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# CORPORATE FINANCE MBAC 6060

GPA 3.82

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This 218 page Class Notes was uploaded by Margot King on Friday October 30, 2015. The Class Notes belongs to MBAC 6060 at University of Colorado at Boulder taught by Jaime Zender in Fall. Since its upload, it has received 16 views. For similar materials see /class/232195/mbac-6060-university-of-colorado-at-boulder in OTHER at University of Colorado at Boulder.

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Date Created: 10/30/15

CHAPTER 18 DIVIDENDS AND OTHER PAYOUTS Solutions to Questions and Problems NOTE All end of chapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answerfor each problem isfound without rounding during any step in the problem 3 a To nd the new shares outstanding we multiply the current shares outstanding times the ratio of new shares to old shares so New shares outstanding 100004 1 40000 The equity accounts are unchanged except that the par value of the stock is changed by the ratio of new shares to old shares so the new par value is New par value 114 025 per share To nd the new shares outstanding we multiply the current shares outstanding times the ratio of new shares to old shares so New shares outstanding 1000015 2000 The equity accounts are unchanged except that the par value of the stock is changed by the ratio of new shares to old shares so the new par value is New par value 151 500 per share 4 To nd the new stock price we multiply the current stock price by the ratio of old shares to new shares so a b 6535 3900 651115 5652 6511425 4561 6574 11375 To nd the new shares outstanding we multiply the current shares outstanding times the ratio of new shares to old shares so a 15000053 250000 b 150000115 172500 p A O c 1500001425 213750 d 15000047 85714 Repurchasing the shares will reduce shareholders equity by 4025 The shares repurchased will be the total purchase amount divided by the stock price so Shares bought 40253500 115 And the new shares outstanding will be New shares outstanding 5000 7 115 4885 After repurchase the new stock price is Share price 1709754885 shares 3500 The repurchase is effectively the same as the cash dividend because you either hold a share worth 3500 or a share worth 3350 and 150 in cash Therefore you participate in the repurchase according to the dividend payout percentage you are unaffected The equity portion of capital outlays is the retained earnings Subtracting dividends from net income we get Equity portion ofcapital outlays 1200 7 480 720 Since the debtequity ratio is 80 we can nd the new borrowings for the company by multiplying the equity investment by the debtequity ratio so New borrowings 80720 576 And the total capital outlay will be the sum of the new equity and the new debt which is Total capital outlays 720 576 1296 a The payout ratio is the dividend per share divided by the earnings per share so Payout ratio 0807 Payout ratio 1143 or 1143 b Under a residual dividend policy the additions to retained earnings which is the equity portion of the planned capital outlays is the retained earnings per share times the number of shares outstanding so Equity portion of capital outlays 7M shares 7 7 80 434M This means the total investment outlay will be Total investment outlay 434M 18M Total investment outlay 614M The debtequity ratio is the new borrowing divided by the new equity so DE ratio 18M434M 4147 Since the company has a debtequity ratio of 3 they can raise 3 in debt for every 1 of equity The maximum capital outlay with no outside equity nancing is Maximum capital outlay 180000 3180000 720000 If planned capital spending is 760000 then no dividend will be paid and new equity will be issued since this exceeds the amount calculated in a No they do not maintain a constant dividend payout because with the strict residual policy the dividend will depend on the investment opportunities and earnings As these two things vary the dividend payout will also vary CORPORATE FINANCE AN INTRODUCTORY COURSE DISCUSSION NOTES MODULE 71 CAPITAL BUDGETING Capital expenditures outlays for xed assets are often large and commit the rm for long time periods If mistakes are made they can have devastating consequences for shareholder wealth Obviously for capital intensive rms those rms with a high percent of xed assets to total assets these decisions are most critical For rms not relying on xed assets to the same degree e g consulting rms or advertising rms the capital budgeting decisions have less impact on shareholder wealth Contrast capital expenditures to operating expenditures Operating expenditures involve outlays for labor and materials used to generate currentperiod revenues These expenditures are not considered longterm Again we shall assume that the relevant opportunity cost the market required rate of return r is known This return re ects the risk of the project under evaluation Therefore our immediate problem is to forecast a project s future cash ows The further the cash ow occurs from t 0 today the more dif cult it is to predict However in the discounting process using the risk adjusted required return r these future cash ows are increasingly quotpenalizedquot in terms of their present values in the discounting process Outlays at t 0 are usually known with considerable prec1slon THE GOLDEN RULES OF CAPITAL BI fDGF TING We emphasize eight quot golden rulesquot that are fundamental to correct capital budgeting analyses We will discuss these rules in detail as the lecture proceeds As an overview however the rules are Rule lCash ows are our concern identify project cash ows Rule 2Do not forget net working capital requirements for a project both out ows and in ows Rule 3Only include a proj ect39s incremental cash ows In other words beware of quotphantomquot allocated cash ows quotSunkquot costs are not relevant Rule 4Do not forget a project s opportunity costs 1 This lecture module is designed to complement Chapter 7 in Ross Wester eld and Jaffe l Rule 5Never neglect taxes Rule 6Do not include nancing costs Rule 7Treat in ation consistently Rule 8Recognize project interactions Let s now turn to an elaboration of these eight rules 1 MuCash lows are our concern identi v project cash ows You are probably getting tired of hearing quotcash ows cash ows cash owsquot Remember Happiness is a positive cash ow However cash ows are our focus in nance Why Cash is the life blood of the rm not accounting pro ts It takes cash to pay dividends pay the bills and keep the rm solvent Accounting procedures do however often affect the size and timing of cash ows For example how we depreciate an asset will affect the size and timing of tax payments Tax payments are a cash out ow Accordingly one must be well versed in accounting procedures We may net cash ows within the same time period For instance if we have in ows of 500 at t 5 and we have out ows of 200 at t 5 we can just net these out and work with a net 300 in ow at t 5 We generally assume that cash ows occur at discrete time intervals eg t 0 t l t 2 etc While we could assume cash in ows and out ows occur continuously throughout a time period the lack of precision in our cash ow estimates typically does not justify such precise mathematics Therefore in this class we will concentrate on annual cash ows for capital budgeting analyses In nance we treat an outlay or in ow when it occurs not when accounting rules recognize the expense or revenue For instance in accounting capital expenditures are capitalized on the balance sheet and written off via depreciation over the life of the asset Example Say a xed asset costs 500 at t 0 and is expensed 100 per year over ve years via depreciation In nance we treat the 500 as an outlay at t 0 This outlay is a cash ow Over time the depreciation expense is not a cash ow but these writeoffs do affect our actual tax expense We will discuss the impact of depreciation on 211 cash ows next Depreciation Again depreciation is not a cash ow but depreciation expense affects cash ows because of its in uence on taxes Tax payments are cash Note that other noncash charges e g amortization losses on disposal of assets etc have the same characteristics as depreciation Example Consider rms A and B While both rms have the same earnings before depreciation and tax they differ in the amount of depreciation expense they writeoff Firm A Firm B Earnings Before Depreciation and Tax 100 100 Depreciation 20 Earnings BeforeTax 80 100 Taxes 0401 32 40 Earnings AfterTax 48 60 Add back Depreciation 20 0 Cash Flow 68 60 Therefore even though Firm A shows lower profits aftertax than B 48 versus 60 Firm A39s cash ow is higher by 8 68 60 Why did this extra 8 occur Firm A had 8 less in tax expense than firm B Therefore Firm A had 8 more in cash ow We added back depreciation to Earnings AfterTax since it is not a cash out ow In general depreciation expense results in tax savings of Amount of Depreciation Expense Tax Rate Tax Savings In this example 20040 8 Depreciation expense or any noncash charge for that matter e g amortization while not a cash ow reduces a real cash out owtaxes Avoiding a cash outflow is just as valuable to a firm as having a cash inflow Rule 2 Do not forget net working capital requirements for a project both outflows anal inflows Adjusting for net working capital changes takes into consideration the differences between accounting recognition of revenues and expenses on the income statement and when the cash ows actually occur For instance credit sales generate accounts receivable While the accountants recognize the credit sale as revenue the cash ow does not occur until the customer pays for the item purchased Accordingly if the only current asset or current liability that changes is an increase in accounts receivable we would subtract this increase from net income to adjust the net income back to a cash ow basis Alternatively credit purchases generate accounts payable If the item purchased is part of cost of goods sold in a given period it ows through the income statement as an expense even though it may not have been paid for during that accounting period Accordingly if the only current asset or current liability that changes is an increase in accounts payable we would add this increase back to net income to adjust net income back to a cash ow basis In aggregate we collect the changes in the current asset accounts in a period and subtract the changes in the current liabilities during the period to get the change in net working capital If net working capital increases we have a cash out ow of this amount during the period If net working capital decreases we have a cash in ow of this amount during the period In summary to adjust for the ali erences between the accounting recognition of revenues anal expenses anal the actual cash flow occurrences of these accounts we aaljust accounting pro ts for the change in net working capital Example Assume that before a project is undertaken the rm has the following current accounts on its balance sheet Current Assets Current Liabilities Cash 100 Accounts Payable 50 Accounts Receivable 150 Taxes Payable 50 Inventory m Wages Payable M Current Assets 450 Current Liabilities 140 The net working capital is current assets current liabilities 450 140 310 Now assume that a capital budgeting project requires increases in current assets but also generates additional current liabilities For instance the project requires increased levels of cash accounts receivable and inventory However this same project results in more purchases which increase accounts payable in addition to taxes and wages payable 1ncreases Assume that the balance sheet after the project is taken is projected to look as follows Current Assets Current Liabilities Cash 150 Accounts Payable 150 Accounts Receivable 300 Taxes Payable 100 Inventory 3M Wages Payable m Current Assets 850 Current Liabilities 350 The net working capital is current assets current liabilities 850 350 500 The net working capital increase due to the adoption of this capital budgeting project has increased by 500 310 190 This commitment offunals to the project isjust as real as the commitment to fixed assets of the project We would include the changes in the net working capital requirements along with the changes in fixed asset requirements in the cash ows for the project Again the inclusion of the net working capital changes adjusts the income statement to re ect actual cash ows The amount of net working capital required to support a project often changes as the revenues generated by the project increase and decrease Therefore from yeartoyear changes in the net working capital are included in the project s other cash ows At the end of the life of a project the project no longer requires the net working capital support and we return the remaining net working capital as an in ow for the project Think about illustrating these concepts using a cash ow time line Rule 3 Only include a project39s incremental cash flows Alternatively beware of quotphantom quot allocated cash flows quotSunkquot costs are not relevant Our goal in analyzing a capital budgeting project is to identify the cash ows that are incrementally associated with the project Accordingly if overhead that would exist with or without the project is allocated to the project it will not be an incremental cash ow because of the project Accordingly it should not be included in the analysis Similarly quotsunkquot costs should be ignored for decisionmaking Some examples should clarify these concepts Example Say a firm has 80 million of existing annual overhead e g top management salaries corporate headquarters expenses etc This overhead is a fixed cost that will not increase or decrease with a 20 percent expansion or contraction of current sales which are 750 million The firm is considering a new manufacturing facility that would expand sales by 10 million in annual sales If this plant is added it would be allocated 1 million annually in existing corporate overhead If the overhead is included as a cash ow outlay to the proposed expansion the NPV turns out to be 8 million If the overhead is not included in the project cash ows the NPV is 4 million Should the company include the overhead in deciding on the expansion No The overhead would be incurred with or without the project It is not incremental to this decision It is what I call a quotphantomquot cash ow if the allocation is made By phantom I mean companywide the overhead cash ows do not change as a result of the project s acceptance or rejection Example Do not include historical events in the evaluation of a project Assume that your rm has already spent 500000 at t 1 doing marketing research to decide whether to introduce a new produce It is now t 0 The 500000 in market research is now water under the bridge This expenditure cannot be recovered if the project is rejected It is a quotsunkquot cost Therefore the marketing research expenditure should not be included in the NPV analysis done at t 0 to determine whether your rm should go ahead with the new product Of course the produce should have had suf cient potential at t 1 to justify the market research in the rst place Example You originally evaluated a project as follows CFO 10000 CF1 5000 CF 10000 Draw the cash ow time line A 10 discount rate is appropriate Therefore the project had a NPV 2810 You accepted the project A year has now passed The project has not worked out as expected M cash ow materialized at the end of the rst year of the project now relabeled as t 0 In fact to get any cash ow in one year t 2 at the time the project was adopted you must make an additional investment of 5000 If you do so you expect to receive 7000 in one year If you make no additional outlay today you will receive nothing in one year Do you spend the 5000 today Before we answer this question let39s return to the time we originally made the project decision or one year ago and assume that we knew what we know today CFO 10000 CF1 5000 CF 7000 Draw the cash ow time line At our 10 required rate this project has a NPV 8760 Of course we would not have taken the project if we d know the quottruequot outcome It is a loser But we did not have a crystal ball a year ago What do we do today Do we throw quot good moneyquot at a quotbad projectquot Using an incremental analysis we ignore the 10000 we originally spent on the project 6 That sum is a quotsunkquot cost Instead we focus on the cash ows that are truly incremental to the decision at hand today the new t 0 CFO 5000 CF1 7000 At 10 the NPV ofthis incremental decision is 1364 As much as we wished we d never seen this project we should make the incremental expenditure today Sometimes it is psychologically very difficult to continue with a project that in retrospect turns out to be a loser However in this example we can recoup some ofthe shareholder losses in wealth if we continue with the project The NPV of the incremental cash ows at this new decision point is positive Rule 4 Do not forget a project39s opportunity costs If a project requires the use of existing assets that have alternative uses or could be sold charge these assets at their opportunity cost to the project being evaluated The opportunity cost is the value of the next best use of the assets A prime example is land already owned Since this land could undoubtedly be leased or sold it is not a quotfreequot good for the project under consideration The opportunity cost e g the replacement cost or the market value of the existing asset should be charged to the proposed project at the time the asset is committed to the project Draw a time line that describes building an office building on land currently owned Note however if these assets have m alternative use or market value ie their opportunity cost is truly zero then they should not be charged to the project An example is surplus square footage in an existing plant that has no current alternative use nor can it be leased out Rule 5 Never neglect taxes All cash ows must be net of taxes for an appropriate NPV analysis Corporations pay estimated taxes quarterly Therefore on average the firm is about one and onehalf months away from a reconciliation with the IRS Taxes on Gains or Losses on Disposal Section 1231 Gains Example An asset originally cost 100000 and was being depreciated straight line over 10 years Five years have passed since the asset was purchased Accordingly the asset has a net book value NBV of 50000 The firm decides to sell this asset The firm has a tax rate of35 Three scenarios exist Scenario 1 The asset sells for 50000 Since the sales price equals the NBV no gains or losses occur on the disposal of the asset Accordingly no taxes are involved The selling price of 50000 is received quotfree and clearquot 7 Scenario 2 The asset sells for 60000 Since the selling price is 10000 over the NBV the rm owes taxes on the quotgainquot on disposal Taxes on the 10000 gain amount to 3500 Accordingly net of taxes the rm receives 60000 3500 56500 Scenario 3 The asset sells for 20000 Since the selling price is 30000 less than the NBV the rm has a quotlossquot on disposal This loss is tax deductible a noncash charge just like depreciation expense Accordingly the loss will lower taxable income by 30000 resulting in a tax savings of 30000035 10500 Accordingly the net cash ow to the rm from this transaction is 20000 selling price ofthe asset 10500 tax savings 30500 Rule 6 Do not include financing costs Including financing costs in the evaluation of a project is one of the most common mistakes made in evaluating a capital budgeting project Correctly conducted the capital budgeting analysis ignores nancing transactions that raise funds for the project as well as ignores any interest expense on debt dividends paid on equity sinking fund payments or any other nancially related cash ows Until later in the course you cannot fully appreciate the reasons for ignoring nancing costs However for now please realize that these nancing costs are included in the discount rate r and including them in the cash ows would be double counting them Once again financing costs are included in a properly determined discount rate Rule 7 Treat in ation consistently Two ways exist for handling a project s cash ows and required discount rate 1 Use nominal cash flows and a nominal discount rate By quotnominalquot cash ows I mean the cash ows are the amounts that will actually occur These cash ows will include all ofthe effects of in ation For example iflabor expenses are expected to grow at 5 per year then the pro forma statements constructed to estimate the future cash ows will include this increase in labor costs The nominal cash ows are then discounted at the nominal interest rate Nominal interest rates are the observable rates in the market place e g the current yields on one year T Bills or the yields on 10 year corporate bonds 2 Use M cash flows and a M discount rate By quotrealquot cash ows I mean to state future cash ows in terms of their purchasing power at t 0 or by quotbackingquot out the impact of in ation on the level of the cash ows These real cash ows are then discounted at the real interest rate Real interest rates are rates over and above the in ation rate Real interest rates are not directly observable in the marketplace real rates must be calculated Real rates are the rates that investors39 quotrealquot wealth or purchasing power increases over time It is the amount earned overandabove in ation To get a quothandlequot on real interest rates we rely on the insights of a famous economist Irving Fisher the same Fisher as in Fisher Separation Fisher developed an equation that relates nominal interest rates expected in ation and real interest rates This equation is as follows 1 nominal rate 1 real rate1 expected in ation Example Let s say the observable nominal rate on a 10year corporate security is 90 per year Further assume that expected in ation is 5 per year Using the above equation we can calculate the real rate 109 1 real rate105 1 real rate 103810 real rate 381 In other words an investor in this security will earn an amount per year that will increase hisher purchasing power by 381 A quotquick and dirtyquot approximation of the real rate is the nominal rate mimus the in ation rate or 009 005 004 or 400 As you can see the quotquick and dirtyquot approximation is not very accurate Let s now work an example capital budgeting problem each way 1 using nominal cash ows and the nominal interest rate and 2 using real cash ows and the real interest rate Example A project has the following nominal cash ows ie these cash ows are at the levels that are expected to materialize at each time period and include the effects of any in ation The observable or nominal interest rate is 92 Expected in ation is 50 CFO 4000 CF1 1700 CFz 3750 Nominal by Nominal NPV 4000 170010921 375010922 702 To calculate the real cash ows we must de ate the expected nominal cash ows and remove the in ationary impact In other words we must find the value of these future cash ows in terms of their t 0 purchasing power We must also calculate the quotrealquot interest rate Using Fisher s equation 1092 1 real rate105 real rate 400 Real by Real NPV 4000 170010511041 375010521042 The cash ows in the numerator are quotde atedquot by the in ation rate to represent t 0 purchasing power The real rate is used to discount these real cash ows NPV 702 Note the NPV s are identical the two methods are equally correct Remember however You must discount nominal cash flows by the nominal rate Q real cash flows by the real rate in order to get the correct answer Which method should you use The authors of your text suggest that you use the method that is easiest I do not disagree with that suggestion However I prefer using the nominalnominal method In my opinion this is more common in the quotreal worldquot and is usually easiest An important point in using the nominalnominal method is that you state the future dollars at the levels that will occur at that future time Revenues labor costs materials costs etc all may have different rates of in ation Depreciation expense on assets purchased in the past is based on their historical costs and will not be in uenced by in ation Serious mistakes will be made in your NPV calculations ifyou discount nominal cash flows by a real rate or ifreal cash flows are discounted by a nominal rate Rule 8 Recognize project interactions Profect Interaction 1 Mutually Exclusive Projects Consider mutually exclusive projects that have the same economic lives In this case choose the alternative with the highest NPV or if the project is compulsory choose the project with the lowest negative NPV ie a cost minimization project Now consider mutually exclusive projects that have different economic lives Example Projects A and B are mutually exclusive Project A has a oneyear life and Project B has athreeyear life The discount rate is 10 The cash ows are as follows Project A B d d d d 10000 12000 10000 5000 5000 5000 10 NPV 909 2434 Case lRegardless of which project you choose you plan to quotget out ofthe businessquot when the chosen proj ect39s life is over Your time in the business corresponds to the project life In this case take Project B This scenario seems unlikely however It does not seem likely that you would cease business based upon the economic lives of the assets involved However this case is a possibility Case 2Regardless of which project you choose you plan to stay in business for a predetermined amount of time For instance suppose you plan to stay in business for three years regardless of which project you take The cash ows of Project B remain as stated above However if we go with Project A we must replace it two times after the original purchase The cash ows are 1Pro39ect d d d d m A 10000 10000 10000 12 000 12 000 12 000 Net CF 10000 2000 2000 12000 2487 In this case you would choose Project A NPVA 2487 gt NPVB 2434 If you planned to stay in business ve years or any other nite period of time you would follow the same procedure including the replacement of the mutually exclusive projects when their economic lives expired You would take the alternative with the higher NPV over the nite time you plan to stay in business Case 3Regardless of which project you choose you plan to stay in business inde nitely or for all practical purposes until in nity In this case you have two alternatives 1 quotRoll the projects cash ows forwardquot until you have the lowest common denominator of their lives In the above example the lowest common denominator of a oneyear and a threeyear project is three years Include all of the relevant cash ows for both projects during this quotcommonlength lifequot period and take the project with the highest NPV We have already completed this alternative in the example above and on this basis we would select Project A In the case of Projects A and B above evaluating their cash ows over the lowest common denominator of their lives three years is no big deal However suppose that A had an economic life of 7 years and B had an economic life of 11 years In this case the lowest common denominator of their lives is 77 years Most of us would prefer to avoid all of the quot grunt workquot associated with these calculations 11 Fortunately a simpler procedure is available 2 Calculate the EquivalentAnnualAnnuity EA1 of both projects and select the best This method involves calculating the NPV for one lifecycle of each mutually exclusive project This NPV for one lifecycle is then converted to an annuity over the economic life of the alternative and having the same NPV as the alternative for one lifecycle The annuities of the alternatives can be compared and the best alternative can then be selected If the mutually exclusive projects are pro t enhancing projects you would select the option with the highest positive annuity or EquivalentAnnual Bene t EAB If the projects are cost minimization projects you would select the option with the lowest negative annuity or lowest annual cost or EquivalentAnnual Cost EAC To best illustrate this approach to selecting mutually exclusive projects let39s take a different example Example Say the revenues generated by two mutually exclusive machines are equal We plan to stay in this business forever The costs of the two machines differ their annual operating costs differ and the machines have different economic lives For the purposes of this illustration we will ignore taxes Since taxes are not a factor depreciation is irrelevant Do you understand why Under these conditions we will make our selection based on cost minimization We want to minimize the NPV of the cash out ows Accordingly we want to choose the alternative with the lowest Equivalent Annual Cost or EAC Project X has athreeyear economic life This machine costs 10000 The annual operating costs for this machine run 6000 Project Y is a mutually exclusive alternative to X Y has a fiveyear economic life This machine costs 20000 and requires annual maintenance of 2000 Draw the cash ow time line for these two options Note that we plan to be in business until infinity Note that the lowest common denominator of years for the two alternatives is 15 years We could discount the relevant cash ows over 15 years for both alternatives and calculate their respective NPV s The machine with the lowest NPV of 15 year cash out ows for the first 15 years also will be the best choice for the second 15 years and so forth Alternatively we could calculate the NPV of both options for onecycle of their economic lives Assume the discount rate is 15 3 NPVX 10000 Z 6000l15t 24000 The EAC for Project X is the threeyear annuity with this NPV or 3 NPVX 24000 Z EACxl15t EACX 10511 tl This EACX is the annual amount of outlay at the end of each year for three years that has the same NPV as the actual cash ows that occur for Project X over its threeyear life 5 NPVY 20000 Z 2000l15t 27000 tl The EAC for Project Y is the veyear annuity with this NPV or 5 NPVY 27000 Z EACYl15t EACY 8055 tl This EACY is the annual amount of outlay at the end of each year for ve years that has the same NPV as the actual cash ows that occur for Project Y We can compare the costs of X and Y on the common time denominator of one year39s EAC These EAC s if incurred annually to in nity would give us the NPV of their respective projects over an in nite horizon Based upon the respective EAC39s we would choose Project Y Note that we make this choice even though Y has a more negative onecycle NPV than Project X However since the projects have different economic lives and we plan to replace them overandover again we cannot compare onecycle NPV s 1 Project Interaction 2 Optimal quot Cycles Imagine that you own a trucking company What is the optimal time to replace your delivery trucks Replace them every year Every other year In fact you could replace the trucks on any cycle up until their maximum economic life when you absolutely have to replace them Imagine that you are in the lumber business and own a forest When should you cut down your trees You could cut them down when they are oneyear old Or cut them when they are twoyears old If fact you could choose any harvest cycle up until the trees blow down from old age or disease This category of nance question is similar to mutually exclusive projects If you replace your trucks every four years you cannot replace them every ve years The replacement cycles are mutually exclusive However we are making mutually exclusive decisions for a single project not comparing across projects Example Assume that you are the nancial manager for Timberland Inc You lease land from the Bureau of Land Management BLMBureau of Logging and Mining and your rm grows and sells a superfast growing hybrid variety of evergreen trees to lumber mills Your rm has no intention of getting out of this business ie its horizon is in nite It costs Timberland 50000 to plant 100 acres ofland The BLM leases 100 acre parcels for 5000 per year payable at the end of each year You are trying to determine the optimal time for your rm to harvest the trees You calculate the appropriate discount rate to be 12 You have assembled the following data the operating cash ows listed below do not include the planting or leasing costs Harvest Cycle Alternative Operating After Tax Cash Flows 1 00039s Cut Trees Every Six Years Trees are rotten and worthless Draw the cash ow time lines for each alternative harvest cycle Calculate the NPV of each option Calculate the EAB for each option Choose the optimal harvest cycle Harvest Cycle Alternative NPVof One Cycle EAB of Cycle Cut Trees Every Six Years Trees are rotten You should persevere until M can duplicate the numbers in the above table If you based the harvest cycle on the NPV of the individual cycles you would harvest the trees every ve years This decision would be incorrect The EAB is maximized with a fouryear harvest cycle Note that this decision is based upon the assumption that Timberland plans to stay in this business into the foreseeable future Proiect Interaction 3 Capital quot We discussed capital rationing earlier Capital rationing involves project interaction because choice of certain projects impacts the choice of others Review the discussion notes on how to deal with capital rationing IV The Consistency of the Finance Objective Function One of the features that I like most about finance is the unambiguous objective function Financial managers should make decisions that maximize the wealth of the shareholders Nothing quotwishy washyquot about this objective In all of the problems that we work in nance this objective underlies our decision Your job in capital budgeting is to identify all of the aftertax project cash ows choose the appropriate discount rate and make wealth maximizing choices V Other Exercises to Accompany this Lecture Module 0 Net Working Capital and Cash Flows The XYZ Company Capital Budgeting Alternative Formats Coleman Coupling Deer Valley Ski Resort CORPORATE FINANCE AN INTRODUCTORY COURSE DISCUSSION NOTES MODULE 191 VAL UA T I ON I Fair Market Value De ned The quotprice at which the property would change hands between a willing buyer and a willing seller when the latter is not under any compulsion to sell and both parties have reasonable knowledge of the relevant facts quot IRS Revenue Ruling 5960 as amended II Factors that are Relevant to Valuation as Suggested by the IRS IRS Revenue Ruling 5960 as amended The nature of the business and the history of the enterprise from its inception The economic outlook in general and the condition and outlook of the specific industry in particular The earning capacity of the company The dividendpaying capacity of the company Sales of the stock and the size of the block of stock to be valued The market price of stocks of corporations engaged in the same or a similar line of business having their stocks actively traded in a free and open market either on an exchange or over thecounter III Valuation Considerations When valuing a company a m financial analysis to identify the firm39s strengths and weaknesses is recommended Study financial ratio trends for the firm and relative to comparable firms To identify comparable firms sources include Value Line not always helpful for small firms Standard amp Poor s OTC Pro les the appropriate Moody s publications and relevant trade publication sources See Appendix B of Copeland et al and Shannon Pratt39s book for additional ideas see footnote 2 below The ideal set of comparable companies will be 1 publiclytraded firms 2 in the same industry 3 in the same size range 4 located in the same general geographic area and 5 of similar capital structure The idea is to hold constant on business risk and financial risk as closely as possible Comparables must be justifiable to the courts and the IRS 1 This lecture module is independent of the Ross Wester eld and Jaffe text An Overview of Valuation Methodologies2 Liquidation Value3 Net Book Value Comparable Price to Earnings Comparable Price to Sales Comparable Price to Book Comparable Dividends to Price Capitalization of lncom e Capitalization of Dividends Recent stock sales Capitalization of quotFree Cash Flow Blended approaches a combination of more than one approach V Discuss the Valuation Approaches in Turn Liquidationslow or fast fire sale It makes a g difference Remember a firm may be worth more quotdead than alive quot Thus it may depend upon the circumstances of the liquidation The criterion for liquidation The NPV of liquidation exceeds the NPV of continuation Net Book Value Assets Liabilities Little logic exists to recommend this approach Perhaps it appeals to accountants because of it conservative nature It is based on historical accounting numbers Why should net book value have a relationship to market value which considers future cash flows and risk For all of the comparable firm approaches calculate the average or more appropriately the J 39 inancia 39 391 eg t e pi39 39 or ratios or the dividend yields dividends per share divided by price per share Establish the current levels of the base parameters eg EPS sales per share book value per share dividends per share for the firm being valued You may want to develop a weighted average of the base parameter levels for the last few years for the firm being valued with heavier weights on recent years 2 See Valuation Measuring and Managing the Value of Companies by T Copeland T Koller and J Murrin 2nd Edition Wiley 1994 This excellent book presents a rigorous discussion of valuation complete with excellent examples Also see Valuipg a Business The Analysis and Appraisal of Closely Held Companies Shannon P Pratt 1988 Dow JonesIrwin This book presents a more quotcookbookquot discussion of valuation but also includes examples Other applicable references include quotA Comparison of Stock Price Predictions Using Court Accepted Formulas Dividend Discount and PE Models K Hickman and G Petry Financral Management Summer 1990 pages 7687 quotCapitalization Rates and Valuation of CloselyHeld Businesses S Choudhury Journal of Financial Management and Analysis JulyDecember 1989 pages 1518 and The Valuation of Cash Flow Forecasts An Empirical Analysis S Kaplan and R Ruback Journal ofFinance September 1995 pages 10591093 3 All other procedures are designed to value the rm on a quotgoing concernquot basis For instance using the PE approach you would multiply the average PE of the comparable firms times the EPS of the firm being valued to come up with an estimate of the equity value of this firm In another example say using dividend yields dividendprice dividend yield you would divide the dividends of the firm being valued by the average dividend yield of the comparable firms to come up with an estimate of the equity value The comparables approaches leave much to be desired For instance they are only loosely based on future cash flows and the risk adjustment is imprecise Any differences in the expected growth between the comparable firms and the firm being valued are not accounted for in these methods I Capitalization of Income approach Equity Value Z Earningstl rs Earnings can be growing at any rate Alternatively use EPS and find share price r5 is the cost of equity capital quotTimes 6 Rule This rule of thumb is occasionally used to estimate the cost of equity or r5 The SBBI small stock return arithmetic average from 1925 through 2000 has been 173 Take the reciprocal of this number or 1 0000173 The result is 578 Round this number to 6 Therefore if the required discount rate of 173 is appropriate for a small firm being valued and you assume that earnings are perpetual zero growth Equity Value Earningsrs or V Earnings1rs 1f 1rs 6 you just multiply the earnings of the firm being valued by 6 to get an estimate of equity value Other ways to estimate r5 1 Gordon Constant Growth Model EP if zero growth 2 Risk Premium or rf 0 where r is an estimate of the riskfree rate and 0 is an ad hoc estimate of the risk premium 3 CAPM 4 SBBI Small Stock Return subjective adj ustm ent Fundamentally this capitalization of income approach leaves much to be desired arnings are not equal to cash flow available to distribute to shareholders What about noncash charges Capital investments Changes in Net WC Capitalization of Dividends Ve Z Divt1r5t This approach is equivalent to the quot free cash flowquot approach discussed below Be sure you understand why Recent stock sales This benchmark can be unreliable particularly if only a small number of shares are intermittently traded by minority shareholders Are they under pressure to sell Do sellers have a liquidity emergency Private transactions by family members A buy out of a shareholder group Is the sale dictated by a death Remember fair market value requires an Aarms length transaction by informed investors without pressure I Capitalization of quotFree Cash Flows FCF This approach is in my opinion the only theoretically defensible and the most frequently used methodology for valuing a firm VL Z FCFtHrwt for n years TVn1rwquot where VL total market value of the firm VL D E where D the market value of interestbearing debt and E the market value of equity Rw is the firms WACC and TVn stands for terminal value at year n TVn estimation procedures This procedure is sometimes based on the current PE of comparable firms I don39t like this approach since EPS is not equal to FCF Alternatively is can be based on a reasonable multiple assumed of FCFn Another choice is based on the Gordon Constant Growth Model Tvn FCFnlgrw g Discuss the composition of FCF FCF should n0t include any financing outlays eg interest or dividends FCF represents Unlevered EATt Depr Capital Expenditurest Changes in Net WCt How big is n 5 to 10 years are accepted by the courts and IRS Beyond 10 years begins to stretch the credibility of the analyst I Blended approaches Some academic studies in extremely weak academic journals show that comparable approaches have merit Courts and IRS have accepted quotblendedquot approaches An example is the Central Trust Case that used the following model Value 050Ave PE EPS 030 DivAve Div Yld 020Ave PriceBook Book 1 Discount What do I think about blended approaches Garbage These approaches are totally ad hoc without theoretical merit They are attempts to play with weights on approaches to replicate observable market values of firms Without theoretical justification I recommend disregarding them entirely I As a parting note on valuation methodologies Beware of any valuation formulas set in corporate bylaws e g 112 times book value per share VI Special Considerations I Discounts Discounts are accepted by the courts and IRS Two discounts have been allowed 1 Marketability and 2 Size The courts and IRS reason that small firms have large risks However if you39ve included an increased risk premium based on small size don39t double count by also using a discount for size If the market is thin sellers have a liquidity problem They may have to wait a long time to find a buyer This discount is a deduction from equity value How much has been allowed for a liquidity discount Discounts up 35 have been allowed by the courts and IRS for lack of marketability VII A Valuation Exercise Using the FCF Approach Teton Valley Corporation You have been hired to value the Teton Valley Corporation TVC of Driggs Idaho TVC manufactures outdoor equipment specifically tents backpacks gaitors and headgear TVC s signature product is its worldfamous extreme weather tent line the WhiteOut Expedition often used in expeditions up Chomolungma Mount Everest K2 and Denali You just completed your WACC calculation for TVC and came up with a 16 rate While publicly held TVC s shares trade only intermittently on the Spokane Stock Exchange Approximately 30 of the firm is held by the family of the founders Because of the lack of a ready market and trading activity you have established that a 20 liquidity discount is appropriate 1000000 shares are currently outstanding TVC has an Employee Stock Option Plan ESOP and is required to perform an annual stock valuation to determine the allowable contribution deduction for tax purposes The firm s current bookvalue balance sheet is simplified as follows TETON VALLEY CORPORATION December 31 2006 Current Assets 250000 NonInterest Bearing Liabilities 170000 Net Fixed Assets 750 000 Aggregate InterestBearing Debt 350000 Equity 480 000 Total Assets 1000000 Total Liabilities Equity 1000000 The firm s most recent simplified income statement for 2006 is as follows TETON VALLEY CORPORATION Twelve Months Ending December 31 2006 Sales 5500000 Cost of Goods Sold 3 575 000 Gross Margin 1925000 GSampA 747500 Depreciation 140 000 EBIT 1037500 Interest 42 000 EBT 995500 Tax 030 298 650 EAT 696850 TVC management has forecast its sales growth to be 10 for the next five years At that point they expect sales to grow at the GDP growth rate or about 4 per year into perpetuity TVC s cost of goods sold runs a reliable 65 of sales GSampA expenses have a fixed component of 500000 plus a variable component of 45 of sales Net fixed assets are expected to grow at 5 per year for the next five years from the current level B capital expenditures less depreciation expense Depreciation expense is expected to run at 20 of beginning net fixed assets Net working capital current assets less current liabilities is currently 80000 and is expected to grow at the same rate as sales for the next five years After year 5 or 2011 free cash flows are expected to grow at 4 in perpetuity relative to the 2011 level What is the total market value of TVC What is its total equity and per share value The current rate on 20 year government bonds is 6 and the market risk premium is estimated at 745 The firm has a target debt to value ratio of 4 can borrow at 10 above government rates due to their high bond rating and currently has an equity beta of 14 How do you feel about your estimate of WACC What other valuation methods would you employ as robustness checks 1quot CORPORATE FINANCE AN INTRODUCTORY COURSE DISCUSSION NOTES MODULE 101 RISK AND RETURN THE CAPITAL ASSET PRICING MODEL CAPM Summary of Key Points The de nition ofrisk is a major issue in nance Risk is a quotslipperyquot concept ie it is not easy to de ne nor does it have a natural unit We assume that investors like expected return Er the more the better and dislike risk the less the better In short investors are risk aversethey must be compensated for bearing risk Therefore the relationship between expected ex ante return and risk must be upward sloping After the fact however realized ex post returns can have upward downward or no relationship with risk I will elaborate on this important distinction between expected and realized returns as a function of risk However over longer holding periods return data which we will discuss shortly illustrate that risk has been rewarded historically with higher realized returns We expect this for a long history Don t M think of Er without thinking about the risk associated with that E r Risk and return go together like hand and glove Practice the following exercise in the shower every morning Toss your bar of soap backandforth between hands and simultaneously say quotrisk versus return risk versus returnquot where one hand with soap represents Er and the other hand with soap represents risk Okay maybe that s a stupid thing to do but do whatever it takes to link these two words in your mind High Er equates to high risk If anyone tells you otherwise beware They are either a fool or a crook Capital markets provide no few free lunches ie high returns with low risk II Er and Volatility of Individual Securities The equation for the expected return on a security for time period t is Ert EPt PM EDivtP1 Note again that the quotEquot represents expectations or an expected but not realized value Assume that at t 0 today a stock costs 60 or PO This stock does not pay dividends You expect one of the following three states of nature to occur in the next year 1 This lecture module is designed to complement Chapter 10 in Ross Wester eld and Jaffe l Statei Probabilit of State 1 1 Recession 13 50 487 Normal Times 13 75 250 Boom Times 100 667 10 The expected price ofthe security at t l is EP1 1350 l375 l3100 75 The expected return on the security in