 79630 7195244 12 Pro 39ectA 743000 23000 1 17900 1 12400 1 9400 1 NPV CPT 750761 743000 7000 1 13800 1 24000 1 26000 1 0 16000 4100 1 11600 1 4516600 1 NPV CPT 500000 CHAPTER 9 B 173 743000 7000 1 13800 1 24000 1 26000 1 NPV CPT 918229 715000 715000 8150 8150 1 1 5050 5050 1 1 6800 6800 1 1 I 15 NPV CPT 37659 48176640 B174 SOLUTIONS 15000 15000 7700 7700 1 1 5150 5150 1 1 7250 7250 1 1 I 0 1 15 NPV CPT NPV CPT 510000 35678 183200 Crossover rate 45000000 78000000 45000000 78000000 1 14000000 1 1 14000000 1 I 10 IRR CPT NPV CPT 5300 1348214286 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 77967 CHAPTER 9 B 175 15 0 0 7300 7300 1 1 6900 6900 1 1 5700 5700 1 1 I 10 I 15 NPV CPT NPV CPT 1662134 1531306 1375849 10 PI 1662134 14000 1187 15 PI 1531306 14000 1094 22 PI 1375849 14000 0983 16 Pro 39ectI CFo 753000 27000 C01 27000 3 F01 3 I 10 I 10 NPV CPT NPV CPT 6714500 1414500 PI 6714500 53000 1267 Pro 39ectH 0 CFo 716000 9100 C01 9100 3 F01 3 I 10 I 10 NPV CPT NPV CPT 2263035 663035 PI 2263035 16000 1414 17 CF04 c Cfo 7300000 7300000 0 C01 20000 20000 20000 F01 1 1 1 C02 50000 50000 50000 F02 2 2 2 C03 390000 390000 390000 F03 1 1 1 I 15 IR CPT NPV CPT 1620 1105807 31105807 PI 31105807 300000 1037 B176 SOLUTIONS 0173 740000 740000 0 19000 19000 19000 1 1 1 12000 12000 12000 1 1 1 18000 18000 18000 1 1 1 10500 10500 10500 1 1 1 NPV CPT 343416 4343416 PI 4343416 40000 1086 f In this instance the NPV criteria implies that you should accept project A while payback period discounted payback pro tability index and IRR imply that you should accept project B The nal decision should be based on the NPV since it does not have the ranking problem associated with the other capital budgeting techniques Therefore you should accept project A 18 7684680 7684680 263279 263279 1 1 294060 294060 1 1 227604 227604 1 1 174356 174356 1 l I 0 IR CPT NPV CPT 1623 274619 CHAPTER 10 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 For tax purposes a firm 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 rm 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 rm s overall capital structure remains 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 in effect each project is assumed to exist over an infinite horizon of Nyear repeating projects Assuming that this type of analysis is valid implies that the project cash ows remain the same forever 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 in ow so the effects of the depreciation tax shield must be added in to get the total incremental aftertax cash flows 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 B178 SOLUTIONS 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 De nitely The damage to Porsche s reputation is de nitely a factor the company needed to consider If the reputation was damaged the company would have lost sales of its existing car lines 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 Porsche would recognize that the outsized pro ts would dwindle as more product comes to market and competition becomes more intense 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 6 million acquisition cost of the land six years ago is a sunk cost The 64 million current aftertax value of the land is an opportunity cost if the land is used rather than sold off The 142 million cash outlay and 890000 grading expenses are the initial xed asset investments needed to get the project going Therefore the proper year zero cash ow to use in evaluating this project is 6400000 14200000 890000 21490000 Sales due solely to the new product line are 1900013000 247000000 Increased sales of the motor home line occur because of the new product line introduction thus 450053000 238500000 in new sales is relevant Erosion of luxury motor coach sales is also due to the new midsize campers thus 9009l000 81900000 loss in sales is relevant The net sales gure to use in evaluating the new line is thus 247000000 238500000 7 81900000 403600000 3 We need to construct a basic income statement The income statement is Sales 830000 Variable costs 498000 Fixed costs 181000 Depreciation 77000 EBT 74000 Taxes3 5 25900 Net income 48100 To nd the OCF we need to complete the income statement as follows Sales 824500 Costs 53 8900 Depreciation 126500 E 159100 Taxes34 54094 Net income 105 006 The OCF for the company is OCF EBIT Depreciation 7 Taxes OCF 159100 7 126500 7 54094 OCF 231506 The depreciation tax shield is the depreciation times the tax rate so Depreciation tax shield thepreciation Depreciation tax shield 34126500 Depreciation tax shield 43010 Sales 108000 Variable costs 51000 Depreciation 6800 EBT 50200 Taxes3 5 17570 Net income 32630 Using the most common nancial calculation for OCF we get OCF EBIT Depreciation 7 Taxes OCF 50200 6800 717570 OCF 39430 CHAPTER 10 B 179 The depreciation tax shield shows us the increase in OCF by being able to expense depreciation To calculate the OCF we first need to calculate net income The income statement is B180 SOLUTIONS The topdown approach to calculating OCF yields OCF Sales 7 Costs 7 Taxes OCF 108000 7 51000 717570 OCF 39430 The taxshield approach is OCF Sales 7 Costs1 7 tc thepreciation OCF 108000 7 510001 7 35 356800 OCF 39430 And the bottomup approach is OCF Net income Depreciation OCF 32630 6800 OCF 39430 All four methods of calculating OCF should always give the same answer The MACRS depreciation schedule is shown in Table 107 The ending book value for any year is the beginning book value minus the depreciation for the year Remember to nd the amount of depreciation for any year you multiply the purchase price of the asset times the MACRS percentage for the year The depreciation schedule for this asset is M Beginning Book Value MACRS Depreciation Ending Book value 1 108000000 01429 15433200 92566800 2 92566800 02449 26449200 66117600 3 66117600 01749 18889200 47228400 4 47228400 01249 13489200 33739200 5 33739200 00893 9644400 24094800 6 24094800 00892 9633600 14461200 7 14461200 00893 9644400 4816800 8 4816800 00446 4816800 0 The asset has an 8 year useful life and we want to nd the BV of the asset after 5 years With straight line depreciation the depreciation each year will be Annual depreciation 548000 8 Annual depreciation 68500 So after ve years the accumulated depreciation will be Accumulated depreciation 568500 Accumulated depreciation 3 42500 9 CHAPTER 10 B 181 The book value at the end of year ve is thus BV5 548000 7 342500 BV5 205500 The asset is sold at a loss to bookvalue so the depreciation tax shield of the loss is recaptured Aftertax salvage value 105000 205500 7 105000035 Aftertax salvage value 140175 To nd the taxes on salvage value remember to use the equation Taxes on salvage value BV 7 MVtc This equation will always give the correct sign for a tax in ow refund or out ow payment To nd the EV at the end of four years we need to nd the accumulated depreciation for the first four years We could calculate a table as in Problem 6 but an easier way is to add the MACRS depreciation amounts for each of the first 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 13v 7 7900000 7 790000002000 03200 01920 01152 13v 7 1365120 The asset is sold at a gain to book value so this gain is taxable Aftertax salvage value 1400000 1365120 7140000035 Aftertax salvage value 1387792 Using the tax shield approach to calculating OCF Remember the approach is irrelevant the nal answer will be the same no matter which of the four methods you use we get OCF Sales 7 Costs1 7tc thepreciation OCF 2650000 7 84000017 035 03539000003 OCF 1631500 Since we have the OCF we can nd the NPV as the initial cash outlay plus the PV of the OCFs which are an annuity so the NPV is NPV 7 73900000 163 1500Pv1FA123 NPV 7 1858771 B182 SOLUTIONS 11 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 m Cash Flow 0 74200000 73900000 7 300000 1 1631500 2 1631500 3 2068000 1631500 300000 210000 0 7 21000035 And the NPV of the project is NPV 7 4200000 163 1500Pv1FA122 2068000 1123 NPV 7 2927979 First we will calculate the annual depreciation for the equipment necessary for the project The depreciation amount each year will be Year 1 depreciation 390000003333 1299870 Year 2 depreciation 390000004445 1733550 Year 3 depreciation 390000001481 577590 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 3900000 7 1299870 1733550 577590 Book value in 3 years 288990 The asset is sold at a loss to book value so this loss is taxable deductible A ertax salvage value 210000 288990 7 210000035 A ertax salvage value 237647 To calculate the OCF we will use the tax shield approach so the cash ow each year is OCF Sales 7 Costs17tc thepreciation M Cash Flow 0 74200000 73900000 7 300000 1 163145450 181000065 0351299870 2 178324250 181000065 0351733550 3 191630300 181000065 035577590 237647 7 300000 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 74200000 1631454501 12 1783242501 122 1916303001 123 NPV 4223243 13 p n A CHAPTER 10 B 183 First we will calculate the annual depreciation of the new equipment It will be Annual depreciation 560000 5 Annual depreciation 112000 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 A ertax salvage value MV BV 7 MVtc 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 aftertax salvage value becomes A ertax salvage value MV 0 7MVtc 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 aftertax salvage value is A ertax salvage value 85000l 7 034 A ertax salvage value 56100 Using the tax shield approach we nd the OCF for the project is OCF 7 16500017 034 034112000 OCF 7 146980 Now we can find the project NPV Notice we include the NWC in the initial cash outlay The recovery of the NWC occurs in Year 5 along with the aftertax salvage value NPV 7560000 7 29000 146980PV1FA1075 56100 29000 1105 NPV 2101024 First we will calculate the annual depreciation of the new equipment It will be Annual depreciation charge 720000 5 Annual depreciation charge 144000 The aftertax salvage value of the equipment is A ertax salvage value 7500017 035 A ertax salvage value 48750 Using the tax shield approach the OCF is OCF 26000017 035 035144000 OCF 219400 B184 SOLUTIONS Now we can nd 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 7 0 7 7720000 110000 219400Pv1FAmR5 48750 7 110000 1IRR5 IRR 2165 To evaluate the project with a 300000 cost savings we need the OCF to compute the NPV Using the tax shield approach the OCF is OCF 30000017 035 035144000 245400 NPV 7720000 110000 245400PVIFA205 48750 7 110000 1205 NPV 9928122 The NPV with a 240000 cost savings is OCF 24000017 035 035144000 OCF 206400 NPV 7720000 110000 206400PVIFA2075 48750 7 110000 1205 NPV 71735266 We would accept the project if cost savings were 300000 and reject the project if the cost savings were 240000 The required pretax cost savings that would make us indifferent about the project is the cost savings that results in a zero NPV The NPV of the project is NPV 0 7720000 110000 OCFPVIFA2075 48750 7 110000 1205 Solving for the OCF we nd the necessary OCF for zero NPV is OCF 21220238 Using the tax shield approach to calculating OCF we get OCF 7 21220238 7 s 7 C17 035 035144000 s 7 C 7 24892673 The cost savings that will make us indifferent is 24892673 16 CHAPTER 10 B 185 To calculate the EAC of the project we first need the NPV of the project Notice that we include the NWC expenditure at the beginning of the project and recover the NWC at the end of the project The NPV of the project is NPV 7270000 7 25000 7 42000PVIFA1175 250001115 743539139 Now we can nd the EAC of the project The EAC is EAC 743539139 PVIFA115 711780398 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 400001 7 035 26000 To calculate the EAC we rst need the OCF and NPV of each option The OCF and NPV for TechronI is OCF 76700017 035 0352900003 7971667 NPV 7290000 7 971667PV1FA1073 260001103 729462973 EAC 729462973 PVIFAmm 711847497 And the OCF and NPV for Techron II is OCF 73500017 035 0355100005 12950 NPV 7510000 12950PVIFA1075 260001105 744476536 EAC 744476536 PVIFAmm 711732798 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 To nd the bid price we need to calculate all other cash ows for the project and then solve for the bid price The aftertax salvage value of the equipment is A ertax salvage value 7000017 035 45500 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 7 0 7 940000 7 75000 OCFPVIFA1275 75000 45500 1125 B186 SOLUTIONS Solving for the OCF we nd the OCF that makes the project NPV equal to zero is OCF 94662506 PVIFA125 26260301 The easiest way to calculate the bid price is the tax shield approach so OCF 26260301 P 7vQ 7 FC17 to th P 7 925185000 7 305000 17 035 0359400005 Intermediate First we will calculate the depreciation each year which will be D1 7 56000002000 7 112000 D2 7 56000003200 7 179200 D3 7 56000001920 7 107520 D4 7 56000001152 7 64512 The book value of the equipment at the end of the project is BV4 560000 7 112000 179200 107520 64512 96768 The asset is sold at a loss to book value so this creates a taX refund Aftertax salvage value 80000 96768 7 80000035 8586880 So the OCF for each year will be OCFl 21000017 035 035112000 172700 OCFZ 21000017 035 035179200 196220 OCF3 21000017 035 035107520 171132 OCF4 21000017 035 03564512 15907920 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 560000 7 20000 7 172700 7 3000109 196220 7 30001092 171132 7 30001093 15907920 7 29000 8586880109quot NPV 7 6981179 20 N p A N N CHAPTER 10 B 187 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 71100001 7 034 0344300004 OCFA 7 736050 NPVA 7430000 7 36050PVIFA1174 NPVA 7541843 17 And the NPV of System B is OCFB 7980001 7 034 0345700006 OCFB 732380 NPVB 7570000 7 32380PVIFA1175 NPVB 7706984 82 If the system will not be replaced when it wears out then System A should be chosen because it has the more positive 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 7541843 17 PVIFAIW EACA 7 717465033 EACB 7 7 70698482 PVIFAHM EACB 716711464 If the conveyor belt system will be continually replaced we should choose System B since it has the more positive EAC 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 Aftertax salvage value 540000l 7 034 Aftertax salvage value 356400 Now we can solve for the necessary OCF that will give the project a zero NPV The current aftertax value of the land is an opportunity cost but