time period lis Er1 l3l67 l3250 l3667 250 or 3 Er1 Z Probir1i where il Er1 is the expected return on the security in period 1 i indexes the three states that may occur Probi the probability of state i occurring and r is the return on the security if state i occurs Note that the probabilities of the states must sum to 10 Alternatively we could calculate the expected return in oneperiod Er1 as above Er1 EP1 P0P0 75 6060 250 note you still have to nd EP1 How might we measure the g of the return over the coming year for this security Candidate measures include Range of outcomes Variance of outcomes Standard Deviation of outcomes Other measures The range of outcomes is l67 to 667 However this measure lacks precision and lacks a relationship to the expected outcome Further probabilities of outcomes are not considered z The var1ance of outcomes 6 1s calculated as N oz Z Probir1i 7 Er12 where i i indexes the state i N represents the number of possible states Prob represents the probability of state i r is the return in state i and Er represents the expected return over all states oz 130167 02502 130250 02502 130667 02502 52 01159 or 11592 in our numerical example The standard deviation of the outcomes 6 is calculated as o 6212 or 01159 2 03405 or 3405 Review the relationship of o to the distribution of r s as per the material in an introductory statistics course ie l o incorporates about 68 of the distribution centered on Er 2039 incorporated about 95 of the distribution 36 includes 99 of the distribution assuming the distribution is normally distributed The other possible measures of risk will be developed below A source of confusion is when to use Prob versus lN versus lNl as the weight in the variance calculation where N equals sample size If all outcomes are equally likely Prob lN Therefore if all of the outcomes here future states are not equally likely then UN is not appropriate to use as the weights in calculating the variance of a distributionuse Prob If you39ll recall from your introductory statistics course if you are analyzing actual realized data and the sample size is small you should use lNl when you have N observations to correct for the small sample size bias problem If the sample is large N gt 30 however the sample size bias problem is negligible using UN is a very close approximation 111 Risk and Diversi cation The Intuition Is the dispersion of the returns actual or expected measured by variance or standard deviation the correct de nition of risk for any single security The answer is quotyesquot and quotnoquot The answer is quotyesquot if you are constrained to hold only one security However if you have the opportunity to diversify your assets the answer is quotnoquot variance is not a good measure of risk for a single asset In our development of risk measures we assume that the great majority of investors have the opportunity to diversify Therefore we must examine the impact diversification has on determining the appropriate measure of risk for an individual security What do we mean by diversification Let s use an intuitive but extreme example We are examining an investment in two companies The Umbrella Company UC and the Sun Tan Oil Company STO Details regarding these investments are as follows States of Nature Probability Umbrella Company Sun Tan Oil Company Sunny Year 05 006 018 Rainy Year 05 018 006 N Er Z Probi ri where il o Er is the expected return on security j in a given time period 0 N is the number of states of nature that can occur in that time period here two 0 Prob is the probability of state i occuring in that time period here two and o ri is the return in state i of the security Eruc the expected return in the Umbrella Company 05006 050 18 012 Ersm the expected return in the Sun Tan Oil Company 05018 05006 012 The probability distributions for the Umbrella Company and the Sun Tan Oil Company look the same and look like this Prob 006 018 ri Note that each security is risky ie the realized return can be either 6 or 18 A priori you don t know what outcome you will get in a given time period Now let s create a portfolio of both stocks with 50 of our investment capital invested in each company The expected return for the portfolio Erp is S Erp Z XjErj where j1 S the number of securities in the portfolio here two Xj weight of securityj in the portfolio here 05 and Erj the expected return of each security over all possible states rain and sun here 12 each Erp 05012 05012 012 What is the risk of this portfolio return The answer is zero as measured by dispersion The probability distribution of returns on the portfolio looks like this Probi 100 012 Er In this example you get a 12 return on your portfolio no matter what state of nature occurs You are able to get this Er without any risk Contrast this situation with what you get with either security alone ie either 6 or 18 with equal probability This example is the general idea of how diversification works when you hold several assets some do well and some do poorly they tend to balance each other out However the above example is an extreme case These two securities have a negative correlation ie when one has high returns the other has low returns How much risk reduction you get depends on the correlation of the returns on the assets included in the p01tfoli0 Just how negative is the correlation between these assets in our example First let39s calculate the covariance between the two securities or Gus N a EProerm 7Eruc rmi 7Erm where F rug and rsmi are the returns to the Umbrella Company and the Sun Tan Oil Company in state i respectively and Erucand Erm are the expected returns on these two firms over all states respectively on 05006 012018 012 05018 012006 012 5 ous 00036 Be able to discuss the intuition of what a covariance represents just a review of the basic statistics course Recall the relationship between covariance ous and correlation pus pus auxaucasto We have calculated ous The standard deviations one and 6510 are both equal 006 You should con rm these values pus 00036006006 100 Recall that all correlation coe icients pij lie in the range 10 perfectly negative correlation to 10 perfect positive correlation The two rms that we have examined have perfect negative correlation In this situation it is possible to choose weights for the securities Xu and X5 which will drive the standard deviation of the portfolio to zero Since we have assumed that investors like Er and dislike risk it is natural for them to seek diversi cation in their investments Diversi cation implies that total risk decreases as you add securities to your portfolio While perfect negatively correlated securities allow you to eliminate risk completely you can reduce risk with any pair of securities as long as the correlation is less than 10 However the less correlated the better in terms of their diversification impact Based upon the above example you should be beginning to see why the variance or standard deviation of a security is not a good measure of the risk of the security With diversi cation much of this standard deviation risk can be eliminated This brilliant insight was rst recognized by Professors Harry Markowitz and James Tobin For their contributions in the elds of economics and nance both were awarded Nobel Prizes in Economics What is the bottom line of this observation Don 39t put all of your eggs in one basket Yes but they demonstrated it rigorously IV Risk and Diversification The Formalities The expected return on a portfolio is S Erp EXjEm where 11 o S is the number of securities in the portfolio 0 Xi is the dollar value proportion of Security i in the portfolio which must sum to 10 6 and o Erj is the expected return on Security j Erp s are easy to calculate they are just the weighted average of the component security returns Example You plan to combine Security A with Security B into a portfolio using 20 of your funds to buy A and 80 of your funds to buy B Securities A and B have expected returns of 10 and 12 respectively Bap 0200 10 080012 0116 or 116 Nothing to it right Unfortunately the variance or standard deviation of a portfolio is not quite so easy The equation for the variance of a portfolio is as follows S S 02PE EXiXJoij where i1j1 S equals the number of securities in the portfolio Xi and Xj are the weighs of securities i and j respectively and oij is the covariance between i and j Be able to discuss in detail what the above equation represents and how it works Given the relationship between covariance and correlation an alternative way to write the above equation is S S ozp E EXiXJpijo ioj i1j1 All we ve done is substitute pijoioj oij Remember pij oijoioj The variance equation for a portfolio looks pretty intimidating doesn39t it However let39s start with the simplest case the case ofa twosecurity portfolio and it won t seem so bad The Case of Two Securities For two securities the portfolio variance equation in terms of oij looks like this sip X21621 xlxzo12 X2X1621 X22622 Since on equals 621 the covariance of security 1 with security 2 equals the covariance of security 2 with security 1 and Xle equals XzXl we can write the middle two terms as 2X1Xzolz Of course we could also write them 2X2X10 21 7 For two securities the portfolio variance equation in terms of pij looks like this 2 7 2 2 2 2 G p X161X1X2p126162 X2X1p216261x 2039 2 Again the two middle terms are equal and can be written 2X1szlzoloz Let s work through an example to make this expansion clear State Prob r r 020 007 012 060 012 010 020 017 008 LANi I At this point you should be able to calculate the Er s for both securities 1 and 2 they are 012 and 010 respectively You should also be able to calculate the variances for assets 1 and 2 as 00010 and 000016 respectively Their standard deviations are 003162 and 001265 respectively The covariance ofthe returns on asset 1 with the returns on asset 2 on equals 00004 Therefore the correlation between 1 and 2 pg equals 10 Check my math Now suppose we want to know the portfolio expected return and variance if we put 30 of our funds in Security 1 and 70 in Security 2 Bap 0300 12 070010 010 106 Using the covariance version of the portfolio variance equation we have 62p 030200010 2030070 00004 0702000016 62p 000 Here is another look at our example ofbeing able to quotsqueezequot all ofthe risk out ofa portfolio if the correlation is perfectly negative 10 Using the correlation based version of the portfolio variance equation we have 62p 030200010 203007010003162001265 0702000016 00 Unfortunately correlations between securities are usually positive Security returns tend to move together to some degree Finding perfectly negatively correlated securities outside of the derivative markets would be a very rare event Let s look at three cases where pairs of securities have correlations of 100 perfectly positive 100 perfectly negative and 000 no correlation These examples include 8 the quotend pointsquot of the correlation range as well as the midpoint of this range We will be putting together three portfolios of two stocks each with weights of X1 and X2 Of course X1 X2 100 since the sum ofthe weights of securities in a portfolio must sum to 100 ofthe portfolio Case 1 Correlation p12 100 Erp X1Er1 X2Er2 This equation represents a straightline linear relationship between the two securities relative to the vertical axis Erp see figure 104 in RWJ 62p X21621 2X1X2p126162 X22622 Since p12 10 we have 2 7 2 2 2 2 o p 7 X lo 1 2X1X2olo2 X 26 2 Th1s expressron can be factored as 2 X X 2 G p 161 262 The standard deviation of this expression is op X161 X262 Again this equation represents a linear relationship between the two securities relative to the horizontal axis op Given these linear relationships can you illustrate the graph of Erp and GP possibilities using securities 1 and 2 by varying the weights of the two securities in the portfolio By combining two securities with perfectly positive correlation we get risk averaging not risk reduction Risk and return are just proportional to the amounts of each security in the portfolio The portfolio39s risk standard deviation of return is a linear function of the weights of the securities in the portfolio X1 and X2 Case 2 Correlation p12 100 Erp X1Er1 X2Er2 as before 52p X21621 2X1X2p12olo2 X22622 Since p12 10 we have 52p X21621 2X1X2olo2 X22622 This expression can be factored as 62p X161 X2602 The standard deviation of this expression is op X161 X262 Again this is a linear relationship Given 61 and 62 it is possible to choose X1 and X2 in such a way as to eliminate all risk 9 or so to get op 0 Remember that X1 X2 must 10 Example Say 61 020 and 62 040 These parameters can be calculated from observable data Assume that p12 100 this is also a parameter that can be calculated from observable data Under these conditions can op X161 X262 0 If so X161 X262 X1X2 6261 Since we know 61 and 62 we can calculate the ratio as 040020 20 Therefore X1X2 20 Therefore X1 20X2 Since X1 X2 10 substituting from above we know 20X2 X2 10 Therefore X2 13 and X1 must 23 These are the portfolio weights that will create a portfolio with zero standard deviation For example if you have 12000 to invest you would put 4000 in Security 2 and 8000 in Security 1 Check Does X161 X262 op 000 23020 13010 0 Therefore these weights do reduce the portfolio standard deviation and so its variance to zero If p12 10 illustrate the following relationship in Er and standard deviation space by varying the weights on the two securities By combining two securities with perfectly negative correlation we can completely eliminate risk by the appropriate choice of X1 and X2 Rememberthe investor chooses the X39s portfolio weights in hisher portfolio With perfectly negative correlation we have diversification at its best Case 3 Correlation p12 000 Erp X1Er1 X2Er2 as before 52p X21621 2X1X2p12olo2 X22622 Since p12 00 we have 62p X21621 X22622 6p leczl X2262212 Given the quotsquaredquot terms we know that this relationship is quadratic or a curved relationship The curve will quotbulgequot to the left Why The SP will be less than when p is 10 but more than when p is 10 the curve will fall between these limiting cases Show Case 3 graphically Again figure 104 in RWJ does this for you but understand it 10 The implications of these three cases where p is 10 10 and 00 illustrate that diversification reduces risk if and only if the correlation is less than 10 You simply get risk averaging if the correlation is 10 With correlations less than 10 you get risk reduction the lower the correlation the greater the risk reduction ie the greater the quotbulgequot to the left The Case of Many Securities Recall the equation for the expected return on a portfolio S Erp Z XiEri where il o S is the number of securities in the portfolio 0 Xi is the dollar value proportion of security i in the portfolio and o Eri is the expected return on security i Erp s continue to be easy to gure for large portfolios they are just the weighted average of the component security returns Unfortunately the variance or standard deviation of a portfolio gets quotmessierquot than in the twosecurity case Again the equation for the variance of a portfolio in terms of covariance is as follows S S 52p Z Z XiXJoij where i1jl o S equals the number of securities in the portfolio 0 Xi and Xj are the weighs of securities i andj and o oij is the covariance between i and j Again we could substitute pijoioj for the covariance term Write out the above variance expression for three securities Thinking of the above variance equation for a large portfolio in terms of a covariance matrix is useful For a portfolio of size S the above equation has S2 terms Think of the equation in terms ofa S x S matrix as below 2 651 o s In this matrix we have cells S of these cells are occupied by variance terms these terms are the diagonal cells from the northwest to the southeast comer of the matrix S2 S of the cells or the remaining cells are occupied by covariance terms Note however that the quotoffdiagonalquot cells have equal values eg on 621 613 631 etc This covariance matrix can easily be converted to a correlation matrix How As S gets very large the impact of the variance terms on the total risk of the portfolio gets smaller and smaller the impact of the covariance terms gets larger and larger To keep it simple and illustrate this point assume the securities all have equal variances VAR Further assume that all of the covariances between pairs of securities are equal or COV and that the weights are all equal so Xi lS 62p S1S2VAR s2 S1S2COV IfS 10 62p 010VAR 090COV IfS 100 62p 001VAR 099COV IfS 1000 62p 0001VAR 0999COV The detail to remember is that as portfolios have more and more assets the variance or standard deviation of securities becomes less and less important and the covariance of the security with all other securities in the portfolio becomes more and more important This observation is additional evidence that a security39s variance or standard deviation is not its relevant measure of risk when an investor has the ability to form portfolios i e diversi v A key point The positive covariances in the market set the limits of diversification If investors dislike risk they will design portfolios that minimize GP for a given Erp What is important is the security39s covariance risk in a portfolio not its individual total risk ozi or 6 Markowitz developed a procedure to select the quotbestquot portfolios from all possible permutations and combinations of portfolios chosen from a set of S individual securities We discuss the definition of quotbestquot next V Portfolio Theory We now have the tools equations to calculate the Erp and UP of portfolios Note it is critical that M are comfortable in making the basic calculations Imagine that you have assembled a database with S securities For each of these securities you have an estimate of its Eri its oi and its covariance with each of the other S1 securities oij Upon inspection you quickly realize that all of your securities have positive covariances with the other securities Accordingly your opportunity to construct portfolios with zero variance or standard deviation is not available Now assume that you throw darts at a listing of your S securities Let s assume that you had 500 securities in your data set You begin to randomly form portfolios from sizes of one security each not really a portfolio up to all S securities You also throw darts at a list of proportions for each security to have in each portfolio from proportions ranging from 000 to 100 Of course the weights for an individual portfolio are properly constrained to sum to 10 Do you understand why You soon grow very tired of throwing darts You next calculate the Erp and GP of each of your portfolios You then plot your results in Erp and GP space Illustrate the results of this exercise graphically It39s a mess right How can we make any possible sense of this graph Dr Markowitz39s procedure a quadratic programming model is as follows Minimize 62p subject to two conditions Erp is a constant solve the problem for a range of values and This model will trace out the quotefficient set of risky portfoliosquot or the set of quotef cient risky portfoliosquot from among all possible permutations and combinations of securities and weights The model determines the X 39s for each security that appears in an individual e icient portfolio Obviously you need a computer to accomplish this task See your text for an illustration of the ef cient set of portfolios with a graph Note that only ef cient risky portfolios are included on the ef cient set These e icient portfolios must satis v two criteria For a given Erp they have minimum 0P For a given up they have maximum Erp 13 In words for each Erp only one minimum op portfolio exists For each op only one maximum Erp portfolio exists These are efficient portfolios Be able to discuss why we ignore or quotthrow awayquot portfolios on the efficient set that lie below the minimum variance point If you draw vertical lines through the efficient set for portfolios below than the minimum variance point you will find another efficient portfolio above it with a higher Erp and the same risk In other words portfolios below this point on the efficient set are dominated no one wants them by efficient portfolios that lie above it on the efficient set and have the same risk op Thus we ignore them Key Questions 0 Why does the e icient set of portfolios quotbulgequot to the left Hint Recall the results of our investigation of two asset portfolios with 10 1 0 and 000 correlation Why doesn39t the efficient set of portfolios intersect the E rp or the y axis Hint you can only completely eliminate risk if you have securities with perfectly negative corrleation How would estimating and using the set of efficient portfolios work in the quotreal world quot Steps 1 Security analysts would provide estimates of Eri s sis and oij s 2 Portfolio managers would take these inputs use Markowitz s quadratic programming procedure and compute the efficient set Again note that the outputs from the computer algorithm are the set of securities to select for each e icient portfolio along with the weights X 39s for each security in the individual e icient portfolio The model effectively quottraces outquot the efficient set of portfolios 3 Investment advisors then present the efficient set to investors Based upon their own tastes and preferences ie their aversion to risk investors quotpickquot an efficient portfolio that meets their needs Note that each investor has the opportunity to choose a unique portfolio As long as the portfolio is a member of the efficient set ie on the quotcurvequot it is an optimal portfolio for that risk level What e icient portfolio would you choose That is a personal decision You consider your own tastes and preferences for risk versus return in making your choice You must decide if you want to quoteat well or sleep well quot No one can tell you which e icient portfolio is best for you But once again whichever portfolio you choose must be an e icient set portfolio if you are risk averse Now that we have explained the rationale for why intelligent riskaverse investors want to diversify and how the most efficient diversification can be achieved the question is how the 14 quotsmallquot investor can achieve this diversi cation Given the high brokerage commissions on small trades is diversi cation practically achievable for most of us Fortunately the answer is yes Even small investors can invest in welldiversified mutual funds with initial investments as small as 1000 or less and subsequent deposits as small as 50 per month If you buy a quotno loa quot mutual fund you do not even have to pay sales commissions A quotno loadquot mutual fund has no sales people you have to write or call the mutual fund for a prospectus fill out an application and mail in your initial investment The prospectus explains the investment philosophy and goals of the fund All types of mutual funds are available from funds that invest in TBills commercial paper and other money market instruments ie money market funds to those that specialize in specific industries sector funds to those that invest only in stocks in a particular country e g Japan You can even buy an quotindex fundquot or a fund that just duplicates an investment in an index such as the SampP 500 Since mutual funds are large institutional investors and trade in large blocks of stock they have much lower transaction costs per trade than the small trader realize Therefore mutual funds are a low cost and practical way for the small investor to begin building an investment portfolio and if selected appropriately achieve quotinstan quot diversification A publication available in the reference section of most libraries Wiesenberger Investment Companies Service summarizes the types of funds commissions charged management fees charged risk etc along with contact information Consensus Opinions Now imagine a world as farfetched as it may seem where all investors have the same estimated values for Eri39s oi s and oij s If these estimates indeed are all the same how many efficient sets of risky portfolios would exist The answer is one 1 which will be explained shortly If all of the input estimates are equal every portfolio manager would come up with exactly the same efficient set of risky portfolios This conclusion has to be true because the location and shape of the efficient set is determined by the input measures which we39ve just assumed to be the same for everyone Obviously every investor will not come up with the same estimates many different estimates of the efficient set will exist at a given time However in aggregate one consensus efficient set will dominate This aggregate consensus efficient set would function as the quotmarket39squot efficient set or as though all investors used the same security estimates Accordingly this quotconsensusquot market efficient set is determined as though investors had homogeneous expectations a fancy term for investors having the same estimates of Eri39s 639s and oij s We will have more to say about the concept of homogeneous expectations later VI Does Diversi cation Work Let s take the data that we discussed above for 500 stocks Now randomly choose 30 single stocks and compute the standard deviations for each security for this group Then average these standard deviations Next randomly choose 30 twostock portfolios and calculate the standard deviation for each portfolio Then average the 30 portfolio standard deviations Repeat the above for 30 threestock portfolios 30 fourstock portfolios etc until you39ve completed thirty 30stock portfolios Next plot your results using average GP for each portfolio size as your yaxis and the number of stocks in the portfolio as your xaxis Illustrate the results with a diagram figure 107 in RWJ Note that quotTotal Portfolio Riskquot for a given portfolio size up quotllarket Riskquot quotUnique Risk quot Also note that the unique risk gets smaller anal smaller as the portfolio size gets larger anal larger Finally note that adding more securities to portfolios that have more than 30 randomly chosen stocks does not significantly contribute to further risk reduction When all possible quotunique or unsystematic or diversifiablequot risk has been quotsqueezedquot out of a portfolio the investor is still left with quotmarket or systematic or nondiversifiablequot risk To recap what we discussed above for the investor that selects a diversified portfolio the total risk 6 of security i is not important What is important is the security39s risk that cannot be diversified away This is the security s contribution to the risk of a diversified portfolio This risk is represented by the security39s covariance risk with the other securities in the portfolio VII Ef cient Risky Portfolios Plus the Risk Free Asset One of Markowitz s PhD students William Sharpe extended his Professor39s work by considering the riskfree asset in combination with the efficient set of risky portfolios Let s revisit our graph of the efficient set using the axes Erp and op Further let s assume homogeneous expectations on the part of investors therefore only one efficient set exists Add the riskfree rate rf to the efficient set diagram or the point on the vertical Erp axis with zero risk or o 0 Think of the riskfree asset as a Treasury Bill Now using this point rf as the origin draw a line that is tangent to the efficient set of risky portfolios Illustrate this procedure graphically figure 109 in RWJ Of course you could have drawn other straight lines from rf to points on the efficient set curve Which of these lines is best If you re a risk averse investor you want the steepest possible sloped line Why You like Er and you dislike risk op Therefore the line with the largest slope is the preferred line this line has the most Er per unit of risk However you are limited in your search for the steepest line by the location of the efficient set of risky portfolios The tangency point M from rf to the efficient set of 16 risky assets is the best you can do ie the highest slope possible The line through the tangency portfolio M is of special signi cance This line represents the new e icient set of all assets risky and risk free Note for any level of risk portfolios formed by combining the riskfree asset and portfolio M dominate all other risky portfolios on the efficient set of just risky portfolios ie have higher Erp Alternatively for any level of Erp all portfolios on the line dominate the set for risky portfolios by having lower risk except for Portfolio M of course Adding the riskfree asset in combination with the efficient set of risky portfolios was a brilliant insight But why is the relationship between the riskfree asset and Portfolio M linear It is because the off is zero and the covariance of M and rf equals zero Example Assume that the expected return on TBills is 70 and the expected return on M is 16 What is the Erp consisting of 30 invested in TBills and 70 in M Using our equation for the Erp we have Erp 03007 07016 0133 What about the variance of the portfolio that we39ve described above Assume that the variance of returns of M is 0044 standard deviation 021 and the variance of the riskfree rate is zero Using our equation for 52p we have 6 03mm 20307cfm 072052 0090 0420 0490044 00256 op 0147 Since the first two terms drop out TBills have zero risk and zero covariance with M the variance of the portfolio is just the weight invested in M squared times the variance of M The standard deviation of the portfolio is op Xmom or 07021 0147 Therefore the Er versus risk op relationship of the riskfree asset and M is a linear relationship in our picture Our new e icient set now extends on a straight line between rf and M in Er and UP space If we invest 100 of our assets in TBills we find ourselves at rf with zero risk If we invest 100 of our assets in M we have an expected return of Erm and risk of om By putting some of our money in TBills and some of our money in M we can position 17 ourselves anywhere on the line between rf and M When we invest some or all of our money in T Bills we are actually quotlendingquot money to the Us Government Therefore any position below M on the line but excluding M is considered to be a quotlending portfolioquot Moreover we can actually move up a linear extension of the rf toM line pastM by borrowing money anal investing our own money plus the borrowed money all in Portfolio M a quotborrowing portfolio quot Example Suppose you have 1000 invested in M Using this investment as collateral you borrow 500 which you also invest in M Assume that you can borrow at the rate rf What is your Erp and risk op Using the numbers from the last example we have Erp 15016 05007 0205 For every 1 of our money we have 150 invested in M Hence the first component of the equation is 150 16 However for every 150 invested we ve borrowed 050 We must pay interest on this loan at 7 This repayment of interest represents the second term 05007 We have quotlevereal upquot the Erm from 16 to 205 by borrowing and investing everything in M NOTE X1Xz 15 05 10 Have we created a money machine By borrowing and investing in Portfolio M you have higher expected return yes but also a higher risk level than the quotpurequot market risk or om 021 The risk of our quotleveredquot portfolio is op Xmom 15021 032 where Xm is the weight invested in Portfolio M Note the substantial increase in risk necessary to get the higher quotleveredquot expected return More risk more expected return No quotfree lunch quot here Illustrate the straight line from rf through M and extended beyond M Once again the separation point between the lending and borrowing portfolio is M The resulting line extending from rf through M and onward is called the Capital Market Line CML Note the risk measure that is relevant for portfolios that lie on the CML efficient portfolios is op This is the relevant measure of risk for efficient portfolios VIII What is Portfolio M Given homogeneous expectations all investors want to hold Portfolio M in some combination with the riskfree asset unless of course they want all of their money in TBills Investors adjust for risk by the percentage held in TBills Lending or borrowing equals being long 18 owning or short borrowing TBills The longshort position in TBills in combination with the ownership percentage of M determines the investor s risk level In short M is a very special portfolio To review once again all investors that are willing to hold some risk in their investment portfolio want to hold M to some degree All assets must be held if no one wants to hold or own an asset it isn39t much of an asset is it Therefore PortfolioM must represent the portfolio of all assets These assets of necessity must be held in market value proportions Does this logic make sense Example Imagine that only four risky assets exist in the economy Assets A B C and D Let s say that the total market value of A is 1000 Similarly B C and D have total market values of 750 500 and 250 respectively The total value of the market portfolio M the sum of all risky assets in the economy is 2500 1000 750 500 250 Everybody wants to hold some of M Why The line defined by M and the riskfree asset represents the efficient set of assets the best risldretum tradeoff you can find Further imagine that the economy has only 10 investors each with 250 they wish to invest in M plus additional amounts they wish to invest in TBills long or short to adjust for their personal risk preferences How will they go about investing in M Since 40 ofthe Market Portfolio M is made up of security A 10002500 each investor will invest 40250 of hisher money in Security A or 100 Similarly each investor will invest 30 of their money in B 20 in C and 10 in D Each investor has invested their 250 in quotmarket value proportionsquot Each investor effectively holds the market portfolio on a quotscaleddownquot basis IX The Capital Market Line CML Let s now write out an equation for the straight line beginning at rf and extending through the Market Portfolio M and beyond Recall the equation for a general straight line is y b mx where y is the unit of measure on the yaxis b is the intercept on the yaxis m is the slope of the line and x is the unit of measure on the xaxis By inspection of the previous diagram the equation of interest to us is as follows Erp rf Erm r omjop Relate the above equation to the equation for a straight line This equation traces out what we have labeled the Capital Market Line CML The CML represents the location of all efficient portfolios The slope ofthis line is Erm r om Note the measure of risk for an e icient portfolio is O39Pl Again this observation is veg important for what follows X Capital Market Theory Let s go back to our efficient set of risky portfolios or the efficient set before we introduced investments combining the optimal risky portfolio M with the riskfree asset Calculating the Markowitz efficient set of risky portfolio is elegant but has one major problem it39s too much work Think about how many estimates the security analyst has to make in order to satisfy the data needs for the efficient set S estimates of Eri39s S estimates of ozi s S2 S2 estimates of quotuniquequot covariance terms oij s For example a portfolio of 100 securities would require 5150 unique estimates 100 100 1002 1002 5150 Accordingly while Markowitz s portfolio theory and efficient set procedure was much admired it was little used because of the enormous data requirements Plus and a BIG plus they didn t have ready access to affordable computers in those days Besides his insight of adding the riskfree asset to the Markowitz s efficient set of risky portfolios Professor William Sharpe further refined Markowitz39s Model to make the estimation requirements and computations more feasible For the significant extensions he made to Markowitz s Portfolio Theory Professor Sharpe was awarded the Nobel Prize in Economics himself Ironically he was awarded this honor jointly with Markowitz in the same yearil990 Sharpe39s idea was to relate all securities to the risk of the Market Portfolio rather than deal with all of the covariance terms necessary to develop the e icient set In other words Sharpe s insight eliminated the necessity for the S2 S2 covariance terms and replaced them with only S terms relating the risk of each security to the risk of the market The general idea is that instead of relating each security to every other security relate all of them to a common security the Market Portfolio which contains theoretically all the securities in market value proportions Thus you really lose nothing you just make it easier to get what you need The relationship between any individual security and the Market Portfolio is captured in a simple regression equation called the Market Model or rj aj jrm 8quot where o rjt is the actual return for security j in time period t 20 lj is the intercept term yaxis in the regression for security j by is the coef cient for security j on the market return rm rm is the actual return on the market in time period t and 811 is the error term in the regression As you may recall from your statistics course the error term is randomly distributed around zero with an expected value of zero In the Market Model by represents the relationship of security j to the market portfolio over time This term represents the contribution of security j to the risk of the market portfolio or quotsystematic risk quot The error term 8quot represents the security39s unique risk or quotunsystematic risk quot 2 Jim 7 m In words beta is the covariance of the security with the market divided by the total risk of the market Therefore Beta is that security39s contribution to the risk of the Market Portfolio Example Let s go through an example of how the market model would be estimated Assume that you have collected monthend price and dividend information for the past 60 months for a security security j and the SampP 500 Index You then calculate the monthly returns for these 60 past months for both the security and the Index In other words you have two vectors of 60 returns aligned in time one for security j and one for the market M0nth 11 11 60 rm60 rj 60 59 rm59 r 59 58 rm58 r 58 l I mrl 1 1 These return data are used to run the simple regression of rj on rm This estimation can be done on the computer with a spreadsheet program with a statistics package or on your handheld calculator Illustrate the graphical relationship between these two variables Beta 5 j is the slope of the relationship of the security return rj on the market return rm Again statistically j cam02m or the covariance of security j with the market divided by the variance of 21 the market Again by is the contribution of security j to the risk of the market portfolio 0 If Bj gt 10 the security is more volatile than the market portfolio it is an quotaggressivequot security As an example if the market return moves up or down by 10 from one period to the next and the security moves up and down by 12 over these same two periods the security has a beta of 12 The security is more risky than the market 0 If Bj lt 10 the security is less volatile than the market portfolio it is a defensive security For example if the market return moves up or down by 10 and the security on average moves up and down by 8 the security has a beta of 08 The security is less risky than the market 0 If Bj 10 the security moves in synchronization with the market eg a 10 up and down movement in the market return would be on average matched by the security Betas B s can easily be calculated using historical data as in the above example or can be obtained from publications such as Value Line or Merrill Lynch39s Beta Book If we convert the market model equation or rjt xj Bjrmt 811 into its variance counterpart take the variance of both sides of the equation we have 02 zjozm 02 Note that 05 is the constant from the regression or the intercept As you will recall the variance of a constant is zero Therefore this term drops out of the equation Therefore the total risk of security j oj is equal to the market risk of security j Bjom plus security j39s unique risk 68 In words 0 Total Risk Market Risk Unique Risk or 0 Total Risk Systematic Risk Unsystematic Risk or 0 Total Risk Nondiversifrable Risk Diversifrable Risk Sorry about that However all of the various terms are used as synonyms in the finance community even though they strictly speaking should not be What will happen when we add securities to a portfolio and regress the portfolio returns against the market returns The dispersion of the data points around the regression line will be quottighterquot with a portfolio than 22 with an individual security Why The answer relates to the fact the error terms of an individual security Sj have an expected value of zero and are quotrandomlyquot distributed around zero As we combine securities into a portfolio the positive error terms of one security will cancel with the negative error terms of another security As the size of the portfolio increases the size of portfolio error term approaches zero Therefore as more and more securities are added to a portfolio the risk that is unique to any security 68 is cancelled out with other securities unique risks 6839s The risk that is not removed is the portfolio risk Bpom In the limit when all of the unique risk has been eliminated the only risk that remains in the portfolio is Bpom The Beta of the Market Portfolio What is the beta of the market portfolio This is the same as asking what will happen if we regress the returns on the market portfolio against the market portfolio I hope the answer is obviousthe beta of the market Bm is equal to 10 The Beta of the RiskFree Asset If you regressed TBill returns against the returns of the market portfolio what relationship do you suppose that you would get Since the riskfree asset has zero covariance with the market portfolio we would get a beta of 00 Does this answer make sense to you Risk and Portfolio SizeOnce Again As we increase the number of assets in a portfolio risk as measured by the standard deviation of the portfolio approaches the following limit PUM this represents the limit of diversification You can costlessly eliminate all of the securities specific unique or unsystematic risk but you cannot eliminate the securities market or systematic risk simply with diversification You have created an e cient risky portfolio if you39ve eliminated all of the security specific risk If investors dislike risk they will diversify and eliminate firm specific risk In other words they will hold efficient portfolios The only way to eliminate systematic risk is to hold securities with lower betas like the riskfree asset What will this do to expected returns This is the cost of reducing systematic risk Since om is common to all securities the relative measure of risk for a security j is beta Bj How do you calculate the beta of a portfolio or Bp Two ways exist 0 Regress the portfolio39s returns against the market s returns or 23 0 Weight the individual security betas by portfolio value weights and add them up Example As an example of the second procedure assume that we plan to add three securities together to form a portfolio in proportions 02 05 and 03 The betas ofthese three securities are 08 09 and 12 respectively N p EXi i where F o N equals the number of securities in the portfolio and o Xi is the proportion of security i in the portfolio Bp 0208 0509 0312 097 A Source of Confusion Refer to a diagram of op versus portfolio size S figure 107 Students often interpret this diagram as meaning that all efficient portfolios converge to the same Bpom limit Not true Different efficient portfolios converge to different limits depending upon their Bp level Say for instance that the BF of one efficient portfolio is 140 and the BF of another efficient portfolio is 080 Both of these portfolios contain no unique risk ie all of the risk is market risk Therefore the quotpurequot market risk in these two portfolios will be 1406m and 0806 respectively XI The Capital Asset Pricing Model Hang in there We39re almost done Now we want to relate all of the above in a quotpricing modelquot for individual securities In this context we are defining a quotpricequot as an expected return For instance the quotprice of moneyquot is the interest rate that you pay to borrow money In a real sense expected returns are prices Prices of assets are determined by discounting by the expected return Remember the Capital Market Line CML Equation or Em r NEW r UmJUP Now what do we know about GP or the total risk of an efficient portfolio If you will refer to the previous diagram we know that op Bpom for an efficient portfolio If we plug in Bpom for GP in the CML equation we get 24 Er1 rf Erm rf 1 This equation is called the CapitalAsset Pricing Model or CAPM The equation expresses the relationship between the expected return on a portfolio and beta Bp The line that this equation de nes is called the SecurityMarketLirie SML Recall the Capital MarketLirie CML gives the relationship between the expected return on efficient portfolios and the risk of the portfolio In this context we defined risk as the standard deviation of the efficient portfolio op The standard deviation or total risk of an efficient portfolio is also equal to the portfolio39s systematic riskeff1cient portfolios contain no unsystematic risk Therefore total risk is an appropriate measure of risk for an efficient portfolio All e icient portfolios plot on the CMU The Security Market Line SML expresses risk as beta B Recall B measures the systematic risk not total risk Accordingly all securities and portfolios both e icient and irie icierit portfolios should plot on the SML Again the equation for the SML or the CAPM equals Erj rf Erm rfBj for any securityj or Erp rf Erm rfBp for any portfolio p efficient or inefficient Illustrate the above graphically figure 1011 in RWJ Example Assume the TBill rate for oneyear TBills is 51 Assume that you39ve calculated the beta on Security A as BA 120 Since you have no insight on the expected market risk premium Erm rf you rely on the historical market risk premium of 9 1 remember the Ibbotson data Your estimate for the ErA 51 91120 160 Lending and Borrowing Revisited As demonstrated above for the CML by adjusting your lending to the government via investing in TBills or by borrowing and investing in M you can position yourself along the CML from rf 100 investment in TBills to positions between rf and M by putting some of your money in TBills and some in the Market Portfolio M to M 100 investment in the Market to points above M by borrowing and investing your money plus borrowed money in M The exact same opportunities exist with respect to the SML 25 Example If you have 1000 to invest and borrow 200 and invest all 1200 in the Market Portfolio what will the beta of your portfolio Bp be S Bp Z XiBi where S equals the number of securities in the portfolio and i1 Bp 02Brr 12Bm of 5p 0200 120Bm Remember the beta of T Bills is 0 The beta of the market is 10 Therefore the answer is Bp 12010 120 What will be your expected return on this portfolio Erp 02rf 12Erm Note that the X s sum to 10 The Implications of the SML The beta of M security or portfolio can be duplicated by combinations of investments in T Bills borrow or lend and the Market Portfolio What happens then if the Er of a security or portfolio does not plot on the SML Specifically what happens if the Er plots above or below the SML as below If securities are not priced to lie on the SML an arbitrage opportunity exists An arbitrage opportunity exists whenever the same commodity sells at different prices If three securities A B and C all have the same risk yet have different expected returns an arbitrage opportunity exists Think about how this works XII CAPM Assumptions Time to fess up CAPM is based upon a set of assumptions some of which I haven t mentioned 0 Capital markets are perfect PCM2 o Investors are risk averse they like Er and dislike risk 2 The Perfect Capital Market PCM assumptions are All traders have equal and costless access to information All buyers and sellers are quotprice takersquot No brokerage fees contracting costs or other transactions