we also need to include the a ertax value of the land in ve years since we can sell the land at that time The equation for the NPV of the project is NPV 0 74100000 7 2700000 7 600000 OCFPVIFA125 50000PVIFA1274 356400 600000 450000 3200000 1125 B188 SOLUTIONS Solving for the OCF we nd the OCF that makes the project NPV equal to zero is OCF 507992911 PVIFAIMS OCF 140922177 The easiest way to calculate the bid price is the tax shield approach so OCF 140922177 P 7vQ 7FC17tcth 140922177 P7 0005100000000 7 95000017 034 03441000005 P 003163 At a given price taking accelerated depreciation compared to straightline depreciation causes the NPV to be higher similarly at a given price lower net working capital investment requirements will cause the NPV to be higher Thus NPV would be zero at a lower price in this situation In the case of a bid price you could submit a lower price and still breakeven or submit the higher price and make a positive NPV 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 Machine B Variable costs 453500000 453000000 Fixed costs 7170000 7130000 Depreciation 4183 333 7566667 EBT 74153333 73696667 Tax 1453667 1293833 Net income 452699667 452402833 Depreciation 483 333 566667 OCF 72216333 71836167 The NPV and EAC for Machine A is vaA 72900000 7 2216333Pv1FA10 NPvA 71255270946 EACA 7 1255270946 PVIFAmm EACA 7288219474 And the NPV and EAC for Machine B is vaB 75100000 7 1836167PVIFA1079 NPvB 71567452756 EACB 7 1567452756 PVIFAlmg EACB 7272173342 You should choose Machine B since it has a more positive EAC CHAPTER 10 B 189 25 A kilowatt hour is 1000 watts for 1 hour A 60watt bulb burning for 500 hours per year uses 2 Ch 30000 watt hours or 30 kilowatt hours Since the cost of a kilowatt hour is 0101 the cost per year is Cost per year 300 101 Cost per year 303 The 60watt bulb will last for 1000 hours which is 2 years of use at 500 hours per year So the NPV of the 60watt bulb is NPV 7050 7 303PV1FA1072 NPV 7576 And the EAC is EAC 7583 PVIFAIMYZ EAC 45332 Now we can nd the EAC for the 15watt CFL A l5watt bulb burning for 500 hours per year uses 7500 watts or 75 kilowatts And since the cost of a kilowatt hour is 0101 the cost per year is Cost per year 750101 Cost per year 07575 The 15watt CFL will last for 12000 hours which is 24 years of use at 500 hours per year So the NPV of the CFL is NPV 7350 7 07575PVIFA10724 NPV 71031 And the EAC is EAC 7581085 PVIFAIOW EAC 115 Thus the CFL is much cheaper But see our next two questions To solve the EAC algebraically for each bulb we can set up the variables as follows W light bulb wattage C cost per kilowatt hour H hours burned per year P price the light bulb The number of watts use by the bulb per hour is WPH W 1000 And the kilowatt hours used per year is KPYWPHXH B190 SOLUTIONS N l The electricity cost per year is therefore ECY KPY X C The NPV of the decision to but the light bulb is NPV 7 P 7 ECYPV1FARJ And the EAC is EAC NPV PVIFARW Substituting we get EAC 7P 7 W 1000 X H X CPVIFAR7 PFIVARW We need to set the EAC of the two light bulbs equal to each other and solve for C the cost per kilowatt hour Doing so we nd 050 7 60 1000 x 500 x CPV1FA1072 Pv1FAwA2 7 350 715 1000 x 500 x CPV1FA10724 PVIFA10724 C 0004509 So unless the cost per kilowatt hour is extremely low it makes sense to use the CFL But when should you replace the incandescent bulb See the next question We are again solving for the breakeven kilowatt hour cost but now the incandescent bulb has only 500 hours of useful life In this case the incandescent bulb has only one year of life left The breakeven electricity cost under these circumstances is 050 7 60 1000 x 500 x CPV1FA101 Pv1FAw1 7 350 715 1000 x 500 x CPv1FA102 Pv1FAmz C 70007l31 Unless the electricity cost is negative Not very likely it does not make financial sense to replace the incandescent bulb until it burns out 28 The debate between incandescent bulbs and CFLs is not just a financial debate but an environmental one as well The numbers below correspond to the numbered items in the question 1 The extra heat generated by an incandescent bulb is waste but not necessarily in a heated structure especially in northem climates 2 Since CFLs last so long from a nancial viewpoint it might make sense to wait if prices are declining 3 Because of the nontrivial health and disposal issues CFLs are not as attractive as our previous analysis suggests 0 CHAPTER 10 B 191 4 From a company s perspective the cost of replacing working incandescent bulbs may outweigh the nancial bene t However since CFLs last longer the cost of replacing the bulbs will be lower in the long run 5 Because incandescent bulbs use more power more coal has to be burned which generates more mercury in the environment potentially offsetting the mercury concern with CFLs 6 As in the previous question if C02 production is an environmental concern the the lower power consumption from CFLs is a bene t 7 CFLs require more energy to make potentially offsetting at least partially the energy savings from their use Worker safety and site contamination are also negatives for CFLs 8 This fact favors the incandescent bulb because the purchasers will only receive part of the bene t from the CFL 9 This fact favors waiting for new technology 10 This fact also favors waiting for new technology While there is always a best answer this question shows that the analysis of the best answer is not always easy and may not be possible because of incomplete data As for how to better legislate the use of CFLs our analysis suggests that requiring them in new construction might make sense Rental properties in general should probably be required to use CFLs why rentals Another piece of legislation that makes sense is requiring the producers of CFLs to supply a disposal kit and proper disposal instructions with each one sold Finally we need much better research on hazards associated with broken bulbs in the home and workplace and proper procedures for dealing with broken bulbs Surprise You should de nitely upgrade the truck Here s why At 10 mpg the truck burns 12000 10 1200 gallons of gas per year The new truck will burn 12000 125 960 gallons of gas per year a savings of 240 gallons per year The car burns 12000 25 480 gallons of gas per year while the new car will burn 12000 40 300 gallons of gas per year a savings of 180 gallons per year so it s not even close This answer may strike you as counterintuitive so let s consider an extreme case Suppose the car gets 6000 mpg and you could upgrade to 12000 mpg Should you upgrade Probably not since you would only save one gallon of gas per year So the reason you should upgrade the truck is that it uses so much more gas in the first place Notice that the answer doesn t depend on the cost of gasoline meaning that if you upgrade you should always upgrade the truck In fact it doesn t depend on the miles driven as long as the miles driven are the same B192 SOLUTIONS 30 Surprise You should de nitely upgrade the truck Here s why At 10 mpg the truck burns 12000 10 1200 Challenge 31 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 7 MVtc Taxes on salvage value 0 7 40000038 Taxes on salvage value 7152000 Market price 400000 Tax on sale 7152000 A ertax salvage value 248000 Now we need to calculate the operating cash ow each year Using the bottom up approach to calculating operating cash ow we nd M Year 1 Year 2 Year 3 Year 4 Revenues 2496000 33 54000 3042000 2184000 Fixed costs 425000 425000 425000 425000 Variable costs 374400 503100 456300 327600 Depreciation 1399860 1866900 622020 3 1 1220 EBT 296740 559000 1538680 1120180 Taxes 112761 212420 584698 425668 Net income 183979 346580 953982 694512 OCF 1583839 2213480 1576002 1005732 Capital spending 74200000 248000 Land 71500000 1600000 NWC 7125000 125000 Total cash ow 75825000 1583839 2213480 1576002 2978732 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 are both 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 The company can sell the land at the end of the project so we need to include that value as well With all the project cash ows we can calculate the NPV which is NPV 75825000 1583839 113 2213480 1132 1576002 1133 2978732 1134 NPV 22926682 N CHAPTER 10 B 193 The company should accept the new product line 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 The initial NWC investment is given and the subsequent NWC investment will be 15 percent of the next year s sales In this case it will be Year 1 sales Realizing we need Year 1 sales to calculate the required NWC capital at time 0 we find that Year 1 sales will be 35340000 So the cash ow required for the project today will be Capital spending 724000000 Initial NWC 71800000 Total cash ow 725800000 Now we can begin the remaining calculations Sales figures 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 1200000 per year To calculate depreciation each year we use the initial equipment cost of 24 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 flows will be discussed below M l 2 i 4 i Ending book value 20570400 14692800 10495200 7497600 5354400 Sales 35340000 39900000 48640000 50920000 33060000 Variable costs 24645000 27825000 33920000 3 5510000 23055000 Fixed costs 1200000 1200000 1200000 1200000 1200000 D J 39 quot 3429600 5877600 4197600 2997600 2143200 EBIT 6065400 4997400 9322400 11212400 6661800 Taxes 2122890 1749090 3262840 3924340 2331630 Net income 3942510 3248310 6059560 7288060 4330170 D J 39 quot 3429600 5877600 4197600 2997600 2143200 Operating cash ow 7372110 9125910 10257160 10285660 6473370 Net cash ows Operating cash ow 7372110 9125910 10257160 10285660 6473370 Change in NWC 7684000 71311000 7342000 2679000 1458000 Capital spending 0 0 0 0 4994040 Total cash ow 6688110 7814910 9915160 12964660 12925410 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 capital is 15 percent of the increase in sales in the next year We will B194 SOLUTIONS 43 43 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 l539900000 7 35340000 Increase in NWC for Year 1 684000 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 ve years will be the purchase price minus the total depreciation So the ending book value is Ending book value 24000000 7 3429600 5877600 4197600 2997600 5 5 Ending book value 5354400 The market value of the used equipment is 20 percent of the purchase price or 48 million so the aftertax salvage value will be A ertax salvage value 4800000 5354400 7 480000035 A ertax salvage value 4994040 The a ertaX 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 725800000 6688110118 78149101182 99151601183 129646601184 129254101185 NPV 385195223 And the IRR is NPV 0 725800000 66881101IRR 78149101 IRR2 99151601 IRR3 129646601 IRR4 129254101IRR5 IRR 2362 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 find the OCF We begin by calculating the depreciation each year using the MACRS depreciation schedule The depreciation each year is D1 61000003333 203313 D2 61000004444 271145 D3 61000001482 90341 D4 61000000741 45201 Using the tax shield approach the OCF each year is OCF1 s 7 C17 035 035203313 CHAPTER 10 B 195 OCF2 s 7 C17 035 035271145 OCF3 s 7 C17 035 03590341 OCF4 s 7 C17 035 03545201 OCFs s 7 C1 7 035 Now we need the aftertax salvage value of the equipment The a ertax salvage value is Aftertax salvage value 40000l 7 035 26000 To nd 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 0 78610000 7 55000 s 7 C065Pv1FAu5 035203313112 2711451122 903411123 452011124 55000 260001125 Solving this equation for the sales minus costs we get s 7 C065PV1FA1275 44728867 s 7 C 19089574 1 This problem is basically the same as Problem 18 except we are given a sales price The cash ow at Time 0 for all three parts of this question will be Capital spending 7940000 Change in NWC 775000 Total cash ow 71015000 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 13 and unitsyear 185000 Year 1 Z i A 2 Sales 2405000 2405000 2405000 2405000 2405000 Variable costs 1711250 1711250 1711250 1711250 1711250 Fixed costs 305000 305000 305000 305000 305000 1 188000 188000 188000 188000 188000 EBIT 200750 200750 200750 200750 200750 Taxes 35 70263 70263 70263 70263 70263 Net Income 130488 130488 130488 130488 130488 I 39 quot 188000 188000 188000 188000 188000 Operating CF 318488 318488 318488 318488 318488 Year 1 Z i 4 2 Operating CF 318488 318488 318488 318488 318488 Change in NWC 0 0 0 0 75000 Capital spending 0 0 0 0 45500 Total CF 318488 318488 318488 318488 438988 B196 SOLUTIONS With these cash ows the NPV of the project is NPV 7940000 7 75000 318488PV1FA1275 75000 45500 1125 NPV 20145110 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 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 7940000 7 75000 OCFPVIFA1275 75000 45500 1125 So the necessary OCF for a zero NPV is OCF 94662506 PVIFA125 26260301 Now we can use the tax shield approach to solve for the minimum quantity as follows OCF 26260301 P 7vQ 7 FC17 to th 26260301 1300 7 925Q 7 305000 17 035 0359400005 Q 162073 As a check we can calculate the NPV of the project with this quantity The calculations are Year 1 Z i A 2 Sales 2106949 2106949 2106949 2106949 2106949 Variable costs 1499176 1499176 1499176 1499176 1499176 Fixed costs 305000 305000 305000 305000 305000 D J 39 quot 188000 188000 188000 188000 188000 EBIT 114774 114774 114774 114774 114774 Taxes 35 40171 40171 40171 40171 40171 Net Income 74603 74603 74603 74603 74603 D J 39 quot 188000 188000 188000 188000 188000 Operating CF 262603 262603 262603 262603 262603 Year 1 Z i 4 2 Operating CF 262603 262603 262603 262603 262603 Change in NWC 0 0 0 0 75000 Capital spending 0 0 0 0 45500 Total CF 262603 262603 262603 262603 383103 NPV 7 940000 7 75000 262603PV1FA1275 75000 45500 1125 u so Note 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 CHAPTER 10 B 197 c To nd the highest level of fixed costs and 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 7940000 7 75000 OCFPVIFA1275 75000 45500 1125 OCF 94662506 PVIFA125 26260301 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 26260301 P7vQ 7FC 1 71c th 26260301 1300 7 925185000 7 FC17 035 0359400005 FC 39097615 As a check we can calculate the NPV of the project with this level of fixed costs The calculations are Year 1 Z i A 2 Sales 2405000 2405000 2405000 2405000 2405000 Variable