costs exists No taxes exist The borrowing rate equals the lending rate for all investors 26 o Investors diversify and hold efficient portfolios o Investors have homogeneous expectations Given these assumptions the CAPM implies that B is the relevant measure of risk for all securities and portfolios efficient and inefficient Er for all securities and portfolios is based upon this measure of risk From a very early discussion we observed that Er rf risk premium CAPM defines the risk premium as Erm TOB Your reaction to the assumptions underlying the CAPM may be to say quotGive me a break How can we believe these assumptionsquot Therefore how can we believe the CAPM A very famous economist Milton Friedman Nobel Prize in Economics once observed that we should judge a theory by how much it helps us understand the world and how much it helps us predict the future not by the realism of its assumptions Just how well does the CAPM work in the quotreal worldquot Note that CAPM is an quotexpectationsquot model ie it gives us the Er Note also that the CAPM is a oneperiod model ie it gives the Er for a single future period This period could be a day a week a year etc If we compare what the CAPM predicts returns Erj to be conditional on the Erm versus what returns actually turned out to be rj and we do this comparison for many time periods and for securities and portfolios with different betas we can get an idea of just how well CAPM works Research has looked into the accuracy of CAPM predictions afterthefact and conditional on the market The comparison between the predicted returns based on the CAPM and given the actual market return can be compared to the plot of actual returns and betas to test the validity of the CAPM This comparison reveals that the predicted returns relative to betas have a lower slope than the actual relative to betas Hopefully the following example will make this discussion clear Example Assume that for period 1 Erm 016 rf 007 and Bj 120 As we enter period one we expect the following return on Security j Erj rf Erm r05 007 016 007120 0178 27 At the end of period 1 we observe that the actual return on the market rm 014 and the actual return on Security j rj 016 The riskfree return was as expected or 007 Given what actually happened in the market afterthefact the CAPM says Security j should have earned E rjlrm rf rm r05 007 014 007120 0154 E rjl rm is read as the expected return on Security j conditional on the actual market return 014 At the start of period 1 we expected Security j to earn 0178 assuming that the market would earn 016 However the market actually returned 014 Therefore conditional on what actually happened in the market we would expect Security j to earn 0154 However Security j actually earned 016 Therefore it outperformed the return predicted by the SML by 016 0154 0006 If we plot what the CAPM predicts a security or portfolio should earn conditional on the market in return and B space and contrast this line to what securities or portfolios actually return as a function of their B s we can test the CAPM What does any discrepancy between predicted and actual results mean 0 It could mean that CAPM is wrong e g omitted variables or o It could mean that CAPM is right but we39ve measured beta or the Market Portfolio incorrectly ie the SampP 500 is not a good proxy for the quottruequot Market Portfolio Very simply tests of the CAPM find it lacking See the Appendix to this module for a more complete discussion However if the CAPM is wrong perhaps the reason is that we39ve omitted some risk factors for which in addition to the market factor the market requires additional return We ill discuss this possibility in Module 11 Recently new advances have allowed it to perform quite well However letl39s not lose perspective CAPM has taken us a long way in understanding the relevant measure of risk and the relationship between Er and that risk Specifically we find that 0 Actual returns are upward sloping with beta as predicted by CAPM and 0 Actual returns are approximately linear as a function of beta as predicted by CAPM Research in asset pricing models continues The CAPM is not the final word in the relationship between Er and risk However as of right now at least in my opinion it is the best theory that we have for practical application In addition it is doubtful that we will ever have a theory that is 28 More intuitive than CAPM More elegant than CAPM and Simpler than CAPM You may have some reservations about this point given what we ve just been through XIII Key Summary Points We assume that investors like Er and dislike risk These behavioral assumptions imply that investors will diversify and hold ef cient portfolios An ef cient portfolio is the portfolio that for a speci c level of risk has the maximum Er or for a speci c Er has minimum risk The set of ef cient risky portfolios dominates all other inef cient portfolios or individual securities The rst two points imply a security39s total risk oi is irrelevant Much of this total risk is unique risk this unique risk can be diversi ed away The portion of total risk that cannot be diversi ed away is systematic risk Only systematic should be rewarded with higher Er It should be a systematic risk 7 expected return relationship If a security has a higher Er based upon its unique risk what would you do You d rush out and buy the security adding it to a welldiversi ed portfolio Therefore you39d quothave your cake while eating it too extra return but no added riskquot But so would other investors In the process the price of the security would increase and its Er would decrease In the nal analysis unique risk will not be rewarded with extra return Therefore why would you or any rational investor bear any unique risk In sum only systematic or market risk is relevant in asset pricing This risk is measured by beta the asset s contribution to the risk of an ef cient portfolio The addition of the riskfree asset to the above points implies the Security Market Line The betas of all securities and portfolios can be duplicated by investments in the risk free asset and the Market Portfolio which becomes the only relevant ef cient risky portfolio plus combinations of lending or borrowing at the riskfree rate Therefore if a security or portfolio does not plot on the SML arbitrage activity drives its price up or down until its Er matches the riskequivalent return speci ed by the CAPM An investor cannot compare returns without holding risk constant This obvious point is violated repeatedly in quotreal worldquot comparisons and decisionmaking You do not compare apples and oranges Therefore do not compare asset returns either expected or realized without adjusting for risk differentials Why have we spent so much time on Portfolio Theory and Capital Market Theory These topics represent the critical cornerstone for nancethe tradeoff between risk 29 and return A word to the wiselearn these principles well 30 APPENDIX CAPM Fact or Fiction Simple tests of the CAPM suggest that expost after the fact the CAPM does not perform as well as we would like Given the realized market return CAPM predictions have a atter slope that actual realized returns Therefore higher beta securities earn returns higher than the CAPM would predict and lower beta securities earn lower returns than the CAPM would predict Is CAPM wrong A major problem in testing the CAPM is that we cannot observe the true riskfree rate or the true market portfolio We use government securities to proxy for the riskfree rate and the SampP 500 Index to proxy for the market portfolio Are CAPM s apparent prediction errors due to the fact that CAPM is wrong or is CAPM the correct model but our proxies are wrong The answer to this question perplexes researchers Ad hoc attempts to add terms to the CAPM have shown the CAPM predictions versus actual realizations are improved when two terms are added Er Rf ERm RfBeta Book ValueMarket Value lFirm Size While these additions to CAPM historically have improved its predictive power and have been adopted in practice in some companies they are ad hoc ie not theoretically justi able They are the result of data mining to search for missing variables in the CAPM equation Remember correlation does not imply causation I believe that these two variables are proxying for true economic variables that remain undiscovered Recent work on conditional asset pricing models ie letting the parameters of the model vary at each point in time work much better than the simple tests whose failures are well known In fact they work just as well as the ad hoc model just described Also adding terms to the Model without theoretical justi cation is dangerous Relationships that worked historically are likely not to work in the future This makes the conditional models much more attractive if much more difficult to implement What to do if you don t want to use CAPM There are alternatives 0 Gordon Constant Growth Model This model has more problems that CAPM and is not applicable for many rms eg hightech fast growth rms or rms that don t pay dividends o APT Arbitrage Pricing Theory We will discuss this Theory in the next Module While elegant and able to expand on the CAPM in an intuitive way this model has its own set of problems However research continues I believe that in a few years we may have a better model than either CAPM or APT In the meantime stay tuned 31 CORPORATE FINANCE AN INTRODUCTORY COURSE DISCUSSION NOTES MODULE 111 RISK AND RETURN THE ARBITRAGE PRICING MODEL APM or APT I A Brief Review of Module 10 Recall the equation for the Capital Asset Pricing Model CAPM Erj rf Erm r j Do you have the intuition of what this equation represents It represents the relationship between the expected return on an asset asset j here and its risk Diversi cation implies that the risk re ected here is only the systematic risk of the asset by in the CAPM Do you understand why CAPM is called a quotpricing modelquot Think of expected return as a price e g the interest rate is the price of money Do you understand why the second term on the righthand side Erm r j is called the quotrisk premiumquot for the security or portfolio under investigation It is the additional compensation for risk or the premium over and above the risk free return The risk free return only compensates you for the time value of money no compensation is included in it for risk since by de nition this is the risk free rate Do you understand what the Bj represents and how it can be estimated Beta is the measure of systematic risk for an individual asset in the CAPM We will have a lot more to say about beta in the notes that follow According to Capital Market Theory the Theory used to develop the CAPM in equilibrium the expected returns on all assets can be plotted as a function of their systematic risk 5 The resulting relationship is a line that has been named the Security Market Line or SML Recall that in an equilibrium situation no pressure exists for an asset s price to change Therefore its expected return Er will not change in equilibrium If an asset s fails to lie on the SML a disequilibrium pressure to change condition exists therefore an arbitrage opportunity exists How should the astute investor capitalize on this arbitrage situation 1 This lecture module is designed to complement Chapter 11 in Ross Wester eld and Jaffe 1 o Astute investors will buy assets that are underpriced their expected returns are too high they plot above the SML o Astute investors will sell or short sell assets that are overpriced their expected returns are too low they plot below the SML In this collective process of buying and selling mispriced assets investors will earn quotabnormal or excess returnsquot on the transactions when the assets prices return to equilibrium ie their returns resume their appropriate place on the SML In other words by identifying assets that are under and overpriced and buying or selling these assets the investor will earn more than the Er that is commensurate with the risk of the asset This buying or selling pressure will move the price and therefore the Er back toward the SML Eventually equilibrium will be restored In a highly competitive market we expect this disequilibrium or arbitrage situation to be corrected very quickly Note that in the CAPM the expected return on the asset Er is determined by the riskfree rate rf and a single factorto adjust for the risk of the security or portfolio Accordingly the CAPM is often referred to as a single factor model In the CAPM Erm rf is the market price for one unit of risk The Market Portfolio has one unit of risk or a i of 10 Therefore Erm rf is the risk premium for the Market Portfolio For assets with less than or more than one unit of 5 risk we multiply their number of units of risk or their B times this market price for one unit of risk Hereafter let39s refer to the risk premium factor in the CAPM as the quotmarketfactorquot II An Alternative to the CAPM If investors demand to be compensated with higher expected returns for common factors other than the market factor we can expand the CAPM into a multifactor model A multifactor model approach is at the heart of what is labeled theArbitrage PricingModel APM The APM is derived fromArbitrage Pricing Theory APT This expanded model is the topic of Chapter 11 However before we jump into the APM let39s develop some background information 111 The Market The quotmarketquot might be visualized as one huge investor who has the resources to set security prices through buying and selling activities with literally millions of small investors In reality market prices are set by the collective buying and selling activities of both individuals and institutions large and small Outstanding securities trade in the quotsecondaryquot market The New York Stock Exchange is an example of a secondary market In addition the supply of securities is affected by corporations and government entities selling new securities in the quotprimaryquot market These new securities generally are marketed by investment bankers who sell directly to their customers Thereafter newly issued securities trade in the secondary market 2 This aggregated buying and selling activity determines prices of securities through the interaction of supply and demand forces However the market works as though one huge investor forms expectations on future cash ows and risks and determines security prices accordingly Therefore when we refer to the market we are referring to the trading activity and the mechanisms through which prices are determined Prices are determined though a quotconsensusquot opinion by participants on the value of future cash ows and the risk of those cash ows Why do security prices and therefore expected returns change Price changes occur for one or both of two reasons 0 Expectations for future cash ows change or 0 Required returns change It is useful to think about these changes using the perpetuity model or PO CFr where P0 is the price ofa security today t 0 CF is the perpetual cash ow received from a security and r is the required return on the security Recall that r rf 6 where 6 is the risk premium Why might the expectations for future cash ows change New competitors enter a firm39s market Tariff changes affect the demand for a firm s product A patent is approved for a new invention New information is released concerning the firm s productivity Why might required returns change The riskfree rate might change For example if in ation is expected to increase riskfree rates will rise Required returns might also change if a firm s required risk premium changes For instance suppose Quaker Oats announces that it is going to diversify into the perfume business an activity about which it presumably knows nothing We would not expect the market to have the same required risk premium for the new business as it had for the original business The material in Chapter 11 relates to required returns We will talk more about changes in cash ows in subsequent chapters The market anticipates the future with respect to cash ows and risk to a certain degree For instance the market may have an opinion on IBM39s next quarterly earnings report The market will have an opinion on what actions the FED will take this week that will in uence interest rates Given the enormous amount of money at stake it is not surprising that many smart and wellendowed individuals and institutions with vast data bases realtime communication channels sophisticated analytic models and powerful computing resources are attempting to predict the future and accordingly form estimates of the intrinsic values of securities Think of the intrinsic value of a security as its true value as if all information was known about future cash ows and risk relevant to the security To the smartest and the fastest players that anticipate changes first go the largest rewards The resources dedicated to these pursuits make the US capital markets the most competitive and ef cient in the world For now define an e icient market as one in which all information is re ected in a security39s price We ll have a lot more to say about this in Chapter 13 IV Announcements and Information To the degree that the market s expectations are realized e g IBM s earnings were as expected the market perfectly anticipated the announcement Therefore when IBM releases its earnings the market will quotyawnquot We would not expect IBM39s stock or bond prices to change because of an announcement that was anticipated The market is said to have quotdiscountedquot the announcement in advance ie the expected and realized level of earnings had already been incorporated in the prices of IBM39s securities Such announcements can hardly be labeled as quotnewsquot They contain no new information Some announcements might be easy to predict in advance other announcements may be impossible to predict in advance Many announcements are partially anticipated but with large uncertainty as to the actual content Announcements that were not or could not be accurately anticipated contain real quotnewsquot ie they totally surprise the market An example may be a plane crash that kills the president of a firm Other announcements deviate to some degree from expectations These announcements also contain an element of quotnewsquot For example IBM s expected dividend increase may be 050 per share but it turns out to be 075 In advance of the announcement the market would have adjusted IBM s stock price for the expected 050 announcement the price will jump at the announcement of 075 but only to adjust to the surprise part of the announcement or the 025 difference Some announcements are firm specific ie the death of the company president Other announcements affect a large number of firms ie the level of actual and expected in ation levels the situation in the Middle East trade relations with significant foreign trading partners etc In summary an announcement may contain two parts 0 An anticipated portion and 0 An unanticipated surprise portion Once again to the extent anticipated events are actually realized the announcement is not really quotnewsquot The announcement simply confirms what had been anticipated Anticipated events are quotpriced outquot re ected in the security s price or already discounted in advance of announcements However if IBM39s dividend announcement were higher or lower than anticipated it contained an unanticipated portion we would expect IBM39s price to rise or fall The unanticipated portion of the announcement is a quotsurprisequot and is accordingly quotnewsquot Once again information is only the surprise part of an announcement Prices will change on the release of news that affects a security 4 Sorry to be so redundant but these points are critical to understanding Chapter 11 Risk relates to surprises If what is expected to happen actually happens and you can rely on expectations to equal outcomes then no risk exists Future events are perfectly anticipated Now let39s see how this discussion relates to the pricing of securities V A Multi Factor Model What if investors demanded compensation via extra returns for factors other than the market factor the CAPM For instance what if investors required expected returns based upon The market factor Erm rf The expected oil price level The Gross Domestic Product GDP and The in ation rate Therefore in this example four risk factors are being considered by investors in setting prices in the marketplace While all four factors in general affect returns on all securities these factors affect the required returns on different securities in different ways Some securities are more sensitive to some factors than are other securities For instance we d expect Texaco s stock price to be more sensitive to oil price changes than Nordstroms s stock price The expected level of these factors will be re ected in the expected return of a security or portfolio However the deviations from the expected levels that actually occur represent the significant risks of securities or portfolios Let s call these deviations surprises of actualfromexpected factor levels F1 through F4 where F1 the marketfactor or the realized level ofF1 less the expected level F1 F2 the oilprz39cefactor or the realized level of F2 less the expected level F2 F3 the GDPfactor or the realized level of F3 less the expected level F3 and F4 the in ationfactor or the realized level of F4 less the expected level F4 Under this model for returns the actual return r will equal the expected return Er plus two categories of surprises o Surprises that occur when actual factor levels deviate from expected factor levels and o Surprises that occur when some company specific news is released Using this model we can write realized returns as ri EI iBlF1BzF2 B3F3 B4F4 8i where ri is the realized return on security i Eri is the expected return on i the F39s are the factors de ned above the B39s are the sensitivities of security i to the four economic factors and s is the company unique or speci c surprise for security i The si risk is often referred to as idiosyncratic risk unique security speci c or unsystematic risk The expected values for F1 through F4 and s are of course zero By de nition surprises are not expected If security i has no sensitivity to a factor say F3 then 53 equals 000 for this security In words sort of I Eri 2 market factor surprises security i s sensitivity to the factors f 1 security i speci c surprises where F equals the number of factors that affect security prices 7 four in this example Accordingly the signi cant risk of a security comes from the surprise factors both the surprise changes in the underlying factor levels that affect all most securities and the surprises unique to a particular security What types of surprises might relate to an individual security but would not affect the market as a whole The company is granted patent protection on a new hot product The company experiences an unexpected labor strike The company39s primary production facility burns down The company39s brilliant RampD scientist has a heart attack and dies The company39s primary competition is shut down because of severe EPA violations What other company speci c surprises can you think of When we move to a multifactor model the graphics occupy F space This makes things hard to draw but the ideas remain the same Example Let s illustrate how the above model works with the simplest possible case a onefactor model for security j Assume that the only relevant factor is the market factor You ll recognize this case as the CAPM Erj expected return rf Erm rfBj rj actual return expected return surprise return rj Erj rm ErmBj sj rj ff 13rm r051 t rm 13rm5j 3139 Let s assume Erm 014 the expected return on the Market Portfolio rm 016 the actual return on the Market Portfolio rf 006 the expected actual return on the riskfree asset Bj 120 the expected m actual sensitivity of security j to the market and Sj 001 the unexpected security speci c news for securityj rj 006 014 006120 016 014120 001 006 0096 0024 001 017 The expected component ofthis actual return is 0156 or 006 0096 The surprise part of the return is 0014 or 0024 001 The surprise consists ofa better than expected return on the market multiplied by security j s sensitivity to the market a pleasant surprise plus the rm speci c surprise for security j an unpleasant surprise We could easily expand our example to include more than the market factor The exibility of the APT is that it allows for the introduction of additional factors However this example is simply trying to convey the intuition of what quotrealquot risk represents VI andU Risks quot Drawing upon your background from Chapter 10 it should be no surprise that the security speci c unexpected news has an expected value of zero and a covariance of zero with other security speci c news items for other securities in the market Given these properties the security speci c risk or idiosyncratic risk can easily be diversi ed away in a reasonably sized portfolio However the other factors are common to all securities in the market in varying degrees of sensitivity These factors pervade the market all security returns more or less are in uenced by these factors Therefore in a multifactor model we see the same decline in total risk 62p as a function of portfolio size as we saw in the CAPM singlefactor development or op Total Risk Systematic Risk Unsystematic Risk VII The APT versus the CAPM The CAPM and the APT are alternative models that relate Er to risk Both have certain advantages and disadvantages In the CAPM a singlefactor model the correlation between securities occurs because they are jointly correlated with the market However movements in the market return per se39 do not quotcausequot movements in the security39s return While market movements are correlated with movements in security returns changes in security returns and in the market returns are caused by the underlying economic factors Remember correlation does not imply causation These quotcausal factorsquot are not speci ed in CAPM In the APT an attempt is made to specify the underlying factors in the economy that do directly affect security returns Again because several factors might be relatedto security returns the APM often is called a multifactor model Correlations between securities occur when securities are affected by the same factor or factors Both the CAPM and the APT imply a positive and linear relationship between expected return and risk The APT simply allows for more factors to be at the root of this total risk Since the APT can incorporate multiple factors ie it is a quotricherquot model This model has the quotpotentialquot to measure Er more accurately than the CAPM Again however note that the CAPM is a subset of the APT a single factor version Having made this observation we note that APT does not specify which economic factors are related to Er It is a plausible description of the world but requires further description of what constitutes the factors or a statistical identification of what each security s factor sensitivity is and what is the premium associated with a statistically identified factor before it can be completely useful To the best of my knowledge the APT is not currently in widespread use in the quotreal worldquot however recent high profile research has hastened its adaptation To be completely true to the theory we must acknowledge that we do not know what the economic factors are nor do we know the sensitivity of individual securities to these factors New work has identified some proxies that make the APT much more useful and more accepted This has caused its adoption by some of the more sophisticated corporations in their decisionmaking processes In contrast the less sophisticated CAPM is theoretically based on the existence of an efficient set of risky portfolios plus the riskfree asset While the assumptions that underlie the derivation of the CAPM may seem unrealistic and its empirical fit with the data can never match that of the APT the CAPM has achieved a considerable almost universal impact on quotreal worldquot decisionmaking We have more on the use of CAPM in the next module CORPORATE FINANCE AN INTRODUCTORY COURSE DISCUSSION NOTES MODULE 31 FINANCIAL MARKETS AND NET PRESENT VALUE I Financial Markets and Net Present Value A Introduction Financial markets make it possible for individuals and rms to borrow or lend over time thereby improving their intertemporal across time periods investment and consumption patterns Think about it Why do individuals invest They invest because they are willing to forego present consumption for future and hopefully larger consumption Why do people borrow or sell assets eg common stocks They want to increase present consumption spend the money now at the sacri ce of future consumption eg repay the debt or not receive future dividends In this context consumption can include purchasing a new car a new home investing in their or their children s education retirement charitable contributions or an inheritance for their heirs Because of the financial markets individuals can choose consumption and investment patterns within the limits of their wealth constraint so to maximize their utility of consumption over time In addition financial markets provide individuals and firms critical feedback regarding required rates of return on investment ie benchmark rates of return Accordingly financial markets are indispensable in a wellfunctioning market based economy as in the Us Investment particularly in quotreal assetsquot is critical in the creation of jobs and increasing the wealth of individuals and in turn society Investment produces future goods and services for our society always at the cost of current consumption The question is always is the sacrifice worthwhile B Market Clearing Financial intermediaries for example banks and other financial institutions perform an important functionthey match up borrowers and lenders This market quotclearsquot when the quantity of money demanded by borrowers equals the amount of money supplied by investors If a surplus of borrowers exists interest rates will rise discouraging some borrowers money will become too expensive and encouraging new lenders the return to lending is higher If a surplus of lenders exists interest rates will fall discouraging lenders and encouraging borrowers The interest rate that causes markets to clear is referred to as the equilibrium market rate of interest2 Interest rates are quite dynamic as the supply and demand of funds varies over time 1 This module is designed to complement Chapter 3 in Ross Wester eld and Jaffe 2 In equilibrium no pressure exists for interest rates to change C PreliminaryAssumptions In the subsequent discussion we will assume the following 1 Perfect certainty Outcomes are known with 100 probability What does perfect certainty look like on a probability diagram Without risk only interest rate will exist in the market for each maturity Differential interest rates result from investments with different risk We temporarily assume risk away 2 Perfect Capital Markets PCM a Information is free and available to all participants that want it b All participants have equal access to the financial markets or rb the borrowing rate r1 the lending rate r c All participants are price takers no investor is large enough to impact the supply or demand for funds d There are no transactions or contracting costs exist and e No distorting taxes exist 3 Investors are Rational in the following sense a Investors prefer more wealth to less wealth b Investors prefer cash sooner than cash later c Investors prefer less risk to more risk 4 We have a one period world t 0 today andt 1 in one time period Do you understand why only one interest rate per maturity can exist in the market if we assume away uncertainty ie no risk exists In a oneperiod world there is then only one interest rate since there can only be one maturity If more than one rate existed you could create a quotmoney machine quot What do I mean by a money machine Borrow low and lend high with the same risk Buy low and sell high with the same risk What is quotarbitragequot in this context Arbitrage opportunities exist when the same item or more generally perfect substitutes sell for two different prices You can buy at the low price and sell at the high price earning a riskless profit Interest rates represent the price of moving money across time If money sells at more than one price ie interest rate you would borrow low low price and lend high high price In a well functioning economy arbitrage opportunities should not persist very long traders will exploit 2 them driving the prices of the substitute items to equality Don t worry I know these are unrealistic assumptions We will begin to relax these assumptions in the next and following lectures However using these assumptions provides us important insights without complicating factors It is a common technique in economics and nance to start with the simplest possible world and then add complications onebyone This procedure allows us to identify the complications that matter the most and precisely what each one does D The Concept ofPresent Value a fundamental business concept Every investor has a utility function de ned with respect to consumption consumption at t 0 again t 0 represents today and at t 1 again t 1 represents one period away The investor39s objective is to maximize the utility of hisher consumption over these two time periods Some of us would prefer to consume more now others would prefer to consume more in one period In other words it is perfectly rational to have different tastes and preferences for consumption from different individuals or at different periods in our lives Tastes and preferences are individualistic a personal choice Consider examples of how your consumption preferences change over your own lifecycle We need a way to relate cash flows that occur at different time periods i e att 0 and t 1 to solve our consumption preference problem For example which would you prefer Case 1100 att 0 or C0 1 and zero dollars att 1 C1 0 or Case 2Zero dollars att 0 or C0 0 and 100 att 1 C1 1 You can t directly compare since the cash arrives at different times Why might you prefer Case 1 Let quotrquot the interest rate eg the rate on TBills What is a TBill quotrquot equals the per period quotmarket interest rate quot Ifr 10 how much is Case 1 100 worth at the end oft 1 Case 1 Case 2 1001r gt 100 for all cases ofrgt0 100110 gt 100 att1 110 att1 gt 100 att1 Since you are quotrationalquot you prefer more to less and it is okay to compare dollar values that occur at the same point in time you prefer Case 1 to Case 2 What ifr 0 What ifr lt 0 The above example implies that money has a time value Ifr gt 0 a dollar received sooner is 3 more valuable than a dollar received later Which of the following cases do you prefer Case 3100 att 0 or C0 1 and zero dollars att 1 so C1 0 or Case 4Zero dollars att 0 or C0 0 and 115 att 1 or C1 115 Since the t 1 payoff for Case 4 is more than 100 this example is not as straightforward as was the comparison of Cases 1 and 2 We cannot answer the question without knowing the market interest rate r Ifr 10 Case 339s 100 grows to 110 at t 1 Therefore you prefer Case 4 which pays 115 at t 1 Ifr 20 Case 339s 100 grows to 120 at t 1 You prefer 3 to 4 In short we need to know r the market rate of interest to make our decision Supplying this benchmark is an important input to decision making provided by the financial markets In our oneperiod world we can always compare values of different investments at t 1 and take the case with the highest t 1 value However for several reasons that will become clear later in finance we prefer to compare alternatives at t 0 versus at later time periods eg t 1 While making comparisons at the same future time period will give us the same answer as comparisons at t 0 we will discover using t 0 is far easier when we move beyond our one period world Let C0 cash att 0 Let C1 cash att 1 What level of C0 and C1 will make us indifferent Indifference implies the same dollar amount at the same time period We saw that Co becomes C01r if we invest at the interest rate r Indifference between our cases above means that C01r C1 A very little algebra gives us the following definitions C01 r C1 C0 is quotcompoundedquot forward for equivalence with C1 C0 C11 r C1 is quotdiscountedquot backward for equivalence with C0 Compounding means take present amounts forward in time Discounting means bringing future amounts back to present values at t 0 Indifference means that Co is the t 0 equivalent of getting C1 at t 1 or getting cash in one period Indifference then means C0 is equal to the quotPresent Value quot orPV of C1 or C0 PVC1 Present value is the value today t 0 of future cash ow in this case C1 4 IfC1 115 andr 10 PVC1 115110 1045 You don t care ifyou have 1045 today or 115 in one period Why don t you care It s not general apathy Here s why 1045110 115 ie ifwe had 1045 att 0 we could invest it at 10 and it would become 115 att 1 Alternatively if you were getting 1 15 at t 1 and the interest rate was 10 a bank would loan you 1045 at t 0 if you promised to pay 1 15 at t 1 Let s take another example If r 15 which case would you prefer Case 5110 today and zero att 1 Case 6125 at t 1 and zero att 0 The PV of Case 5 PVC0 1 10 Why The present value PV ofa dollar today att 0 is a dollar PV of Case 6 125115 1087 PVC1 Therefore you prefer Case 5 its PV is higher Comparing present values is comparing dollars today Thus very simply which ever is higher is more preferable for rational investors What is the most that you39d pay for Case 6 getting 115 in one period 1087 If you paid this amount at t 0 you would earn 15 on your investment Why What market interest rate r would make you indifferent between Case 5 and Case 6 110 1251 r Solve for r 1 r 125110 11364 r 01364 or 1364 Ifr lt 1364 you prefer Case 6 Ifr gt 1364 you prefer Case 5 Ifr 1364 you are indifferent the future t1 and present t0 values of the two cases are equal Test yourself to make sure that you understand these statements E The Concept of NetPresent Value NPV An investment costs 100 at t 0 and pays off with certainty 125 at t 1 Illustrate this situation using a cash ow diagram as shown in the RWJ text Is this investment a good deal You can t tell yet the answer depends on r Where do we get an r From observing the market The market interest rate r provides a benchmark rate by which we can judge an investment as good accept or bad reject Our general rule can be stated as We want to accept investments that are worth more than they 5 cost We reject investments that are worth less than they cost To evaluate the investment just described we must compare the PV of 125 at received at t 1 to 100 which is the PV ofthe cost of generating the future payout at t 0 Ifthe PV of 125 is greater than 100 the PV of what you get is greater than what you pay we should accept the investment Net Present Value NPV PVinflows PVout ows If NPV gt 0 accept the project Our wealth is increased by making the investment If NPV lt 0 reject the project Our wealth is decreased If NPV 0 we are indifferent Our wealth is unchanged NPV PVCl PVCO PVin ows PVout ows NPV 1251 r 100 Say that r 12 NPV 125112 100 11161 100 1161 Since NPV is positive accept the project What does 1161 represent It represents the wealth increase from undertaking the project Your current wealth increases from 100 to 11161 Since we prefer more wealth to less the investment makes us better off and so we accept the project An alternative way to look at this same decision is that if we invested 100 at t 0 at the market rate r 12 we d have 1001 12 112 att 1 We can obtain this t 1 amount by investing at the market rate eg buying a government security Thus this 112 is what you lose if you invest in the project the future opportunity cost of investing the 100 in the project If you invest in the project you can t spend the same 100 to buy the government bond Contrast this amount to what you have if you invested the 100 at t 0 in the project t 0 t 1 If you invest in the market and not the project 100 112 If you do invest in the project and not the market 100 125 With the project we are 13 better off at t 1 than without the project What is the PV of this 13 difference att 1 PV 13112 1161 Therefore the project increases our wealth by 1161 at t 0 This amount is the NPV of the project NPV accounts for the time value sacri ce of your 39 It 391 the u 39 J cost of passing up alternative investments at the market rate We can always choose to invest 6 at the market rate of interest instead of investing in the project eg go out and buy a US government security Thus it is a meaningful comparison for a riskless investment project If a rm was contemplating the above investment what would happen to the common stock value of the rm when this project was accepted Assume that the rm has one share of common stock outstanding and its only asset is a 100 bill The rm was worth 100 After the investment it would be worth 11161 The stock price would increase by 1161 If 10 shares were outstanding each share would go up by the NPV divided by the number ofshares 116110 shares 1161share from 10 to 11161 per share Review the basic equations again NPV PVC1 PVC0 NPV C11 r C0 Example Say an investment would cost 10000 at t 0 and would return 11500 at t 1 The market rate of return is 12 Should you take the investment NPV 115001 12 10000 NPV 268 Since the NPV is positive accept the investment What is the actual rate of return on this investment The rate of return which we will soon be referring to as the internal rate afretnrn or the yield is that rate that drives the NPV to zero Set the NPV to zero and solve for this return r 0 115001 r 10000 Thus rig 015 or 15 Note in a one period world you can easily solve for this rate algebraically or using your calculator Better yet use a spreadsheet r9 is the internal rate afretnrn 0n the investment The internal rate afretnrn is a rate intrinsic to the investment It should not be confused with r the market rate afretnrn In this case r 12 Since the investment returns 15 you would accept it r gt r This just says that the investment gives a higher return than will the opportunity available in the market Take an investment only if its yield is greater than the market rate This is equivalent to the NPV rule Borrowing Note that if you were borrowing money to be repaid in one period you would have a positive cash ow at t 0 and a negative cash ow at t 1 when you repay the loan 7 Say you borrow 10000 at t 0 and must repay 11250 in one year What is the effective interest rate You need to nd the interest rate that equates the PV in ow with the PV out ow Therefore at this interest rate the NPV will equal zero or NPV PVC0 PVC1 Note the signs are reversed on the cash ows for the quotborrowingquot problem 0 10000 112501 E The rate rquot on the loan is 0125 or 125 Take this arrangement if the market rate is greater than the yield r It s better to borrow at the lower rate just as it is better to invest at the higher rate Note that the NPV rule doesn t change 111 Comparative Statics NPV C11 r Co for a typical oneperiod investment problem ie negative cash out ow at t 0 followed by a positive cash in ow at t 1 What happens to the NPV if all else equal C1 increases C1 decreases C0 increases C0 decreases r increases r decreases In economics this type of questioning is called quotcomparative staticsquot You should think about the results until they are intuitive to you For the most part for the rest of the course we 39ll put projects39 cash flows all on a t 0 common time denominator i e we will compare projects on the basis of their net present values It is important to remember that we can directly compare cash flows only if they occur at the same point in time e g you cannot directly compare a dollar at t 0 to a dollar at t 1 Understand why Bringing future cash ows back to present values is called discounting to time zero t 0 IV Analysis A One Period Market Opportunities Let s assume you have a labor income stream of 1000 att 0 or C0 1000 and 1000 at t 1 or C1 1000 You have no other wealth Both cash ows are in ows so both are positive W0 the current value of your total wealth income stream evaluated at t 0 is W0 C0 C11 r Further assume that no financial markets exist ie you have no borrowing or lending opportunities It39s like you are stranded on a desert island and cash shows up in a bottle BC comic at t 0 and at t 1 A beer stand is located on the island and it is open only at t 0 and t 1 How much beer can you consume at t 0 and t 1 8 t0 tl 1000 maximum 1000 minimum money is useless later and you like beer 500 1500 0 minimum 2000 maximum Now a nancial market opens on the island The market rate of interest is announced as 10 which is an equilibrium rate The market allows for borrowing and lending At t 0 how much could you borrow and be able to pay it off with your labor income at t 1 10001 10 909 This is your maximum borrowing power given your certain income at t 1 W0 1000 10001 10 1909 This represents the PV of your total wealth at t 0 Your maximum beer consumption at t 0 is now 1909 Of course if you borrowed and consumed the maximum how much could you consume at t 1 Zero All of your t 1 income must go to repay the loan of 909 or 9091 10 1000 909 in principal and 91 in interest Your maximum beer consumption at t 1 is 2100 How did we get this amount W1 or your total wealth evaluated at t 1 a future value is W1 C01 r C1 W1 10001 10 1000 1100 1000 2100 For maximum t 1 consumption you invest all of your income at t 0 which will grow to 1100 at t 1 This amount plus your income at t 1 allows you to consume 2100 at t 1 Obviously to accomplish this maximum consumption at t 1 you must consume nothing at t 0 With this bank in operation do you care what your income pattern is during t 0 and t 1 Q long as the PV of the income is 1909 NO With access to the nancial market any pattern of income at t 0 and t 1 with this present value is equivalent to any other With nancial markets you can consume anywhere along the market opportunity line With nancial markets you have more choices of consumption now versus consumption later Therefore nancial markets are socially desirable more choices In economics and nance more choices dominate fewer choices Financial markets allow us to move dollars and therefore consumption across time periods W1W0 21001909 110 1 r where r is the rate of transformation of current for future consumption In sum given C0 and C1 and r you can calculate your current wealth W0 In essence with nancial markets only W0 matters As long as any pair of C0 and C1 has the same PV your wealth is the same Draw a twoperiod diagram just with W0 positioned on the xaxis Next draw the market opportunity line beginning at W0 and extending toward the yaxis with a slope of 1 r All pairs of C0 and C1 with a PV W0 lie on this line With the right choice of borrowing or lending you can consume anywhere on this line Your constraint is W0 which is your wealth at t 0 based on your present and future income All that matters with respect to your consumption opportunities is W0 and r Dissect the market opportunity line into the region of borrowing and the region of lending If you want to consume more than 1000 at t 0 you must borrow If you want to consume more than 1000 at t 1 you must lend at t 0 so you receive the repayment at t 1 You as an individual bring your tastes and preferences for consumption into the picture and determine your t 0 versus t 1 consumption levels What happens if r decreases Those with preference for t 0 consumption borrowers are happy Those with preference for t 1 consumption lenders are unhappy If r increases the reverse is true What happens to the above analysis if you have an initial endowment of wealth in addition to C0 and C1 Take the above example but say you have 1000 at t 0 in addition to your labor income at t 0 and t 1 Just add the 1000 to the previous 1909 so 2909 Then proceed with the analysis given this new wealth constraint B One Period Productive and Market Opportunities In addition to being able to lend or borrow at the market rate of interest along the market opportunity line assume that you have discovered three productive investment opportunities in real assets The Market rate r 10 Again a major advantage of having a financial market is that it provides a benchmark return against which to judge real investments in productive opportunities t 0 t 1 Project Total Investment Out ow In ow Return 1 NPV NPV Il 636 1272 100 520 520 Iz 636 954 50 231 751 13 636 668 5 29 722 1908 2894 Where r the project return C1 C0C0 and NPV PVC1 C0 C1110 7 C0 Recall with labor income in t 0 and t 1 of 1000 each your W0 1909 Mark the t 0 aXis with W0 At this point all we care about is W0 and r Why Consider 11 If we take this project our wealth becomes W0 1909 636 12721 10 2429 versus the 1909 wealth we had before we took the project Therefore our wealth has increased by 2429 1909 520 which equals the NPV of this investment By the NPVRule take the project the NPV is positive rquot 1272 636636 10 or 100 Therefore 11 is desirable 1 r gt 1 r Using the Rate of Return Rule take projects where r gt r By this rule we accept the project 100 gt 10 By investing in 11 we are transforming present dollars for future dollars at a higher rate than if we invest in the market Our new t 0 wealth after taking 11 increases by 520 Now we can move up and down a higher market line Desirable Of course We are rational investors Draw the new market opportunity line Consider 1 If we take this project in addition to 11 our wealth becomes W0 1909 636 636 1272110 954110 2661 Over and above our wealth increase from taking 11 12 increases our wealth 2661 2429 232 we have a 1 rounding error versus the NPV listed in the table Since the NPV increase is positive we take the project by the NPV Rule rig 954 636636 050 50 r gt r so accept the project by the Rate of Return Rule From both investments we add 520 232 752 to our original wealth of 1909 Our new wealth after taking both investments is 1909 520 232 2661 Consider 1 If we take this project our wealth becomes W0 1909 636 636 636 1272110 954110 668110 2631 However with just the first two investments our wealth was 2661 Therefore by taking 13 our wealth has decreased by 2631 2661 or 30 the NPV ofthis project By the NPV Rule we would reject the project rig 668 636636 005 or 5 Therefore 13 is undesirable by the Rate of Return Rule or r lt r We would be better off investing in the market at 10 than in 13 at 5 In sum take 11 and 12 These projects increase our wealth Reject project 13 Taking this project decreases our wealth Summa 1 Take projects if the NPVis positive the NPVRule 2 Take normal projects if the r is greater than r the Rate of Return Rule As long as the project represents a better opportunity than does the market the project has a positive NPV Now consider a very large number of projects The productive opportunity line can be represented by a curve Draw the diagram Take all projects with r gt r or with NPV gt 0 Stop at the tangency point on the productive opportunity line with the market opportunity line The tangency point is where the slope of the market opportunity line equals the slope of the productive opportunity line We continue to invest in productive opportunities up to this point Therefore the tangency point has major signi cance in our investment decisions C Fisher Separation Financial managers make their firms39 investment decisions Recall their objective is to maximize the wealth of their shareholders However how does the nancial manager take into consideration the tastes and preferences of shareholders for consumption now versus consumption later in making the firm39s investment decision If the financial manager had to worry about shareholder tastes and preferences before making an investment decision heshe would have a real mess Different shareholders will have different consumption preferences What would the manager do Would they send out a questionnaire before making each investment Fortunately this situation is easily handled The financial manager just takes all projects that pass the NPV Rule or in our simple world the Rate of Return Rule This action will maximize the value of the firm and accordingly the wealth of the shareholders Once their wealth is maximized shareholders are free to individually choose how much to consume versus how much to invest If they want to consume more they simply sell some of their stock If they want to consume less they maintain or increase their investment in the firm Regardless all are better off with a higher stock price By taking wealthincreasing projects the financial manager has maximized the opportunities for the shareholder to consume more now or later The investment decision is therefore separate 12 from the investors39 consumption decision Once their wealth is maximized shareholders can engage in market quot to ma imi e their 39 J39 39J 39 utilities This separation of the investment and consumption decisions is called Fisher Separation after the famous economist Irving Fisher In sum what can nancial managers do for their shareholders Maximize their wealth or W0 They can leave the rest to the shareholders themselves CORPORATE FINANCE AN INTRODUCTORY COURSE DISCUSSION NOTES MODULE 61 INVESTMENT CRITERIA The objective of these discussion notes is to evaluate different techniques used to analyze the desirability of longterm asset acquisitions Capital budgeting is the process of making long term xed asset investment decisions For capitalintensive rms rms with high percentages of xed assets to total assets these are the most critical decisions that the rm makes Mistakes are painful to rectify The phrase throwing good money after bad did not arise in a vacuum To be a welleducated business student you must understand the decisionmaking options that a rm may use to make project choices Importantly you must understand the strengths and weaknesses ofthese methods The capital budgeting analytic techniques we will discuss in turn are NPV The Pro tability Index Internal Rate of Return IRR Payback and Discounted Payback and Accounting Rate of Return The techniques above the line discount all project cash ows to make assessments about the merits of the project All three of these methods also use marketdetermined discount rates instead of using quotad hocquot cutoff levels for project acceptance These methods are referred to as Discounted Cash Flow Techniques or DCF Techniques DCF techniques are the most frequently used decision making tools in corporations today and large rms are more likely to use these tools than are smaller rms as are rms whose CEOs have MBAs The latter two methods below the dotted line do not consider or inadequately consider in the case of Discounted Payback the time value of money There are other problems with these approaches that we will discuss later DCF methods properly focus on the required rate of return for a project r or alternatively the opportunity cost of capital for the proiect Therefore these techniques acknowledge the time value of money The cash flows of a project should cover the 1 This lecture module is designed to complement Chapter 6 in Ross Wester eld and Jaffe l outlays for the project plus the return you could earn elsewhere instead of this project i e cover the opportunity costs of the investment However as we will also discover not all DCF methods are created equally The NPVmethod will emerge as the clear winner when all of the quotdust settles quot NonDCF methods ignore the opportunity cost of capital or the time value of money This is their main source of error With respect to the Accounting Rate of Return method this method quotadds insult to injuryquot in that cash ows are not even used in the analysis Proper methods of analyzing capital budget projects should value more highly More cash versus less cash Nearer cash versus later cash and Less risky cash versus more risky cash As we will discover only the NPV Rule consistently satis es all of these criteria NPVis the method of ch oice for project evaluation among welltrained nancial managers Having made the above point you may wonder why we are taking valuable class time to discuss the other methods Good question Say you go to work for a rm that does not use NPV Without insights into the strengths and weakness of the method being used and the ability to contrast this method to the virtues of the NPV approach you will not be able to present a very effective case for changing capital budgeting evaluation methods A NPV NPVDecision Rule Take Positive NP VProjects Reject Negative NP VProjects NPV PV in ows PV out ows where all cash ows are discounted at the appropriate rate of return r Since we ve discussed using NPV in calculating project desirability extensively we will only brie y review the mechanics of this method Example A project has a required rate of return of 10 and the following cash ows Time 0 1 2 3 4 Cash Flow 5000 2000 2500 3000 2000 NPV 5000 20001101 25001102 30001103 20001104 NPV 2504 The NPV is and so collective shareholder wealth increases by 2504 if the project is accepted Therefore take the project If 1000 shares are outstanding the price of each share should increase by 2504 when the rm announces the project NPV measures the immediate increase in wealth of shareholders in dollars If the NPV is positive the project has earned more than the opportunity cost of funds r NPV focuses on cash flows the timing of the cash flows and the risk of the cash flows the higher the risk the higher the discount rate r NPVis expressed in dollars Measuring project desirability in dollars is consistent with our target objective maximizing shareholder wealth After all isn39t wealth expressed in dollars or pounds yen deutsche marks francs krona pesos krone lira peseta etc 7 B Pro tability Index 1P1 Profitability Index Decision Rule Take Projects when the P gt 10 Reject Projects when the P lt 10 In government jargon the PI method is referred to as BenefitCostAnalysis However it is calculated in the same manner T PI z CFt1rtC0 H PI PV of Cash Flows after t0 divided by the negative of the Cash Flow at t 0 Discuss this equation If PI gt 10 the PV of the in ows is greater that the PV of the out ows Hence NPV is when PI gt 10 For an individual project the decision made using the PI method always will equal the decision made using the NPV method When one method signals acceptance so will the other method When one method signals rejection so will the other method Can you see why However PI can give us acceptreject signals that con ict with the NPV Rule when we must rank projects against one another When must we rank projects Projects are mutually exclusive andor The firm is capital rationed By mutually exclusive we mean you can take one project or the other but not both Say you own a comer lot You can build a gas station or you can build a ower shop but you cannot build both buildings These options are mutually exclusive By capital rationing we mean that the rm has more good projects than it has capital to fund the projects this requires some capital market imperfection On the personal level most of us nd ourselves capital rationed most of the time That s not a capital market imperfection that s greed ExampleMutuallv Exclusive Proiects Let s take two mutually exclusive projects and demonstrate how NPV and PI can give us con icting answers CF stands for Cash Flow PROJECT CF t 0 CF t 1 NPV 10 P A 1000 1500 364 136 B 10000 13000 1818 118 By both decision criteria these are good projects Do you see why If we could we would take both This example illustrates that if we are considering projects independently the NPV Rule and the PI Rule give us the same acceptrej ect decisions However these projects are mutually exclusive we can only take one By the NPV Rule we should take Project B it has a higher NPV We can think of project A s B s NPV as part ofthe opportunity cost of taking project B A Thus we must take the project with the higher NPV ie find the project s NPV in the normal way and subtract the NPV of the other project to account for the opportunity cost However by the PI Rule we should take Project A it has a larger PI Accordingly the two procedures give us con icting signals However remember our goal is to maximize shareholder wealth Our shareholders can t spend an index they need cash Accordingly we should choose Project B The con ict occurs because the PI does not properly consider the scale ofthe project As an index it gives the profit per dollar of invested capital Larger projects may often have smaller PI s than smaller projects but the larger project may increase shareholder wealth by a larger amount think about a 100 increase on 1000 versus a 1 increase on 1 Billion ExampleCapital Rationing I rarely disagree with the authors of the text if I had there is some chance Steve would not have given me my degree However Itake issue with them on their recommendation to use PI for ranking projects if the firm is capital rationed Say your rm has a capital budget of 25 million that it cannot exceed The interest rate is 15 You have identi ed six projects that have positive NPV s and PI s that exceed 10 None of the projects are mutually exclusive Therefore you39d like to accept all six projects but you don t have the funds to do so The data are as follows all dollars in millions PROJECT OUTLAY t0 NPV PI 5 356 E 80 E 241 Without capital rationing you would accept all six projects at a t 0 cost of 80 million and a combined NPV 241 However you have a budget constraint of 25 million If you rank the projects based on P1 you would take projects F E and D for a total outlay of 20 million Why Those are the highest ranked projects within the budget constraint of 25 million After F and E you39d like to take A the 3 ranked project However you only have 15 million left after taking the first and second ranked projects so you cannot afford A which costs 25 million Similarly you can t take B after taking F and E Our analysis assumes that you cannot take 3 5ths of project A or 34ths of project B If you can freely take fractional projects without a ecting their pro tability per dollar invested the PI rank is a valid approach What is the combined NPV ofthese three projects 356 313 142 811 million Is this PI ranked set really the best set of projects from the shareholders39 perspective NO Is ranking the projects by their NPV s the best solution What projects would you take Project A it has the highest NPV 852 million but it exhausts our 25 million budget However this choice is still not optimal from the shareholders perspective Consider all possible subsets of these six projects such that each subset is within the budget constraint Sum the NPV s within each subset The results are as follows DEF 20 811 This analysis suggests that Projects B and F are the shareholders39 preferred choice The two projects are within the budget constraint and have a total NPV of 925 million Note that this choice is different from and preferred to the choices made by ranking projects with either the PI ranking method or the individual project NPV ranking method Accordingly if a firm is capital rationed it should find all possible subsets of projects that individually have positive NPV39s and fit within the budget constraint Choose the subset of projects that has the highest combined NPV2 In the above example note what capital rationing costs the shareholders in foregone wealth Without capital rationing all siX projects would be chosen which have a combined NPV of 241 million With capital rationing only Projects B and F are chosen with a combined NPV of 925 million What is the loss of wealth because of capital rationing 241 925 1485 million This loss is a very steep price to pay for being capital rationed Should the firm impose capital rationing upon itself I contend that the firm should try very hard to nd the money to take all positive NPV projects Theoretically in a perfect capital market a firm should never be capital rationed If you have good projects some investors should be very willing to supply the capital However a very high percentage of firms self impose some form of capital rationing yearafter year The reasons for this observed phenomenon are not well understood Perhaps the real reason firms turn down positive NPV projects has more to do with not being able to find qualified employees to manage the projects or perhaps they suffer shortages in required raw materials to initiate the projects Another reason some firms give for imposing capital rationing is to force managers to prioritize among projects While prioritization may be desirable sacrificing shareholder wealth is not C Internal Rate of Return gIRR 2 This problem can be solved using a variant of Linear Programming called Integer Programming IRR Decision Rule Take Projects with IRR39s gt r RejectProjects with IRR39s lt r where r is the market required rate of return for the project Note In our oneperiod world discussion we used r as the rate of return generated by the project From now on we will use IRR to represent the project s rate of return Also recall that in our oneperiod world discussion we examined both the NPV Rule and the Rate of Return Rule now the IRR Rule to determine project acceptability If the NPV of a project was positive negative or if its Rate of Return was greater than less than the market rate we learned that the project should not be accepted In the one period world the NPVRule and the IRR Rule always gave us the same acceptreject answer for any individual project Unfortunately in a multi period world we discover that sometimes the NPVRule and the IRR Rule will give us con icting answers regarding project acceptancerejection One rule might indicate the project should be accepted while the other rule might signal rejection and vice versa The IRR on a project is calculated just like the yieldtomaturity YTM on a bond is calculated ie what is the rate of return that will drive the NPV of the cash out ows and in ows to zero People like it because it is a way to summarize the project in a single number that is derived only from information speci c to the project Unlike the NPV which uses the market determined discount rate Of course to w the IRR rule you must compare the IRR to this same market rate The IRR is internal to the project It is just that rate that corresponds to a zero NPV The IRR is not the market rate of return or r We will compare the IRR to r Why The IRR is in essence the return on invested capital provided by the project The market determined r is the opportunity cost of capital Unless the project returns more than its opportunity cost it s a dog Example review of problem considered above A project has a required rate of return of 10 and the following cash ows Time 0 1 2 3 4 Cash Flow 5000 2000 2500 3000 2000 NPV 5000 20001101 25001102 30001103 20001104 2504 Since the NPV is shareholder wealth will increase if the project is accepted Therefore take the project As stated above if 1000 shares are outstanding the price of each share should 3 NOTE This conclusion is true only for normal projects which we will discuss momentarily increase by 2504 when the rm announces the project If we calculate the IRR of this project we nd it is 3097 Those of you without quotsmartquot calculators can arrive at this answer through the quottrial and errorquot method It may take a bit of time but you can try various IRR39s until you get a zero NPV Again spreadsheets will do the trial and error for you If r 10 the IRR Rule says take this project The IRR of 3097 is greater than the market rate of 10 The IRR Rule will still give us the same acceptreject decision on an individual project as the NP VRule if the project has quotnormal quot cash flows By quotnormal quot we mean the project has negative cash outflows followed by positive cash in ows Let s calculate aNPVProfile diagram using this same numerical example In a graph put NPV on the vertical axis and on the horizontal axis put various discount rates 5000 1272 You should con rm these NPV39s using these various discount rates Draw the NPV Pro le Explain what the diagram represents and that the intersection on the X Axis represents the IRR or the rate where the NPV equals zero The NPV Pro le will always decline smoothly from left to right if the project has quotnormalquot cash ows ie out ows followed by in ows However many projects do not meet this requirement In ows followed by out ows Examples Consulting retainer Manufacturing advances Cash ows ip op between positive and negative values Examples Strip mines Projects whose physical plants require major overhauls during their lives Draw a NPV Pro le for in ows followed by out ows Note that in these cases the line increases from negative to positive NPV upward from left to right Think about the reversed IRR acceptance criteria for cash in ows followed by cash outflows ie accept projects with r gt IRR reject projects with r lt IRR Why does this change occur The NPV Pro le for projects with cash ows that alternate more than once in sign eg out ows followed by in ows followed by more out ows gets interesting try one We get multiple IRR39s for cases where cash flows change sign more than once Descartes39Rule of Signs for n3911 degree polynomial expressions tells us that quotnquot real quotrootsquot or solutions can exist In general a project can have as many IRR39s as it has changes in the signs of its cash flows over time In our numerical example given above there was only one sign change the negative C0 followed by a positive C1 After that all were positive so no more changes In addition to the above two cases where the IRR Rule and the NPV Rule may give us different answers we can also get conflicting signals between the two methods ifwe must rank projects Again why would we rank projects Two common reasons 1 Mutually exclusive projects must be ranked 2 We must rank projects if we are capital rationed Scale is again the issue Example Projects A and B are mutually exclusive PROJECT CF t 0 CF t 1 NPV 10 IR A 100 150 36 50 B 1000 1200 91 20 Which project would you prefer Why If you prefer more wealth to less wealth you should pick Project B Note however that ranking by IRR would suggest A is the preferred choice Even in a one period world IRR gives the wrong advice if the projects must be ranked Example Let s say I give you the following options Option A Give me 100 now and I39ll give you 150 at the end of class IRR on this option A 50 return over about 75 minutes Option B Give me 1000 now and I ll give you 1100 at the end of class The IRR on this option A 10 return over about 75 minutes Using IRR39s Option A appears superior What option will you pick Again if you are a wealth maXimizer and assuming that you have 1000 you should pick Option B Why You39d prefer a 100 increase in your wealth to a 050 increase in your wealth Summary NP V VERSUS IRR For one period individual projects the NPVRule and the IRR Rule will always agree however see the third item in this list For multi period projects the NPVRule and the IRR Rule will give us the same acceptreject decisions when we evaluate individual projects with quotnormal quot cash flows i e out owsfollowed by in ows For projects either single period or multi period with in ows followed by outlays the normal decision criteria of the IRR Rule are reversed i e take projects with IRR39s lt r and reject projects with IRR39s gt r For multi period projects with several changes in the signs of cash flows multiple IRR39s will exist Which IRR should you use Theory does not give us an answer For ranking projects either because they are mutually exclusive or because the firm is capital rationed the IRR Rule can give us conflicting signals relative to the NPVRule even in a one period world Since NPVmaximizes shareholder wealth versus a quotpaper quot rate of return the NPVRule is superior Making PI and IRR Calculations that Agree with NPV Answers The authors of your textbook discuss how you can calculate the PI and IRR on the quotincrementalquot cash ows of two projects that must be ranked and make the PI and IRR methods compatible with the answers that NPV provides However my question is why bother Why should we quotstand on our headsquot to make PI and IRR answers consistent with NPV answers Why not just calculate project NPV s in the rst place Also what if there are more than two projects to rank D Payback Payback Decision Rule Take Projects with Payback periods lt The Required Payback Period Reject Projects with Payback periods gt The Required PaybackPeriod The required payback period is specified by management and is not market determined e g a project must have a payback in say three years or less to be acceptable Why 3 years Payback has two virtues 1 It is easy and 2 It uses project cash ows Other than these positive points little can be said to recommend its use Note Sometimes managers say payback has a third virtuepayback favors projects with a rapid return of capital or investment outlay This observation is true However at what cost in shareholder wealth If you buy a T Bill today and sell it for the same price tomorrow you have a payback of one day Did you increase your wealth in the process 10 Achieving the rapid return of capital is not what shareholders hire managers to do They want managers to maximize their wealth The timing of cash ows can be adjusted by using the capital market Example Look at Table 61 in RWJ In this context be sure you can discuss the problems associated with Payback The Payback method does not satisfy the three criteria for judging capital budgeting techniques identi ed earlier4 The weaknesses of Payback are that it 1 Ignores the time value of money within the payback period 2 Ignores cash ows beyond the payback period 3 Does not consider risk and 4 Uses required payback cut offs that are arbitrarily determined ie these periods are not determined using feedback from the market but rather by management fiat What about Discounted Payback Does Discounted Payback solve the problems It solves one of the problems stated in the previous paragraph but not the rest This method still suffers from problems 2 through 4 listed above Plus as long as you re discounting cash ows in the first place why not do it right and discount all cash ows at a market determined riskadjusted rate ie use NPV E Accounting Rate of Return ARR ARR Decision Rule Take Projects with ARR39s gt Required Accounting Rate Reject Projects with ARR 39s lt Required Accounting Rate The required accounting rate is speci ed by management it is not market determined e g take projects only if they have anARR greater than 20 One of my accounting friends yes I have some once remarked to me quotWhy is it that every time you quotfinance typesquot find some really stupid decision rule you attach quotaccountingquot to its titlequot Accounting Rate of Return is simply the average net income over the life of a project divided by the average investment in the project Example Say a project with a fiveyear life has earnings aftertax EAT of 50 60 60 50 and 40 The five year average EAT is 52 Say this project requires assets with a net book value of 200 180 160 140 and 120 at the start of each year over its fiveyear life The average book value investment is 160 4 Recall our earlier discussion 1 We prefer more cash to less cash 2 We prefer cash sooner versus later and 3 We prefer less risk to more risk Superior capital budgeting evaluation techniques must recognize all three of these dimensions 11 The ARR equals Average EATAverage Book Value Investment 52 160 0325 or 325 If management specifies that acceptable projects must earn a 25 return this project would be accepted The weaknesses of the ARR are that it 1 Does not focus on cash ows even payback uses cash ows 2 Does not consider the time value of money even for accounting pro ts 3 Does not adjust for risk and 4 Uses an arbitrarily speci ed cut off rate ie it is not a market determined required rate of return Concluding Comments My comments on the varying capital budgeting evaluation procedures are completed After our discussion and carefully reading Chapter 6 you should feel comfortable discussing the strengths and weaknesses of each of the five procedures used by corporations to make capital budgeting decisions Again the methods we ve discussed are The NPV Rule The Profitability Index The Internal Rate of Return Rule IRR Rule Payback and Discounted Payback and Accounting Rate of Return You should be able to make an intelligent and persuasive defense for using the NPV approach to evaluating capital budgeting proposals CORPORATE FINANCE AN INTRODUCTORY COURSE DISCUSSION NOTES MODULE 11 COURSE INTRODUCTION NOTE TO THE READER Reading these discussion notes should not be viewed as a substitute for attending class and carefully taking your own notes These discussion notes represent an annotated outline of the discussion that actually took place While the notes are designed to cover most of the highlights of a given class many of the more subtle points are quotfleshed outquot in the discussion as the result of questions that come up in class or alternative ways that I find to explain the materials in quotreal time quot Typically additional examples will be covered in class On the other hand these notes may contain materials that consider basic some of the following discussion has not been formally covered in class In combination carefully reading the chapters reviewing your own class notes reviewing these discussion notes and working and understanding the assigned problems and cases should provide you with an in depth understanding of the materials in Corporate Financial Management Good luck I COURSE OVERVIEW PURPOSE AND FOCUS A Carefully review syllabus ie course objectives learning goals course prerequisites course materials library materials course policies grading guidelines review sessions and the course scheduleassignments B Use the balance sheet framework and a valuation framework to explain where we are going in MBAC 6060 The major topical coverage in financial management is the following 1 Financial Analysis and Planning 2 Capital BudgetingThe LongTerm Investment Decisions 3 Capital Markets Risk Versus Return and Asset Pricing 4 Capital StructureThe Financing Decisions and we often highlight 5 Working Capital ManagementThe ShortTerm Current Decisions Brie y elaborate on these topics in the context of the Balance Sheet Equation 1 These lecture notes presume the reader has completed an introductory nancial accounting class and an introductory statistics class In addition while not absolutely essential the completion of an introductory microeconomics class is helpful Algebra is the only math required to master the mateIial in these notes These modules are designed to complement the Ross Wester eld J affe text Corporate Finance IIwinMcGrawHill ISBN 0072503637 Subsequently references to this text will be indicated using the authors initials RWJ This particular module is designed to complement Chapter 1 inRWJ l Assets Liabilities Equity In nance we generally think of the value of assets liabilities and equity accounts based upon market values not necessarily accounting historical values Therefore it is necessary to understand how the market prices both real and nancial assets To develop this understanding we must understand how to measure risk all other things equal the higher the risk the lower the market value Initially we will implicitly consider risk without formally measuring it Later we will develop both a measure for risk and an equation that relates the measured risk to required returns and ultimately to prices Back to the balance sheet format Assets B for bonds S for stock Assets B S Real Assets Financial Assets tangible amp intangible paper assets Market Value Market Value Discuss tangible bricks and mortar and intangible human capital reputation based or perhaps an existing competitive advantage assets 11 CORPORATE FINANCE DECISIONS 0 Financial Analysis and Planning In this section we assess the strengths and weaknesses of the firm via the Statement of Cash Flow financial ratio analysis and common sized financial statements We will look at the timeseries performance of the firm or trend analysis and compare the firm to appropriate comparable firms Finally using the best available information along with the f1rm s goals and strategies we will project financial statements into the future or pro forma statements to evaluate the consequences and feasibility of the f1rm s goals and strategies If the goals and strategies do not seem financially viable we must cycle back and revise the f1rm s goals and strategies e g the f1rm s sales growth rate objective Important outcomes of this analysis will be forecasts of cash ows expected by the firm and a forecast of the required additional external financing 0 Capital Budgeting These decisions involve what longterm assets xed assets the rm should acquire Using our balance sheet model these are lefthand side decisions These decisions are the quotinvestmentquot decisions They determine the future generation of cash In order to make these decisions as well as the other basic nance decisions an quotobjective functionquot must be speci ed for the nancial manager By an objective function I mean what the manager wants to accomplish We then turn this objective into quotrulesquot that the manager should follow in deciding what assets to acquire We discuss possible rules for the nancial manager below To be properly made the capital budgeting decision requires estimating the future cash ows that an asset will generate either through revenue enhancement cost reduction or both Once these future cash ows are estimated we must place a value upon them ie how much would we pay to receive this stream of cash ows We will see that the value of m real asset real or nancial is a function of three factors 0 The size of the future cash ows 0 The timing of these cash ows and o The risk of these cash ows Value f size timing and risk of cash ows f means is a function of How would you predict that asset value would be in uenced by these three factors one byone A preliminary answer to this question relies on common sense We would expect an asset to be more valuable everything else equal ceteris paribus if 0 its future cash ows are larger 0 its future cash ows occur sooner and 0 its future cash ows are less risky A more complete or precise answer will take some development We will devote a large portion of our efforts in the class on assessing these factors See below for a sneak peek Note that nance is preoccupied with cash ows not accounting numbers eg earnings per share Cash is the life blood of the firm It takes cash to pay bills make investments pay off creditors and distribute dividends Cash makes the world go around Everybody likes cash This is not to say that accounting numbers are not valuable in making some decisions Over the next few lectures and assignments we will gain practice in determining and estimating cash ows Once we determine what an asset is worth to the rm ie its value we simply compare this value to what the asset costs If the asset is worth more than it costs the rm should acquire the asset Note this is the de nition of creating value If the asset is worth less 3 than it costs the rm should reject the opportunity to acquire the asset The size and composition of the rm39s longterm asset base are determined by capital budgeting decisions For capitalintensive businesses the capital budgeting decisions are usually the most important decisions that the rm makes whether the rm is a quotmom and popquot machine tool manufacturer or a gigantic multinational corporation A liberal interpretation of the term capital budgeting extends this conclusion to all businesses 0 Capital Markets and Risk Versus Return A key issue in nance is the study of risk First how do we measure risk The answer to this question is not straightforward Once we understand how to measure risk the next question is how expected returns are related to risk This relationship gives rise to asset pricing models We need a model of asset pricing to determine the rates of expected return investors require in order for them to be willing to hold purchase risky assets Since this compensation for risk from the investor s perspective is a cost from the rm s perspective we will also need such a model in order to design a rm s capital structure see below 0 Capital Structure These decisions involve the righthand side of the rm s balance sheet ie how the rm structures its sources of nancing These decisions are the quotfinancingquot decisions These decisions include how much money the rm should borrow from its bank whether to borrow shortterm or longterm whether to sell bonds preferred stock how much equity to issue etc The general goal is to sell securities debt and equity that are worth more to the buyer than their cost to the rm The task is how to blend the quotportfolioquot of debt and equity securities together that has the lowest possible quotcostquot to the rm De ne a portfolio a portfolio is a collection of assets The assets can be real or nancial assets so the idea of thinking of the rm as a portfolio of assets extends to the lefthand side of the balance sheet as well The sources of funds that reside on the righthand side of the balance sheet are nancial securities The value of these nancial securities is determined by the cash ows generated by the quotreal assetsquot that reside on the lefthand side of the balance sheet Let s consider two examples Why would a banker lend a rm money The banker lends a rm money when heshe believes the rm will repay the loan plus adequate interest to make the loan quotpro tablequot for the bank Where does the money to repay the loan come from It is generated by revenues from sale of goods or services generated by the quotreal assetsquot on the lefthand side of the balance sheet combined with operating decisions and the talents of the 4 employees of the rm These employees truly are quotassetsquot of the rm However they are not listed with the assets shown on the balance sheet This omission is one of many reasons that the book value of assets doesn t equal the market value of assets Similarly why would an investor buy stock in a rm A rational investor must expect to realize more from future dividends and or selling the stock than the original cost of the stock To buy stock without this expectation would not make sense for any quotrationalquot human being Of course expectations are not always realized The stock price will not likely increase in value unless the rm manages its quotreal assets ie the lefthand side assetsquot effectively so that they generate more cash than their initial cost The quotdividend decisionquot the rm makes how much to pay out to shareholders is part of the nancing decision All else equal the more generous the rm39s dividend policy the more funds must be raised by retaining earnings selling bonds or issuing stock Financial assets e g loans bonds preferred stock and common stock are considered to be quotcontingent claimsquot They are called contingent claims because the nancial asset derives its value from its claim on the cash ows generated by the real assets on the left hand side of the balance sheet ie it is contingent upon the real assets Without real assets backing the nancial assets the nancial assets are merely quotpieces of paperquot ie they are worthless The real value of the rm is determined by the ability of the real assets to generate cash ows along with the risk of those cash ows Consider the Chapter 1 diagram in Ross Wester eld Jaffe See footnote 1 that expresses the value of debt and equity as a function of terminal rm value for a one period rm In this context discuss the term quotcontingent claimquot Since the valuation of nancial assets also is determined by cash ows and risk the nancial manager must also understand how nancial securities are valued in the marketplace 0 Working Capital Management The working capital management decision is actually just a subset of the investment and nancing decisions of the rm Working capital management involves the management of the current assets and current liabilities of the rm For example how much cash should the rm hold what credit policies should the rm adopt or how much inventory should be maintained Working capital management decisions also affect the accounts payable balance ie how much trade credit should the rm use wage payments and tax payments Again working capital management involves shortterm assets and shortterm liabilities and the management of the level of these accounts Net working capital refers to the difference between current assets and current liabilities or Net Working Capital CurrentAssets Current Liabilities The larger the net working capital balance the more the rm has to raise in terms of longterm nancing eg bonds and stock in order to nance this balance Refer to the balance sheet representation of net working capital and nancing the difference through longterm sources 111 CORPORATE FINANCE OBJECTIVES It is useful to think of managers as agents ie they are the agents for the owners of the rm Agents make decisions for principals Managers specialize in decision making Correspondingly think of stockholders as principals ie they are the quotownersquot of the rm Stockholders specialize in risk healing through their ownership of the quotresidual claimsquot of the cash ows of the rm or the common stock Stockholders are quotresidual ownersquot in that they get paid last ie after the other claims against the rm39s cash ows are paid off eg suppliers the bank loans the bondholders and the preferred stockholders Keep in mind that the shareholders quotownquot the rm Without their equity capital the rm would not exist You don t borrow money without an equity base Accordingly the managers quotwork forquot the stockholders The shareholders elect the board of directors The board of directors hires compensates promotes and far too infrequently res the managers Given the above discussion what do you think the managers should have as their objective A long list of objectives is possible Here is a list of fatally flawed objectives 1 Maximize pro ts Problems Not cash ow oriented No time horizon No incorporation of risk 2 Maximize earnings per share Same problems as above 3 Size maximization either book value of assets or sales Size may not be in shareholders best interest size doesn t equal value Size doesn t incorporate future cash ows or risk 4 Market share maximization Market share at what price Cash ows Risk This objective may not be in shareholders best interest 5 Compensation manager maximization Obviously a awed objective from the shareholders perspective We choose as the appropriate objective function for the nancial manager as shareholder wealth maximization Can you discuss and defend or refute this objective Think about maximizing shareholder wealth ie share price maximization Shareholders are after all the owners of the rm Is shareholder wealth maximization what shareholders want managers to do YES This objective is logical If managers make decisions that maximize shareholder wealth shareholders can make their own optimal investment and consumption decisions If they choose to give to charities they can have a bigger impact if they have more wealth By wealth maximization we maximize shareholders wherewithal Is shareholder wealth maximization socially irresponsible NO Why What will happen to share price if managers quotrip offquot 1 Employees 2 Suppliers 3 Customers 4 Government 5 Society In the long run shareholder wealth maximization must consider these other stakeholders in the rm Share price will suffer when the penalties associated with corporate rip offs of other stakeholders are assessed Future cash ows and risk will be damaged if the rm is not a responsible citizen This argument however assumes that these markets are well functioning which may not always be the case consider pollution IV AGENCY PROBLEMS Hopefully we all feel comfortable that the normative theoretically appropriate objective function for managers is to maximize shareholder wealth However in the quotreal worldquot do managers strive for this objective Could they be more concerned about good old 1 than they are about the shareholders Con icts of interest arise naturally between managers and shareholders We would be kidding ourselves if we thought otherwise Left to their own devices managers will probably act to maximize their own selfinterests While you yourself may be an exception this selfinterested behavior is consistent with most observations of human nature The existence of these con icts of interest has led to a stream of finance and economic literature that is referred to as agency theory What are the sources of con ict agency costs between managers and shareholders Managerial choice of effort Managerial versus shareholder risk aversion Managerial versus shareholders time horizons and Managerial tendency to overretain funds within the rm that should be distributed to shareholders How are con icts resolved 0 Market control mechanisms e g management gets ousted in a hostile takeover o Managerial labor market eg good managers make more money reputation is a valuable asset 0 Product markets e g inef cient rms go out of business 0 Firm s internal monitoring e g internal accounting system monitoring by other managers board of directors 0 Firm s contracts eg contracts with suppliers of capital customers corporate charters management compensation contracts equitybased rewards o Firm s outside monitoring e g SEC outside auditors o Shareholder lawsuits e g classaction lawsuits and 0 Monitoring by large outside block holders eg institutional investors with large equity positions While far from a perfect assumption we will assume that managers make decisions that maximize shareholder value Under this assumption we can formulate decision rules for managers to follow You must know the objective and how to keep score before you can formulate the rules of the game This assumption also establishes an important benchmark Before you can know how much money any con icts of interest are costing you you need to know what performance could be achieved in their absence The assumption that managers work in the best interest of the shareholders despite recent examples directly to the contrary is also not totally off the wall The best way for managers to keep their jobs is to maximize shareholder wealth If managers perform poorly share price will fall Shareholders will become unhappy Managers will be more likely to get red or the firm might be taken over in a hostile takeover by a better management team that will increase share price with better decisionmaking In this course we will assume that the disciplinary mechanisms discussed above will keep managers quoton their toesquot You will revisit this issue in other nance elective classes that I hope you will take V A SNEAK PREVIEW In our discussion of the major decisions facing the rm we danced around a very important goal of this course understanding how one values an asset This is by no means the only issue we will address but much of what we do will either relate directly to valuation or can be understood by reference to this topic I have also found that students ask a lot fewer questions of the why are we doing this variety if they can see this particular goal written down So here is one of the answers we seek The value of an asset real asset financial asset or even a firm is given by V C C C C071 2 243m 1r 1r 1r where the C s are future expected cash ows that will come in each of the future periods of the asset s life the r is what is called a discount rate and V just means value This is a discounted cash ow valuation formula No I don t expect many of you will have seen this type of equation before But it does tell us many things that we will be doing This equation basically says that if we are going to value a firm or asset we need some forecast of future cash ows that will be generated by ownership of the firm asset and an understanding of when they will be coming We also need some idea of an appropriate discount rate r This is going to be most in uenced by the riskiness of the cash ow We turn rst to understanding how we will develop forecasts of the cash ows in the numerators of this expression VI FINANCE AND THE OTHER FUNCTIONAL AREAS OF THE FIRM As we move through