costs 1711250 1711250 1711250 1711250 1711250 Fixed costs 390976 390976 390976 390976 390976 D J 39 quot 188000 188000 188000 188000 188000 EBIT 114774 114774 114774 114774 114774 Taxes 35 40171 40171 40171 40171 40171 Net Income 74603 74603 74603 74603 74603 D J 39 quot 188000 188000 188000 188000 188000 Operating CF 262603 262603 262603 262603 262603 Year 1 Z i 4 2 Operating CF 262603 262603 262603 262603 262603 Change in NWC 0 0 0 0 75000 Capital spending 0 0 0 0 45500 Total CF 262603 262603 262603 262603 383103 NPV 7940000 7 75000 262603Pv1FAu5 75000 45500 1125 z 0 35 We need to find 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 1 2 3 4 Sales 855000 1710000 2280000 1425000 Variable costs 525000 1050000 1400000 875000 EBT 330000 660000 880000 550000 Tax 132000 264000 352000 220000 Net income and OCF 198000 3 96000 528000 330000 B198 SOLUTIONS So the addition to NPV of these market sales is NPV ofmarket sales 7 198000113 7 3960001132 5280001133 330000113 NPV of market sales 105367299 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 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 2750001 7 40 165000 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 73400000 7 95000 105367299 OCF PVIFAIMA 165000 95000 1134 Solving for the OCF we get OCF 238186414 PVIFADW 80076890 Now we can solve for the bid price as follows OCF 80076890 P 7vQ 7 FC17tcth 80076890 7 P 7 17517500 7 60000017 040 7 04034000004 P 7 25317 1 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 1450001738780000 538 149180 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 a ertax salvage value which is A ertax salvage value 1500001 7 38 93000 Now we can calculate the NPV of the new computer as NPV 7 7780000 7 149180PVIFA1275 7 93000 1125 CHAPTER 10 B 199 NPV 718946879 And the EAC of the new computer is EAC 718946879 PVIFAIMYS 75256049 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 a ertax salvage value of the old computer is A ertax salvage value 210000 390000 7 21000038 377200 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 60000 130000 7 6000038 86600 Now we can calculate the NPV of the old computer as NPV 7377200 49400PVIFA1172 136000 1122 NPV 722464749 And the EAC of the old computer is EAC 722467449 PVIFAIZW 713293947 Even if we are going to replace the system in two years no matter what our decision today we should replace it today since the EAC is more positive B200 SOLUTIONS 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 flows 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 7377200 7402800 1 149180 49400 99780 2 149180 136000 13180 3 149180 0 149180 4 149180 0 149180 5 242180 0 242180 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 of the decision to replace ignoring what will happen in two years is NPV 7402800 99780112 131801122 1491801143 1491801144 2421801145 NPV 3520570 If we are not concerned with what will happen in two years we should replace the old computer system CHAPTER 11 PROJECT ANALYSIS AND EVALUATION 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 It makes wages and salaries a fixed cost driving up operating leverage Fixed costs are relatively high because airlines are relatively capital intensive and airplanes are expensive Skilled employees such as pilots and mechanics mean relatively high wages which because of union agreements are relatively xed Maintenance expenses are signi cant and relatively xed as well 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 after this ordering will always apply The cash breakeven is achieved rst 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 Soft capital rationing implies that the rm as a whole isn t short of capital but the division or project does not have the necessary capital The implication is that the rm is passing up positive NPV projects With hard capital rationing the rm is unable to raise capital for a project under any circumstances Probably the most common reason for hard capital rationing is nancial distress meaning bankruptcy is a possibility The implication is that they will face hard capital rationing B202 SOLUTIONS 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 1 a The total variable cost per unit is the sum of the two variable costs so Total variable costs per unit 543 313 Total variable costs per unit 856 b The total costs include all variable costs and xed costs We need to make sure we are including all variable costs for the number of units produced so Total costs Variable costs Fixed costs Total costs 856280000 720000 Total costs 3116800 c The cash breakeven that is the point where cash ow is zero is Qc 720000 1999 7 856 QC 6299213 units And the accounting breakeven is QA 720000 220000 1999 7 856 QA 8223972 units 2 The total costs include all variable costs and xed costs We need to make sure we are including all variable costs for the number of units produced so Total costs 2486 1408120000 1550000 Total costs 6222800 The marginal cost or cost of producing one more unit is the total variable cost per unit so Marginal cost 2486 1408 Marginal cost 3894 CHAPTER 11 B 203 The average cost per unit is the total cost of production divided by the quantity produced so Average cost Total cost Total quantity Average cost 6222800 120000 Average cost 5186 Minimum acceptable total revenue 50003894 Minimum acceptable total revenue 194700 Additional units should be produced only if the cost of producing those units can be recovered The basecase bestcase and worstcase values are shown below Remember that in the bestcase sales and price increase while costs decrease In the worstcase sales and price decrease and costs increase Unit Scenario Unit Sales Unit Price Variable Cost Fixed Costs Base 95000 190000 24000 4800000 Best 109250 218500 20400 4080000 Worst 80750 161500 27600 5520000 An estimate for the impact of changes in price on the pro tability of the project can be found from the sensitivity of NPV with respect to price ANPVAP This measure can be calculated by nding the NPV at any two different price levels and forming the ratio of the changes in these parameters Whenever a sensitivity analysis is performed all other variables are held constant at their basecase values a To calculate the accounting breakeven we rst need to nd the depreciation for each year The depreciation is Depreciation 724000 8 Depreciation 90500 per year And the accounting breakeven is QA 780000 9050043 7 29 QA 62179 units To calculate the accounting breakeven we must realize at this point and only this point the OCF is equal to depreciation So the DOL at the accounting breakeven is DOL 1 FCOCF 1 FCD DOL 1 78000090500 DOL 9919 We will use the tax shield approach to calculate the OCF The OCF is OCFbase P VQ FC1 to t th OCFbse 43 7 2990000 7 780000065 03590500 OCFbase 343675 B204 SOLUTIONS Now we can calculate the NPV using our basecase projections There is no salvage value or NWC so the NPV is Nvase 7724000 343675Pv1FA158 Nvase 81818022 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 95000 units The NPV at this sales level is OCFnew 43 7 2995000 7 780000065 03590500 OCFnew 389175 And the NPV is NPvnew 7724000 389175PVIFA1578 NPvnew 102235335 So the change in NPV for every unit change in sales is ANPVAS 102235335 7 8181802295000 7 90000 ANPVAS 40835 If sales were to drop by 500 units then NPV would drop by NPV drop 40835500 2041731 You may wonder why we chose 95000 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 30 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 30 is OCFnew 43 7 3090000 7 780000065 03590500 OCFnew 285175 So the change in OCF for a 1 change in variable costs is AOCFAv 285175450 7 34367530 7 29 AOCFAv 758500 If variable costs decrease by 1 then OCF would increase by 5 8500 CHAPTER 11 B 205 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 OCFbest 7 431 1 7 2909900001 1 7 78000009065 03590500 OCFbest 7 939595 The bestcase NPV is NPVbes 7 724000 939595PV1FA1578 NPVbes 7 349226485 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 OCFworsc 4309 7 29119000009 7 78000011065 03590500 OCFW 7 168005 The worstcase NPV is NPVWO 7 724000 7 168005PV1FA1578 NPVWO 7 147789245 The cash breakeven equation is QC FCP 7V And the accounting breakeven equation is QA FC t DP 7quot Using these equations we nd the following cash and accounting breakeven points 1 QC 7 141W3020 7 2275 QA 7 14M 7 65M3020 7 2275 QC 7 18792 QA 7 27517 2 QC 7 7300038 7 27 QA 73000 15000038 7 27 Qc 6636 QA 7 20273 3 QC 7 1200117 4 QA 7 1200 8401174 QC 7171 QA 7 291 B206 SOLUTIONS 9 We can use the accounting breakeven equation QA FC DP 7V to solve for the unknown variable in each case Doing so we nd 1 QA 7 112800 7 820000 D417 30 D 7 420800 2 QA 165000 32M 115l1P7 43 P 6936 3 QA 4385 160000 10500098 7V V 3757 The accounting breakeven for the project is QA 6000 18000457 7 32 A 540 And the cash breakeven is Qc 900057 7 32 QC 360 At the nancial breakeven the project will have a zero NPV Since this is true the initial cost of the project must be equal to the PV of the cash ows of the project Using this relationship we can find the OCF of the project must be NPV 0 implies 18000 OCFPVIFA1274 OCF 592622 Using this OCF we can find the financial breakeven is QF 9000 59262257 7 32 597 And the DOL of the project is DOL 1 9000592622 2519 In order to calculate the financial breakeven we need the OCF of the project We can use the cash and accounting breakeven points to find this First we will use the cash breakeven to find the price of the product as follows QC FC P 7 V 13200 140000P 7 24 P 3461 H N CHAPTER 11 B207 Now that we know the product price we can use the accounting breakeven equation to nd the depreciation Doing so we nd the annual depreciation must be QA PC DP7v 15500 140000 D34617 24 Depreciation 24394 We now know the annual depreciation amount Assuming straightline depreciation is used the initial investment in equipment must be ve times the annual depreciation or Initial investment 524394 121970 The PV of the OCF must be equal to this value at the nancial breakeven since the NPV is zero so 121970 OCFPVIFA1575 OCF 3725069 We can now use this OCF in the nancial breakeven equation to find the nancial breakeven sales quantity is QF 7 140000 372506934617 24 QF 7 16712 We know that the DOL is the percentage change in OCF divided by the percentage change in quantity sold Since we have the original and new quantity sold we can use the DOL equation to nd the percentage change in OCF Doing so we nd DOL AOCF AQ Solving for the percentage change in OCF we get AOCF DOLAQ AOCF 34070000 7 6500065000 AOCF 2615 or 2615 The new level of operating leverage is lower since FCOCF is smaller Using the DOL equation we nd DOL1FCOCF 340 1 130000OCF OCF 54167 The percentage change in quantity sold at 58000 units is AQ 58000 7 65000 65000 AQ 71077 or 71077 B208 SOLUTIONS p n 43 So using the same equation as in the previous problem we nd AOCF 34071077 AQ 73662 So the new OCF level will be New OCF 1 7 366254167 New OCF 34333 And the new DOL will be New DOL 1 13000034333 New DOL 4786 The DOL of the project is DOL 1 7300087500 DOL 18343 If the quantity sold changes to 8500 units the percentage change in quantity sold is AQ 8500 7 80008000 AQ 0625 or 625 So the OCF at 8500 units sold is AOCF DOLAQ AOCF 183430625 AOCF 1146 or 1146 This makes the new OCF New OCF 8750011146 New OCF 97531 And the DOL at 8500 units is DOL 1 7300097531 DOL 17485 We can use the equation for DOL to calculate xed costs The xed cost must be DOL 235 1 FCOCF FC 235 7141000 FC 5 8080 If the output rises to 11000 units the percentage change in quantity sold is AQ 11000 71000010000 AQ 10 or 1000 CHAPTER 11 B209 The percentage change in OCF is AOCF 7 23510 AOCF 7 2350 or 2350 So the operating cash ow at this level of sales will be OCF 7 430001235 OCF 7 53105 If the output falls to 9000 units the percentage change in quantity sold is AQ 9000 7 1000010000 AQ 710 or 71000 The percentage change in OCF is AOCF 7 2357 10 AOCF 7 72350 or 72350 So the operating cash ow at this level of sales will be OCF 430001 7 235 OCF 32897 Using the equation for DOL we get DOL 1 FCOCF At 11000 units DOL 1 5805053105 DOL 20931 At 9000 units DOL 1 5805032895 DOL 27647 Intermediate 1 At the accounting breakeven the IRR 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 7 I 1NPVIFARYN 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 B210 SOLUTIONS 17 p A on 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 flows 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 110000 units will be OCF P 7vQ 7 FC17tctcD OCF 32 7 19110000 7 210000066 0344900004 OCF 846850 We will calculate the OCF at 111000 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 32 7 19111000 7 210000066 0344900004 OCF 7 855430 The sensitivity of the OCF to changes in the quantity sold is Sensitivity AOCFAQ 846850 7 855430110000 7 111000 AOCFAQ 858 OCF will increase by 528 for every additional unit sold At 110000 units the DOL is DOL 1 FCOCF DOL 1 210000846850 DOL 12480 The accounting breakeven is QA PC DP 7v QA 210000 490000432 7 19 Q A 25576 And at the accounting breakeven level the DOL is DOL 1 2100004900004 DOL 27143 CHAPTER 11 B 211 The basecase bestcase and worstcase values are shown below Remember that in the bestcase sales and price increase while costs decrease In the worstcase sales and price decrease and costs increase Scenario Unit sales Variable cost Fixed costs Base 190 11200 410000 Best 209 10080 369000 Worst 171 12320 451000 Using the tax shield approach the OCF and NPV for the base case estimate is OCFbase 18000 7 11200190 7 410000065 03517000004 OCFbase 722050 NPVbse 71700000 722050PVIFA1274 NPVbse 49311810 The OCF and NPV for the worst case estimate are OCFworst 18000 7 12320171 7 451000065 03517000004 0013mm 486932 NPVWO 71700000 486932PV1FA1274 NPVWO 722101741 And the OCF and NPV for the best case estimate are OCFbest 18000 7 10080209 7 369000065 03517000004 OCFbest 984832 NPVbes 71700000 984832Pv1FA12 NPVbes 129127883 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 420000 The OCF using this level of xed costs and the other base case values with the tax shield approach we get OCF 18000 7 11200190 7 410000065 03517000004 OCF 715550 And the NPV is NPV 71700000 715550Pv1FA12 NPV 47337532 The sensitivity of NPV to changes in fixed costs is ANPVAFC 49311810 7 47337532410000 7 420000 ANPVAFC 71974 For every dollar F C increase NPV falls by 1974 B212 SOLUTIONS c The cash breakeven is QC FCP 7v Qc 41000018000 711200 QC 60 d The accounting breakeven is QA FC DP7v QA 410000 1700000418000 711200 Q A 123 At the accounting breakeven the DOL is DOL 1 FCOCF DOL 1 410000425000 19647 For each 1 increase in unit sales OCF will increase by 19647 20 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 x 51000 38250000 Exp clubs 1200 x 711000 713200000 Cheap clubs 420 x 9500 3990000 29040000 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 anymore we will save these variable costs which is an in ow So Var costs New clubs 7330 x 51000 716830000 Exp clubs 7650 x 711000 7150000 Cheap clubs 7190 x 9500 71805000 11485000 The pro forma income statement will be Sales 29040000 Variable costs 11485000 Costs 8100000 Depreciation 3200000 EBT 625 5000 Taxes 2502000 Net income 3753000 H CHAPTER 11 B 213 Using the bottom up OCF calculation we get OCF 7 N1 Depreciation 7 3753000 3200000 OCF 7 6953000 So the payback period is Payback period 3 2791000 6953000 Payback period 3401 years The NPV is NPV 7 2400000 7 1250000 7 6953000Pv1FA107 12500001107 NPV 7 1084156369 And the IRR is IRR 722400000 7 1250000 6953000PVIFAmR7 12500001RR7 IRR 2264 The best case and worst cases for the variables are Base Case Best Case Worst Case Unit sales new 51000 56100 45900 Price new 750 825 675 VC new 330 297 363 Fixed costs 8100000 7290000 8910000 Sales lost expensive 11000 9900 12100 Sales gained cheap 9500 10450 8550 Bestcase 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 750 x 56100 46282500 Exp clubs 1200 x 79900 7 11880000 Cheap clubs 420 x 10450 4389000 38791500 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 anymore we will save these variable costs which is an in ow So Var costs New clubs 7297 x 56100 716661700 Exp clubs 7650 x 79900 6435000 Cheap clubs 7190 x 10450 7 1985500 712212200 B214 SOLUTIONS The pro forma income statement will be Sales 38791500 Variable costs 12212200 Costs 7290000 Depreciation 3200000 16089300 Taxes 643 5720 Net income 9653580 Using the bottom up OCF calculation we get OCF Net income Depreciation 9653580 3200000 OCF 12853580 And the bestcase NPV is NPV 7 2400000 7 1250000 12853580PVIFA1077 12500001107 NPV 7 3956805839 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 Sales New clubs 675 x 45900 30982500 Exp clubs 1200 x 7 12100 7 14520000 Cheap clubs 420 x 8550 3591000 20053500 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 anymore we will save these variable costs which is an in ow So Var costs New clubs 7363 x 45900 716661700 Exp clubs 7650 x 712100 7865000 Cheap clubs 7190 x 8550 7 1624500 710421200 The pro forma income statement will be Sales 20053500 Variable costs 10421200 Costs 8910000 Depreciation 3200000 7 2477700 Taxes 991080 assumes a tax credit Net income 71486620 N Using the bottom up OCF calculation we get OCF N1 Depreciation 1486620 3200000 OCF 1713380 And the worstcase NPV is NPV 722400000 7 1250000 1713380PVIFA1077 12500001107 NPV 71466710092 CHAPTER 11 B 215 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 800 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 800 x 51000 40800000 Exp clubs 1200 gtlt 711000 713200000 Cheap clubs 420 x 9500 3990000 3 15 90000 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 anymore we will save these variable costs which is an in ow So Var costs New clubs 7330 x 51000 716830000 Exp clubs 7650 x 711000 7150000 Cheap clubs 7190 x 9500 71805000 711485000 The pro forma income statement will be Sales 31590000 Variable costs 11485000 Costs 8100000 Depreciation 3200000 EBT 8805000 Taxes 3 522 000 Net income i 5283000 Using the bottom up OCF calculation we get OCF N1 Depreciation 5283000 3200000 OCF 8483000 B216 SOLUTIONS And the NPV is NPV 722400000 7 1250000 8483000PVIFA1077 12500001107 NPV 1829024448 So the sensitivity of the NPV to changes in the price of the new club is ANPVAP 1084156369 7 1829024448750 7 800 ANPVAP 14897362 For every dollar increase decrease in the price of the clubs the NPV increases decreases by 14897362 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 52000 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 52000 39000000 Exp clubs 1200 x 711000 713200000 Cheap clubs 420 x 9500 3990000 29790000 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 anymore we will save these variable costs which is an in ow So Var costs New clubs 7330 x 52000 717160000 Exp clubs 7650 x 711000 7150000 Cheap clubs 7190 x 9500 71805000 711815000 The pro forma income statement will be Sales 29790000 Variable costs 11815000 Costs 8100000 Depreciation 3200000 EBT 6675000 Taxes 2670000 Net income i 4005000 CHAPTER 11 B 217 Using the bottom up OCF calculation we get OCF N1 Depreciation 4005000 3200000 OCF 7205000 The NPV at this quantity is NPV 722400000 7 1250000 7205000PVIFA1077 12500001107 NPV 1206840523 So the sensitivity of the NPV to changes in the quantity sold is ANPVAQ 1084156369 7 120684052351000 7 52000 ANPVAQ 122684 For an increase decrease of one set of clubs sold per year the NPV increases decreases by 122684 a First we need to determine the total additional cost of the hybrid The hybrid costs more to purchase and more each year so the total additional cost is Total additional cost 5450 6400 Total additional cost 7850 Next we need to determine the cost per mile for each vehicle The cost per mile is the cost per gallon of gasoline divided by the miles per gallon or Cost per mile for traditional 36023 Cost per mile for traditional 0156522 Cost per mile for hybrid 360 25 Cost per mile for hybrid 0 144000 So the savings per mile driven for the hybrid will be Savings per mile 0 156522 7 0144000 Savings per mile 0012522 We can now determine the breakeven point by dividing the total additional cost by the savings per mile which is Total breakeven miles 7850 0012522 Total breakeven miles 626910 So the miles you would need to drive per year is the total breakeven miles divided by the number of years of ownership or Miles per year 626910 miles 6 years Miles per year 104485 milesyear B218 SOLUTIONS b First we need to determine the total miles driven over the life of either vehicle which will be Total miles driven 615000 Total miles driven 90000 Since we know the total additional cost of the hybrid from part a we can determine the necessary savings per mile to make the hybrid financially attractive The necessary cost savings per mile will be Cost savings needed per mile 7850 90000 Cost savings needed per mile 008722 Now we can nd the price per gallon for the miles driven If we let P be the price per gallon the necessary price per gallon will be P23 7P25 008722 P123 7125 008722 P 8 c To nd the number of miles it is necessary to drive we need the present value of the costs and savings to be equal to zero If we let MDPY equal the miles driven per year the breakeven equation for the hybrid car as Cost 0 5450 7 400PVIFA1076 001252214DPYPVIFA106 The savings per mile driven 0012522 is the same as we calculated in part 1 Solving this equation for the number of miles driven per year we nd 0012522MDPYPVIFA1076 719210 MDPYPVIFA1076 57436944 Miles driven per year 131879 To nd the cost per gallon of gasoline necessary to make the hybrid break even in a financial sense if we let CSPG equal the cost savings per gallon of gas the cost equation is Cost 0 5450 7 400PVIFA1076 CSPG15000PV1FA1076 Solving this equation for the cost savings per gallon of gas necessary for the hybrid to breakeven from a financial sense we nd CSPG15000PVIFA105 719210 CSPGPVIFA1076 047947 Cost savings per gallon of gas 0110091 Now we can nd the price per gallon for the miles driven If we let P be the price per gallon the necessary price per gallon will be P23 7P25 0110091 P123 7125 0110091 P 3165 CHAPTER 11 B 219 The implicit assumption in the previous analysis is that each car depreciates by the same dollar amount The cash ow per plane is the initial cost divided by the breakeven number of planes or Cash ow per plane l3000000000 249 Cash ow per plane 52208835 In this case the cash ows are a perpetuity Since we know the cash ow per plane we need to determine the annual cash ow necessary to deliver a 20 percent return Using the perpetuity equation we nd PV C R 13000000000 C 20 C 2600000000 This is the total cash ow so the number of planes that must be sold is the total cash ow divided by the cash ow per plane or Number ofplanes 2600000000 52208835 Number of planes 4980 or about 50 planes per year In this case the cash ows are an annuity Since we know the cash ow per plane we need to determine the annual cash ow necessary to deliver a 20 percent return Using the present value of an annuity equation we nd PV CPVIFA2010 13000000000 CPVIFA2010 c 3100795839 This is the total cash ow so the number of planes that must be sold is the total cash ow divided by the cash ow per plane or Number ofplanes 3100795839 52208835 Number of planes 5939 or about 60 planes per year Challenge 25 a The tax shield de nition of OCF is OCF PivQ iFC17tcth Rearranging and solving for Q we nd OCF itCD17tc Piv Q irc Q FC t OCF tCD1 tclP V B220 SOLUTIONS b The cash breakeven is 5 1 Qc 50000040000 7 20000 Qc And the accounting breakeven is QA 7 500000 700000 7 70000003806240000 7 20000 A 7 60 The nancial breakeven is the point at which the NPV is zero so OCFF 7 3500000PVIFA2MS OCFF 7 117032896 So QF FC OCF 7tc X Dl 7tcP 7V QF 500000 117032896 7 38700000l 7 3840000 7 20000 QF 9793 m 98 At the accounting breakeven point the net income is zero This using the bottom up definition of OCF OCF N1 D We can see that OCF must be equal to depreciation So the accounting breakeven is QA FC t 13 7th17tP 7V QA FC DP 7V QA FC OCFP7V The tax rate has cancelled out in this case 26 The DOL is expressed as DOL 7 AOCF AQ DOL OCF17 OCFoOCFo Q1 7 QoQo The OCF for the initial period and the rst period is OCFI P7VQ1 7FC1 7tc tCD OCFo P VQ0 7 FC17tcth The difference between these two cash ows is OCFI OCFO P VXI tCXQI Q0 CHAPTER 11 B 221 Dividing both sides by the initial OCF we get OCFI 7 OCF0OCF0 P7v 17 tcQ1 7 Q0 OCFO Rearranging we get OCF1 7 OCFoOCFoQ1 7 QoQo P V1 tcQolOCF0 OCFo 7 th t FC1 7 tOCF0 DOL 1 FC1 7 t 7 thOCF0 27 1 Using the tax shield approach the OCF is OCF 7 230 7 18535000 7 450000062 03832000005 OCF 7 940700 And the NPV is NPV 73200000 7 360000 940700PVIFA1375 360000 50000017 381135 NPV 11230860 b In the worstcase the OCF is OCFworst 7 23009 7 18535000 7 450000062 03836800005 OCFworst 7 478080 And the worstcase NPV is NPVWOm 73680000 7 360000l05 478080PVIFA1375 360000105 50000008517381135 NPVWOm 7202830158 The bestcase OCF is OCFbest 7 2301 1 7 18535000 7 450000062 03827200005 OCFbest 7 1403320 And the bestcase NPV is NPVbest 7 2720000 7 360000095 1403320PV1FA1375 360000095 5000001151738l135 NPVbest 225291879 28 To calculate the sensitivity to changes in quantity sold we will choose a quantity of 36000 The OCF at this level of sale is OCF 7 230 7 18536000 7 450000062 03832000005 OCF 7 968600 B222 SOLUTIONS The sensitivity of changes in the OCF to quantity sold is AOCFAQ 968600 7 94070036000 7 35000 AOCFAQ 2790 The NPV at this level of sales is NPV 73200000 7 360000 968600PVIFA1375 360000 500000l 7381135 NPV 21043936 And the sensitivity of NPV to changes in the quantity sold is ANPVAQ 21043936 7 1123086036000 7 35000 ANPVAQ 9813 You wouldn t want the quantity to fall below the point where the NPV is zero We know the NPV changes 9813 for every unit sale so we can divide the NPV for 35000 units by the sensitivity to get a change in quantity Doing so we get 11230860 9813AQ AQ 1144 For a zero NPV we need to decrease sales by 1144 units so the minimum quantity is QM 35000 71144 QMm 33856 At the cash breakeven the OCF is zero Setting the tax shield equation equal to zero and solving for the quantity we get OCF 0 230 7 185Qc 7 450000062 03832000005 QC 1283 The accounting breakeven is QA 450000 32000005230 7 185 QA 24222 From Problem 28 we know the nancial breakeven is 33856 units CHAPTER 11 B 223 30 Using the tax shield approach to calculate the OCF the DOL is DOL 1 45000017 038 7 03832000005 940700 DOL 103806 Thus a 1 rise leads to a l03806 rise in OCF IfQ rises to 36000 then The percentage change in quantity is AQ 36000 7 3500035000 02857 or 2857 So the percentage change in OCF is AOCF 2857 103806 AOCF 29659 From Problem 26 AOCFOCF 968600 7 940700940700 AOCFOCF 0029659 In general ifQ rises by 1000 units OCF rises by 29659 CHAPTER 12 SOME LESSONS FROM CAPITAL 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 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 semistrong form efficiency Ignoring trading costs on average such investors merely earn what the market offers stock investments all have a 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 financial 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 CHAPTER 12 B 225 b Under 2 if the market is not semistrong form ef cient then this information could be used to buy the stock cheap before the rest of the market discovers the nancial 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 profit opportunity exists c Under 3 if the market is not strong form ef cient 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 pro t 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 profit opportunity exists Under 4 this information does not signal any profit opportunity for traders any pertinent information the managerinsiders may have is fully re ected in the current share prrce 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 1 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 102 791 240 91 1473 or 1473 2 The dividend yield is the dividend divided by price at the beginning of the period price so Dividend yield 240 91 0264 or 264 And the capital gains yield is the increase in price divided by the initial price so Capital gains yield 102 7 91 91 1209 or 1209 3 Using the equation for total return we nd R 83 7 91 240 91 70615 or 415 And the dividend yield and capital gains yield are Dividend yield 240 91 0264 or 264 Capital gains yield 83 791 91 70879 or 7879 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 B226 SOLUTIONS 4 The total dollar return is the increase in price