the topics of Corporate Finance it is important to understand how finance interacts with the other functional areas of the firm ie accounting marketing production and operations management human resource management etc Just as the financial manager must be well versed in the basics of accounting marketing etc so must the other functional area managers understand the basics of finance Do you aspire to own your own business Are you entrepreneurial If so to be successful a basic understanding of financial management is indispensable Statistics illustrate that poor financial management is the leading cause of firm failure Finance is not an easy subject for most of us nevertheless if you are to be a successful business person or even a successful lawyer you must master the basics of finance Further you will discover that understanding finance principles will also be of great benefit in your personal life e g planning for major 1 your 39 portfolio planning for retirement etc A little knowledge applied correctly will go a long way 139 1 39 39 Many of us consider finance to be a fascinating subject We hope you will share this fascination CORPORATE FINANCE AN INTRODUCTORY COURSE DISCUSSION NOTES MODULE 81 THE JOURNEY CONTINUES I A PAUSE TO REVIEW REFLECT QUESTION AND INDICATE EXTENSIONS At this point about midway through the term let us pause to re ect on some of the implications and extensions of topics we ve already covered Recall from our discussion at the start of the semester that nancial managers are preoccupied with two major types of decisions 0 The Investment Decision or decisions involving the size and composition of assets for the left hand side of the firm s balance sheet and o The Financing Decision or the composition of financing sources on the right hand size of the balance sheet The nancial manager is also involved with two important secondorder decisions Decisions involving net working capital management or how the firm should manage its day to day cash account other current assets and current liabilities In this sense working capital management decisions involve elements of both the investment and financing decisions Decisions involving dividend policy which is a subset of the financing decision Also recall that in making these decisions nancial managers should adopt decision rules that have one primary objective how will the decisions affect the wealth of the owners of the rm or the stockholders 0 Financial managers objective should be to maximize the wealth of the shareholders In the context of making the investment and nancing decisions with the objective of maximizing shareholder wealth you were introduced to the concepts of asset valuation both real assets lefthand side assets both tangible and intangible and nancial assets righthand side assets 1 This lecture module has been designed to complement Chapter 8 in Ross Wester eld and J affe l Recall that asset valuation both real and nancial is tied to the following conceptual equation Asset Value f Size Timing anal Risk of the Cash Flows Generated by the Asset Accordingly you learned how to discount an asset s aftertax future cash in ows and compared the present value of the in ows to the asset s cost to establish whether or not to acquire the asset If the PVs of the in ows exceed the PVs of the out ows or the asset has a positive NPV the asset should be acquired Shareholder wealth will increase by the amount of the NPV II BUT WHERE DO POSITIVE NPV PROJECTS COME FROM In Chapter 7 of RWJ and Module 7 you learned about NPV and Capital Budgeting However you may not have spent much time thinking about how positive NPV projects can exist In a perfectly competitive market for productive goods would you expect to find positive NPV projects lying about in neat little piles just waiting to be plucked by the casual manager Think about this question Why haven t a firm s competitors already taken similar projects or why haven t these same competitors bid up the prices associated with the inputs for the project e g materials and labor such that the project earns a zero NPV After all this is the equilibrium process in a perfectly competitive market Prices adjust until no surplus rents exist in the language of economics or finance until the rate of return actually earned is the minimum required rate of return which or course gives rise to a zero NPV project One explanation for why positive NPV opportunities exist in real asset markets relates to the relative efficiency of the markets in which these real assets trade I contend that real asset markets are much less efficient than the capital markets where financial assets trade Barriers to entry high transaction costs patent protection regulatory restrictions imperfect information oligopolistic industries asset illiquidity etc are common to real asset markets As we will soon see these market imperfections are much less severe or nonexistent in the capital markets However upon reviewing a project that apparently has a positive NPV managers should ask themselves what it is about this project that produces a positive NPV In general if the firm participates in a competitive market managers should be suspicious of calculations that indicate a project has a positive NPV The managers should ask what are the possible sources of value in the project Skepticism should rule the day We will conclude that manager s can and do find positive NPV investment projects Uncovering these investment opportunities in real assets is the financial manager s main contribution to creating value wealth for their shareholders Conceptually the idea is that managers should take actions to shift the investment opportunity curve upward anal to the right creating incremental wealth for the shareholders Here are some specific ways that positive NPV projects can exist or be created 0 Hire innovative employees and provide then with incentives to be creative in developing ideas for revenue enhancement cost reduction product innovation research and development etc eg General Electric Introduce a new product e g Henry Ford s introduction of the internal combustion engine for automobiles Apple Computer s introduction of the personal computer or Jake Burton s introduction of snowboards Develop a core technology e g Honda s mastery of smallmotor technology or 3 M s mastery of sandpaper technology which later developed into products which range from abrasive wheels to the ubiquitous postit notes Create barriers to entry eg Alcoa ATampT Polaroid and Merck monopoly positions often resulting from patient protection Introduce tweaks to an existing product that increase demand e g Royal Crown Cola the first diet cola or Chrysler s introduction of the minivan Create product differentiation by aggressive advertising and marketing e g Coke s It s the real thing and Chevrolet s Like a Rock Utilize organizational innovation eg Motorola s use of justintime inventory management Amway International Tupperware and Mary Kay Cosmetics innovative multilevel marketing systems Other Keep in mind however that we don t expect positive NP V projects to come easily for afirm in a competitive industry As mentioned above managers should be skeptical when presented with an attractive NPV project Is the source of the positive NPV forecaster optimism computational error naivete dishonesty or vested interest Or is the source of the positive NPV a real wealth enhancing opportunity The decisionmaker should constantly be examining these questions before undertaking any significant investment project III DOES THE MARKET REACT TO INVESTMENT OPPORTUNITIES Theoretically if a project has a positive NPV of 5 million and the firm has 10 million shares outstanding the price per share should increase by 050 at the announcement of the project This market reaction should occur since the NPV represents the wealth increase to the shareholders ie the return that is above the return required on a project of comparable risk in the market place John McConnell and Chris Muscarella Journal ofFinancial Economics vl43 399422 studied the stock market reaction to announcements of increased and decreased capital expenditures by corporations On average they found positive negative stock price changes to increases decreases in capital expenditures Their evidence suggests that the market does react to the availability of wealth increasing projects We should find their evidence comforting in 3 that the market pays attention to rms capital expenditure programs and their impact on rm value As we have repeatedly stressed our emphasis is on a projects cash ows and the risk of those cash ows This emphasis is often at odds with the often stated but ill advised corporate objective of maximizing sales pro ts or earnings per share EPS What does the market seem to value cash ows or accounting pro ts Several studies examine the market s reaction to rm announcements that increase cash ows but at the same time decrease accounting pro ts and EPS at least in the short run Examples include the decision to switch from FIFO to LIFO during in ationary times announcements of increases in RampD expenditures and increases in marketing expenditures All of these corporate decisions decrease pro ts in the shortrun However on average the market reacts positively to these earnings decreasing announcements providing support for the claim that the market does value the rm based upon cash ows versus accounting numbers Academic studies also highlight that rms with higher RampD expenses capital expenditures and marketing expenses have higher markettobook and priceeamings ratios than rms with lower outlays on these items In summary the collective research is at odds with the folklore that suggests the capital markets are myopic shortsighted and that US managers manage for the short run at the expense of the longrun as is sometimes suggested in the popular nancial press IV ANALYZING PROJECT OPTIONS Standard NPV analysis is often static dynamic alterations to the project over the course of its life often are not considered Corporations however make decisions in a dynamic environment and accordingly they often have investment options or real options to re ect real asset investment decisions to represent the options associated with the investment These options should be considered in project evaluation Two prominent examples include o The Option to Expand The option to increase the scale of the project when economic prospects improve relative to forecasts has value This option is valuable and can potentially increase the NPV of a project versus the original forecast o The Option to Abandon The option to close a facility or drop a project when market expectations are not met may have value This option can potentially limit the adverse downside risk of undertaking a project that does not pan out However the value of these real options is often dif cult to quantify frequently they are just considered in a qualitative sense A complete discussion of the topic of real options is outside of the scope of this introductory class For example several books have been written to deal solely with this speci c topic2 2 For instance see Real Options A Practitioners Guide by T Copeland and V Antikarov Texere Press 2001 ISBN 158799 0288 V REFINEMENTS TO CAPITAL BUDGETING ANALYSES3 Several analytic techniques can improve the capital budgeting procedures that you learned in Chapter 7 of RWJ and Module 7 These techniques can improve managers assessment of the risk of projects As a preface to a discussion of these techniques don t ever forget the old adage Garbage In Garbage Out GIGO An analysis can never be any better that the quality of the numbers and the assumptions that go into the calculations While not discussed in detail here a list follows to give you a flavor of some of the re nement techniques to the basecase capital budgeting analyses which we have been emphasizing Decisions Trees Decision trees are a convenient way of representing sequential decisions over time Such decisions often arise when the uncertainty surrounding an investment can be reduced by some initial information gathering such as test marketing a new product or preparing a feasibility study Sensitivity Analysis Sensitivity analysis looks at the sensitivity of a project s NPV to varying outcomes of a single variable eg sales revenue Scenario Analysis In contrast to sensitivity analysis scenario analysis allows several variables to change at once in an attempt to identify outcomes characteristic of say a most likely or best guess an optimistic and a pessimistic scenario Simulation When the input variables of a capital budgeting problem are described as probability distributions as opposed to singlepoint observations a simulation can be conducted The simulation software draws repeated observations from the input probability distributions and creates a probability distribution of NPV outcomes The probability that the NPV will fall below zero can be calculated and this risk evaluated against the potential rewards Break Even Analysis This technique provides an estimate of the scale of operation necessary for the project to achieve a zero NPV VI SO WHAT S NEXT We are now about to take up topics that you ve been thoughtfully inquiring about thus far in the course For instance so far we have completely finessed the issue of measuring asset risk You have been given a discount rate or a required rate of return that re ected the risk of the asset However the game s up We must take this discount rate out of its black box We have side slipped this critical risk issue long enough One of our primary goals in the next segment of financial management is to understand risk and to attempt to quantify this risk for asset valuation and management decisionmaking Specifically we will turn our attention to the topics of 3 See Chapter 8 in RWJ for more speci c coverage Lessons from capital market history on risk versus return Measuring risk Asset pricing models that relate expected return to risk Market ef ciency Capital structure design Measuring a rm s weightedaverage cost of capital Formulating diVidend policy Investment banking and Mergers and acquisitions CORPORATE FINANCE AN INTRODUCTORY COURSE DISCUSSION NOTES MODULE 51 SECURITY VALUATION I Security Valuation Security prices are simply the present value of their expected future cash ows discounted at a rate appropriate for the risk of these cash ows This basic principle is true regardless of the type of security eg bonds preferred stock common stock convertible securities warrants etc II Security Valuation Bonds Let s begin with the simplest type of security and proceed to the more complex ie from quotpure discountquot bonds to common stock Valuing Bonds Pure Discount Bonds Pure discount bonds pay no interest The investor earns hisher quotinterestquot by buying the bonds at a discount a price below the bonds39 face values and receiving the face value at maturity ie paying 925 for a 1000 bond Pure discount bonds also are called zeros for zero interest payments or bullets all of the return is received in a final bullet A synthetic form of this is a strip This is created when someone buys a coupon bearing Treasury security and sells the coupon stream and the final payment of the face value separately Real world examples of pure discount bonds are TBills issued in three six and twelvemonth maturities by the Us government TBills are auctioned off on Mondays They come with a face value maturity value of 10000 In the process of auctioning off the TBills the government will sell enough bills to satisfy their needs to the highest bidders Assume that you want to enter a quotbidquot the buy price for a oneyear TBill Further you do not want to buy the bill unless you can earn at least five percent interest You are willing to settle for such a low interest rate because TBills are considered to be quotdefaultfreequot securities ie the Us government is assumed to pay off the face value of its obligations at maturity without risk What would you bid in the above situation 1 This lecture module is designed to complement Chapter 5 in Ross Wester eld and J affe 1 Your Bid Price 100001051 952381 If your bid is accepted you ll pay your bid price and receive 10000 in one year The capital appreciation from bid to face value is 10000 952381 47519 This price appreciation earns you your required five percent return 47519952381 005 or 500 If other bidders are willing to accept a rate of return lower than 5 ie they are willing to bid a higher price for the TBills your bid will not be accepted How much would a bidder who was willing to accept a 412 return bid for a TBill Bid Price 1000010451 956938 Obviously the US government would prefer to sell its bills for 956938 than for your bid price of 952381 In that case you lose C onsol Bonds We ve discussed these securities before Consol bonds do not have a maturity date Accordingly they can be valued using the perpetuity formula or PV Cr where PV is the price of the bond C is the annual interest payment and r is the required discount rate Say a consol bond pays 60 every year and the required rate of return is 8 per year What would be the price of this bond PV 60008 750 Level Coupon Bonds Let s now examine the most common type of bond the levelpayment coupon bond The coupon is the rate of return paid on the quotface valuequot or maturity value of the bond If a bond has a coupon of 800 and a face value of 1000 the owner ofthe bond will receive 80 per year in interest until the bond matures At maturity the owner will receive the last interest payment plus the face value of the bond Most corporate bonds in the US have face values of 1000 Typically these bonds pay interest semiannually Therefore in the above case instead of receiving 80 at the end of each year the owner receives 40 every six months This introduces atechnical complication What is it The equation for valuing a bond paying interest semiannually is T PV Z SixMonth Coupon Payment1 r2t 1000l r2T t l where T is the number of sixmonth periods until maturity r is the stated required annual rate and 1000 is the face value Some people will set T equal to the number ofyears so 2T is the number of 6 month periods but it s just a matter of convention Let s take a real example from a recent October 24 2002 page Cl3 Wall Street Journal The numbers represent the closing from the prior trading day or October 23 2002 Cur Net Bonds Yld Vol Close Chg ATT 707 82 1381 9450 025 Interpret this quote This bond is a New York Stock Exchange NYSE listed ATampT bond with a coupon rate of 7 7750 of interest per year on a 1000 face value and paid in two semiannual installments of 3875 each and maturing in the year 2007 07 are the last two digits of the maturity year The current yield is 82 The current yield is the annual interest payment 7750 divided by the closing price on October 23 or 9450 or 82 to the nearest tenth l38l bonds sold on October 23 The closing price was 9450 or 945 of face value of 1000 Bonds are quoted as a percentage oftheir face value The closing price was up 025 from the closing price on October 22 NYSE exchange traded bonds like stocks are currently quoted in decimals Assume that this bond just paid its semiannual coupon payment of 3875 and that the bonds mature on October24 year 2007 You could go to the library and look this bond up in M oody39s Bond Guide to determine the exact coupon payment dates and the maturity date If you had purchased this bond for its closing price of 9450 and you held the bond to maturity what would be your quottruequot rate of return This return is called the yield to maturity Of course we are assuming that ATampT doesn39t default on its promise to pay interest plus principal Using our familiar PV equation 10 PV z 38751 r2t 10001 r210 where t 1 10 is the number of sixmonth intervals until maturity r 2 is the semiannual interest rate and 1000 is the face value of the bond at maturity We solve for r 2 Note that in RWJ the authors use y instead of my choice of r2 to depict the semiannual yieldto maturity Enter the appropriate values into your calculator or spread sheet and come up with r2 4572 Note that r 2 is the semiannual rate We entered the number of semiannual periods We entered the semiannual coupon payment Accordingly we solved for a semiannual rate The e ective annual rate is 1 r22 1045722 1 009353 or 9353 and 2X04572 9144 is the stated annual rate YTM is commonly given on a stated annual basis Notice that this effective annual rate is not equal to the 82 current yielal stated in the bond quote See above The current yield is usually not an accurate approximation for the e quotective rate for a bond maturing relatively soon e g in less than ten years The current yielal technically is only accurate if the bond is a perpetuity Do you understand why You should Make sure you do However the current yield is easy to calculate and for a longerterm bond ie a bond with a maturity over 10 years it is a reasonably good approximation for the effective annual rate Also note that the bond is yielding more than its coupon rate of 775 The reason is that the bond is selling at a discount ie below its face value Accordingly the owner if holding the bond to maturity will receive a capital gain of 1000 945 55 which enhances the 775 coupon payment The net result is a return that is above the coupon rate What would be your return relative to the coupon rate if the bond were selling at a premium ie the bond was selling for more than its face value of 1000 Your effective annual rate would be less than the coupon rate Prove this statement to yourself assuming that the ATampT bond was selling for 103 or 103 offace value 1030 r2 003514 or 3514 2X003514 1 007028 or 7028 this percentage is less than the coupon rate of 775 Now the capital loss suffered offsets the 7 75 coupon rate An Old Example The WSJ indicates a bond has a closing price of 12758 s Therefore the closing price of each bond is 1276251000 1276251000 127625 The bond matures in exactly 10 years and has a coupon of 12 payable semiannually This bond was issued when interest rates were veg high What is the yield to maturity on this bond 20 127625 z 60001 r2t 10001 r220 t 1 r2 39723 The annual effective yield is 103972 1 0081024 81024 A omment About 39 Rate Risk Any bond except a quotpure discount bondquot is subject to reinvestment rate risk Reinvestment rate risk is a form of interest rate risk Even if the bond is quotdefault freequot ie it is guaranteed to make the promised interest and face value payments reinvestment rate risk is still a real risk the investor must consider By reinvestment rate risk we mean the uncertainty about the rate of return at which you will be able to reinvest the coupon payments received from the bond Since interest rates change through time you will not know the reinvestment rate in advance Yield to maturity assumes that you can reinvest at the original rate Example Let s take a simple example to illustrate reinvestment rate risk Say you buy a quotdefault freequot bond for 99500 that matures in exactly one year and pays interest semiannually The coupon rate is 800 and the bond has a face value of 1000 Accordingly you will receive 40 in six months and another 40 plus the face value of 1000 in one year The semiannual yield on this bond is 42661 resulting in an annual yieldtomaturity of 87142 However whether you actually earn this rate over the whole year depends upon the rate at which you reinvest the first 40 coupon payment in six months Say you are able to reinvest the first coupon payment at the semiannual rate of 42661 In this case at the end of one year you will receive FV 400010426611 4000 100000 10817064 Therefore your oneyear rate of return is PV FVl r1 2 99500 108170641 r1 r 87142 or the same value that we calculated as the yieldtomaturity on the bond However what if interest rates fall dramatically during the first six months that you own this bond Say at the end of six months stated annual interest rates are 40 Therefore the rate at which you can invest your first 40 coupon payment is 20 for the rest of the year At the end of one year you have FV 400010201 4000 100000 108080 Therefore your quottruequot oneyear rate of return is 99500 1080801 r1 r 86231 not the yieldtomaturity that we calculated of 87142 This difference in actual versus expected yieldtomaturity may not seem like a big deal to you but it is The longer the bond is to maturity and the more money that you have invested in bonds the bigger is the impact on your wealth Investment bankers prosper or fail based on a few basis points or 1100th of one percentage point 100 basis points 1 Therefore implicitly when we calculate the yield to maturity on a bond we are assuming that we can reinvest the coupon payments at this rate If actual reinvestment rates alijfkr from this rate our future wealth will be either less than or more than the yielal to maturity would indicate This risk is referred to as reinvestment rate risk a form of interest rate risk Notice that investors are subject to reinvestment rate risk ifany coupons are paid even if the bond is default free A Comment About Price Risk The other type of interest rate risk relates to price changes in fixedincome securities caused by interest rate changes or price risk If you do not hold the security to maturity you are subject to this type of risk Example You buy a bond for 1142 that matures in exactly 10 years has a 9 coupon a 1000 face value and pays interest semiannually This bond is considered free of default risk ie the interest and face value payments are considered certain If you hold this bond until maturity you will earn an effective annual rate of7 12 or about 7 on a stated annual basis check my math However after 5 years you find your self in a financial jam and need to sell the bond to pay a bill Over your five year holding period interest rates have increased to a stated rate of 10 per year Therefore when you sell the bond you receive only 96139 check my math Therefore you have suffered a 18061 capital loss Your actual return over your five year holding period was 513 check this far less that the 712 you would have earned if you d held the bond to maturity The lesson in this example is that if interest rates rise bond prices fall and viceversa Therefore if you do not hold a fixedincome security to maturity even if it is defaultfree you may suffer a capital loss Again this form of interest rate risk is referred to as price risk 111 Security Valuation C0mm0n Stock In the June 1 2001 Wall Street Journal page C3 I observed the following stock price quote for American Telephone amp Telegraph YTD 52 Weeks YLD VOL NET CHG Hi L STOCK SYJVI DIV PE 100 s LAST CHG 227 3775 1650 ATampT T 015 07 dd 22935 2117 056 Interpret this Quote YTD CHG percent price change for the calendar year to date SYM is the ticker symbol DIV is the most recent dividend annualized YLD is DIV Current Price PE is the PriceEarnings ratio dd indicates a loss in the most recent four quarters VOL 100s is the number of shares traded the prior business day in 100s of shares LAST is the previous business day s closing price and NET CHG is the price change from the closing two business days earlier Our task is to explore the valuation of a share of common stock like ATampT What factors give rise to the closing price observed or 2117 If you buy a stock how long do you plan to hold it To infinity Unlikely Assume you re an investor with a oneyear holding period horizon and you expect to be able to sell a share of stock for P1 10000 just after receiving an annual dividend DIV1 of 300 Assume that the required return r 15 for this stock What would you pay for this stock at t 0 or what is P0 P0 DIVIJ r1 PIJ r EQ 1 3001151 100001131 8956 The capital gain on your investment is 10000 8956 1044 Note that r 3008956 10000 89568956 00335 01165 01500 or the total return r equals the dividend yield plus the capital gain yield Therefore total return has two components In a world without taxes you don t care about the miX of the components ie the dividend yield and the capital gain yield as long as they have the same total Return Return Return Why would anyone buy the stock from you at t 1 for 100 The future cash ows must justify this price unless you believe in the quotbigger foolquot theory Assume that the t 1 investor you will sell your share to likewise has a oneyear holding horizon ie she plans to sell the stock at t 2 She would be willing to pay P1 DIVzI r1 P21 r EQ 2 What combinations of DIV and P2 justify a 100 price at t 1 ifr continues to equal 015 Obviously an infinite number of combinations of DIV and P2 could have a PV 100 However assume that she forecasts DIVz 330 and P2 11170 P1 3301151 111701151 10000 Plug EQ 2 the value for P1 into EQ 1 P0 D1v11 r1DIVzl r1 P21 r11 r1 P0 DIV1 r DIV1 r2 P21 r2 EQ 3 Similarly P2 D1V31 r P31 r EQ 4 Ifwe quotplugquot EQ 4 into EQ 3 we get P0 D1v11 r1 DIVz1 02 DIV31 o3 P31 r3 If we proceed by calculating P3 in terms of DIV4 and P4 and so forth we end up with a general equation P0 DIVl1r1 DIVz1r2 DIV31r3 DIV41r4 DIVN1rN PN1rN 01 N P0 z DIVtl 01 PNl 0N tl As N approaches in nity stock has no maturity PNl rN approaches zero and 00 P0 EDIV1 r EQ 5 t 1 E Q 5 is our basic common stock valuation model Pg is a function of future dividends to current shareholders and only the dividends to stockholders from t 0 determine P0 This model simply says that a common stock is the present value of its future cash ows discounted at the required rate of return r How do shareholders get dollars out of the rm Dividends Actually the rm can also repurchase stock from stockholders and distribute cash that way However that s just a special case of a dividend A major way that investors get cash out of rms is in MampA activity again a special form of dividend Also a rm might liquidate and make a liquidating distribution to the shareholders However that form of distribution is also just a special case ofa dividend Since dividends are the source of cash to the holder of a share of common stock dividends are the source of value If the dividends grow at a constant rate forever ie a growing perpetuity what form does EQ 5 take Recall that this situation is just a growing perpetuity or P0 Dlvlr g EQ 6 where P0 is the t 0 share price DIV1 is the expected dividend cash ow in one period r is the required rate of return per period and g is the constant growth rate per period of dividends to in nity Examples Case 1What is the current price P0 of a stock that just paid a 300 dividend if the market required return is 15 and the dividends are expected to grow at 10 forever P0 3001101015 010 6600 The next dividend the rst in the growing perpetuity is expected to be 330 Case 2What if the dividends are not expected to grow ie g 0 Now we have our perpetuity situation or P0 DIVoI Plugging in the above numbers we have P0 300015 2000 far less than the price appropriate for a stock with a positive growth rate in dividends Case 3What if dividends are expected to decrease at a constant rate of 4 Note we can still use the equation we used in Case 1 however the growth rate g equals 4 P0 300104015 04 288015 004 288019 1516 far less than even the price with a zero growth rate I mportantPoint The dividend growth rate can be negative as well as positive Review 00 P0 z DIVtl r1 EQ 5 t1 Again EQ 5 is our basic common stock valuation model P0 is a function of future dividends to current shareholders and only the dividends that accrue to stockholders at t 0 determine P0 ie yesterday s dividends do not contribute to today s price Since dividends are the source of cash to the holder of a share of common stock dividends are the source of value But you ask what about capital gains Who plans to hold the stock until in nity Have we left out capital gains What if I plan to sell shares in the future and realize capital gains in addition to dividends Why don t capital gains show up in EQ 5 I39ll give you the quotquickquot answer now and then I ll justify the answer The expected capital gains of the stock have not been left out of EQ 5 Expected capital gains are a function of caused by expected dividend growth Dividend growth is included in EQ 5 Therefore capital gains are included in EQ 5 If you remember how we arrived at EQ 5 you can see we haven t left anything 10 Let me demonstrate the above statement with an example Stock A has an expected DIV1 of 10 and an expected dividend growth of zero forever Stock B has an expected DIV1 of 10 and an expected dividend growth of 10 forever Both A and B have a required return of 15 Future expected dividends are t1 t2 t3 t4 t5 m m m mg m StockA 1000 1000 1000 1000 1000 StockB 1000 1100 1210 1331 1464 Since both stocks have a constant growth rate we can use the growing perpetuity equation or P0 DIVIr g EQ 6 EQ 6 is often called the Constant Growth Equation or the Gordon Model after Myron Gordon University of Toronto who is often said to have been the first to derive the equation J B Williams dissertation at Harvard in the 1930 s however contains the mathematics of the model Using this equation we calculate the stock price of A and B at each point in time as t0 tI t2 t3 t4 t5 0 2L 2 EL 24 2 Stock A 6667 6667 6667 6667 6667 6667 StockB 20000 22000 24200 26620 29282 32210 What is Stock A39s capital gain growth per year Zero exactly its dividend growth rate per year What is Stock B s capital gain growth per year 10 exactly its dividend growth rate per year The bottom line of this example Capital gains growth is quotcaused byquot the dividend growth EQ 5 includes the dividend growth by including the specific DIVt s for each year to infinity Since dividend growth is included capital gains growth is likewise included Capital gains are quotfoldedquot into the dividend stream Note EQ 5 is a quotgeneralquot model of stock valuation Any dividend growth pattern can be accommodated by this equation e g rising dividends falling dividends level dividends dividends growing at any irregular rate no dividends for some years followed by some dividends in later years etc EQ 6 is a simplification that can only be used in very special circumstances it however has been often abused 11 Rearranging EQ 6 and solving for r we have r DIVjPg g EQ 7 where g is the growth rate in dividends which as we saw equals the growth rate in stock price For Stock A att 0 r 10006667 0 015 15 For Stock B att 0 r 100020000 010 005 010 015 15 The current dividend yield is 5 and the dividend and capital gain growth rate is 10 As we will see later EQ 7 is an equation that can be and has been used to estimate the required return appropriate discount rate r for common stock The tough part of using EQ 6 and EQ 7 is coming up with an estimate of g or the dividend growth rate Four methods of estimating g are often used 1 The past growth rate of dividends 2 Analysts forecast of dividends and earnings eg SampP Earnings Forecaster IBES forecasts 3 The Sustainable Growth Rate SGR ROE1 Payout Rate and 4 Your own estimate of g based upon private information or personal research Where possible I recommend using the fourth method Absent this estimate my alternative choice is the second method Past dividend growth may not be a good predictor of future dividend growth The SGR equation is often quite unstable from yeartoyear EQ 6 and EQ 7 are often misapplied in security valuation Where would this equation be most appropriate For a mature steady growth company such as a public utility firm Where would this model be least likely to be appropriate For a fast growing young company that will undoubtedly be unable to maintain its current growth rate What if a company currently pays no dividends Suppose you are looking at a young high growth company that currently pays no dividend Then using EQ 7 implies that r g Here g is very hard to estimate and is clearly unstable However remember we arrived at EQ 7 by assuming that the firm s dividends would grow at a constant rate To use a formula that assumes a constant growth rate forever in this situation is the height of folly IV Stock Price Volatility 12 Let s assume that a stock quali es for valuation via the Gordon Model or PO DIVlr g What are the assumptions you must make to use this stock price valuation model 0 Dividend growth g must be constant forever and o r gtg otherwise we would calculate either an unde ned or a negative share price As we know r is the market required return on the stock This market required rate of return is determined by r rf 6 where rf riskfree rate of return e g the TBill rate and 6 the risk premium associated with this stock39s level of risk Stocks with varying levels of risk will have varying s If the T Bill rate is 45 and a firm39s risk premium 6 is 85 the appropriate quotrquot for the stock is 130 There will be more a lot more about 6 in subsequent discussions We can further decompose rf as equal to rf EI Real Rate an approximation2 where EI the expected in ation rate and Real Rate the rate of quotreal returnquot the market requires overandabove earning the expected in ation rate If rf just equaled EI investors in T Bills would have a quotrealquot return of zero In other words price increases caused by in ation would exactly offset the returns on TBills investors39 real wealth would not increase if they invested in TBills They would just quothold their ownquot versus increases in the cost of living Historically T Bills have earned about 05 more than the actual in ation rate Therefore a reasonable forecast for real rates equals 05 Having provided this background why do stock prices change from daytoday Again assuming the Gordon Model of stock valuation or 2 The actual formula is that 1rf l Real Ratel El People frequently use the approximation because it is easier for most to think about 13 P0 DIVlr g what changes in the situation of a rm and or in the economy can cause stock prices to change What happens if EI goes up all else equal Goes down What happens if a rm39s expected growth goes up all else equal Goes down What happens if the market suddenly revises its forecast of DIV1 Up Down What if the market risk of the rm or of the entire equity market goes up Down Given the above discussion it is not di icult to understand why stock prices change change frequently and often change dramatically Note that if r increases but g also increases and by enough a stock39s price can increase in spite of the increase in interest rates Since stocks do not promise speci c cash dividends and since expected growth rates in future dividends for a stock can change frequently the interaction between r and g will determine how a stock39s price will react to a change in interest rates Contrast this result for stocks to that for bonds A bond promises contractual cash ows Therefore the standard corporate bonds future cash ows will not increase with changes in interest rates Therefore a bond s price will fall as interest rates rise and viceversa V Stock Prices and PE Ratios PE ratios current share price divided by either trailing annual EPS usually the case or forecast annual EPS are often afforded some quotmagicalquot importance by some nancial commentators What this ratio represents is often misunderstood It is sometimes suggested that PE can be used as a tool or a quottrading rulequot to quotpickquot stocks eg invest in high or low PE stocks and quotget richquot Think about PE In the numerator we have a stock valued as the discounted stream of future dividends The quotPquot therefore is determined by the future cash ows to the owner of a share and the risk of these cash ows P is a market determined number In the denominator we have earnings per share EPS is income after 1 depreciation 2 interest 3 taxes and 4 preferred dividends EPS could be paid out as dividend or retained in the rm as part of retained earnings EPS is a historically determined accounting number When you divide a market number by an accounting number what do you expect to get Anything of value Why should P E relate to P0 14 As we know stock price is affected if a rm invests in a project at a rate different than the market rate r with luck we invested at a higher rate than the required return and price went up Let r the project s rate of return If r gt r P0 should increase the project has a NPV Ifr lt r P0 should decrease the project has a NPV If r r P0 should not change the project has a 0 NPV A rm with a large number of NPV projects both now and in the future will have higher future earnings or EPS as the projects quotcome on linequot These firms should be expected to have a high current price relative to current earnings or EPS Therefore growth firms ie firms with NPV projects in general will have high PE39s We observe this relationship in the market place High Growth Firms Tend to Have High PE39s Similarly firms with few NPV projects have low expected growth and therefore future earnings or EPS will not be substantially higher than EPS today These firms should be expected to have lower current prices relative to current earnings or EPS Therefore low growth firms those with few NPV projects in general will have low PE39s We observe this relationship in the market place Low Growth Firms Tend to H ave Lower PE39s Example Think of a firm with no NPV projects However it has current assets that are generating operating cash ow For algebraic simplicity say the firm reinvests its depreciation expense to sustain these current assets they maintain their perpetual earning power Since the firm has no NPV projects it is paying out all of its net income as dividends The value of this hypothetical firm is V0 Net Incomer where V0 is the current market value of all of the rm39s equity Alternatively on a per share basis P0 EPSr DIVr where P0 is the current price per share and EPS DIV a perpetuity Now assume the rm39s engineering group suddenly comes up with a new product The product is analyzed and has a NPV The project has not yet been undertaken but has been announced to the public What is the new value of the rm V0 Net Incomer NPV 15 Pg EPSr NPV Shares Out EQ 8 Think of EPSr as the quotcash cowquot part of the company or the cash owing to shareholders from assets in place100 payout Think of the NPV at t 0 as the NPV of future growth Qpportunities or NPVGO Divided EQ 8 through by EPS or PoEPS 1r NPVGO Shares OutEPS PE 1r NPVGO Shares OutEPS This equation illustrates that the PE ratio is a positive function of future growth opportunities It is also a function of 1r so less risky rms all else equal will have higher PE ratios Sometimes EP the reciprocal of PE is used as an estimate of r the required rate of return on a stock By inspection of the previous equation you can see when this estimate may or may not be a decent approximation If the firm has zero NPVGO and it pays our close to 100 of earnings then perhaps EP is not a bad approximation of r However the probability of all of these conditions being met is veg small Accordingly I do not recommend estimating r using the EP ratio Can PE be used to make pro table investment decisions NO Iflife were this easy we d all be rich While high growth stocks tend to have high PEs this empirical regularity cannot be utilized to get rich Example Stock A and Stock B both require a 15 return Stock A has EPS 400 and just paid a 400 dividend A has an expected dividend growth rate of 5 forever PA 4001051015 005 4200 PE for A 4200400 105 Consider Stock A39s total return of 15 10 of As return is from its dividend yield 5 of As return is from its capital gain yield As we know from before capital gains yield is equal to the expected growth rate of future dividends Stock B has EPS 400 and just paid a 400 dividend B has an expected dividend growth rate of 12 forever The higher growth of B relative to A is because B has more NPV projects than A can identify 16 PB 4001121015 012 14933 PE for B 14933400 373 Consider Stock B s total return of 15 3 of BS return is from its dividend yield 12 of BS return is from its capital gain yield You would pay more for Stock B but you would get larger future dividends that justify this high stock price You pay less for Stock A but you get smaller future dividends that justify this lower stock price Once again what is your return on each stock 15 Do you care which stock you own If we ignore any tax issues the answer is NO Both stocks provide the same rate of return So if they have equivalent risk you are indeed indifferent You pay for the extra growth of B By paying a higher share price the expected returns from owning both stocks is made the same Therefore the P E ratio cannot be used to quotpick stocksquot that will perform better than other stocks with the same risk All stocks with the same risk will be quotpricedquot to earn the same expected return In general a high PE can be the result of 0 Good investment opportunities or high NPVGO o Conservative accounting practices which lower EPS 0 Low current earnings relative to expected future earnings or 0 Low market required rate of return r due to a low risk premium 6 The market prices stocks to earn their required rate of return PE will not assist you in becoming quotrichquot 17 CORPORATE FINANCE AN INTRODUCTORY COURSE DISCUSSION NOTES MODULE 161 CAPITAL STRUCTURE PART II In addition to corporate taxes considered above the additional market imperfections we will consider are Costs of Financial Distress CFD including the Agency Costs of Debt Personal Taxes and Agency Costs of Equity After we consider these additional market imperfections we will address other factors that seem to affect a firm39s capital structure specifically The Probability of Using the Interest Tax Shield NonDebt TaxShield Substitutes and Financial Slack We will then explore some of the systematic differences in capital structure found across industries and countries along with how they might be explained We will discuss some empirical studies of abnormal returns surrounding increases and decreases in leverage as a clue to how some of the market imperfections might quottrade off39 against other imperfections e g tax savings versus CFD Unfortunately we will discover that we cannot develop a precise equation for use by the financial manager in determining the optimal capital structure for hisher firm Too many of the relevant parameters defy precise quantification e g the costs of financial distress However as a partial substitute for an explicit answer on how to design a firm39s capital structure we develop a checklist on how managers might go about making the capital structure decision for their firm V The Costs of Financial Distress CFD As we have observed we do not see firms as highly leveraged as suggested by the relation VL VU tcB One obvious reason relates to the costs of financial distress CFD As firms become more highly levered the chances that they will run into financial difficulties increase We will categorize the types of problems that firms can have in financial distress and the costs associated with these problems Obviously the ultimate case of financial distress is bankruptcy However financial stress sets in well before a firm might be forced into filing for bankruptcy 1 This lecture module is designed to complement Chapter 16 in Ross Wester eld and Jaffe If a rm les for bankruptcy it seeks court protection from its creditors The law related to bankruptcy is contained in various quotchaptersquot in the Federal Bankruptcy Code The two most important chapters of the Code for corporations are Chapter 11 and Chapter 7 In Chapter 11 a rm seeks to restructure its nancial obligations while under court protection from creditors l Indirect Costs of Financial Distress Prior to hitting the extreme limit of nancial distress or bankruptcy rms may gradually feel the quotnoosequot tighten around their necks when they begin having dif culties meeting their nancial obligations e g making payments to trade creditors For instance Management may become preoccupied with survival When managers are worried about making their debt payments they are not attending to the main business of the rm Accordingly the daytoday affairs of the rm may go unattended as managers scramble to keep the rm a oat nancially Management distractions can have serious short and longterm negative consequences Relationships with trade creditors deteriorate As the rm becomes shaky nancially trade creditors may curtail or cancel normal