plus the coupon payment so Total dollar return 1070 7 1040 70 100 The total percentage return of the bond is R 1070 7 1040 70 1040 0962 or 962 Notice here that we could have simply used the total dollar return of 100 in the numerator of this equation Using the Fisher equation the real return was 1R1r1h r 10962 104 7 1 0540 or 540 5 The nominal return is the stated return which is 1230 percent Using the Fisher equation the real return was 1R1r1h r 1123103171 0892 or 892 6 Using the Fisher equation the real returns for longterm government and corporate bonds were 1R1r1h rG 10581031 7 1 0262 or 262 rc 106210317 1 0301 or 301 7 The average return is the sum of the returns divided by the number of returns The average return for each stock was N 1be Nwo7gom7go 11 CHAPTER 12 B 227 N 7 Nw14600r1460 y 5 11 Remembering back to sadistics we calculate the variance of each stock as 0 gm ifzN71 a kosamsy 2170782 1770782 71670782 0970782020670 a 1671462 3871462 1471462 72171462 2671462 048680 The standard deviation is the square root of the variance so the standard deviation of each stock is 0 020670 2 1438 or 1438 Ty 048680 2 2206 or 2206 We will calculate the sum of the returns for each asset and the observed risk premium rst Doing so we get Year Large co stock return Tbill return Risk premium 1970 394 650 7256 1971 1430 436 994 1972 1899 423 1476 1973 71469 729 72198 1974 72647 799 73 446 1975 m M M 3330 3624 7294 a The average return for large company stocks over this period was Large company stocks average return 3330 6 555 And the average return for Tbills over this period was Tbills average return 3624 6 604 B228 SOLUTIONS b Using the equation for variance we nd the variance for large company stocks over this period was Variance 150394 7 05552 1430 7 05552 1899 7 05552 71469 7 05552 72647 7 05552 3723 7 0555Z Variance 0053967 And the standard deviation for large company stocks over this period was Standard deviation 0053967 2 02323 or 2323 Using the equation for variance we nd the variance for Tbills over this period was Variance 150650 7 06042 0436 7 06042 0423 7 06042 0729 7 06042 0799 7 06042 0587 7 06042 Variance 0000234 And the standard deviation for Tbills over this period was Standard deviation 0000234 2 00153 or 153 c The average observed risk premium over this period was Average observed risk premium 7294 6 449 The variance of the observed risk premium was Variance 1570256 7 700492 09947 700492 1476 7 700492 72198 7 700492 73446 7 700492 3136 7 700492 Variance 0059517 And the standard deviation of the observed risk premium was Standard deviation 0059517 2 02440 or 2440 d 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 9 a To nd the average return we sum all the returns and divide by the number of returns so Average return 07 712 1138 145 1160 or 1160 11 p A N CHAPTER 12 B 229 b Using the equation to calculate variance we nd Variance 140771162 71271162 1171162 38 7 1162 14 7 1162 Variance 0032030 So the standard deviation is Standard deviation 003230 2 01790 or 1790 a To calculate the average real return we can use the average return of the asset and the average in ation in the Fisher equation Doing so we nd 1R1r1h F 116010357 1 0783 or 783 b The average risk premium is simply the average return of the asset minus the average riskfree rate so the average risk premium for this asset would be 7 if 7 1160 7 042 7 0740 or 740 We can nd the average real riskfree rate using the Fisher equation The average real riskfree rate was 1R1r1h if 10421035 7 1 0068 or 068 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 F1747 E1 7837068 715 Tbill rates were highest in the early eighties This was during a period of high in ation and is consistent with the Fisher effect B230 SOLUTIONS p A 43 Intermediate To nd the real return we first need to nd the nominal return which means we need the current price of the bond Going back to the chapter on pricing bonds we nd the current price is P1 80PVIFA776 1000PVIF776 104767 So the nominal return is R 104767 7 1030 801030 0948 or 948 And using the Fisher equation we nd the real return is 17R1r1h r 109481042 7 1 0507 or 507 Here we know the average stock return and four of the five 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 525 077127 18 19R R 205 or 205 The missing return has to be 205 percent Now we can use the equation for the variance to nd Variance 1407 7 1052 7127 1052 18 7 1052 19 7 1052 205 7 1052 Variance 0018675 And the standard deviation is Standard deviation 0018675 2 01367 or 1367 The arithmetic average return is the sum of the known returns divided by the number of returns so Arithmetic average return 03 38 21 7 15 29 7 13 6 Arithmetic average return 1050 or 1050 Using the equation for the geometric return we nd Geometric average return 1R1 X 1 R2 gtlt X 1 R7llT 1 Geometric average return 7 1031381 2117151 291713 6L 1 Geometric average return 0860 or 860 Remember the geometric average return will always be less than the arithmetic average return if the returns have any variation CHAPTER 12 B231 16 To calculate the arithmetic and geometric average returns we must first calculate the return for each year The return for each year is R1 7366 7 6018 060 6018 2340 or 2340 R2 9418 7 7366 064 7366 2873 or 2873 R3 8935 7 9418 072 9418 70436 or 7436 R4 78497 8935 080 8935 71126 or 1126 Rs 9505 7 7849 120 7849 2263 or 1263 The arithmetic average return was RA 02340 02873 700436 7 01126 022635 01183 or 1183 And the geometric average return was RG 12340128731704361 711261 226315 71 01058 or 1058 17 Looking at the longterm corporate bond return history in Figure 1210 we see that the mean return was 62 percent with a standard deviation of 84 percent In the normal probability distribution approximately 23 of the observations are within one standard deviation of the mean This means that 1 3 of the observations are outside one standard deviation away from the mean Or PrRlt 722 or Rgt 146 m 13 But we are only interested in one tail here that is returns less than 722 percent so PrRlt 722 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 722 7 6284 7100 Looking at the Ztable this gives a probability of 1587 or PrRlt 722 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 t 20 62i 284 71060 to 2300 B232 SOLUTIONS 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 u at 30 62 i 384 71900 to 3140 18 The mean return for small company stocks was 171 percent with a standard deviation of 326 percent Doubling your money is a 100 return so if the return distribution is normal we can use the Z statistic So Z X 7 uc Z 100 7 171326 2543 standard deviations above the mean This corresponds to a probability of m 055 or once every 200 years Tripling your money would be Z 200 7 171326 5610 standard deviations above the mean This corresponds to a probability of about 000001 or about once every 1 million years 19 It is impossible to lose more than 100 percent of your investment Therefore retum distributions are truncated on the lower tail at 7100 percent 20 To nd the best forecast we apply Blume s formula as follows R5 53 91 x 119 39 x 153 1495 R10 E x119 w x 153 1452 39 39 201 4020 R20 7 X119 7 X 153 1364 39 39 21 The best forecast for a one year return is the arithmetic average which is 123 percent The geometric average found in Table 124 is 104 percent To nd the best forecast for other periods we apply Blume s formula as follows R57 5391 x104 E x12371221 821 821 R20 20391 x 104 H x 123 1185 821 821 R30 30391 x 104 H x 123 1162 821 821 CHAPTER 12 B233 22 To nd the real return we need to use the Fisher equation Re writing the Fisher equation to solve for the real return we get r1R1h71 So the real return each year was b M Tbill return In ation Real return 1973 00729 00871 00131 1974 00799 01234 00387 1975 00587 0 0694 00100 1976 00507 0 0486 00020 1977 0 0545 0 0670 00117 1978 00764 0 0902 00127 1979 01056 0 1329 00241 1980 01210 0 1252 00037 06197 07438 01120 The average return for Tbills over this period was Average return 0619 8 Average return 0775 or 775 And the average in ation rate was Average in ation 07438 8 Average in ation 0930 or 930 Using the equation for variance we nd the variance for Tbills over this period was Variance 7 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 12347 09302 0694 7 09302 0486 7 09302 0670 7 09302 0902 7 09302 1329 7 09302 1252 7 09302 Variance 0000971 And the standard deviation of in ation was Standard deviation 7 0000971 2 Standard deviation 00312 or 312 B234 SOLUTIONS 23 N A The average observed real return over this period was Average observed real return 71122 8 Average observed real return 70140 or 7140 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 in ation risk ie the purchasing power of the investment can actually decline over time even if the investor is earning a positive return Challenge Using the Z statistic we find Z X7u6 Z 0 7123200 41615 PrR 0 7 2693 For each of the questions asked here we need to use the Zstatistic which is Z X7u6 a 21 10 7 6284 04524 This z statistic 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 PrR210 17PrR 10 176745 m 3255 For a return greater than 0 percent 22 0 7 6284 417381 PrR210 17PrR 10 17 7698 m 2302 The probability that Tbill returns will be greater than 10 percent is 23 7 1073831 2 PrR210 7 1 7PrR 10 7 1 7 9772 u 228 CHAPTER 12 B 235 And the probability that Tbill returns will be less than 0 percent is Z4 0 7 383 1 712258 PrR 0 m 1101 c The probability that the return on longterm corporate bonds will be less than 4118 percent is 25 41 18 7 6284 712357 PrR 74 18 m 1083 And the probability that Tbill returns will be greater than 1056 percent is 25 1056 7 3831 21806 PrR21056 7 17PrR 1056 7 179823 z 146 CHAPTER 13 RISK RETURN AND THE SECURITY MARKET LINE 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 If the market expected the growth rate in the coming year to be 2 percent then there would be no change in security prices if this expectation had been fully anticipated and priced However if the market had been expecting a growth rate other than 2 percent and the expectation was incorporated into security prices then the govemment s announcement would most likely cause security prices in general to change prices would drop if the anticipated growth rate had been more than 2 percent and prices would rise if the anticipated growth rate had been less than 2 percent a systematic b unsystematic 0 both probably mostly systematic d unsystematic e unsystematic f systematic a a change in systematic risk has occurred market prices in general will most likely decline b no change in unsystematic risk company price will most likely stay constant c no change in systematic risk market prices in general will most likely stay constant d a change in unsystematic risk has occurred company price will most likely decline 8 no change in systematic risk market prices in general will most likely stay constant No to both questions The portfolio expected return is a weighted average of the asset 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 CHAPTER 13 B237 Such layoffs generally occur in the context of corporate restructurings To the extent that the market views a restructuring as valuecreating stock prices will rise So it s not layoffs per se that are being cheered on Nonetheless Wall Street does encourage corporations to takes actions to create value even if such actions involve layoffs Earnings contain information about recent sales and costs This information is useful for projecting future growth rates and cash ows Thus unexpectedly low earnings often lead market participants to reduce estimates of future growth rates and cash flows price drops are the result The reverse is often true for unexpectedly high earnings 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 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 18045 14027 11880 The portfolio weight for each stock is WeightA 1804511880 6818 WeightB 1402711880 3182 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 2950 3700 6650 So the expected return of this portfolio is ERp 29506650011 37006650015 1323 or 1323 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 6009 2517 1513 1160 or 1160 B238 SOLUTIONS 4 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 124 14wX 10517wX We can now solve this equation for the weight of Stock X as 124 14wX 105 7105wX 019 035wX wX 0542857 So the dollar amount invested in Stock X is the weight of Stock X times the total portfolio value or Investment in X 054285710000 542857 And the dollar amount invested in Stock Y is Investment in Y 1 7 054285710000 457443 5 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 25708 7 7521 1375 or 1375 6 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 20705 7 5012 3025 1250 or 1250 7 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 1505 6508 2013 0855 or 855 ERB 15717 6512 2029 1105 or 1105 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 then add all of these up The result is the variance So the variance and standard deviation of each stock is 5A2 1505 7 08552 6508 7 08552 2013 7 08552 00060 5A 00060 2 0246 or 246 CHAPTER 13 B 239 c552 157 17 7 11052 65 12 7 11052 20297 11052 01830 GB 01830 2 1353 or 1353 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 2508 5515 2024 1505 or 1505 If we own this portfolio we would expect to get a return of 1505 percent a To nd 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 1s 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 nd ERp 351833 650333 0858 or 858 This portfolio does not have an equal weight in each asset We still need to find 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 2007 20 15 6033 2420 or 2420 Bust ERp 2013 2003 60706 70040 or 040 And the expected return of the portfolio is ERp 352420 657004 0821 or 821 To nd the variance we find 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 a 352420 7 08212 6570040 7 08212 013767 B240 SOLUTIONS 10 11 p n N a This portfolio does not have an equal weight in each asset We first 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 40715 30705 70720 or 7720 Bust ERp 30706 40730 30709 71650 or 71650 And the expected return of the portfolio is ERp 153690 451210 3570720 0571650 0764 or 764 b To calculate the standard deviation we first need to calculate the variance To nd the variance we find 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 a 153690 7 07642 451210 7 07642 3570720 7 07642 0571650 7 07642 2 7 c 7 02436 c 02436 2 1561 or 1561 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 0p 2584 20117 15111 40136 115 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 1301313813Bx Solving for the beta of Stock X we get BX 162 p A 43 CHAPTER 13 B 241 CAPM states the relationship between the risk of an asset and its expected return CAPM is ERa Rf t ERM Rf X Ba Substituting the values we are given we nd ER 052 117052105 1129 or 1129 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 ER 102 045 0858 B 067 Here we need to nd the expected return of the market using the CAPM Substituting the values given and solving for the expected return of the market we nd ERi 135 055 ERM 7 055117 ERM 1234 or 1234 Here we need to find the riskfree rate using the CAPM Substituting the values given and solving for the riskfree rate we nd E01 14 Rf 115 in145 14 Rf 16675 7145Rf Rf 0594 or 594 1 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 16 0482 1040 or 1040 B242 SOLUTIONS b We need to nd the portfolio weights that result in a portfolio with a B of 095 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 095 ws135 17ws0 095 135ws 0 7 0ws ws 095135 wS 7037 And the weight of the riskfree asset is WM 17 7037 2963 c We need to find the portfolio weights that result in a portfolio with an expected return of 8 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 08 16ws 04817ws 08 16ws 048 7 048ws 032 ll2ws ws 7 2857 So the B ofthe portfolio will be Bp 28571351728570 0386 d Solving for the B of the portfolio as we did in part a we find Bp 270 ws13517ws0 ws 270135 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 nd 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 1510 7 wWl25 1 7wW0 7 l25ww 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 CHAPTER 13 B 243 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 1s ERW 152 053 MRP125 MRP 099125 0792 or 792 So now we know the CAPM equation for any stock is ERp 053 0793310 The slope of the SML is equal to the market risk premium which is 00792 Using these equations to ll in the table we get the following results WW ERp Bp 000 530 0000 2500 778 0313 5000 1025 0625 7500 1273 0938 10000 1520 1250 12500 1768 1563 15000 2015 1875 There are two ways to correctly answer this question We will work through both First we can use the CAPM Substituting in the value we are given for each stock we nd ERy 08 075130 1775 or 1775 It is given in the problem that the expected return of Stock Y is 185 percent but according to the CAPM the return of the stock based on its level of risk the expected return should be 1775 percent This means the stock return is too high given its level of risk Stock Y plots above the SML and is undervalued In other words its price must increase to reduce the expected return to 1775 percent For Stock Z we find ERz 08 075070 1325 or 1325 The return given for Stock Z is 121 percent but according to the CAPM the expected return of the stock should be 1325 percent based on its level of risk Stock Z plots below the SML and is overvalued In other words its price must decrease to increase the expected return to 1325 percent B244 SOLUTIONS N p A We can also answer this question using the rewardtorisk ratio All assets must have the same reward torisk ratio The rewardtorisk ratio is the risk premium of the asset divided by its B 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 0075 or 75 percent Calculating the rewardtorisk ratio for Stock Y we nd Rewardtorisk ratio Y 185 708 130 0808 The rewardtorisk ratio for Stock Y 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 reward torisk ratio For Stock Z we find Rewardtorisk ratio Z 121 7 08 70 0586 The rewardtorisk ratio for Stock Z 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 reward torisk ratio We need to set the rewardtorisk ratios of the two assets equal to each other which is 185 7 Rf130 1217Rf070 We can cross multiply to get 070185 7Rf 1301217Rf Solving for the riskfree rate we find 01295 7 070Rf 01573 7 130Rf Rf 0463 or 463 Intermediate For a portfolio that is equally invested in largecompany stocks and longterm bonds Return 1230 7 5802 905 For a portfolio that is equally invested in small stocks and Treasury bills Return 1710 3802 1045 CHAPTER 13 B 245 22 We know that the rewardtorisk ratios for all assets must be equal This can be expressed as ERA erBA ERB RfBB The numerator of each equation is the risk premium of the asset so RPABA 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 11 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 424 436 255 3500 or 3500 Normal ERp 417 413 209 1380 or 1380 Bust ERp 400 4728 2745 72020 or 72020 And the expected return of the portfolio is ERp 3535 50138 157202 1612 or 1612 To calculate the standard deviation we rst need to calculate the variance To nd the variance we find 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 3535 7 16122 50138 7 16122 157202 7 16122 62p 03253 c 03253 2 1804 or 1804 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 in 1612 7 0380 1232 or 1232 B246 SOLUTIONS c The approximate expected real return is the expected nominal return minus the in ation rate so Approximate expected real return 1612 7 035 1262 or 1262 To nd the exact real return we will use the Fisher equation Doing so we get 1 ERa 1 t h1 600 11612 103501 eri er1161210357 1 1219 or 1219 The approximate real risk premium is the expected return minus the riskfree rate so Approximate expected real risk premium 1612 7 038 1232 or 1232 The exact expected real risk premium is the approximate expected real risk premium divided by one plus the in ation rate so Exact expected real risk premium 11681035 1190 or 1190 24 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 find the weight of the third stock Doing so we find Bp 10 wA85 wB120 wc135 wa0 Solving for the weight of Stock C we nd wC 324074 So the dollar investment in Stock C must be Invest in Stock C 3240741000000 32407407 We know the total portfolio value and the investment of two stocks in the portfolio so we can find the weight of these two stocks The weights of Stock A and Stock B are wA 7 210000 1000000 7 210 wB 3200001000000 320 5quot CHAPTER 13 B 247 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 1 wA wB wC wa 17 210 73207 3240747wa wa 145 926 So the dollar investment in the riskfree asset must be Invest in riskfree asset 1459261000000 14592593 Challenge We are given the expected return of the assets in the portfolio We also know the sum of the weights of each asset must be equal to one Using this relationship we can express the expected return of the portfolio as ERp 185 wX172 136 185 wX17217Wx136 185 172wX 136 7 136wX 049 036wX wX 136111 And the weight of Stock Y is wy17136111 wy736111 The amount to invest in Stock Y is Investment in Stock Y 736111100000 Investment in Stock Y 73611111 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 To nd the beta of the portfolio we can multiply the portfolio weight of each asset times its beta and sum So the beta of the portfolio is 31 136111140736111095 BI 156 B248 SOLUTIONS 26 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 2511 5029 25l3 2050 or 2050 Using the CAPM to find the B of Stock 1 we find 2050 04 0881 B 206 The total risk of the asset is measured by its standard deviation so we need to calculate the standard deviation of Stock 1 Beginning with the calculation of the stock s variance we nd 02 2511720502 50297 20502 2513 7 20502 012 00728 01 0072812 0853 or 853 Using the same procedure for Stock 11 we nd the expected return to be ERH 25740 5010 2556 0900 Using the CAPM to find the B of Stock 11 we nd 0900 04 08811 in 063 And the standard deviation of Stock 11 is 6112 25740 7 09002 50 10 7 09002 2556 7 09002 on 11530 N on 11530 2 3396 or 3396 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 diversified away I is actually the riskier stock despite the lack of volatility in its retums StockI will have a higher risk premium and a greater expected return 27 28 CHAPTER 13 B249 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 risk premiums of the assets equal to each other and solving for the riskfree mte we nd 1327Rf135101712930 3001327120 135101 71 1056 78m 13635 7 13512f 5512703075 1270559 or 559 Now using CAPM to ndthe expected return on the market withboth stocks we nd 101 0559 80RM 70559 RM 1123 or 1123 132 0559 135RM 7 0559 RM 1123 or 1123 a 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 15708 7013 1548 1510 or 1510 E02 15705 7014 1529 1340 or 1340 b We can use the expected returns we calculated to nd 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 Expected Return Beta SlopeSML Rise Run SlopeSML Increase in expected return Increase inbeta SlopegML 7 1510 7 1340 25 SlopeSML 0680 or 680 B250 SOLUTIONS Since the market s beta is l and the riskfree rate has a beta of zero the slope of the Security Market 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 ERA 151 Rf 3 25MRP ERB 134 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 MRP 068 or 68 which is the same answer as our preVious result CHAPTER 4 COST OF CAPITAL Answers to Concepts Review and Critical Thinking Questions 1 It is the minimum rate of return the rm must earn overall on its existing assets If it earns more than this value is created Book values for debt are likely to be much closer to market values than are equity book values 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 aftertax equity costs The primary advantage of the DCF model is its simplicity The method is disadvantaged in that l the model is applicable only to firms that actually pay dividends many do not 2 even if a firm does pay dividends the DCF model requires a constant dividend growth rate forever 3 the estimated cost of equity from this method is very sensitive to changes in g which is a very uncertain parameter and 4 the model does not explicitly consider risk although risk is implicitly considered to the extent that the market has impounded the relevant risk of the stock into its market price While the share price and most recent dividend can be observed in the market the dividend growth rate must be estimated Two common methods of estimating g are to use analysts earnings and payout forecasts or to determine some appropriate average historical g from the rm s available data 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 dividend discount model 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 rm and the market s return data or by using beta estimates provided by analysts and investment firms The appropriate a eitax 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 firm 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 firms with the same credit rating assuming the film s private debt is rated or 3 consulting analysts and investment bankers Even if the debt is publicly traded an additional complication is when the firm has more than one issue outstanding these issues rarely have the same yield because no two issues are ever completely homogeneous B252 SOLUTIONS 9 This only considers the dividend yield component of the required return on equity 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 c Equity is inherently more risky 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 Iftaxes are considered in this case it can be seen that at reasonable tax rates the cost of equity does exceed the cost of debt 9 9 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 difficult 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 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 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 Here we have information to calculate the cost of equity using the CAPM The cost of equity is RE 053 105127 053 1234 or 1234 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 CHAPTER 14 B 253 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 two percent higher than average and the estimate from the dividend growth model is about one percent higher than the historical average so we cannot de nitively say one of the estimates is incorrect Given this we will use the average of the two so RE 1180 10582 1119 or 1119 To use the dividend growth model we first need to find the growth rate in dividends So the increase in dividends each year was g1 1127105105 0667 or 667 g2 119 7112112 0625 or 625 g3 130 7119119 0924 or 924 g 143 7130130 1000 or 1000 So the average arithmetic growth rate in dividends was g 0667 0625 0924 10004 0804 or 804 Using this growth rate in the dividend growth model we find the cost of equity is RE 143108044500 0804 1147 or 1147 Calculating the geometric growth rate in dividends we nd 143 7 1051 g4 g 0803 or 803 The cost of equity using the geometric dividend grth rate is RE 143108034500 0803 1146 or 1146 The cost of preferred stock is the dividend payment divided by the price so RP 696 0625 or 625 The pretax cost of debt is the YTM of the company s bonds so P0 1070 35PVIFAR730 1000PVIFR730 R 3137 YTM 2 X 3137 627 And the aftertax cost of debt is RD 06271 7 35 0408 or 408 B254 SOLUTIONS 7 a The pretax cost of debt is the YTM of the company s bonds so P0 950 40PV1FAR746 1000PVIFR746 R 4249 YTM 2 X 4249 850 b The aftertax cost of debt is RD 08501735 0552 or 552 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 80000000 35000000 115000000 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 in MVD 9580000000 6135000000 MVD 76000000 21350000 MVD 97350000 The YTM of the zero coupon bonds is PZ 610 1000PVIFR714 R 3594 YTM 2 X 3594 719 So the aftertax cost of the zero coupon bonds is RZ 07191 7 35 0467 or 467 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 0552769735 046721359735 0534 or 534 1 Using the equation to calculate the WACC we find WACC 6014 050635081735 1052 or 1052 b Since interest is tax deductible and dividends are not we must look at the aftertax cost of debt which is 081735 0520 or 520 Hence on an aftertax basis debt is cheaper than the preferred stock 10 H H CHAPTER 14 B 255 Here we need to use the debtequity ratio to calculate the WACC Doing so we find WACC 151165 09651651 7 35 1140 or 1140 Here