trade credit provisions In the extreme they may put the rm on a quotcash onlyquot basis This curtailment of normal trade credit will hamper the rm s normal production process as material shortages occur Customers may become concerned that the business will survive As customers observe the rm39s dif culties they begin to worry about the rm39s ability to be a reliable supplier or its ability to live up to product guarantees or warranties In the process they may shift their business elsewhere Valuable employees may seek jobs elsewhere for fear the rm will go out of business All else equal few want to work for a rm that is about to or actually declares bankruptcy The Agency Costs of Debt also adversely impact rms that are in nancial distress By Agency Costs of Debt we are referring to the costs of the con icts of interest that arise between managersstockholders versus bondholders when a rm is in trouble nancially Managers are the quotagentsquot of the shareholders hence the term agency costs Managers on behalf of the shareholders may begin to make decisions that favor shareholders at the expense of bondholders These temptations are especially large under conditions of financial distress Examples of these activities include o Overinvestment ie take NPV projects 0 Underinvestment ie reject NPV projects 0 Taking the money anal running ie selling important assets and paying out a large dividend quotmilking the propertyquot to use the label in the textbook Ex ante at the time the debt is issued bondholders realize the possibility that shareholders and managers might collude and take detrimental actions if the rm gets into nancial distress The objective is to emerge from bankruptcy as a nancially restructured and viable rm In Chapter 7 the assets of a rm are liquidated in an orderly fashion and the proceeds used to pay off claimholders in the order of their priority to the rm s assets Seeking compensation in advance for these detrimental activities bondholders will demand higher rates of return before they will bear these risks To the extent that debt agency costs can be anticipated therefore shareholders will pay for the agency costs of debt in advance by paying higher interest rates If shareholders wish to avoid the quotpenaltyquot of higher interest rates they will be willing to write protective covenants into their contract with the bondholders or the bond indenture These contractual provisions can limit the adverse activities of the shareholdermanagers in times of nancial distress thus protecting the bondholders39 interests In some indenture agreements up to 30 protective covenants have been observed Examples of these covenants include provision to keep debttoequity levels below certain levels keeping current ratios above certain levels limits on asset sales without bondholder approval restrictions on dividend payments etc However to summarize well before declaring bankruptcy firms may suffer significant losses due to financial distress Correspondingly firm value may plummet as bond and stock prices fall with the increased financial risk of the firm and the impaired cash ows Often firms try to obtain concessions from creditors when their financial plight becomes serious These arrangements are called out of court restructurings However under the Trust Indenture Act of 1939 firms cannot change the quotcorequot terms of their publiclytraded debt ie the interest rate maturity date or principal value without permission from 100 of the bondholders a practical impossibility Therefore outofcourt restructuring involve a voluntary exchange of securities in which creditors exchange their old debt claims for equity or new debt claims that have either a lower interest rate a longer maturity a lower face value or some combination of these adjustments The court is not involved in these exchanges However for the exchange to significantly reduce the distressed firm39s liabilities it is necessary that most creditors participate in the transaction Creditors who do not participate retain their original claims they free ride on the concessions of other creditors If the outofcourt restructuring fails the distressed firm typically les a Chapter 11 bankruptcy petition Direct Costs of Financial Distress In a traditional Chapter 11 bankruptcy either the distressed firm voluntary filing or less commonly a creditor of the firm involuntary filing files a bankruptcy petition with a regional Bankruptcy Court Once under the supervision of the Bankruptcy Court the debtor firm receives protection against creditors and has the exclusive right to propose a plan of reorganization which specifies the cash and securities to be received by all claimholders in the reorganization Once the plan is filed and the bankruptcy court determines that the firm has made adequate disclosure to allow claimholders to assess the merits of the plan claimholders vote on whether to accept the plan For voting purposes claimholders are grouped into homogeneous classes based upon the type and priority of their claim e g secured debtholders unsecured debtholders preferred stock etc Confirmation approval of the plan by the Court requires approval by twothirds in dollar amount and more than onehalf in number by each class of claimholders It is common for more than one plan of reorganization to be filed before a plan is confirmed If claimholders cannot agree to a plan of reorganization the bankruptcy court can cram down a plan on dissenting claimholders In a Chapter 11 reorganization all claimholders must exchange their old securities in accordance with the terms of the plan Obviously the above process is expensive Bills from lawyers expert witnesses accountants etc can be enormous In addition the court costs can be signi cant The quot J of Indirect and Direct Costs of Financial Distress Ed Altman a Finance Professor at NYU has attempted to estimate the combined indirect and direct costs of nancial distress CFD While his methodology is subject to some controversy he comes up with costs in the range of 20 to 25 of the total market value of the rm before it becomes nancially distressed This number is too large to be ignored Problems Re ardin the CFD At this point we can rewrite our equation for the value of the levered rm as VL VU tcB CFD where the last term is a deduction from the value of the levered rm for the costs of nancial distress This relationship between the tradeoff between the tax advantages of debt and the cost of nancial distress has been labeled the TradeOff Theory of Capital Structure Exhibit I illustrates this relationship where DebtEquity is the quotoptimal debtequity ratio of the rm The value of the levered rm at rst increases because of the tax savings associated with debt However at some point the CFD begins to overwhelm the tax advantages forcing the value downward The point of highest rm value corresponds to the optimal capital structure that of course is also the minimum WACC level presuming we can take these costs accurately into consideration in the WACC equation Therefore why do not we just specify the functional form of CFD and give it to nancial managers to use in designing their capital structures The problem is we do not know how to specify CFD Different rms can have very different costs associated with nancial distress Take as an example the costs of nancial distress that might be incurred by two rms with distinctly different assets the Marriott Corporation and HewlettPackard The majority of Marriott39s assets are tangible bricks and mortar If Marriott gets into nancial dif culty the value of its assets will not disappear or dissipate nearly to the degree of HP39s assets HP39s assets are largely intangible human capital type assets If HP gets into nancial dif culties and loses its talented pool of engineers and technical personal its value will plummet to a much larger degree than would Marriott s Therefore HP has a much higher CFD for any debt equity ratio than does Marriott By extrapolation you can see the dif culty in specifying a quotone size ts allquot expression for CFD The costs of nancial distress are very rm speci c In my opinion we will never have a precise equation to estimate CFD in determining a rm s optimal capital structure VI Personal Taxes Professor Merton Miller was troubled by his observations that seemed inconsistent with the received tradeoff theory of capital structure Again the tradeoff is between the bene ts of taX savings and the costs of nancial distress First based upon the work of one of his students Miller did not believe the CFD were as high as they were commonly thought to be This student Jerry Warner wrote his dissertation on the direct costs of nancial distress for failed railroads eg lawyers fees expertwitness fees court costs etc His evidence suggested that the direct costs amounted to only 5 of the market value of the railroad at the time bankruptcy was declared and amounted to only 1 of the rm s value 5 years prior to the bankruptcy ling These numbers are surprisingly low Miller used a parable of Horse and Rabbit Stew recipe to make his point The recipe for the stew is one horse and one rabbit Miller s point is that if the taX advantages of debt are very large ie the horse and the costs of nancial distress are relatively small ie the rabbit it seems like the tax advantages would overwhelm the CFD disadvantages and we would see rms with extremely high leverage levels Professor Miller went on to reason that if CFD were truly high much higher than the surprising low level found by Warner why do we not observe more income bonds What is an income bond An income bond has all of the taxadvantages of regular bonds ie taX deductibility without the attendant risk of bankruptcy if the rm defaults on the payments Interest and principal are due only if the rm can pay them Accordingly income bonds have all of the advantages of regular debt without the disadvantages of CFD However we observe very few issues of income bonds Why One investment banker told Miller we don t observe more income bonds because they have the smell of death about them However Miller reasoned via an old Latin proverb Money Has No Odor If the CFD were truly high and these costs could be avoided by using income bonds we would certainly see rms that would cover their noses and issue them Second on Miller s list of puzzles was the fact that rms had debt before the existence of corporate income taxes rst instituted in the US in 1913 Since the primary advantage of debt is taX savings offset by the primary disadvantage of CFD why would a rm have debt if the advantages were not present but the disadvantages remained As a third question Miller wondered why rms did not increasedecrease their debt levels as corporate taX rates have increaseddecreased through time Over time DebtEquity levels have been remarkable constant If the advantage of debt is tcB you would expect debt to increase with the taX rate and viceversa Finally Miller observed that many successful companies have very low levels of debt eg pharmaceutical companies eg Merck Obviously these successful companies do not have quotstupidquot managers Why do not they take advantage of the taXshield provided by debt Professor Miller suggested that the personal tax structure may explain at least some of these puzzles He noted that while rms can deduct interest payments and save taxes individuals that receive these interest payments must pay taxes on them at their personal ordinary tax rates The tax rate on ordinary income is higher than the tax rate on capital gains Since a larger percentage of bond payments to investors are subject to ordinary income tax relative to stocks more of stocks39 returns come from capital gains than do bonds returns bond returns have a tax disadvantage at the personal level versus stock returns Example Assume that you are a bond investor You can either invest in a taxfree municipal bond or a taxable corporate bond Assume that both bonds have the same risk level You require a 70 return aftertax on both bonds You are in the 30 tax bracket If you invest in the muni you get 7 0 taxfree Since the risk is the same you demand a 70 return on the corporate bond aftertax What will the pretax return on the corporate bond have to be to give you 7 0 aftertax 70 rb1 tb where rb is the beforetax return on the corporate bond and tb is the tax rate on corporate bond income rb 701 030 100 In other words a 100 rate on the taxable corporate bond equals the 70 return on the nontaxable municipal bond The message contained in this story Investors will demand higher returns on corporate bonds to compensate them for the higher effective tax rate on debt versus equity This higher required rate olTsets at least to some degree the tax advantage of debt at the corporate level This requirement will narrow the beforetax differential between debt and equity and make debt quotrelativelyquot more expensive for the firm relative to what we believed before we considered the effect of personal taxes This intuition is part of Miller39s genius He went on to derive a model for valuing the perpetually levered firm that contained both corporate and personal taxes I will spare you that derivation although it parallels the leveredfirm value derivation done in the corporate tax only case you will find in the text Value of the Firm with only Corporate Taxes vL vU tcB CFD 1 Value of the Firm with Corporate Taxes anal Personal Taxes vL vU 1 1 tc1 ts1 tbB CFD where 2 tC the corporate tax rate ts the personal taX rate on stock and tb the personal tax rate on bonds The bold portion of the above two equations 1 and 2 represent the taxsavings value added to the levered rm Note the following in comparing the two equations If tC 0 and t5 and tb equal zero then VL VU This result is the original notaX case IftC gt 0 and t5 and tb equal zero then VL VU tcB This result is the corporate taX only case IftC gt 0 and t5 and tb are gt than zero but are equal ts tb then VL VU tcB This result is equivalent to the corporate taX only case no differential taxation at the personal level is equivalent to the case of no personal taxes If tC gt 0 and tb gt ts then the bolded portion the taX advantage of debt of equation 1 is greater than that of equation 2 It can be the case that ltbltltclts This means that the taX disadvantage of debt on the personal side outweighs the advantage on the corporate side and a levered firm would be less valuable than an unlevered firm Example Let us assume some taX parameters that are reasonably consistent with the current taX code Remember that capital gains can be deferred until the stock is sold Therefore given the time value of money even if the capital gains taX rate equals the ordinary income taX rate the deferability of capital gains effectively lowers the present value of this taX rate ts tc 035 ts 020 for longterm over 12 months capital gains tb 040 actually 0396 for highest individual taxbracket investors Plugging in these rates into equations 1 and 2 results in VL VU tcB CFD vL vU 0353 CFD 1 vL vU 1 1 tc1 ts1 tbB CFD VL VU 1 1 0351 0201 040B CFD VL VU 013B CFD 2 Note that the taX benefit debt in 1 without personal taX is 035 per dollar of debt With personal taxes 2 the taX benefit of debt is reduced to only 013 per dollar of debt What is the bottom line In the presence of personal tax the tax bene t of corporate debt is significantly reduced While under plausible tax scenarios some tax bene t to corporate debt still exists it is not the overwhelming advantage that it seemed to be when only corporate taxes were considered Miller strikes again VII The Agency Costs of Equity Another form of agency costs arises between managers and shareholders ie the Agency Costs of Equity Managers like other selfinterested parties tend to look out for 1 or themselves All things equal many managers would prefer to Have more leisure time versus less Have higher salariesbonuses versus lower Have more perquisites or perks versus fewer Have lower rm risk versus more and Pay fewer dividends rather than more Theoretically the Board of Directors oversees managers and keeps them from taking advantage of the shareholders with respect to the above list and any other issues However this oversight does not always work as intended The board is composed of top managers of the rm and outside members Sometimes these outside members are friends or have business dealings with the managers Therefore objective decisionmaking by the board may be compromised Further the outside members may not be well informed or conscientious Under these conditions board decisions in the best interest of the shareholders may not always rule the day These agency costs of equity nancing can be reduced by the proper choice of capital structure If managers have a signi cant equity stake in the rm they will be more concerned with the impact of their decisions on shareholder value If a higher debt level increases the ownership concentration of equity holders especially increasing the managers shareholdings the agency costs of equity can decrease with increases in the Debt Equity ratio Further high debt levels must be refunded periodically The refunding forces managers to the capital markets which scrutinize the performance of the rm and its managers This scrutiny by the capital markets increases the likelihood that managers will run a quottighter shipquot This capital market scrutiny is missing in an allequity rm that nances its activities by internally generated capital In addition with large interest payments managers may have to quotstay on their toesquot more to meet the rms nancial obligations than if they are largely equity nanced With equity nancing compulsory nancing out ows are not necessary One CFO ofa midsized rm once told me quotHaving a high level of debt is like sleeping on a sword it keeps you awake at night worrying about the next interest payment You re figuring out how to operate more ef ciently to raise the cash On the other hand high levels of equity are like sleeping on a goose down pillow you sleep like a baby without a concern about having to run hard to meet the nancial obligations You become more complacent about the ef ciency of generating cash quot When we compare the trade off of the equity costs of debt and equity alone in determining capital structure we see the relationship shown in Exhibit II Again DebtEquity represents the optimal capital structure However notice that this diagram implies an optimal capital structure even without the presence of tax or CFD imperfections VIII Other Factors in the Capital Structure Decision Before we can conclude the discussion on the design of capital structure we must consider three other factors that can in uence the firm s choice of how much debt it should add Probability osting the Tax Shield Depending upon the volatility of their EBIT s over time some firms may be able to take advantage of the taxsavings of debt better than other firms Assume that two firms have equal interest payments of 10000 per year In addition over the next two years they have the same average EBIT 21000 Both have a 35 tax rate However the volatility of their EBIT s differ considerably Time t Time tl Firm 1 EBIT 20000 22000 Firm 2 EBIT 35000 7000 While Firm 1 will be able to take full advantage of the taxshield of interest 10000035 3500 in both years Firm 2 will have tax savings of 3500 at time t but reduced tax benefits at time tl At time t1 Firm 2 will have an EBT of 3000 so no taxes will be due Of course Firm 2 can carry losses forward to reduce future taxes but the present value of these tax savings will be less than for Firm 1 In short the probability of being able to use the tax shield will affect the amount of debt that a firm should add to its capital structure Non Debt Tax Shield Substitutes Other taxshields exist besides interest deductions For instance some firms will have more depreciation taxloss carryforwards pension plan deductions losses on disposal etc than other firms Accordingly these firms will ceteris paribus all else equal be able to use less interest taxshield deductions than firms without these other deductions Therefore the existence of nondebt tax shields can affect the appropriate amount of debt to add to the capital structure The Value of Financial Slack Debt can be raised more quickly and for lower transactions costs than equity Accordingly if a rm needs to raise cash quickly its managers may prefer to quotback o the DebtEquity ratio they perceive to be optimal in order to preserve their exibility and ability to move quickly in raising debt capital This demand for quot nancial slackquot will vary across rms Accordingly some rms may have more or less debt depending upon their need for nancial exibility IX Differences in DebtEquity Ratios across Industries We observe wide variations in capital structure across industries For instance the DebtEquity ratio in the steel industry is 17 while it is only 01 in the pharmaceutical industry Why Several empirical regularities seem to exist The more pro table the rm is the less debt it uses More pro table rms generate more funds internally Given the quotpecking order hypothesisquot and the preference rms have for internal versus external funds this inclination may explain this observation3 Firms with higher levels of intangible assets use less debt This observation may be explained by the higher CFD s for these rms relative to rms with more tangible assets e g the HewlettPackard versus Marriott example Firms with more promising projects ie a backlog of potential NPV projects both present and future use less debt This observation also may relate to higher CFD s for growth rms if they miss out on these projects due to nancial dif culties Firms with heavy levels of advertising and RampD expenditures tend to have less debt This regularity also seems to relate to growth prospects and rms relative CFD Several studies have examined DebtEquity ratios of rms both within and across industries In general these studies nd that within an industry rms tend to have similar DebtEquity ratios but the differences across industries are often signi cant In addition rms that differ from the average DebtEquity ratio in their industry tend to move back toward that industry average over time Other evidence suggests that rms behave as though the have a quottargetquot DebtEquity Deviations from this target are reversed over time This tendency may re ect the economies of scale in 3 The pecking order hypothesis first proposed by Stewart Myers at MIT suggests that at the top of the pecking order firms prefer to finance projects with internal sources or operating cash ows Next firms prefer to raise funds by accessing lowcost debt sources As a third source firms will raise cash needed to finance projects with riskier debt At the bottom of the pecking order is external equity financing The financing decision therefore amounts to working progressively down this pecking order in search of the first feasible source of financing Myers suggests that the observed capital structure of these pecking order firms are less the result of rational balancing of the pluses and minuses of debt relative to equity but more the result of the firm s profitability relative to its investment needs Accordingly highm argin modestly growing firms can finance with little debt and no external equity while lower margin more rapidly growing firms may be forced to live with higher leverage ratios and eventually new issues of outside equity financing lO issuing securities Because ofthe xed costs of selling securities it makes sense to raise external capital with a disproportionate amount of debt one year followed by a disproportionate amount of equity in a subsequent year Accordingly rms will move back and forth across their quottargetquot DebtEquity ratio over time in an attempt to capture the economies of scale in issuing securities X Differences in DebtEquity Ratios across Countries Firms in the Us have low DebtEquity ratios relative to other industrialized countries Why Tax differences may explain part of the difference The fact that foreign governments have at least in the past guaranteed the debt of companies eg Japan may explain part of the observed differences Further regulatory restrictions in the Us limit how much debt and equity nancial institutions can hold in a given rm If a nancial institution can balance equity and debt holdings in a given rm they may be willing to lend more since the agency costs of debt and equity can be reduced Finally the CFD may be much higher in the Us than in other countries The US does in fact have about nine times as many lawyers per capita than does Japan XI Market Reactions to Capital Structure Changes Under certain sets of tax rates the Miller model with corporate and personal taxes implies the capital structure decision is irrelevant ie rm value is not a function of the tax environment or VL VU However empirical evidence clearly shows that market values change signi cantly when rms make nancial transactions that either increase or decrease their leverage levels A number of studies document average positive abnormal returns to equity when rms increase their leverage Negative abnormal returns are documented on average when rms decrease their nancial leverage This evidence is consistent with the positive tax and agency cost of equity effects more than offsetting the increased CFD including the agency costs of debt effects4 In any case market reactions suggest that capital structure is clearly not an irrelevant corporate decision variable XII Managerial Check List for Establishing a Capital Structure By this time it should be clear that a precise equation for determining capital structure is not to be provided to you While research into this perplexing question continues we have made much progress in identifying the market imperfections that cause the capital structure decision to be relevant Speci cally corporate and personal taxes CFD and the agency costs of equity are imperfections that impact a rm s choice of its DebtEquity ratio Since a complete theory is lacking the best that I can do for you is to provide a quotcheck listquot of factors to consider in making the capital structure decision 4 This evidence also is consistent with certain quotsignalingquot theories of capital structure such as proposed by Steve Ross currently at MIT The bottom line of these theories is that managers can signal their inside information by adjusting the capital structure of the rm If they increase leverage managers must have con dence that they can meet the xed charge requirements of the debt Therefore they are signaling that good times are in the future However if they decrease leverage perhaps they are pessimistic about their ability to service existing debt The market would interpret this quotsignalquot negatively ll Capital Structure Check List5 Tax Rateceteris paribus all other factors equal the higher the corporate taX rate the higher the DebtEquity ratio Agency Costs of Equity iceteris paribus the higher the potential for these costs the more debt will assist in lowering these costs Probability of Using Tax Shield ceteris paribus the more reliably the rm can use the interest taXshield ie the more stable its EBIT the more debt it should have Tax Shield Substitutesnceteris paribus the more taXshield substitutes a rm has e g depreciation the lower the Debt Equity ratio should be Relative Tax Ratesceteris paribus the greater the tC versus tb and the more ts is equal to tb the more debt the rm should have Business Riskceteris paribus the more business risk the rm has the lower its DebtEquity ratio should be Costs of Financial Distressnceteris paribus the higher the CFD the lower the debt ratio Undoubtedly this factor will be related to the nature of the rm39s assets The more intangible the assets and the more future growth opportunities that are present future NPV projects the less debt the rm should have Financial Slackceteris paribus the more the rm needs nancial slack the lower the debt ratio In addition to the selfexamination of the rm based upon the above checklist managers must turn to external sources to make their nal target capital structure decision Since lenders bondholders stockholders bond rating agencies commercial bankers and investment bankers look at a rm39s DebtEquity ratio relative to its industry departing too far from the industry norm may not be feasible This observation is a fact of life even though managers may believe that they should have signi cantly more leverage than the average rm in their industries If people will not lend you money where will you get the debt Therefore use the industry average as a starting point Secondly talk to your investment banker and commercial banker for guidance These individuals are selling securities and making loans every day As a nancial manager you are engaging in these nancial transactions only intermittently Accordingly these individuals will be much more in touch with the nancial markets than you will be their insights will be useful input in making the capital structure decision 5 This check list has been derived inpart from Ross Wester eld and Jaffe See footnote 4 for the complete reference 12 Exhibit I Firm Value VU I D 39 ebt Equity DebtEquity Exhibit 11 Agency Costs Total Agency Costs Agency Agency C osts of Equity DebtEquity 7 DebtEquity NET WORKING CAPITAL AND CASH FLOWS The following example is designed to shed some light on the correct handling of net working capital cash ows in project valuation ie NPV analysis You are the proprietor of a mythical rm that has atwoday life You sell beer on the beach You hire labor for 50 per day You want 1000 in inventory at the beginning of each day and to sell your entire inventory daily You prefer to stock out of inventory rather than providing storage overnight Your inventory must be paid for in cash at delivery at the start of each day You have a 100 markup At the start of the first day t0 you pay for your first 1000 in inventory This sum represents your net working capital outlay at t0 note no current liabilities are created to fund this increase in inventory e g no accounts payable The end of day 1 and the beginning of day two are both labeled t1 The end of day two is t2 Your daily income statements are as follows Day 1 Day 2 Sales 2000 2000 E M 1000 GM 1000 1000 Labor j j EBT 950 950 MGM E E EAT 475 475 Your daily cash ow pattern looks as follows t0 t1 t2 Cash In ows 2000 Cash Sales 2000 Cash Sales Cash Disbursements 1000 Inv 1000 Inv 50 Wages 50 Wages 475 Tax 475 Tax Net Cash 1000 475 1475 Flow Your daily net working capital cash ow pattern is as follows Net WC Out Net WC In t0 t1 t2 Net Working 1000 0 1000 Capital Cash Flow ANWC At time t0 we invested in Net Working Capital in the amount of 1000 At t2 we recapture this investment in Net Working Capital as our inventory is changed from 1000 at time tl to 0 at time t2 Note that if we now combine the income statements and the negative of the changes in Net Working Capital we can recreate the actual cash ow pattern for times t0 tl and t2 The above example extrapolates to the general procedure that we will follow of owing out the net working capital for a project at t0 and flowing in the investment in net working capital at the conclusion of the projects life in this case t2 This will make more sense later in the class It is important to introduce it now when we rst start to talk about nding FCF so we don t create early confusions Note however that if we could charge the inventory for one day ie create an accounts payable for one day the cash ow picture would drastically change The effective net working capital would be zero ie current assets or inventory would equal current liabilities or accounts payable Therefore the net cash ows would be 475 at both tl and t2 with no cash ow at t0 What would be the net cash ow by period if we could charge onehalf of the inventory for each time period As a check your answer should be as follows t0 500 tl 475 t2 975 My experience indicates that the correct handling of net working capital cash ows for project or firm valuation is a significant mystery for students Please see me if this example does not clear up the puzzle for you CHAPTER 9 SOME LESSONS FROM CAPITAL MARKET HISTORY Solutions to Questions and Problems NOTE All end of chapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answerfor each problem isfound without rounding during any step in the problem 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 R9178314083 R1133 or1133 The dividend yield is the dividend diVided by price at the beginning of the period so Dividend yield 140 83 Dividend yield 0169 or 169 And the capital gains yield is the increase in price diVided by the initial price so Capital gains yield 91 7 83 83 Capital gains yield 0964 or 964 Using the equation for total return we nd R 76 7 83 140 83 R 70675 or 7675 And the dividend yield and capital gains yield are Dividend yield 140 83 Dividend yield 0169 or 169 Capital gains yield 76 7 83 83 Capital gains yield 70843 or 7843 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 Remember the Buffett bond s we discussed in the bond chapter The nominal return is the stated return which is 1240 percent Using the Fisher equation the real return was 1R1r1h r 112401031 71 r 0902 or 902 The average return is the sum of the returns divided by the number of returns The average return for each stock was N 2le Viszwom 1000 11 N Y Z Nwmzommo y 5 11 We calculate the variance of each stock as s 81 ifzN71 s kniioof 0671002 70871002 2871002 1371002016850 s kssamf 70771622 2171622 71271622 4371622061670 The standard deviation is the square root of the variance so the standard deviation of each stock is sX 016850 2 sX 1298 or 1298 sy 061670 2 sy 2483 or 2483 We will calculate the sum of the returns for each asset and the observed risk premium rst Doing so we get M Large co stock return Tbill return Risk premium 1973 71469 729 72198 1974 72647 799 73446 1975 3723 587 3136 1976 2393 507 1886 1977 7716 545 71261 1978 657 764 7107 1941 3931 71990 a The average return for large company stocks over this period was Large company stock average return 1941 6 Large company stock average return 324 And the average return for Tbills over this period was Tbills average return 3931 6 Tbills average return 655 b Using the equation for variance we nd the variance for large company stocks over this period was Variance 1571469 7 03242 72647 7 03242 3723 7 03242 2393 7 03242 70716 7 03242 0657 7 03242 Variance 0058136 And the standard deviation for large company stocks over this period was Standard deviation 7 0058136 Standard deviation 02411 or 2411 Using the equation for variance we nd the variance for Tbills over this period was Variance 150729 7 06552 0799 7 06552 0587 7 06552 0507 7 06552 0545 7 06552 0764 7 06552 Variance 0000153 And the standard deviation for Tbills over this period was Standard deviation 7 0000153 2 Standard deviation 00124 or 124 c The average observed risk premium over this period was Average observed risk premium 71990 6 Average observed risk premium 7332 The variance of the observed risk premium was Variance 1572198 7 03322 73446 7 03322 3136 7 03322 1886 7 03322 71261703322 70107 7 03322 Variance 0062078 And the standard deviation of the observed risk premium was Standard deviation 006278 2 Standard deviation 02492 or 2492 9 a To nd the average return we sum all the returns and divide by the number of returns so Arithmetic average return 216 21 04 16 195 Arithmetic average return 5520 or 5520 b Using the equation to calculate variance we nd Variance 14216 7 5522 2175522 04 7 5522 16 7 5522 19 7 5522 Variance 0081237 So the standard deviation is Standard deviation 081237 2 Standard deviation 09013 or 9013 10 a To calculate the average real return we can use the average return of the asset and the average in ation rate in the Fisher equation Doing so we nd 1R1r1h 7 155201042 71 7 4894 or 4894 u u 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 i 7 if 5520 7 0510 5010 or 5010 l l 1 11 We can nd the average real riskfree rate using the Fisher equation The average real risk free rate was 1R1r1h Ef 10511042 71 if 0086 or 086 And to calculate the average real risk premium we can subtract the average riskfree rate from the average real return So the average real risk premium was 47 if 4417 086 E 7355 13 To nd the return on the zero coupon bond we rst need to nd the price of the bond today Since one year has elapsed the bond now has 19 years to maturity so the price today is P1 100011019 P1 16351 There are no intermediate cash ows on a zero coupon bond so the return is the capital gains or R 16351 7 15237 15237 R 0731 or 731 p A J The return of any asset is the increase in price plus any dividends or cash ows all diVided by the initial price This preferred stock paid a diVidend of 5 so the return for the year was R 8027 7 8412 500 8412 R 0137 or 137 CORPORATE FINANCE AN INTRODUCTORY COURSE DISCUSSION NOTES MODULE 91 RISK AND RETURN I Review of Financial Management Concepts 0 An investment should be accepted if and only if it earns at least the as much as comparable alternatives ie with the same risk available in the capital markets 0 The required rate of return for projects or securities is determined in the capital markets Assets with the same risk should earn the same expected return r o NPV PV in ows PV out ows when discounted at the required rate of return r Accept positive NPV projects reject negative NPV projects This acceptancerejection rule the NPV Rule implies that when everything works nicely the asset being valued is accepted rejected if the estimated return IRR on the assets exceeds is less than the market required return Until this point we have nessed the topic of risk speci cally the risk of the cash ows of a project The discount rate used was either given or taken to be the riskfree rate rf If the project s cash ows are certain ie will occur with probability 1 100 the riskfree rate is in fact appropriate In the November 5 2001 Wall Street Journal WSJ we nd that the TBill rate our progv for the riskfree rate was 197 Consider a project that costs 1000 at t 0 and returns 1050 at t 1 Ifthis project is risk free would you accept it NPV 105010197 1000 2971 Yes take the project Taking the project will increase wealth by 2971 Alternatively if we invested the 1000 in the capital market at 197 we would have only 101970 at t 1 versus the 1050 generated by the project Conceptually it s very simple The capital market is the benchmark do better or don t do it 1 This lecture module is designed to complement Chapter 9 in Ross Wester eld and Jaffe 1 11 But What About Risk The question of the hour What if the above project does not pay off 1050 at t 1 with certainty What if the project is riskier than a TBill For instance maybe the payoffs at t1 could be as low as 1000 or as high as 1200 ie was risky Would you require a return greater than rf A quotyesquot answer means you are a typical riskaverse investor ie you dislike risk and must be compensated in the form of a higher expected return for exposing your wealth to it bearing risk Do we observe riskaverse behavior by individuals and institutions Yes People buy insurance even if they are not required by law to be insured Individuals and institutions invest in portfolios of securities they do not invest in a single security or put all of their eggs in one basketquot If people were not riskaverse they would not voluntarily buy insurance nor would they diversify their investments In short we observe many phenomena that lead us to believe individual and institutional investors are riskaverse Riskaverse investors require higher expected returns to compensate them for higher risk The assumption of riskaverse investors is critical to the development of our risk versus return relationship With this assumption we can draw an upwardsloping relation between expected return Er and Risk where the vertical intercept zero risk rf is the riskfree rate of return Er Risk What if our proposed investment is risky and requires a 50 return versus 197 for risk free investments NPV 10501051 1000 0 Therefore we are indifferent to this project Why This return can easily be duplicated in the capital markets However what if the project is very risky and requires a return of 150 NPV 10501151 1000 87 Reject the project It earns less than the required rate of return of 15 Rate of return criterion 5 lt 15 so reject Observe that the relationship between Er and NPV is an inverse one The higher the Er or the riskadjusted required rate of return for a given project the lower the NPV In our example we observed that Er m 197 2971 50 0 150 87 Raising the discount rate Er is a way to quotpenalizequot a project that has more risk In other words you require more return for more risk The required return for a risky project can be expressed as Er ff risk premium or Er If t 9 where 6 is the symbol for the risk premium The simple notion which we will soon complicate is that if you are to invest in a risky asset you demand an expected return at least equal to what you could get holding a risk free asset plus some compensation for bearing the risk 9 A primary goal in the forthcoming material is to give you the intuition and analytic tools to estimate this risk premium 9 We need to understand how E r and 9 are related We require measures of expected return in order to evaluate capital budgeting projects to understand how securities are priced in the capital markets to design firms39 capital structures etc 111 Returns Returns and More Returns A Realized Returns In turn we will discuss Returns Returns Holding Period Returns Compound Annual Returns and Arithmetic or Average Returns Assume that an investor bought 100 shares of a 10 exdividend price stock at the end of 1996 The investor has maintained this position and reinvested all dividend payments to acquire additional shares of the stock The following information is relevant to this investment End of Period 1996 1997 1998 1999 2000 End of Period End of Period Shares With Investment Stock Price DividendShare Q Full Reinvestment Level IL LLL 1000 1000000 100000 1125 050 1750 1044444 117500 11750 1300 075 2222 1104701 143611 12222 1050 050 1538 1157305 121517 08462 600 040 2190 2468917 148135 12190 r P PM DivPH Capital Gain Retum Dividend Retum Total Return or 1125 1000 0501000 01750 for 1997 050100 Shares 50 received at the end of 1994 in dividends 501125 44444 additional shares are purchased with the dividends Therefore the total new position is 100 shares original position 44444 new shares or 1044444 shares Notice that this ratio ofinvestment levels 1 r 21 Stock Split during 2000 Therefore the position size in shares doubles while the share price at the time ofthe split all else equal will drop in half Remember a stock split isjust apaper transaction Nothing ofvalue is created in a stock split the Jture cash flows and risk of those cash ows remains unaltere Given the above information what are total dollar returns the investor received in 1997 Capital Gains Dollar Returns Dividend Dollar Returns 1125 1000100 Shares 050100 Shares 125 50 175 or 175share The periodbvperiod annual returns are 39 39 A using the equation rt Pt PM DivtPt1 This equation can be rewritten as rt Pt PLOPH DivtPH or the capital gain percentage return the dividend percentage return For 1997 we have rt 1125 10001000 0501000 01250 00500 01750 or 1750 The holdingperiod return er for the four years the stock has been held 1 er 1 r11 rz1 r31 r4 where r1 the percent return for 1997 r2 the percent return for 1998 etc 1 er 11750122220846212190 14813 er 048813 or 4881 Again this holdingperiod return is a fouryear return and 39 39J J into 39 shares A JA39 assumes 39 of J What is the compound annual return rc over this fouryear period 1 rc4 1 W 1 rc4 14813 1 rc1481314 1 rc 11032 rc 1032year compound annual return This return represents the w of wealth increase over this period assuming reinvestment of dividends into shares of common stock 110324 14813 The average return or the arithmetic return ra equals ra r1 r r3 r44 1750 2222 1538 21904 ra 1156 You should persevere until all of the above returns are second nature to you Make up your own examples and see those provided in RWJ When do you use the compound annual return and when do you use arithmetic return Use rc if the question is what has been the annual increase in wealth over time When thinking about holding periods this is usually the choice Use ra if the question is what has been the return in a typical year This is also a good answer to what do you expect for next year unless you have reason to believe that next year won t be typical Using ra versus rC can be very misleading Example Assume that no dividends are being paid Ending Percent 1 Stock Price Annual Return 1 100 2 200 100 3 100 50 ra 100 502 25 rc 11001050 2 10 2000500 2 10 0 Your true wealth increase has been 0 not an increase of 25 In this instance using ra to represent your average annual wealth increase is veg misleading B Expected Returns To calculate expected returns or yields versus the realized returns discussed above you simply substitute the expected future price HR and the expected future dividend EDivt into the return equation The need to work with expected returns comes when we introduce risk When prices or cash ows are risky you don t know what return will occur until after it is too late The equation is Ert EPt PM EDivtPt1 For instance assume the current price of a stock is 15 Further assume that you quotexpectquot the price in one period of be 17 and that the stock will pay a 1 dividend at this future time Ert 17 15 115 020 or 20 IV Risk Revisited Chapter 9 in RWJ bypasses a detailed definition of the risk for a security beta or B in an attempt to give you some preliminary intuition about risk At this point be aware that the variance or standard deviation of a security is not a good measure of risk for a security unless it is held alone Variance or standard deviation is always a good measure of risk for a welldiversified portfolio of securities it is a terrible measure of risk of an individual security As you will see later the security39s beta B is the appropriate measure of risk in these situations Beta measures a security39s contribution to the risk of a well diversified portfolio of securities Returning to the discussion of risk recall that Ert rf 6 where rf represents the return on a riskfree security and 6 represents the required risk premium In subsequent discussions we will be developing a formal relationship for this equation or the Capital Asset Pricing Model CAPM The CAPM relates expected return to risk as follows Erj rf BjErm rf Where Erj is the expected return on security j rf is the current TBill rate Bj is the beta of security j Erm is the expected return on a quotmarket basketquot of securities or the quotmarket portfolio In words the CAPM equation indicates that the expected return on a security starts with the current TBill rate plus a term that compensates the investor for the risk of the security This risk adjustment is made up of a measure of the amount of risk the security has the security s beta B times the price or compensation provided for holding one unit of risk expected difference in the return on a quotmarket basketquot of securities the beta of such a portfolio is 10 and TBills In the CAPM BLErm rf is the measure of the risk premium or what we have called 9 The graphical representation of this equation is Er All that has changed from the last picture is that we replaced risk with beta We will be learning a great deal more about the CAPM in subsequent discussions Therefore do not be surprised or disturbed ifeverythirig is not clear at this point V Lessons from the History of Capital Markets Has risk been rewarded with higher realized rates of return over past time periods If the answer to this question is quotnoquot then nancial economists must return to the quotdrawing boardquot in search of a better theory to explain the relationship between risk and return Fortunately however the answer to this question is a resounding quotyes quot Holding risk has been on average well rewarded over time At this point we will discuss the comprehensive risk and return data provided by Ibbotson Associates a Chicago consulting