we have the WACC and need to nd the debtequity ratio of the company Setting up the WACC equation we nd WACC 0890 12EV 079DV1 7 35 Rearranging the equation we find 0890VE 12 07965DE Now we must realize that the VE is just the equity multiplier which is equal to VE 1 D E 0890DE 1 12 05135DE Now we can solve for DE as 06765DE 031 DE 8234 a 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 110000006 66000000 BVD 70000000 55000000 125000000 So the total value of the company is V 66000000 125000000 191000000 And the book value weights of equity and debt are EV 66000000191000000 3455 DV liEV 6545 b The market value of equity is the share price times the number of shares so MVE 1100000068 748000000 Using the relationship that the total market value of debt is the price quote times the par value of the bond we nd the market value of debt is MvD 9370000000 10455000000 122300000 B256 SOLUTIONS 13 14 This makes the total market value of the company V 748000000 122300000 870300000 And the market value weights of equity and debt are EV 748000000870300000 8595 DV 17EV 1405 c The market value weights are more relevant 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 dividend growth model so RE 41010668 06 1239 or 1239 Next we need to nd the YTM on both bond issues Doing so we nd P1 930 35PV1FAR742 1000PVIFR742 R 3838 YTM 3838 X 2 768 P2 1040 40PVIFAR712 1000PVIFR712 R 3584 YTM 3584 X 2 717 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 wD1 9370000000122300000 5323 wD2 10455000000122300000 4677 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 173553230768 46770717 0484 or 484 Using these costs we have found and the weight of debt we calculated earlier the WACC is WACC 85951239 14050484 1133 or 1133 1 Using the equation to calculate WACC we find WACC 094 1205 14 1052051 7 35RD RD 0772 or 772 15 CHAPTER 14 B 257 b Using the equation to calculate WACC we find WACC 094 1205RE 105205068 RE 1213 or 1213 We will begin by nding the market value of each type of nancing We nd MVD 80001000092 7360000 MVE 25000057 14250000 MVP 1500093 1395000 And the total market value of the firm is V 7360000 14250000 1395000 23005000 Now we can nd the cost of equity using the CAPM The cost of equity is RE 045 10508 1290 or 1290 The cost of debt is the YTM of the bonds so P0 920 3250PV1FAR740 1000PV1FR740 R 363 2 YTM 3632 X 2 726 And the a ertax cost of debt is RD 17350726 0472 or 472 The cost of preferred stock is RP 593 0538 or 538 Now we have all of the components to calculate the WACC The WACC is WACC 047273623005 1290142523005 0538139523005 0983 or 983 Notice that we didn t include the l 7 tc term in the WACC equation We 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 nancing We nd MVD 1050001000093 97650000 MVE 900000034 306000000 MVP 25000091 22750000 And the total market value of the rm is v 97650000 306000000 22750000 426400000 B258 SOLUTIONS 18 C 1 So the market value weights of the company s nancing is DV 97650000426400000 2290 PV 22750000426400000 0534 EN 306000000426400000 7176 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 125085 1563 or 1563 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 17350833 0541 or 541 The cost of preferred stock is RP 691 0659 or 659 Now we can calculate the WACC as WACC 05412290 15637176 06590534 1280 or 1280 Projects X Y and Z Using the CAPM to consider the projects we need to calculate the expected return of the 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 8011705 0980 lt 10 so accept W EX 05 9011705 1040 lt 12 so acceth EY 05 14511705 1370 gt 13 so reject Y EZ 05 16011705 1460 lt 15 so acceptZ Project W would be incorrectly rejected Project Y would be incorrectly accepted He should look at the weighted average otation cost not just the debt cost 19 N G N p A CHAPTER 14 B 259 b The weighted average oatation cost is the weighted average of the oatation costs for debt and equity so fT 0575175 081175 0671 or 671 c The total cost of the equipment including oatation costs is Amount raised1 7 0671 20000000 Amount raised 200000001 7 0671 21439510 Even if the speci c funds are actually being raised completely from debt the otation costs and hence true investment cost should be valued as if the lm s target capital structure is used We rst need to nd the weighted average oatation cost Doing so we nd fT 6509 0506 3003 071 or 71 And the total cost of the equipment including oatation costs is Amount raised1 7071 45000000 Amount raised 4500000017071 48413125 Intermediate Using the debtequity ratio to calculate the WACC we nd WACC 7 90190048 1190 13 7 0912 or 912 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 912 200 1112 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 Ifyou remember the equation for the PV of a growing perpetuity is the same as the dividend growth equation so PV of future CF 27000001112 7 04 37943787 The project should only be undertaken if its cost is less than 37943787 since costs less than this amount will result in a positive NPV The total cost of the equipment including oatation costs was Total costs 7 15000000 850000 7 15850000 B260 SOLUTIONS N N Using the equation to calculate the total cost including oatation costs we get Amount raised1 if r Amount needed after oatation costs 1585000017 fT 15000000 fT 0536 or 536 Now we know the weighted average oatation cost The equation to calculate the percentage oatation costs is fT 0536 07EV 03DV We can solve this equation to nd the debtequity ratio as follows 0536VE 07 03DE We must recognize that the VE term is the equity multiplier which is 1 DE so 0536DE 1 08 03D 3 DE 06929 To nd the aftertax cost of debt for the company we need to nd the weighted average of the four debt issues We will begin by calculating the market value of each debt issue which is le 10340000000 le 41200000 Mv2 10835000000 Mv2 37800000 MV3 09755000000 MV3 53500000 MV4 11140000000 MV4 55500000 So the total market value of the company s debt is MVD 41200000 37800000 53350000 55500000 MVD 187850000 The weight of each debt issue is wl 41200000187850000 wl 2193 or 2193 wz 37800000187850000 wz 2012 or 2012 W 53500000187850000 w3 2840 or 2840 CHAPTER 14 B261 w4 55500000187850000 W4 2954 or 2954 Next we need to nd the YTM for each bond issue The YTM for each issue is P1 1030 835Pv1FAWO 1000PVIFR710 R1 2768 YTM1 3146 x 2 YTM1 629 P2 1080 4250PV1FAR716 1000PV1FR715 R2 3584 YTM2 3584 X 2 YTM2 717 P3 970 41PV1FAR731 1000PVIFR731 R3 3654 YTM3 4276 X 2 YTM3 854 P4 1110 49PVIFAR750 1000PVIFR750 R4 4356 YTM4 4356 X 2 YTM4 871 The weighted average YTM of the company s debt is thus YTM 21930629 2012 0717 28400854 29540871 YTM 0782 or 782 And the aftertax cost of debt is RD 078217034 RD 0516 or 516 1 Using the dividend discount model the cost of equity is 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 B262 SOLUTIONS 24 Challenge 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 11714 07172012WACC 112081735 Solving for WACC we find 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 oatation cost except we will use the oatation cost for each form of financing Doing so we get Flotation costs 11708 071720120 ll204 0608 or 608 The total amount we need to raise to fund the new equipment will be Amount raised cost 45000000l 7 0608 Amount raised 47912317 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 7479l23 17 6200000 1076 NPV 9719777 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 17 5455 4545 or 4545 Ch CHAPTER 14 B263 Now we can calculate the weighted average oatation costs for the various percentages of internally raised equity To nd the portion of equity oatation 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 oatation costs are fT 7 05455081 7 0 04545035 fT 7 0555 or 555 The initial cash out ow for the project needs to be adjusted for the oatation costs To account for the oatation costs Amount raised1 7 0555 145000000 Amount raised 14500000017 0555 Amount raised 153512993 If the company uses 60 percent intemally generated equity the oatation cost is fT 7 05455 81706004545035 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 oatation cost is fT 054550817104545035 fT 7 0191 or 191 And the initial cash ow will be Amount raised1 70191 145000000 Amount raised 145000000170191 Amount raised 147822057 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 7 2400001000094 7 225600000 MvE 7 900000071 7 639000000 MVP 7 40000081 7 32400000 The total market value of the company is v 7 225600000 7 639000000 7 32400000 7 897000000 B264 SOLUTIONS 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 Po 7 940 7 3750PV1FAR740 1000Pv1FR40 R 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 11 55081 0679 or 679 The weighted average oatation cost is the sum of the weight of each source of funds in the capital structure of the company times the oatation costs so fT 63989708 32489706 225689704 0692 or 692 The initial cash outflow for the project needs to be adjusted for the oatation costs To account for the oatation costs Amount raised1 70692 35000000 Amount raised 35000000l 7 0692 37602765 So the cash ow at time zero will be CFO 75100000 7 37602765 7 13000000 744002765 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 oatation 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 oatation costs so Total cost of NWC including oatation costs 13000001 7 0692 1396674 This would make the total initial cash ow CF0 7 75100000 7 37602765 7 1396674 7 744099439 CHAPTER 14 B265 To nd the required return on this project we rst need to calculate the WACC for the company The company s WACC is 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 a ertax salvage value will be A ertax salvage value 6000000 3513125000 7 6000000 8493750 Using the tax shield approach the OCF for this project is OCF 7 P 7vQ 7FC1 7 t th OCF 7 10900 7940018000 7 70000001 7 35 35350000008 7 14531250 The accounting breakeven sales gure for this project is QA FC DP 7v 7000000 437500010900 7 9400 7583 units We have calculated all cash ows of the project We just need to make sure that in Year 5 we add back the a ertax 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 7 744002765 14531250Pv1FA13974 30325000113975 NPV 7 1413071381 B266 SOLUTIONS And the IRR is NPV 0 44002765 14531250PVIFARR4 303250001 IRR5 IRR 2525 If the initial NWC 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 14531250PV1FA13974 30325000113975 NPV 1403403967 And the IRR is NPV 0 44099439 14531250PV1FARR4 303250001 IRR5 IRR 2515 CHAPTER 15 RAISING CAPITAL 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 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 dif cult 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 dif cult 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 B268 SOLUTIONS 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 1 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 50000081 6000070 44700000 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 500000 old shares 60000 new shares 833 rights per new share c 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 13X 44700000500000 60000 79 82 d The value of the right Value ofa right 8100 7 7982 118 e A rights offering usually costs less it protects the proportionate interests of existing shareholders and also protects against underpricing 2 a The maximum subscription price is the current stock price or 53 The minimum price is anything greater than 0 b The number of new shares will be the amount raised divided by the subscription price so Number of new shares 40000000 48 833333 shares And the number of rights needed to buy one share will be the current shares outstanding divided by the number of new shares offered so Number of rights needed 4100000 shares outstanding 833333 new shares 492 CHAPTER 15 B269 c A shareholder can buy 492 rights on shares for 49253 26076 The shareholder can exercise these rights for 48 at a total cost of 26076 48 30876 The investor will then have EXrights shares 1 492 EXrights shares 592 The eXrights price per share is PX 49253 48592 5216 So the value ofa right is Value ofa right 53 7 5216 084 d Before the offer a shareholder will have the shares owned at the current market price or Portfolio value 1000 shares53 53000 After the rights offer the share price will fall but the shareholder will also hold the rights so Portfolio value 1000 shares52 16 1000 rights084 53000 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 PX 7480 N81 40N 1 N 5613 The number of new shares is the amount raised divided by the pershare subscription price so Number of new shares 20000000 40 500000 And the number of old shares is the number of new shares times the number of shares eXrights so Number of old shares 5613500000 2806452 B270 SOLUTIONS 4 If you receive 1000 shares of each the pro t is Pro t 10007 7 10005 2000 Since you will only receive onehalf of the shares of the oversubscribed issue your pro t will be Expected pro t 5007 7 l0005 7l500 This is an example of the winner s curse 5 Using X to stand for the required sale proceeds the equation to calculate the total sale proceeds including oatation costs is X1 7 09 60000000 X 65934066 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 65934066 21 3139717 6 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 09 60000000 90000 X 66923077 required total proceeds from sale Number of shares offered 6692307721 3186813 7 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 Net amount raised 10000000 shares1820 7 900000 7 320000 180780000 The company received 180780000 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 20 per share and the company received 1820 per share The difference which is the underwriters spread is also a direct cost The total direct costs were Total direct costs 7 900000 20 7182010000000 shares 18900000 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 320000 2560 7 2010000000 shares 56320000 CHAPTER 15 B271 This makes the total costs Total costs 18900000 56320000 75220000 The oatation costs as a percentage of the amount raised is the total cost divided by the amount raised S0 Flotation cost percentage 75220000180780000 4161 or 4161 The number of rights needed per new share is Number of rights needed 120000 old shares25000 new shares 48 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 NPRO PsiN 1 a b C PX 4894 