rm Ibbotson Associates produces various types of historical return information and produces an annual yearbook entitled Stocks Bonds Bills and In ation SBBI Among the many categories of the data they provide are the historical returns on seven indices of market performance going back in time to January 1 1926 Through December 31 2000 these data encompass 75 years of market history Obviously 1926 is a very long time ago Accordingly the lessons from the capital markets over this interval should provide great insights into how the market has rewarded investors for bearing risk The seven market indices that Ibbotson Associates examines are Large company common stock returns the SampP 500 Index Small company common stock returns a small rm index Longterm high quality corporate bond returns AAA bonds with 20year maturities Longterm government bond returns with 20year maturities Intermediateterm government bond returns with 5year maturities TBill returns or shortterm government security returns The CPI consumer price index returns as a measure of in ation A summary of the performance of these indices is provided below BASIC SERIES SUMMARY STATISTICS OF ANNUAL RETURNS STOCKS BONDS BILLS AND INFLATION SBBI Ibbotson Associates 2001 Yearbook Returns from Jan 1 1926 through Dec 31 2000 SampP 500 Small Company 20 Year Aaa 20 Year 5 Year 1 Year CPI What do the lessons from capital market history tell us Risk has been well reward with higher returns over past time periods Intuitively we would expect T Bills to be the least risky of these asset categories shortterm and defaultfree followed by government bonds longterm and defaultfree corporate bonds longterm and default free large company stocks m maturity and n0t defaultfree and small company stocks m maturity and n0t defaultfree Do you recall why TBills are less risky than government bonds Both asset classes are guaranteed by the Us government ie defaultfree The answer relates to interestrate risk For T Bills interest rate risk is negligible relative to longerterm government bonds Why Examination of the standard deviations of these asset classes supports this intuitive risk assessment The returns realized have been positively correlated with risk TBills have the lowest realized average return 39 with a standard deviation of 32 and small company stocks have had the highest average return 173 with a standard deviation of 334 SBBI provide a graph reproduced in the text which illustrates how 100 invested on January 1 1926 in each of the indices would have grown if the investment had been maintained through December 31 2000 75 years All dividend and interest payments are assumed to have been reinvested in the respective index Recall the example above where dividends were reinvested 9 The 100 TBill investment grew to 1656 a compound growth rate of38 percent per year see the geometric mean in the above table The small company stock investment of 100 grew to 640223 over the same period a compound growth rate geometric mean of 124 percent per year The SBBI data suggest quotno free lunchquot has existed in the capital markets To earn higher returns you had to bear the higher risk As an example look at the standard deviations in the above table In 21 of the 75 years since 1925 large company stocks had negative returns 28 percent of the years Results of 2001 remind us that despite the experience of the 1990 s stocks are risky The Standard and Poor s 500 Index SampP 500 which is the basis for the large company stock index in the SBBI data is an index of 500 of the largest stocks traded in the Us This index is often used as a proxy for the quotmarket portfolioquot a term made precise in Chapter 10 The market portfolio is used to describe the return on a welldiversified basket of all types of securities or the quotmarketquot While the SampP 500 Index does not include bonds real estate and other nonequity assets this index is highly correlated with more inclusive indices that include a larger variety of assets Since the SampP 500 Index is accessible and since the index does move very closely with the quotlargerquot market of different assets it is a standard choice to represent the market portfolio The SBBI data allows us to calculate the quotmarket riskquot premium over this historical period By market risk premium we mean the average difference between returns on the SampP 500 and TBills Over this 75 year period the market risk premium has been 91 the 130 return on the SampP 500 index less the 39 return on TBills In other words investing in the quotmarket portfolioquot has earned investors an average of9 1 more than investing in TBills or the riskfree asset This market risk premium can supply us valuable insights with respect to estimating the CAPM Let s reexamine the CAPM expected return equation for any asset 139 its expected return is Erj ff BJErm ff Two of the parameters in this model rf and Erm are marketbased parameters ie they are the same for all assets Only the B is specific to the security in question To estimate the model we can look up the current TBill rate in the WS our estimate of rf Procedures to estimate security is beta Bi will be outlined in subsequent chapters For now just think of B as security is contribution to the risk of the market portfolio Estimating the quotmarket risk premiumquot Erm rf is a challenge This premium is not directly observable in the marketplace To estimate this parameter we can either supply our own forecast for the future of the market s return Erm or we can assume that history will repeat itself For instance if we assume the last 75 years are a good representation of different future states of the world eg wars recessions boom periods depressions etc then we might consider using the historical market risk premium 91 as our estimate Let s consider an example using this estimate Assume that you are trying to estimate the expected return on a security You looked up the security39s beta in a published source it is estimated to be 125 The current TBill rate is 197 percent Plugging these data into the CAPM we have Eri197 12591 1335 which is an estimate of the security39s expected return for the coming 12months Let s use the CAPM to predict the return on the quotmarket portfolio or the SampP 500 for the next 12 months As we will soon discover the beta on the quotmarket portfolioquot is 100 Therefore I expect the market return over the next 12 months to be Ea 197 10091 or 1107 Please check my prediction in one year to see how close 1107 is to the actual market return However if my estimate is off don t ask for a tuition refund Recent research shows that we can do better than this estimate of the market portfolio s expected return by conditioning our forecast on current economic data In other words if we are really trying to estimate what the return on the market portfolio will be next year we know that it tends to be higher during economic expansions and lower during contractions However if we are trying to nd a discount rate to use in valuing a project that will have cash ows stretched out over the next 20 years over several business cycles it will be surprisingly difficult to increase the precision of this simple estimate 2 In fact I can guarantee that the probability of my prediction being completely accurate approaches zero 11 COLEMAN COUPLING Suggested Solution 1 Phantom Cash Flows Students are often tempted to include quotphantomquot cash ows in their analysis of Coleman Coupling By phantom cash ows I mean changes in allocated costs that do not result in any changes in overall corporate cash ows or potential changes in cash ows with little basis in fact Coleman has examples ofboth categories of phantom cash ows ie l The reallocation of utilities with the different plant layout 2 The reallocation of general overhead costs due to the change in direct labor hours 3 The higher capacity of the new equipment which may or may not translate into higher sales and 4 The intangible bene ts that result from the elimination of the graveyard shift While the quotfreed upquot oor space is a qualitative plus no alternative use currently exists for this space Accordingly no quothard numbersquot can be attached to this feature of the new equipment With higher capacity higher sales may be possible down the road However since no information on increased sales is provided in the case this feature is another qualitative plus in favor of the new equipment Finally the elimination of the graveyard shift may decrease a number of operating problems e g a reduction in work related accidents However the quothardquot benefits of eliminating this shift are captured in the reduction of direct labor hours A note on these issues are that often times these issues can be distorted as divisional managers in a large corporation look to maximize the value or size of their division rather than thinking about total rm value 11 Analysis A Depreciation tax savings New equipment 15000012 12500year in depreciation for years 1 through 12 Overhaul equipment 16000 37000 8 6625 in depreciation for years 1 through 8 Differential depreciation must be taken into consideration B Investment Tax Credit An ITC of 7 is available on the new equipment which provides 150000 007 10500 oftax savings in the next tax paying period here assumed to be at time 0 C Alternatives Note that the quotstatus quoquot is not an option given in this case Accordingly the only two options considered are quotoverhaul of existing equipmentquot or quotnew equipmen quot The analysis that follows treats the two options separately However a single quotdifferentialquot analysis works just as well I have also used the table format for illustration D Overhaul option Item Amount 1Outlay 37000 2Depr Tax Savings 6625 E New equipment option Il Amount 1Outlay 150000 2Depr Tax Savings 12500 3Work Cap 10000 4Work Cap 10000 5Loss on Disposal 16000 6ITC 10500 6Labor Savings 24000 7Maintenance Savings 6000 8Salvage 10000 F Decision Time Tax Factor None Tax Factor None 034 None None 034 None 066 066 Amount A er Tax 3 7000 2225 Amount A er Tax 150000 4250 10000 10000 5440 10500 15840 3960 6600 14 m 3 7000 10 449 26551 14 m 150000 89659 22415 Since both alternatives have NPV39s should both be rejected No We have not included the common cash ows to these alternatives eg the revenue stream from coupling sales which is presumed to be the same for both alternatives However absent the total revenue and cost information we cannot say for sure Coleman should stay in the coupling business We can only say that if they do stay in this business the new equipment is the better way to manufacture couplings How much are the shareholders better off with the new equipment alternative Instead of bearing 26551 in present value impact for the overhaul option they will absorb 4484 in present value reduction with the new equipment Therefore the net wealth increase for the shareholders is 22067 with the new equipment versus the overhaul decision Also the qualitative quotsoftquot factors unanimously favor the new equipment e g less space is consumed a higher capacity exists if greater sales materialize and the graveyard shift is eliminated This observation reinforces the conclusion drawn with the quothardquot data used in the analysis G Other issues 1 Sensitivity tests could be run on the discount rate or other variables Note a higher discount rate will disadvantage the new equipment alternative versus the overhaul option and viceversa The cash ows are spread over alonger time period for the new equipment option which will lower their present values more dramatically with a discount rate increase 2 We have made no allowances for possible labor or maintenance wage increases as a function of time Do labor contracts dictate automatic pay increases or are labor negotiations on the horizon If these contractual cost increases can be predicted they should be included in the analysis If these cost increases are a factor the new equipment option will be even more attractive since this decision is less labor intensive 3 Along the same lines as 2 we have not included unexpected cost increases in materials used in maintenance Unexpected increases in in ation also favorthe new equipment over the overhaul alternative since the new machinery option is less maintenance intensive CORPORATE FINANCE AN INTRODUCTORY COURSE DISCUSSION NOTES MODULE 151 CAPITAL STRUCTURE PART I I An Initial Perspective As a business student one of the features that I liked about nance was the speci cation of an unambiguous obiective function for managerial decisions The goal of financial decision making is to take actions that maximize the wealth of shareholders Nothing wishy washy here The criterion for making the capital structure decision continues in this framework of wealth maximization How do we design the capital structure or the righthand side of a rm s balance sheet to maximize shareholder wealth This discussion begins our journey to answer this question The rm39s menu of different nancing choices is bewildering even to nance professionals and professors Therefore we will keep this discussion relatively simple We will categorize the rm39s choice of capital structure as the choice between generic debt and equity In the notation used by many texts we will use B to represent bonds debt and S to represent stock equity However many texts and applications use D and E to represent these securities In lecture I will use both notations to help you become comfortable with either presentation II The Capital Structure Decision in Perfect Capital Markets PCM As in other areas of economics nancial economics begins most investigations of atopic by assuming the most simple of all environments environments quotpurequot of contaminating quotreal worldquot factors In this context we will begin by assuming Perfect Capital Markets PCM to address the question of how the choice of capital structures affects shareholder wealth2 Once we have answered the question in this environment we will begin relaxing our PCM assumptions onebyone to see how the quotreal worldquot existence of market imperfections in uences our capital structure choice 1 This lecture module is designed to complement Chapter 15 in Ross Wester eld and Jaffe As a review under PCM the following conditions are assumed 1 Information is free and available to everyone on an equal basis 2 No distorting taxes exist 3 Flotation and transactions costs are nonexistent 4 No contracting costs or agency costs exist therefore managers make decision to maximize shareholder wealth and they do not attempt to exploit bondholders 5 All investors and rms are pricetakers ie they do not exert enough power in the markets to in uence the market price of securities 6 Individuals and rms all have equal access to the nancial markets and on the same terms eg interest rates 1 We will conclude that if it were not for market imperfections shareholders would be totally indifferent to the managers39 decisions about capital structure ie one capital structure would be just as good as another To put it a slightly different way no capital structure would increase shareholder wealth relative to another capital structure Why do we begin with the unrealistic world of PCM As we will discover market imperfections are so complicated that if we threw them all into a capital structure model at once we would become hopelessly confused Accordingly we relax the assumptions and add imperfections one at atime to get quotcleanerquot picture of what imperfections might quotcausequot capital structure to matter ie impact shareholder wealth in what way While we will relax most of our assumptions about PCM we will not relax two additional assumgtions o The rm will make all real investment decisions using the NPV Rule ie take all projects that have positive NPV39s and o The rm will hold its dividend policy constant ie do not change the dividend policy of the rm as a function of the capital structure decision At this point in the course you should be comfortable with the fact that managers can increase shareholder wealth by taking positive NPV projects While the evidence is much less compelling we will later learn that it is possible that managers may be able to in uence shareholder wealth by their choice of dividend policies Accordingly to separately examine the in uence of capital structure on shareholder wealth we will hold these other two decisions constant In our simpli ed model of capital structure the value of the unlevered all equity nanced rm V is simply the market value of the rm s equity or VU S where S is the aggregate value of the equity or the price per share of common stock times the number of shares of common stock outstanding Correspondingly the value of the levered rm VL is the market value of the rm s debt plus the market value of its equity or VL B S where B is the aggregate value of the debt or the price of each bond times the number of bonds outstanding and S is de ned as above Again our goal as nancial managers is to maximize shareholder wealth While we will not 3 Perhaps it goes without saying but without debt a rm is quotunleveredquot With debt a rm has leverage is levered or more speci cally has nancial leverage mathematically prove that maximizing shareholder wealth is equivalent to maximizing rm value this conclusion holds under the assumptions we maintain Maximizing rm value also will maximize sh areh older wealth4 Under these conditions the nancial manager strives to choose the capital structure that maximizes rm value Since all rms must have some equity ie some residual owners must exists the real question is whether by adding debt we can design a capital structure so that VL gt VU or the value of the rm if it were levered is greater than the rm s value is when it is unlevered Also if we can get this inequality to hold what is the optimal amount of debt to add ie the amount of debt that will maximize VL Modigliani and Miller Without doubt the most famous names in corporate nance are Franco Modigliani lH T andMertonMiller University of Chicago These nancial economists often referred to as MampM have published several seminal academic papers together and both have won the Nobel Prize in Economics 1988 and 1990 respectively Part of the reason for their winning this prestigious award was because of their work in the area of capital structure theory Under the assumptions given above speci cally PCM MampM prove that the choice of capital structure is irrelevant ie changes in capital structure will not change rm value or the wealth of the shareholders Why is this result important Not because it is a good description of the world Rather because it allows us to concentrate on those market imperfections if any that in uence the impact of capital structure on shareholder wealth Various market imperfections differentially impact rm value Accordingly depending upon their susceptibility to market imperfections different rms will choose dissimilar capital structures However we are getting ahead of ourselves Let us rst prove that under PCM the capital structure decision is irrelevant Capital Structure Irrelevance Under PCM To simplify the algebra let us assume that a rm39s Earnings Before Interest and Tax EBIT equal cash ow Further for mathematical convenience let us assume that EBIT is perpetual Finally assume that the rm s investment in assets equals its depreciation expens6e each year This reinvestment rate is presumed suf cient to generate the perpetual EBIT Let 4 This conclusions holds under the conditions that default on the rm39s debt is not likely Most textbooks illustrate this condition For instance see Ross Wester eld Jaffee Corporate Finance Irwin McGrawHill 2001 6th Ed Section 152 5 Imagine a rm with 100 debt nancing ie only debt is used to nance the rms assets Who bears the residual risk Debtholders In this case the debtholders are actually the equityholders This 100 debt rm is really an allequity rm 6 We do not need to assume a perpetual EBIT or 100 reinvestment to make the points that follow However the algebra becomes considerably messier if we allow for a more general case 3 VU market value of an unlevered rm VU S VL the market value of the levered rm VL B S B the market value of the rm s bonds the number of bonds times the market price of each bond S the market value of the rm39s stock the number of shares times the market price of each share rB the required return on the rm39s bonds the cost of debt rU the required return of the rms shares ifthe rm is unlevered an allequity rm rs the required return on the rms shares if the rm has leverage the cost of equity rWAcc the Weighted Average Cost of Capital WACC7 and EBIT the perpetual operating cash ow As we discussed before the WACC or rWAcc is the required portfolio return for the rm s nancial securities or in our world of PCM in particular no taxes rwacc BB SrB SB Srs Example Let us assume PCM and an allequity rm with 100 shares outstanding each valued at 10 VU S 10010 1000 Now assume the rm decides to add debt to its capital structure by selling 500 worth of bonds and repurchasing shares worth 500 we hold the total capitalization constant sot that we don t change the lefthand side of the balance sheet If the stock price does not change how many shares can be retired 50010 50 shares retired 50 shares remain outstanding The new value of the levered rm is vL B s 500 5010 500 500 1000 Under these conditions VU VL As a shareholder would you care if the rm changed its capital structure No If you sold shares back to the rm you received 10 per share If you did not sell your shares are worth 10 Either way your wealth has not changed m many believe that by adding leverage the value of the rm will increase 8 In this case suppose that when the rm announced that it was adding 500 worth of debt to its capital structure and retiring equity worth 500 that the share price increased to 1250 Note that m share price change will serve our example just as well 7 Note that rU equals rS equals rwacc if the rm is unlevered 8 The rational for the view is that rs does not change as the rm begins to add quotjudiciousquot amounts of leverage Since rs must be greater than rB what happens to rWAcc See the prior footnote The rU becomes rS as the rst 1 of debt is added As debt is added rWAcc at rst begins to drop What happens to the total value of the rm ier falls Given our perpetuity example and PCM rm value rises V EBITrWAcc EBIT is a perpetuity rWAcc is falling therefore V must be increasing In this case 500 1250 40 shares would be retired 60 shares would still be outstanding VL B S 500 601250 500 750 1250 In this case VL gt VU Would the shareholders be happy Of course Regardless of whether they sold their shares back to the rm or retained their shares they would be 250 per share better off Let us think about the above example Has the rm done anything to the asset side of its balance sheet in this nancial transaction No The assets remain unchanged Therefore the operating cash ows generated by the assets remain the same The business risk of the firm the risk of the operating cash ows remains the same Therefore what could be the source of the value increase This increase in shareholder wealth should leave you a little uneasy It seems too easy Can changing the capital structure of the rm create an increase in shareholder wealth At the time MampM were studying capital structure most academics and nancial managers believed rm value could be enhanced with the addition of some judicious amounts of leverage MampM did not buy the story that in perfect capital markets rms can increase their value by adding leverage In order for the rm value and share price to increase an economic rationale must eXist They could not identify an economic rationale Accordingly MampM did not believe that the share price would increase If they are correct the result is M ampM 39s famous Proposition 1 or VL VU See Exhibit I 10 Proposition I says that a rm39s value is independent of its debtequity ratio To prove their claim ask this question What would you do if you observed two rms with identical EBIT s and business risk selling for two prices VU 1000 and VL 1250 You could buy either rm in its entirety and be entitled to the entire operating cash ow To buy the unlevered rm you would buy all of the stock To buy the levered rm you would buy all of the bonds and all of the stock Assume the EBIT of the two rms was 150 per year in perpetuity What would be your returns in the two rms rU the return on the unlevered rm 1501000 015 or 15 rWAcc the return on the levered rm 1501250 012 or 12 Which return would you prefer Remember the business risk of the two rms is equal But 9 Remember we are assuming PCM Recall that in PCM taxes equal zero and all individuals and rms have equal access to the nancial markets ie they can borrow or lend on the exact same terms 10 All exhibits are located at the end of the note what about the nancial risk Recall in our example you bought all of the debt and all of the equity of the levered rm Effectively you unlevered this rm in your personal portfolio Therefore the nancial risk is equal for the two transactions described in the above example If this situation of VL gt VU existed an arbitrage oggortunigg exists The sameEBI T with the same risk is selling at two different prices Buy low and sell high Investors would be selling stock in the leveredfirm to get the higher return available from owning the unlevered firm This buyingselling pressure would drive the firm values back to VU VL This implies that the stock price in our example cannot change In a much more rigorous fashion MampM proved this result Despite the screams of protest from people that wanted to believe a levered rm was more valuable given their assumptions MampM39s logic has withstood the test of time constant attack and been shown to be robust to many extensions beyond their original simple framework Beyond dogmatic fervor it is not clear why people wanted to disprove the result so much This is just an extension of the zero pro t competitive equilibrium from economics in a friction free market The Cost of Capital Let s reconsider the above example We illustrated that VU VL for all ranges of leverage or all possible DebtEquity ratios The value of the unlevered rm is determined by VU EBITI LL where rU or r A is the cost of equity for the unlevered rm Remember for the unlevered rm no interest I is paid Similarly in a PCM there are no taxes The valued of the levered rm is determined by VL EBITTWAcc where rwacc is the overall cost of capital or the WACC Since VU VL and EBIT EBIT what is the relationship between rU and rWAcc They must be equal See Exhibit II Therefore the fact that VU VL implies that rU rWAcc The Cost of Equity Again recall the equation for rU or rWAcc or I WAcc I U BBS1 B SBSI s Solving this equation for rs we nd rs rU rU r3 BS MampMPr0p0siti0n II This relation between rs andBS or the DebtEquity ratio isM ampM Proposition II The cost of equity rs is an increasing function of the amount of leverage in a firm 393 capital structure Does this relation make sense As nancial leverage increases the equity becomes riskier Remember debtholders have priority to the rm s operating cash ows Accordingly the more debt in the rm s capital structure the greater nancial risk equity holders bear Given PCM which includes zero bankruptcy costs we can take the cost of debt rB to be invariant to the DebtEquity ratio If the rm misses an interest payment the debtholders can costlessly take control of the rm and given the assumptions about information they knowjust when it is necessary to do so In Exhibit III we see the relationship between rB rWAcc rs rU and the debt equity ratio Example Assume PCM and that a rm s EBIT is a perpetuity equal to 150 The cost of unlevered equity rUY is 15 What is the value of the unlevered rm What is the rm s rWAcc VU EBIT rU 150015 1000 I WAcc I U I BBBS I sSBS 1 B0 rs11 TS In this case rU also equals rs Now assume that the rm issues 500 in debt and retires equity The cost of debt rB is 8 With PCM rB is invariant to the DebtEquity ratio We already know that VL will equal VU Further we know that the Weighted Average Cost of Capital rU rWAccy will not change However what will be the cost of equity rs under this new capital structure rs rU rU rBBS rs 015 015 008500500 022 If you will recall our discussion on the impact of nancial leverage on a rm s beta g the fact that rs increases with leverage should not surprise you Recall the following equation ls Al Since is is an increasing function of the Debt Equity in the CAPM you should get a higher rs In other words MampM Proposition 11 is completely consistent with Capital Market Theory which was developed about 15 years after MampM s research insight III The Capital Structure Decision in Imperfect Capital Markets Add Corporate Taxes In the real world we observe that rms within industries generally have similar capital structures Often DebtEquity ratios are radically different across industries If the capital structure decision is irrelevant as suggested by MampM Proposition I then this pattern of behavior is puzzling If capital structure does not matter rms should have random DebtEquity ratios within and across industries Given the observed patterns rms behave as though capital structure matters What have we left out of the above analysis that may cause capital structure to matter Corporate Taxes One of the features of the taX code is that rms can deduct interest payments before computing their taX payments In other words debt payments are subsidized by the taX code For example if a corporation has a 100 interest payment it writes a check for 100 Without any tax consideration the 100 is the cost of the debt for that period However what if the interest on debt is taX deductible Assume that a rm has EBIT of 500 The rm has ataX rate of35 The rm pays out all of its EAT or earnings aftertax to shareholders Contrast the rms if they do or do not have a 100 interest payment No Interest Payment Interest Payment EBIT 500 500 391 0 EBT 500 400 TaX035 EAT 325 260 Dividends 325 260 Even though the interest payments requires a 100 check to be written to the bondholders the EAT only falls by 65 325260 This drop is because the tax payment was reduced by the deductibility of interest by 35 175140 Accordingly the IRS effectively picked up 35 of the cost of the 100 interest payment What are the total distributions to claimholders in the above two situations With no debt and no interest 325 is available to distribute to the shareholders In the case of debt and a 100 interest payment the debtholders get 100 and the stockholders get 260 for a total of 360 The higher payment in the second case again higher by 35 is because the IRS gets a smaller slice ofthe EBIT pie Should all else equal a rm with greater distributions to its claimholders sell for a higher price The answer is a resounding YES A Formal Proof The value of the unlevered rmi Again let us assume our rm with level EBIT in perpetuity We will assume that 100 of earnings are paid out as dividends Initially our rm is allequity nanced What is the value of this rm First we must calculate the cash ows to the stockholders of this unlevered rm From the EBIT the rm must pay taxes Since no interest is paid EBIT equals EBT Aftertax the EAT equals EBIT1 tc which is paid out in dividends How do we value this perpetual cash ow to shareholders We must compute its present value by discounting the cash ow at the required rate of return Since the rm has no leverage the shareholders require rU as the discount rate or VU EBIT1 tc rU The value of the levered rm Now assume the above rm issues B dollars of debt at a rate of rB and uses the proceeds to repurchase equity What is the value of this levered rm First we must calculate the rm s cash ows that go to both the debtholders and the equityholders Note that the payment to the bondholders is the cost of debt times the amount of debt or rBB interest This amount is deducted from EBIT before guring EBT The equityholders will receive EBIT rBBl tc We have EBT in the rst set of parentheses If we expand this expression we get EBIT1 tc rBB rBBtc 1 The bondholders will receive TB B 2 or the interest payment If we add 1 and 2 we get the total payments to the claimholders of the levered rm or EBIT1 tc rBBtc 3 How do we value this perpetual cash ow to both claimholders We must discount the cash ows by the required rates of return What discount rate will the market require for the rst term in equation 3 This cash ow is exactly the same cash ow as discussed above in the unlevered case The EBIT has not changed the tax rate has not changed and the business risk of the rm has not changed 9 Therefore this term should be discounted at rU just as above What discount rate will the market require for the second term in equation 3 Recall rBB is the interest payment Bondholders require rB on their investment This annual tax shield is as risky as the debt Accordingly we should discount the second term by rB Therefore the value of the levered rm VL equals VL EBIT1 tc rU rBBtcrB The second term is important In the numerator we have the interest payment rBB times the tax rate tc This product represents the tax savings generated by the interest payment When we divide this perpetual tax savings by rB we get the present value of the tax savings Noting that the rB s cancel out we get VL EBIT1 tc rU Btc The rst term equals the value of the unlevered rm VU Accordingly VL VU 13 4 Therefore the value of the levereal rm equals the value of the unlevereal rm plus the present value of the tax savings generated by the tax deductibility of the interest payment Accordingly VL gt VU It is critical that you understand the intuition of this very important result Do you See Exhibit I Vfor the graph of the value of the levered rm VL and the level of leverage or DebtEquity For every dollar of debt we add the value of the rm increases by Btc or ltc If tC equals 35 rm value goes up by 035 for every 1 in debt Since rm value is going up with debt and the EBIT remains constant what is happening to the rm39s cost of capital Recall VL EBIT1 IcI WAcc SO I WAcc EBIT1 IcVL Therefore rWAcc must decline as rm value increases as a function of its Debt Equity ratio See Exhibit V The implication of equation 4 is that rms should be almost 100 debt nanced For every dollar of debt added rm value goes up by 1 tc In the limit we would nance rms 10 to the point where only one share of stock is still left outstanding A J39 of Kev 7 quot for Corporate Tax Before we leave this topic let us update two key equations with the inclusion of corporate taxes speci cally the equation for the cost of equity rs MampM Proposition 11 and the equation for the overall cost of capital or rWAcc With corporate taxes we must adjust two prior equations as follows Vs TU f m r3 1 to BS and rWAcc BBts VB 1 to t SBts ms Note the difference in these two equations relative to the notaX versions is the inclusion of a l tc term to account for the taX deductibility of interest Comprehensive Example Assume that we have an all equity perpetual rm with EBIT 76923 Say tC 35 and rU 10 VU EBIT1 tcrU VU 76923l 035010 5000 Assume that 100 shares are outstanding therefore each share trades for 50 Now assume that the rm decides to increase its Debt Equity ratio by selling 2500 in debt and repurchasing equity The bonds carry a rate rB of 6 What is the new value of the rm What is the new cost of capital for the rm VL VU Bquot tc VL 5000 2500035 5875 Since the total rm value increased by 875 the price per share increases by 875 100 shares 875 per share The new share price will equal 50 875 5875 Since we sold 2500 in bonds the rm can buy back 25005875 4255 shares 100 shares 4255 shares 5745 shares remaining S 58755745 shares 3375 Alternatively VLBS 5875 2500 S S 3375 rs rU rU rB1 tcBS 010 010 0061 03525003375 0119259 1 am carrying a lot of decimal places to avoid a rounding problem rWAcc BBSrB1 tc SBSrs rWAcc 25002500 33750061 035 33752500 33750119259 0085106 Since we have rWAcc we can also value the rm as VL EBIT1 tcrWAcc VL 769231 0350085106 5875 A Summag of the Results The existence of corporate taxes explains why rm value should increase by the present value of the tax savings provided by interest deductibility However taken to the limit we observe that considering the tax savings associated with debt alone ie considering no other imperfections implies that rms should be nanced just shy of since all rms need some equity to exist 100 debt VL VU tcB This conclusion should leave us with an uneasy feeling however We do not observe rms with anywhere near this level of debt in their capital structures Why Perhaps we need to examine additional market imperfections Don t worry we will Exhibit I Firm Value V W Debt Equity A Exhibit 11 Percent Return n Debt Equity Exhibit III A Percent Return n 39 Debt Equity Exhibit IV Finn Value VU Debt Equity Exhibit V AK Percent Return 139 at 1391 139b Debt Equity CORPORATE FINANCE AN INTRODUCTORY COURSE DISCUSSION NOTES MODULE 131 EFFICIENT CAPITAL MARKETS I An Initial Perspective We now turn our attention to the righthand side of the balance sheet This lecture begins our quest to answer the question quothow should the rm structure its sources of fundsquot We refer to this issue as the capital structure alecision2 We will keep this question relatively simple In this course we will simply address the question of nancing tradeoffs between debt and equity 7 gardenvariety debt and equity at that We do not even get into the issue of shortterm versus longterm debt just undifferentiated debt Further we will not begin to address more complex sources of funds ie fmancial securities such as putable bonds convertible debt convertible preferred stock warrants etc The finance elective courses cover some of these more exotic securities As we turn to the capital structure decision we initially take two other financial decisions as given 0 We will hold the investment capital budgeting decision constant In other words with respect to the decisions on the lefthand side of the balance sheet we will assume the firm follows the NPV Rule ie take all real investments with positive NPV s 3 We will examine possible interactions between investment decisions and capital structure decisions later in the course We will hold the dividend policy decision constant We will take up the topic of dividend policy in Chapter 18 Both the investment and the dividend policy decisions can change firm value We therefore hold these decisions constant so that we can isolate the impact of the capital structure decision on firm value and therefore on shareholder wealth We will approach the question of the choice of an optimal capital structure from the same perspective that we maintain through the coursewe want to make decisions that will maximize shareholder wealth 1 This lecture module is designed to complement Chapter 13 in Ross Wester eld and Jaffe Use of the terms nancial structure and capital structure are o en confusing For our purposes we will refer to all of the accounts sources of funds on the righthandside of the balance sheet as the nancial structure Those sources that have explicit costs are referred to as the capital structure The difference between the nancial structure and capital structure accordingly would be noninterest bearing liabilities such as accounts payable wages payable or taxes payable 3 Remember quotrealquot assets reside on the le hand side of the balance sheet Financial assets reside on the righthand side 1 11 Prices of Financial Securities The prices of all assets real and nancial are determined using the discounted future cash ow model or T P0 z CFt 1 r where 71 P0 the price of an asset at t0 or today CFt the aftertax cash ow at time t N the period of the nal cash ow and r the appropriate riskadjusted required rate of return After our work with capital budgeting you should feel comfortable with this equation Likewise you should feel comfortable about the determinants of r Note for equity T approaches infinity In capital budgeting the evaluation of projects real assets we hope to find positive NPV opportunities Investing in real assets with positive NPV39s will increase shareholders39 wealth by the amount of the NPV Market imperfections in the product real asset markets are the source of shareholder wealth increases NPV s for capital budgeting decisions these represent deviations from longrun competitive equilibrium Some of the sources of market imperfections for investments in real assets you should recall include Creating andor exploiting barriers to entry such as patent or trademark protection government regulations economies of scope or scale etc Producing goods or services at a lower cost than competitors Convincing the consumer that your products are of higher quality you have better selection your service is superior you have better credit terms etc Being the first in your market to develop a new product or service Alternatively you may locate an unsatisfied demand for a product or service If real asset product markets were perfectly competitive all positive NPV projects would be quotbid awayquot If any positive NPV projects existed competitors would ood the market lowering prices and increasing expenses to the point that NPV39s would approach zero In economics we referred to this condition as a zero profit competitive equilibrium Under this environment you would earn a quotfairquot rate of return on investments or the rate of return commensurate with the risk as described by the SML This is an idealized situation and we don t expect that it will be exactly true very often However we do expect to be able to identify markets for which it will be approximately true How We have the same objective with respect to our decisions on the righthand side of the balance sheet f1nd the combination of financial securities that will maximize shareholder wealth In order to increase shareholder wealth we must find positive NPV opportunities that exist because of market imperfections in the capital markets such as being able to o Issuing securities that are worth less than the proceeds the rm receives and 0 Buying securities that are worth more than they cost Similarly to the real asset markets with perfect capital markets PCM a rm would be unsuccessful in creating value for its shareholders by changing its capital structure Any capital structure would be as good as another We will demonstrate this clearly in Chapter 15 Therefore if capital structure is to matter ie to affect shareholder wealth it must be because of market imperfections III Real Asset Markets versus Capital Markets Evidence strongly suggests that the opportunities to nd and exploit market imperfections as they relate to investment opportunities in the product markets on the lefthand side of the balance sheet far outnumber opportunities to nd and exploit market imperfections that affect the pricing of nancial assets on the righthand side of the balance sheet The reason for this imbalance of opportunities relates to the greater competition in the capital markets the existence of or the ability to create close substitutes for nancial assets and the existence of fewer market imperfections Many of the sources of imperfections as they relate to investment opportunities in real assets either do not exist in the capital markets or their impacts are signi cantly smaller If capital markets were perfect all nancial securities would be priced quotfairlyquot By being priced fairly we mean that prices would re ect all that is known and knowable about the future cash ows and risk of the security In such a world all nancial securities would plot on the SML If securities plot on the SML the NPV from investing or selling such securities is zero In such a world the capital structure decision would be irrelevant ie you could not increase shareholder wealth by changing from one capital structure to another any such change would be a zero NPV transaction If prices are quotrightquot ie if security returns all lie on the SML capital markets are said to be ef cient If all that is known and knowable is re ected in the prices of securities the capital markets are said to be quot efficient capital markets quot The hypothesis that states that capital markets are efficient is labeled the quotefficient markets hypothesis quot By de nition if capital markets are perfect PCM capital markets are ef cient Therefore market inef ciency implies imperfections in the capital markets No other proposition in economics and nance has been more extensively studied than the ef cient markets hypothesis EMH Given the tremendous amount of work that has gone into attempts to discredit it with remarkably few exceptions the evidence supports the EMH IV Information and Market Ef ciency A market is ef cient with respect to an information set pt if it is impossible to make economic pro ts NPV39s at tl by trading on the basis of information contained in p17 at time t The EMH has been subdivided into three distinct levels for testing 3 o The Weak form of the E icientMarkets Hypothesis 0 The Semi Strong form of the E icientMarkets Hypothesis and o The Strong form of the E icientMarkets Hypothesis The EMH is concerned with how the market digests information and reacts to new information based on three different sets of information Therefore the focus of tests of the EMH is on the market39s ability to process particular sets of information efficiently Your textbook has an excellent gure showing the relationships of the three information sets pt for the three levels of EMH tests Be sure that you understand Figure 134 Weak Form Tests of the EMH The Weak form of the EMH concerns the ability of investors to make economic profits based upon historical trading information The information set pt used to test the Weak Form version of the EMH therefore is all historical information on security prices trading volumes etc Any informationdata relating to past trading is included in this information set Investors that make buy and sell decisions based upon this information set are said to be technical analysts or technicians Technical analysts follow technical trading rules If patterns in past data allow investors to trade securities and earn positive abnormal returns i e NP V39s the market is said to be ine icient with respect to the Weak form of the EMH Imagine that by plotting past prices and studying the pattern that you can predict future prices If you can project the past and predict the future you have the basis of a trading rule A trading rule is a quotrule of thum quot that tells you when to buy or sell securities that subsequently will earn you abnormal returns In a market exhibiting Weak Form efficiency sequential security price changes are said to follow a random wal with a positive drift Consider the following equation Pt Pt11 130 1 Pt1 H 8t This equation says the price of a security at time t HR is the price at tl PH plus the expected return on the security PH gtlt Er plus an error term SL The error term has an expected value of zero and is randomly distributed about zero This means that the best guess at what the price will be at time t is what it was at tl plus an increase to account for the asset s expected return Ethe timeseries of error terms has a significant positive or negative correlation serial correlation so we could make a different and better guess than PM u this would be evidence against the Weak form of market efficiency While the term random walk has complicated mathematical properties the intuition behind this term is quite simple Once we adjust for the positive expected return a security s price changes plus or minus are random Prices are just as likely to increase as decrease The text contains an excellent summary of the statistical tests that evaluate this level of market efficiency The extensive literature on Weak Form market ef ciency suggests that based on historical information the markets are extremely efficient In other words trading on the basis of past patterns does not garner the investor abnormal returns versus a quotbuy and holdquot strategy especially if transactions costs are included To the extent that trading on historical information generates high transactions costs this trading strategy can leave the investor signi cantly