94480 1 9400 No change PX 4894 904 80 1 9331 Price drops by 069 per share PX 4894 85480 1 9245 Price drops by 155 per share Intermediate 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 3500000050 8700000 Since the book value per share is 18 the old book value of the shares is the current number of shares outstanding times 18 From the previous solution we can see the company will sell 700000 shares and these will have a book value of 50 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 nd the new book value per share will be New book value per share 800000018 700000508700000 2353 The current EPS for the company is EPS0 NTOShares0 170000008000000 shares 213 per share And the current PE is PE0 50213 2353 B272 SOLUTIONS If the net income increases by 1100000 the new EPS will be EPSI NIlsharesl 181000008700000 shares 208 per share Assuming the PE remains constant the new share price will be P1 PE0EPS1 2353208 4895 The current markettobook ratio is Current markettobook 5018 2778 Using the new share price and book value per share the new markettobook ratio will be New markettobook 48952057 2379 Accounting dilution has occurred because new shares were issued when the markettobook ratio was less than one market value dilution has occurred because the firm nanced a negative NPV project The cost of the project is given at 35 million The NPV of the project is the cost of the new project plus the new market value of the rm minus the current market value of the rm or NPV 735000000 87000004895 7 800000050 79117647 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 8700000 shares2 13 per share 18487500 10 The total equity of the company is total assets minus total liabilities or Equity 8000000 7 3400000 Equity 4600000 So the current ROE of the company is ROE0 NIOTEO 9000004600000 1957 or 1957 The new net income will be the ROE times the new total equity or N11 ROE0TE1 19574600000 850000 1066304 The company s current earnings per share are EPS0 NTOShares outstanding0 90000030000 shares 3000 The number of shares the company will offer is the cost of the investment divided by the current share price so Number ofnew shares 85000084 10119 H CHAPTER 15 B273 The earnings per share after the stock offer will be EPSI 106630440119 shares 2658 The current PE ratio is PE0 842658 2800 Assuming the PE remains constant the new stock price will be P1 28002658 7442 The current book value per share and the new book value per share are BVPSO TEDshareso 460000030000 shares 15333 per share BVPSI TElsharesl 4600000 85000040119 shares 13585 per share So the current and new markettobook ratios are Markettobooko 8415333 05478 Markettobook1 744213585 05478 The NPV of the project is the cost of the project plus the new market value of the rm minus the current market value of the rm or NPV 850000 744240119 4 8430000 384348 Accounting dilution takes place here because the markettobook ratio is less than one Market value dilution has occurred since the rm is investing in a negative NPV project Using the PE ratio to nd the necessary EPS after the stock issue we get P1 84 2800EPS1 EPSI 3000 The additional net income level must be the EPS times the new shares outstanding so NI 3010119 shares 303571 And the new ROE is ROEI 303571850000 3571 or 3571 B274 SOLUTIONS p A N p A 43 Next we need to nd the NPV of the project The NPV of the project is the cost of the project plus the new market value of the rm minus the current market value of the rm or NPV 7 7850000 8440119 7 8430000 7 0 Accounting dilution still takes place as BVPS still falls from 15333 to 13585 but no market dilution takes place because the rm is investing in a zero NPV project 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 1900000060000000Ps N 003167Ps We know the equation for the eXrights stock price is PX NPRO 4 PsiN 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 71 N76 PsN 1 71 7 76003167Ps Ps003167Ps 1 71 7 240667PS1 003167Ps PS 7 2748 Using PRO 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 And the equation for the value of a right is Value of a right PRO 7 PX Substituting the eXrights price equation into the equation for the value of a right and rearranging we get Value ofa right PRO 7 NPR0 PsN 1 Value ofa right N lPRo 7 NPRO 7PsNl Value ofa right PRO 7PsN l 14 5quot CHAPTER 15 B275 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 231 7 06 2162 per share So to raise the required funds the company must sell New shares offered 56000002162 259019 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 650000 old shares259019 new shares Number of rights needed 251 rights per share The eXrights stock price will be PX NPRO PslN 1 PX 25150 23251 l 4231 So the value ofa right is Value ofa right 50 7 4231 769 And your proceeds from selling your rights will be Proceeds from selling rights 5000769 3846741 Using the equation for valuing a stock eXrights we nd PX NPRO PslN T 1 PX 460 354 l 55 The stock is correctly priced Calculating the value of a right we nd Value of a right PRO 7 FX Value ofa right 60 7 53 7 So the rights are underpriced You can create an immediate pro t on the eXrights day if the stock is selling for 53 and the rights are selling for 3 by executing the following transactions Buy 4 rights in the market for 43 12 Use these rights to purchase a new share at the subscription price of 35 Immediately sell this share in the market for 53 creating an instant 6 pro t CHAPTER 6 FINANCIAL LEVERAGE AND CAPITAL STRUCTURE POLICY Answers to Concepts Review and Critical Thinking Questions 1 Business risk is the equity risk arising from the nature of the rm s operating activity and is directly related to the systematic risk of the rm s assets Financial risk is the equity risk that is due entirely to the lm s chosen capital structure As nancial leverage or the use of debt nancing increases so does nancial risk and hence the overall risk of the equity Thus Film 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 Because many relevant factors such as bankruptcy costs tax asymmetries and agency costs cannot easily be identi ed or quanti ed it s practically impossible to determine the precise debtequity ratio that maximizes the value of the fum However if the rm s cost of new debt suddenly becomes much more expensive it s probably true that the firm is too highly leveraged The more capital intensive industries such as airlines cable television and electric utilities 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 rms Overall the general tendency is for rms with identi able tangible assets and relatively more predictable future earnings to use more debt nancing These are typically the rms with the greatest need for external nancing and the greatest likelihood of bene ting from the interest tax shelter It s called leverage or gearing in the UK because it magnifies gains or losses Homemade leverage refers to the use of borrowing on the personal level as opposed to the corporate eve 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 filing 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 CHAPTER 16 B277 9 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 10 The basic goal is to minimize the value of nonmarketed claims 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 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 14000 28000 36400 Interest 0 0 14 000 28000 36400 EPS 280 560 728 AEPS 750 7 30 b If the company undergoes the proposed recapitalization it will repurchase Share price Equity Shares outstanding Share price 2500005000 Share price 50 Shares repurchased Debt issued Share price Shares repurchased 90000 50 Shares repurchased l800 B278 SOLUTIONS The interest payment each year under all three scenarios will be Interest payment 9000007 6300 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 14000 28000 36400 Interest 6300 6300 6300 NI 7700 21700 30100 EPS 241 678 941 AEPS 452 7 3871 2 a A table outlining the income statement with taxes for the three possible states of the economy is shown below The share price is still 50 and there are still 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 14000 28000 36400 Interest 0 0 0 Taxes 4900 9850 12740 NI 9100 18 200 23 600 EPS 182 364 473 AEPS 750 7 30 b 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 14000 28000 36400 Interest 63 00 6300 6300 Taxes 2695 7595 10535 NI 5005 14105 19565 EPS 156 441 611 AEPS 452 7 3871 Notice that the percentage change in EPS is the same both with and without taxes CHAPTER 16 B279 Since the company has a markettobook ratio of 10 the total equity of the rm is equal to the market value of equity Using the equation for ROE ROE NI250000 The ROE for each state of the economy under the current capital structure and no taxes is Recession Normal Expansion ROE 0560 1120 1456 AROE 750 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 250000 7 90000 Equity 160000 So the ROE for each state of the economy is ROE NT160000 Recession Normal Expansion ROE 0481 1356 1881 AROE 452 7 3871 If there are corporate taxes and the company maintains its current capital structure the ROE is ROE 0364 0728 0946 AROE 750 7 30 If the company undertakes the proposed recapitalization and there are corporate taxes the ROE for each state of the economy is ROE 0313 0882 1223 AROE 452 7 3871 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 350000160000 shares EPS 219 B280 SOLUTIONS 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 500000 7 082800000 NI 126000 And the EPS will be EPS 126000 80000 shares EPS 158 PlanI has the higher EPS when EBIT is 350000 b Under Plan I the net income is 500000 and the EPS is EPS 500000160000 shares EPS 313 Under Plan H the net income is NI 500000 7 082800000 NI 276000 And the EPS is EPS 276000 80000 shares EPS 345 Plan II has the higher EPS when EBIT is 500000 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 EBIT160000 EBIT 7 08280000080000 EBIT 448000 5 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 2800000 160000 7 80000 Share price 3500 per share The value of the company under the allequity plan is V 3500160000 shares 5600000 And the value of the company under the levered plan is v 350080000 shares 2800000 debt 5600000 CHAPTER 16 B 281 6 a The income statement for each capitalization plan is I II Allequity EBIT 39000 39000 39000 Interest 16000 24000 0 N1 239 0 15000 39 000 EPS 329 300 355 The allequity plan Plan 11 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 RDD Shares 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 allequity capital structure the interest term is zero To nd 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 EBIT11000 EBIT 7 101600007000 EBIT 44000 And the breakeven EBIT between the allequity capital structure and Plan 11 is EBIT11000 EBIT 7 102400005000 EBIT 44000 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 101600007000 7 EBIT 7 102400005000 EBIT 7 44000 This breakeven level of EBIT is the same as in part b again because of MampM Proposition 1 B282 SOLUTIONS d The income statement for each capitalization plan with corporate income taxes is I Allequity I EBIT 39000 39000 39000 Interest 16000 24000 0 Taxes 9 200 6 000 15 600 NI 13800 900 23 00 EPS 197 180 213 The allequity plan still has the highest EPS Plan II still has the lowest EPS We can calculate the EPS as EPS EBIT 7 RDDl 7 tc Shares outstanding This is similar to the equation we used before except 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 e 4011000 EBIT e 101600001 e 407000 EBIT 44000 The breakeven EBIT between the allequity plan and Plan II is EBIT1 e 4011000 EBIT e 102400001 e 405000 EBIT 44000 And the breakeven between Plan I and Plan II is EBIT e 101600001 e 407000 EBIT e 102400001 e 405000 EBIT 44000 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 7 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 So under Plan I the value per share is P 16000011000 7 7000 shares P 40 per share And under Plan II the value per share is P 24000011000 7 5000 shares P 40 per share This shows that when there are no corporate taxes the stockholder does not care about the capital structure decision of the rm This is MampM PropositionI without taxes CHAPTER 16 B 283 8 a The earnings per share are EPS 320008000 shares EPS 400 So the cash ow for the company is Cash ow 400100 shares Cash ow 400 b To determine the cash ow to the shareholder we need to determine the EPS of the rm under the proposed capital structure The market value of the rm is v 558000 v 440000 Under the proposed capital structure the rm will raise new debt in the amount of D 035440000 D 154000 in debt This means the number of shares repurchased will be Shares repurchased 154000 55 Shares repurchased 2800 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 32000 7 08154000 NI 19680 This means the EPS under the new capital structure will be EPS 196808000 7 2800 shares EPS 37846 Since all earnings are paid as dividends the shareholder will receive Shareholder cash ow 37846 100 shares Shareholder cash ow 37846 c To replicate the proposed capital structure the shareholder should sell 35 percent of their shares or 35 shares and lend the proceeds at 8 percent The shareholder will have an interest cash ow of Interest cash ow 355508 Interest cash ow 154 B284 SOLUTIONS The shareholder will receive dividend payments on the remaining 65 shares so the dividends received will be Dividends received 3784665 shares Dividends received 246 The total cash ow for the shareholder under these assumptions will be Total cash ow 154 246 Total cash ow 400 This is the same cash ow we calculated in part a d The capital structure is irrelevant because shareholders can create their own leverage or unlever the stock to create the payoff they desire regardless of the capital structure the rm actually chooses 9 a 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 80000 7 08300000 NI 56 000 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 5600030000 300000 Dividends received 5600 So the return the shareholder expects is R 560030000 R 1867 or 1867 b 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 08730000 Interest cash ow 72400 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 8000060000 600000 Dividends received 8000

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