poorer than simply adopting a quotbuy and holdquot strategy Semi Strong Tests of the EMH The SemiStrong form of the EMH concerns the ability of investors to make positive abnormal returns based upon all currently available public information The information set used to test the SemiStrong version of the EMH pt therefore includes all historical information instantaneous new public information annual reports SEC filings projections regarding firms industries the economy expected in ation rates etc In short any informationdata that is publicly available is included in this information set the instant it becomes public Investors that make buy and sell decisions based upon this publicly available data set are labeled fundamental analysts Most securities analysts are fundamental analysts They make buy and sell recommendations based upon analyses of this more comprehensive data set Trading rules therefore are not as mechanical such as those used by technical analysts These trades require judgement after digesting a great deal of contemporary information If investors can earn abnormal returns based upon fundamental analysis the market is said to be ine icient with respect to the Semi Strong form of the EMH Imagine that by trading instantly on news releases ie a good or bad news announcements about a firm you can earn abnormal returns You keep your eye on real time price quotes and your ear to the financial news network Once you receive news you buy on good news or sell or shortsell on bad news instantly through a computer hookup to the oor of stock exchanges or the OTC market At night you read everything that you can about securities and markets In short you have no life If you discover some clue about a security being under or overpriced you submit a trade order to some market that may still be open ie Tokyo Hong Kong Frankfort London If you can earn positive abnormal returns following these strategies then markets are not semistrong form efficient Most of the evidence on the SemiStrong form ef ciency of the market is developed in event 4 A book that I highly recommend that uses this term isA Random Walk Down Wall Street by Burton G Malkiel published by Norton Press in New York studies Event studies look at the market39s reaction to the public release of information concerning rms Event studies measure abnormal returns and cumulative abnormal returns CAR 39s around these announcement dates We will label the time period surrounding the event under study as the event period For instance say you are studying the markets reaction to merger announcements Because of the possibility the news hit the market before it was officially reported by the electronic or printed news media you may want to look a few days before and after the first day the event was reported in the Wall Street Journal say plus or minus two days Abnormal returns are defined as the actual daily return of a security minus some benchmark for the daily return that is expected Depending upon the type of event study benchmark returns can be 1 an average of daily returns for the firm surrounding the event period 2 the daily market return 3 the daily market model return and 4 the daily CAPMadjusted return SML Regardless of how they are calculated abnormal returns can be positive or negative If the benchmark is the daily returns surrounding the event period you just average daily returns for the security for a period say 20 days before and after the event period This average return is then subtracted from the actual returns during the event period to calculate the abnormal return This procedure is easy and uses the security s own return as the benchmark ostensibly eliminating the need for riskadjustment However if riskshifts exist because of the announcement this may result in over or understating the abnormal return If the benchmark is the market return the return on a market index e g the SampP 500 for the event period is subtracted from the actual return on the security or portfolio for the event period The difference is the abnormal return for the study period This benchmark is easy to calculate All you need to know is the security return for period t say the daily return for a security and the market return for that same period However this benchmark suffers because it does not adjust for the risk of the security or portfolio from the market level of risk which has a beta equal 10 If the benchmark is the market model lj Bjrmt is subtracted from the actual return on security j rjt in time period t to calculate the abnormal return for time t The market return rmt is the realized return on the market in time period t This benchmark is just based on the regression equation used to calculate a security s beta This benchmark has the disadvantage of being more complicated than using the market return benchmark It has the advantage of adjusting for the individual risk of the security relative to the market ie Bj If the benchmark is the CAPM rf rmt rfBj is subtracted from the actual return on security j rjt in time period t to calculate the abnormal return for time t Obviously this benchmark also adjusts for the risk of the security relative to the market These latter two benchmarks the market model and the CAPM are for all practical purposes equivalent5 Cumulative abnormal returns CAR s are just daily abnormal returns AR s added up over 5 Studies have shown that the results from event studies using any of these four benchmarks are remarkably similar ie the results are robust to the choice of benchmarks time T CAR Z ARL where t1 CAR the cumulative abnormal return T the number of time periods ARt the abnormal return in period t These CAR s form patterns around a news release and allows us to make inferences regarding market efficiency at the SemiStrong level Example Assume that we have observed actual returns rjt on security j and expected returns conditional on what actually happened in the market rf rmt rfBj for security j over four time periods This later return is what the CAPM would predict the return to security j to be given what actually happened to the market This level of return lies on the SML Given these data we can calculate the abnormal returns for each time period ARt and the cumulative abnormal return CAR Time 1 Z 1 4 Actual Return rjt 011 007 013 012 Predicted Return rfrmt rfBj 010 008 012 014 Abnormal Return ARt 001 001 001 002 Cumulative Abnormal Return CAR 001 000 001 001 Draw a plot of the CAR s in this example as a function of time In this example the CAR s seem to be randomly distributed around zero Accordingly nothing quotabnormalquot seems to be going on with this security Examples of event studies include quotgoodquot and quotbadquot news announcements news on capital expenditure levels capital structure shifts mergers earnings dividends stock splits and accounting procedure changes Various trading rules were analyzed for their effectiveness around these announcements The key issue is how fast prices adjust to the release of new information If prices or equivalently returns adjust with a lag then the evidence argues 7 against market efficiency Researchers have also examined the performance of mutual funds which trade based upon public information as a clue to the efficiency of the market at the SemiStrong level With all of their resources computer power instantaneous access to information and army of smart analysts we would expect to see mutual funds consistently earning abnormal profits if any investor could What does this evidence suggest For the most part mutual funds earn quotnormalquot returns ie returns that we would expect given the SML The extensive literature on Semi Strong Form market ef ciency while not as unanimous as the WeakForm evidence suggests that based upon all publicly available information the markets are remarkably e icient In other words trading on the basis of public information both past and current does not garner investors abnormal returns versus a quotbuy and holdquot the market strategy To the extent that trading on public information generates high transactions costs this trading strategy will leave the investor significantly poorer than simply adopting the quotbuy and holdquot strategy When you think about this evidence on Semi Strong efficiency I hope that you are not too surprised Why The capital markets are m competitive Capital market imperfections are few relative to real goods markets eg barriers to entry are low transactions costs are low information is abundantly available at low cost economies of scale are not large etc On top of these factors some of the smartest best financed and most aggressive people make their living by trading in the capital markets If a dollar ie an abnormal return is lying on the table someone is going to pick it up very guickly Of all the markets that we can examine the capital markets come closest to being the most perfectly competitive with the fewest imperfections Strong Form Tests of the EMH The Strong form of the EMH concerns the ability of investors to make abnormal returns based upon the information set of all information pt historic information public information and private information Private or inside information is that information known to only one or a small group of individuals Accordingly this information has not been impounded or re ected in the current price of the security Obviously this level of test is the most extreme and severe test of the EMH Let us keep this discussion short and sweet The evidence clearly indicates that if investors have inside information they can earn abnormal returns on the securities they trade Examples of those earning these abnormal profits are corporate insiders ie top managers who trade before news is released to the public However trading on the basis of inside information is illegal You can go to jail for trading on inside information ask Martha The literature on Strong form market ef ciency suggests that based on trades re ecting inside information the markets are inefficient The EMH clearly breaks down at the Strong form level When people refer to the E icientMarket Hypothesis without being specific as to whether they are referring to the Weak form Semi Strong form or Strong form versions they are usually referring to the Semi Strong form of the EMH The Semi Strong form is the standard benchmark for market ef ciency Should You Throw Darts to Pick Your Portfolio If markets are perfectly efficient all securities will plot on the SML earning a fair riskadjusted return Under these conditions securities will not be under or overpriced Accordingly should you just throw darts to pick your investment portfolio The answer is a resounding amp As an investor and at a minimum you should have a target level of risk for your investment portfolio p and a goal of efficient diversi cation ie no unsystematic or diversi able risk in your portfolio Throwing darts is not likely to achieve either objective If you enjoy investing and want to manage your own portfolio ne One study on individual investors and their ability to match mutual fund returns if they had buyandhold strategies thus avoiding excessive turnover transaction costs and taxes suggests that it s not that difficult to do However you need a fairly signi cant number of securities to eliminate unsystematic risk probably 20 to 30 securities if chosen carefully ie not highly correlated securities If you don t have the resources to take a reasonably signi cant position in this number ofsecurities say 1000 invested in each then I recommend going with noload mutual funds that match your risk target If you don t have the time money energy or interest to manage your own welldiversi ed portfolio then I suggest choosing a few no load funds preferably with some of your money in a broad market index fund p 10 that are tailored to jointly meet your risk and diversi cation preferences I would include at a minimum an international fund and a money market fund for liquidity in addition to the index fund Maybe for a little spice you could throw in a smallcap tech fund or a smallcap index fund It s up to you 7 you know your own trade offs between eating and sleeping A Paradox One of the interesting paradoxes of market efficiency is that the m people that believe it the ef cient the markets will become If you believe the markets are ef cient you will not bother to do security analysis you will not seek out under or overpriced securities Why Because you believe that everything that is known and knowable is already re ected in the prices of securities Therefore why waste your energy nding mispriced securities Accordingly under these conditions security prices will not be scrutinized and can become mispriced Hence if eve one believes the markets are efficient they will become inefficient Therefore individuals that seek out mispriced securities are providing a valuable service they are making the markets more efficient V Anomalous Evidence and Market Ef ciency It would not be fair to omit some cites of puzzling evidence that seems to contradict the SemiStrong Form evidence that markets are ef cient These anomalies include 9 the rm size effect the PE effect the January effect the dayoftheweek effect and momentum effects Remember however in assessing market efficiency we must assume a benchmark against which we measure abnormal returns In other words tests for market efficiencies are really joint testswe are testing a benchmark for market efficiency and we are measuring actual returns against this benchmark and drawing conclusions about market efficiency Therefore if markets appear inefficient it could be because Markets really are inefficient or The benchmark for market efficiency is wrong or The benchmark ie the CAPM is right but our estimates of the parameters e g the market portfolio is misspecificed ie the betas are wrong Given the evidence on the CAPM we know that the CAPM fails to completely explain realized returns Therefore since our most sophisticated tests of market efficiency use the CAPM as a benchmark we must be cautious in drawing our conclusions VI The Implications of Market Ef ciency You should review the table in your text for the summary of the EMH specifically what it means and what it does not mean However let us stress the implications of market efficiency as it relates to the financial manager If the capital markets are perfect the capital markets also will be efficient all securities will plot on the SML Accordingly all securities will be properly priced In such a world the financial manager will not find any positive NPV opportunities on the righthand side of the balance sheet In this world the capital structure decision will be irrelevant With respect to shareholder wealth one capital structure will be just as good as the next Financial managers for the most part do not behave as though they believe that capital markets are efficient They seem to sell stock after stock prices rise Perhaps they think that their stock price is too high and now is the time to sell This behavior implies that they think that they can quottimequot the market ie buy low and sell high Given the patterns of securities issues over time the evidence suggests that managers attempt to time the market If managers make this selling decision based upon inside information e g they know some bad news that the market does not know so they better sell new stock fast then they can obviously time the market However remember that it is illegal to transact in securities based upon inside information Such managers would undoubtedly be sued by new shareholders that bought the stock at a high price only to realize the stock was overpriced relative to managers39 private information 10 However in our world with market imperfections e g taxes and transaction costs it is possible for managers to design capital structures that can increase shareholder wealth even if capital markets are ef cient We will study these opportunities in chapters 15 and 16 VII Some Personal Re ections Regarding Market Ef ciency for Y0u The Investor The topic of market ef ciency generates heated debates and passionate feelings Money managers and security analysts do not like the notion of ef cient markets Obviously if markets are ef cient the value of what they sell superior performance is called into question However money managers e g mutual fund managers can provide socially useful functions even if markets are ef cient Given funds39 ability to diversify effortlessly their economies of scale their lower transactions costs the liquidity they provide investors the taxtailoring of investments that some funds provide speci c investor clienteles and their record keeping services mutual funds provide valuable services that justify some level of management fees I believe that assuming markets are ef cient is a good starting place for assessing an investment opportunity The rst thing to think about if an investment appears to have the potential for abnormal returns is what the source of the market imperfection driving the abnormal return might be eg taxes Additionally what special market insights and credentials does the seller of the investment have Be wary of any sales person that quotpromisesquot you any return greater than the risk free return Unless a security is risk free uncertainty eXists Therefore promises of returns higher than the risk free rate are not believable True the expected return should be higher than the risk free rate but no promises can be made about future actual returns of risky securities If sales commissions transactions costs and management fees are high you can even earn less than the risk free rate on a risk free investment Some questions to ask an individual that is trying to sell you an investment based upon hisher past performance record include Are the returns you are citing risk adjusted If so how are they risk adjusted Are the returns that you are citing net of transaction costs Selling commissions Management fees Do any quothiddenquot costscharges exist Are the returns that you are citing net of taxes Is your track record for superior abnormal returns over a long time period Throughout both quotbullquot and quotbearquot markets Are you prepared to quotprove quot to me your af rmative answers to the above questions If someone offers you an investment opportunity that seems quottoo good to be truequot it probably is A fool and hisher money are soon parted I f you only learn one thing in your finance classes I hope it is that free lunches if they ever exist do not exist for very long Too many smart people are looking to snap up free lunches that may exist So we expect them to exist only briefly if at all before they become fairly priced In short be skeptical After all it39s your money If you want to earn positive abnormal returns on investments stay away from markets that are the most competitive e g the NYSE Competitive markets are most likely to be efficient markets You cannot earn positive abnormal returns in e cient markets Research on individual investor performances levels on the NYSE the ASE on regional stock exchanges and in the OTC markets shows that your opportunities to earn abnormal returns get better the farther you get away from the NYSE However a downside exists to this strategy of trading in quotless ef cientquot and less liquid markets Remember inefficiency means that securities can be overpriced just as easily as they can be under priced Accordingly if you are going to trade in an inefficient market you better be prepared to become an quotexpertquot in the characteristics of securities that trade in these inefficient markets Then you had better be prepared for hard work in seeking out mispriced securities Also remember the costs that you bear to becoming an expert in a market and then the time you spend analyzing the prices of securities that trade in an inefficient market Therefore adjust your abnormal returns downward to re ect your opportunity costs ie what you could earn with your best alternative use of time If you want to earn quotfair returnsquot returns that re ect the risk and lie on the SML the best strategy is to trade in relatively efficient and liquid markets using a longterm buyandhold strategy If you recall the Ibbotson data this strategy has paid off in longterm performance where risk has been rewarded CORPORATE FINANCE AN INTRODUCTORY COURSE DISCUSSION NOTES MODULE 41 INVESTMENT IN A MULTIPERIOD WORLD I The Multiperiod World We now drop the oneperiod world assumption ie projects can have cash ows that extend beyond t0 and t 1 We will also acknowledge that a project s future cash ows are generally not known with certainty ie they usually are risky However we will leave the formal analysis of risk to a later lecture For now I will simply claim that for risky cash flows the appropriate approach is to use the expected cash flow in place of the certain value and to adjust the discount rate to account for risk I will also simply give you the rate of return required appropriate discount rate for a project risky or completely certain The rate for risky projects will exceed the market rate r on riskless securities e g a government T Bill Rational investors require a higher rate of return for bearing the risk of a risky project relative to a risk free investment Later you will learn how to estimate risk adjusted required rate of returns for risky projects One thing that you will want to do with the examples here is to use them to become familiar with doing these kinds of problems on a calculator or better yet a spreadsheet One way to do this is to con rm my calculations on all the examples in this note It is also usually helpful to draw pictures of the cash ow streams that are being discussed The diagrams introduced in the text are useful devices for visualizing the ows What happens to the PV of a typical project s future cash ows as the discount rate increases By typical we mean that a project s cash out ows precede the proj ect s in ows What happens to the project s NPV A project is penalized with progressively higher discount rates that re ect increasing levels of risk Is should be intuitive that NPV decreases as the discount rate increases for the typical investment project the project is compared to a higher hurdle A The Time Exponent PVC1 Cl1 r1 Note the exponent 1 We haven t been bothering to include this exponent since 1 r l r However in a multiperiod world we must use exponents 1e we must consider t 2 t 3 t N 1 This module is designed to complement Chapter 4 in Ross Wester eld and Jaffe l In general for a multiperiod world PVC1 Ctl r PVC1 is the present value value at t 0 of cash ow Ct that is actually received at the end of year t An equivalent way to write the above equation is PVC1 Ct 11 rt Discounting Example What is 3426 received in 16 years worth today if the market interest rate r is 8 PVC15 3426 110816 or PVC16 342610816 1000 Notation 11 r discount factor Before our quotsmartquot calculators were affordable or even available at any price we used tables to find discount factors See Table A1 in Appendix A of RWJ For 8 and 16 years the discount factor is 02919 This amount is the present value of 1 in 16 years at 8 ie 29 Since we receive 3426 at t 16 it current or present value is 342602919 1000 Compounding Example What is 1000 worth in 16 years at 8 FV15C0 C01 rt The subscript to FV of 16 reminds us the future referred to is t 16 I will often omit this subscript if the meaning is clear FV15C0 100010816 3426 Does this answer surprise you given the previous example It shouldn t I r compoundfactor See Table A3 in Appendix A of RWJ For 8 and 16 years the compound factor is 34259 This is the future value of 1 in 16 years at 8 Since we have 1000 att 0 the future amount FVC0 100034259 3426 Note that the discount factors 1 1 r1 are just the reciprocals of the compound factors 1 r Simple Interest Versus Compound Interest What is the simple interest on 100 at 12 over 50 years The interest is 12 per year for 50 years or 600 With simple interest you do not earn interest on interest It is as if you withdrew the interest you earn each year and kept it in a cookie jar At the end of 50 years you would have your 100 principle plus 600 in interest paid at the end of each year for a total of 700 Contrast this sum to the balance if interest is compounded annually 10011250 2890022 Of this amount 100 is principal and 2880022 is compound interest This is the interest on principal as with simple interest the interest on interest and the interest on the interest from the interest Needless to say there is quite a difference from simple interest if the time horizon is very long Multiperiod Cash Flows If cash ows are over several time periods and the cash ows occur at the m of each period the present values can be calculated individually and the total PV is just the sum of the individual PVs Why is it that after we discount them and only after can we just add the discounted values together Be w you can answer this question We label this value additivity PV Cl1 r1 C21 02 C31 o3 CN1 0N where N is the last period in which there is cash ow The shorthand way of writing this equation is PV 2 CL 1 rt Be able to discuss this expression and de ne all the terms t 1 Recall that NPV PVin ows PVout ows PV means that all cash ows are expressed at their t 0 equivalence Example Say r 10 A project costs 200 at t 0 and requires another outlay of 100 at t 4 The project will bring in 100 att 1 200 at t 2 75 at t 3 and 300 at t 5 NPV 200 1001101 2001102 751103 1001104 3001105 NPV 200 91 165 56 68 186 NPV 230 Since the NPV is positive this project expands our opportunities creates value and so should be accepted Do you understand what the 230 means to the shareholders of a rm evaluating this project II Perpetuities Consider an investment that provides cash in ows that never cease ie they continue into in nity Further each periodic cash ow is of equal size In other words C1 C2 C3 CN where N equals in nity Does an investment like this sound preposterous In fact it shouldn39t Real world examples include preferred stock dividends or the interest payments from a quotConsolquot bond a bond that promises the owner a set interest payment forever yet no return of principal Neither security has a maturity date the payment each period to the owner of the security is identical How would you nd the PV of the perpetual cash in ows Note that your calculator does not have an in nity key Therefore nding the PV of a perpetuity sounds like a veg tough or very long job Fortunately with a little Algebra it is not really all that dif cult 1 PV C1r1 C1 02 C1 o3 C1 rN39 Multiply both sides of equation 1 by 1 r1 2 PV1 01 c C1 r1 C1 02 C1 rN391 Subtract equation 1 from equation 2 PV1 r1 PV C C1 0N Recall that for a perpetuity N increases to in nity Under this condition what is the value of the last term Zero Why Expanding the equation we have PVPVrPVC PV1r1C PVr C PV Cr Example What would you pay to receive 100 at the end of each year forever ifr 10 PV 100010 1000 Think about it ifyou invested 1000 at 10 you could draw off 100 per year and never touch your principal of 1000 Therefore it makes sense that the value of this investment is 1000 With a 1000 investment you can exactly quotconstructquot your own 100 perpetuity Thus this must be the value of the perpetuity III Annuities An annuity is an equal sum of money received at regular intervals an annuity is a perpetuity that has a nite life it matures as opposed to an in nite life it never matures A quotregular annuity quot has the payments received at the end of each time period An quotannuity duequot is an annuity where the payments are received at the start of each time period Regglar Annuities From above we know that if an annuity never ended it would have a PV of PV Cr However suppose the last payment in the annuity is received at the end oft 5 quotPretendquot that you are at t 5 and just received the last payment of C What would be the PV at t 5 of the remaining payments if the annuity continued without ever ending ie it was a perpetuity PV5 Cr PV5 is the PV at t 5 of receiving C from the end of year 6 to in nity What is the PV at t 0 ofa perpetuity from year 6 to in nity Discount the PV formula for 5 years PVO Crll r5 This is the PV of the part of a perpetuity that we would not receive if we instead really had a veyear annuity Therefore the PV of the veyear annuity is the PV of a perpetuity minus the part of the perpetuity that will not be received or PV Cr Crl1r5 If we factor out the C from both terms we have C times the annuity factor PV Clr lrlr5 For any general annuity we can rewrite the above as PV Crl llrt where t is the length ofthe annuity Note This equation looks a bit different from the annuity equation in RWJ Chapter 4 I f you inspect the two equations closely you will see that they are equivalent This way of writing the formula is easier for some to remember Remember how we got here and you ll never fail Example What is the PV of an 8year annuity of 150 per year to be received at the m of each year The interest rate is 12 Using the above equation we have PV 150012111128 74515 See Table A2 in RWJ This table gives the PV of 1 to be received for various times and interest rates For the above problem the table factor is 49676 Again this amount is the PV of 1 per year for 8 years at 12 The above annuity is for 150 per year Therefore multiply the table factor by 150 for the answer or 1504 9676 74514 We have a 001 rounding error relative to the table Alternatively we could deposit 74515 at 12 and draw out 150 at the end of each year for eight years If you did this you would have a zero balance in our account at the end of the eight years You should confirm for yourself that this statement is true Annuity Due In an annuity due the cash ows are received at the start of each period not at the m of each period for a regular annuity How would you find the PV of an annuity due Example What is the PV of an annuity due for of 200 for five years at 9 PV 200 plus the PV of a four year regular annuity Do you see why PV 200 64794 84794 Displaced Annuities A displaced annuity is an annuity that begins with a delay Say 100 per year is expected from the end of t 4 through t 7 or for four years The interest rate is 8 The PV of this annuity is 7 PV z 1001 r t4 Think of three ways to solve this problem 1 Bring back each year individually or PV 7350 6806 6302 5835 26293 This procedure while correct represents too much work 2 Find the PV of a sevenyear annuity and subtract the PV of a threeyear annuity or the part of the sevenyear annuity not received 52064 25771 26293 3 At the end of t 3 nd the PV of a fouryear annuity and bring this amount back to time 0 or 3312111083 26293 33121 is the PV ofa fouryear annuity at the end of t 3 The next term the discount factor brings this amount back to t 0 Take your pick of the three ways Be advised however that the rst way can become very tedious Imagine nding the PV of a displaced 20year annuity IV Growing Perpetuities Recall that the PV of a perpetuity is PV Cr However what if the cash ows are growing at a constant rate g Say at the end of the rst year you receive C1 Then at the end of the second year you will receive C11 g1 at the end ofthe third year you ll receive C11 g2 and so on Make sure you understand why we cannot use Cr to value this growing perpetuity The basic equation is N PV ZC11 gt391l r1 Cl1 r1C11 g1l 02 C11 gN391l 0N t1 If you multiply through this equation by 1 rl g subtract the above equation from the resulting equation and let N approach in nity the same approach we used before you get PV Clr g Again this is the equation for a growing perpetuity Also note that C1 is the first cash ow received ie at the w of the first year Think of this as we did before If you put PV in the bank now and want to fund an infinite stream of constantly growing payments how big does PV have to be The interest generated on the investment is PVr We want the principal to grow at rate g to ensure the payments can as well Thus from the interest generated the first year we subtract PVg and payout this amount If we do this the principal we carry forward will be PV1g and we can do it all over again next year PVr 7 PVg C1 determines the exact size of the initial principal balance required to start the payment stream at C1 and let it grow at rate g Inspect the above equation Use of this equation requires 1 r gt g Why 2 C1 grows at the rate g forever Example You expect a 300 dividend at the m of this year You expect this dividend to grow at 6 forever The discount rate is 15 How much would you pay for a share of common stock with these characteristics PV Clr g 300015 006 3333 You have just had your quot rst taste quot of valuing a nancial security V Growing Annuities at a Constant Rate g Take the above example but assume you receive the growing cash ow for only 15 years not forever How would you value this cash ow PV clr g 011 g rg11r PV C1r g1 1 t g1 rDi PV 300015 006110611515 2352 Contrast this PV to the PV for the growing perpetuity or 3333 Since the growing annuity has a shorter life than the growing perpetuity the value is less However note that even though the perpetuity is forever about 23rds of the total value comes in during the first 15 years VI Growing Perpetuities or Annuities at a Non Constant Rate You want to nd the value of an investment opportunity The investment just paid 300 The required rate of return on the investment is 11 You expect the cash ows from the investment to grow at the following rates g 15 for years t 1 through t 3 g 10 for three years t 4 through t 6 g 5 thereafter from t 7 until in nity It is very helpful for this example to draw the cash ow growth diagram 3 6 PV Z 300115t111t Z 3001153110t393111t i 1 i 4 300115311031051011 00511116 PV 3451111 3971112 4561113 5021114 5521115 6071116 638011 00511116 7634 This amount is the maximum that you could pay for the investment to realize an 11 rate of return At 11 this project would have a zero NPV if you paid 7634 If you could acquire the investment for less than 7634 your return would be higher than 11 ie the project would have a positive NPV If the last part of this example held a growing annuity instead of a growing perpetuity you would just substitute the growing annuity formula for the growing perpetuity formula in the example VII More Examples Growth Rate Problem At the end of 1989 a firm had an EPS 150 At the end of2001 the EPS 650 What is the compound annual growth rate of EPS Note 12 years of growth occur FV PV1 g1 650 1501 g Solve for g 1300 Doubling Your Money How long would it take you to double your money at 9 FV PV1 g1 200 100109 Note doubling you money just requires that FV is twice as large as PV You could use 40 and 20 50 and 25 etc for FV and PV and you will get the same answer Solve for t t 804 years A Useful quotRule of ThumbquotThe Rule of 72 Divide 72 by the whole percentage rate in the above problem or 9 729 800 years The quotRule of 72quot is a quotroughquot way to estimate the amount oftime it takes to double your money 72 is a constant You supply the interest rate as a quotwhole percentage not a decimal percentage VIII Putting It All Together A Comprehensive Problem You plan to pay your niece39s tuition through four years of college In today39s dollars tuition is 8000 per year at the university she wishes to attend She will begin college at the beginning of year 8 which is equal to the m of year 7 Tuition is due at the start of each year You expect tuition to increase with the rate of in ation You estimate future in ation at 5 per year The interest rate is 10 How much do you need to save at the end of each year years 1 through 7 in order to provide her with exactly enough money to pay her tuition Draw a cash ow time line it will help The beginning of year 8 when she starts college and makes her first tuition payment is the same as the end of year 7 The future value and present value of each tuition payment is as follows End of Year FV of Tuition PV of Tuition 10 80001057 11257 5777 8 80001058 11820 5514 9 80001059 12411 5263 10 800010510 13013 5 024 21579 In other words if you had 21579 invested today t 0 at 10 you would have just exactly enough to make the four tuition payments After the last payment your account would have a 10 zero balance If you save the same amount each year how much must you save at the end of each year for the next 7 years to have the necessary amount The present value of the savings contributions must just equal the 21579 found above 7 PV 21579 Z Cl10t where C is the annual endof year savings for 7 years t l C is the annual endof year deposit C 443245 IX Compounding Intervals NOTE Interest rates are stated on an annual basis unless speci cally noted The stated annual interest rate often called the quotnominalquot interest rate is the annual rate without compounding The quote ectivequot interest rate is the annual interest rate that includes the e ects of intra year within the year compounding The effective interest rate is the actual rate of growth in a year opening balance To this point in our development of nancial mathematics we have assumed annual cash ows and annual interest rates FVL PV1 rt where r is the stated annual rate and t is the number of annual periods or years Now let s let compounding occur quotmquot times per year Fvt PV1 rm Note that rm corresponds to the quotperiodic interest ratequot and mt is the total number of periods The rate and number of periods must be in agreement If the rate is quarterly the number of periods must be four Common values for m include 2 4 12 54 365 and in nitely small A review of the quotold daysquot Life under Regulation Q Regulation Q limited the interest banks and SampL39s could pay on passbook savings accounts The rate was 5l4 for banks and 5l2 for SampL s However while the FED limited the stated or nominal annual rate they neglected to specify the compounding interval ll 1st Banker The first banker was an unimaginative individual This banker compounded interest annually Accordingly after one year a 1 deposit would grow to FV1 1001 00525111 10525 Therefore the investors wealth in the first bank grows by 525 In this case the effective interest rate equals the stated or nominal rate 2nd Banker A more perceptive second banker recognized that the compounding interval affected the e ective interest rate and decided to compound semiannually Accordingly a 1 deposit in the second bank in one year would grow to FV1 1001 00525221 10532 Therefore the investors wealth in the second bank grows by 532 the e ective rate of 532 exceeds the stated rate of 525 The second banker takes out an ad in the local newspaper proudly announcing that his bank has a higher e ective interest rate than the first The compounding war has begun 3rd Banker The third banker saw the second banker39s ad in newspaper She thinks quotI can top thatquot She decides to compound monthly Therefore after one year a 1 deposit in the third bank will grow to FV1 1001 0052512 1 10538 Since the investors39 wealth grows from 100 to 10538 the effective rate is 538 Needless to say this rate is advertised in the local paper Savers ock in to make their deposits 4th Banker Finally a clever fourth banker who has studied Module 4 of these discussion notes decides to go all out This banker understands continuous compounding of course The ad for the fourth bank reads quotWE PAY THE HIGHEST INTEREST RATEALLOWED BYLA W NOBODY CAN TOP US With continuous compounding a 1 deposit grows in one year to Fv1 1001 00525In nity1quot quotity 100ert 3910030525quot1 105390256 The effective rate is 539 The other three banks lose all of their depositors go bankrupt and end up working for the fourth banker Values of ert are tabulated in A5 of RWJ Values of Me or e39 are listed in A6 Table A5 is use to find continuously compounded future values A6 is used to find continuously discounted 12 present values Examples If the stated annual interest rate is 85 what is the effective rate if compounding is continuous ert e0081 10887 The effective rate is therefore 887 What is the future value of 10000 today at the end of 10 years if the stated annual interest rate is 85 and compounding is done annually 10000108510 2260983 What is the future value of 10000 today at the end of 10 years if the stated annual interest rate is 85 and compounding is done continuously 10000e 1000060085quot10 2339647 What is the present value of 5000 to be received at the end of 7 years if the stated interest rate is 8 but discounting is done continuously PV Fv7ert 5000603956 5000175067 285605 What is the present value of 5000 to be received at the end of 7 years if the stated interest rate is 8 but discounting is done quarterly PV FV71 r447 500010228 5000174102 287188 13 CHAPTER 4 DISCOUNTED CASH FLOW VALUATION Solutions to Questions and Problems NOTE All end of chapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answerfor each problem isfound without rounding during any step in the problem 1 The simple interest per year is 5000 X 07 350 So after 10 years you will have 350 X 10 3500 in interest The total balance will be 5000 3500 8500 With compound interest we use the future value formula FV PV1 r FV 5000107l 983576 The difference is 983576 7 8500 133576 To nd the FV ofa lump sum we use FV PV1 r a FV 10001051 162889 b FV 100010710 196715 c FV 10001052 265330 d Because interest compounds on the interest already earned the future value in part c is more than twice the future value in part a With compound interest future values grow exponentially To nd the PV ofa lump sum we use PVFV1r PV 15451 1056 1152977 Pv515571119 2015491 Pv 886073 11618 6126687 Pv 550164 119 1006728 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 L 1 Fv 307 2651 r2 r 307 265 2 71 763 Fv 896 3601 r9 r 896 360 71 1066 FV 162181 390001 r r 162181 39000 15 71 997 Fv 483500 465231 r30 r 483500 46523 0 71 812 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 fort we get t 1nFV PV ln1 r FV 1284 62510839 t ln1284 625 In 108 936 yrs FV 4341 81010739 t ln4341 810 In 107 2481 yrs FV 402662 18400121 t ln402662 18400 1n 121 1619 yrs FV 173439 21500129 t ln173439 21500 1n 129 820 yrs 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 fort we get t 1nFV PV ln1 r The length of time to double your money is Fv 2 1107t t In 2 In 107 1024 years The length of time to quadruple your money is O F FV 4 1107 t In 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 nd the PV of a lump sum we use PvFV1r Pv 800000000 109520 13025895912 To nd the future value with continuous compounding we use the equation Fv PVeRt a Fv 1000e12 5 182212 b Fv 10009103 134986 0 EV 1000e05 10 164872 d Fv 10009W8 175067 Here we are given the EAR and need to nd the APR Using the equation for discrete compounding EAR 1 APR m39quot 71 We can now solve for the APR Doing so we get APR ml EAR 71 EAR 081 1 APR 22 41 APR 21081 2 41 794 EAR 076 1 APR 1212 71 APR 121076 12 71 735 EAR 168 1 APR525271 APR 52116815271 1555 Solving the continuous compounding EAR equation EAR eq 7 1 We get APR ln1 EAR APR ln1 262 APR 2327 17 N O N 03 For discrete compounding to nd the EAR we use the equation EAR 1 APR m39quot 71 So for each bank the EAR is First National EAR 1 122 1212 71 1291 First United EAR 1 124 22 71 1278 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 Here we are trying to nd the interest rate when we know the PV and FV Using the FV equation FV PV1 r 4 31 r r 43 71 3333 per week The interest rate is 3333 per week To nd the APR we multiply this rate by the number ofweeks in a year so APR 523333 173333 And using the equation to nd the EAR EAR 1 APRm quot 71 EAR 1 333352 71 31391651569 Intermediate We need to nd the annuity payment in retirement Our retirement saVings ends at the same time the retirement withdrawals begin so the PV of the retirement withdrawals will be the FV of the retirement saVings So we nd the FV of the stock account and the FV of the bond account and add the two FVs Stock account FVA 7001111236 711112 196316382 Bond account FVA 3001 071236 710712 36599130 So the total amount saved at retirement is 196316382 36599130 232915511 SolVing for the withdrawal amount in retirement using the PVA equation gives us PVA 232915511 C171109123 0912 C 232915511 1191616 1954619 withdrawal per month 24 N 0 Since we are looking to triple 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 00 161 r3 We need to nd the present value of an annuity Using the PVA equation and the 15 percent interest rate we get PVA C1711 rt r PVA 5001 7 11 1515 15 PVA 292369 This is the value of the annuity in Year 5 one pe1iod before the rst payment Finding the value of this amount today we nd Pv FVl r Pv 2923691 125 PV 165898 The amount borrowed is the value of the home times one minus the down payment or Amount borrowed 400000l 7 20 Amount borrowed 320000 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 320000 Cl 7 1 1 0812360 0812 C 234805 Now at time 8 we need to nd the PV of the payments which have not been made The balloon payment will be PVA 2348051 7 1 1 0812 0812 PVA 29125663 The company would be indifferent at the interest rate that makes the present value of the cash ows equal to the cost today Since the cash ows are a perpetuity we can use the PV of a perpetuity equation Doing so we nd PV C r 240000 21000 r r 21000 240000 r 0875 or 875 33 The company will accept the project if the present value of the increased cash ows is greater than the cost The cash ows are a growing perpetuity so the present value is PV C 1reg Mr 137 X 1 g1 01 PV 7 120001117067111706 X1061115 PV 7 4939878 The company should not accept the project since the cost is greater than the increased cash ows 35 The relationship between the PVA and the interest rate is PVA falls as r increases and PVA rises as r decreases FVA rises as r increases and FVA falls as r decreases The present values of 5000 per year for 10 years at the various interest rates given are PVA10 50001 7 11101 10 3072284 PVA5 50001 7 11051 05 3860867 PVA15 50001 7 11151 15 2509384 36 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 12511012 711012 Solving for t we get 1008332 1 200001012125 t In 233333 ln 100833 10210 payments 4 O To solve this problem we simply need to nd the PV of each lump sum and add them together It is important to note that the rst cash ow of 1 million occurs today so we do not need to discount that cash ow The PV of the lottery winnings is 1000000 1400000110 18000001102 22000001103 26000001104 7 30000001105 34000001106 38000001107 42000001108 7 46000001109 500000011010 1875893079 4 03 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 17 payments so the PV of the annuity is PVA 20001 7 111217 12 1423926 Since this is an ordinary annuity equation this is the PV one period before the rst payment so it is the PV at t 8 To nd the value today we nd the PV of this lump sum The value today is 48 UI O PV 1423926 1128 575100 The cash ows in this problem are semiannual so we need the effective semiannual rate interest rate given is the APR so the monthly interest rate is Monthly rate 12 12 01 To get the semiannual interest rate we can use the EAR equation but instead of using 12 months as the exponent we will use 6 months The effective semiannual rate is Semiannual rate 7 1016 71 7 615 We can now use this rate to nd the PV of the annuity The PV of the annuity is PVA t 9 60001 7 1 106151 0615 4384421 Note that this is the value one period siX months before the rst payment so it is the value at t 9 So the value at the various times the questions asked for uses this value 9 years from now PV t 5 4384421106158 2719483 Note that you can also calculate this present value as well as the remaining present values using the number of years To do this you need the EAR The EAR is EAR 7 1 0112 711268 So we can nd the PV at t 5 using the following method as well PV t 5 4384421 112684 2719483 The value of the annuity at the other times in the problem is PV t 3 43844211061512 2141772 PV t 3 4384421112686 2141772 PV t 0 43844211061518 1496938 PV t 0 4384421112689 1496938 We need to use the PVA due equation that is PVAd e 1 r PVA Using this equation PVAd e 56000 1 081512 X C1711081512quot8081512 5562223 C171108151248081512 C 136182 p A N Notice that 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 Challenge The monthly interest rate is the annual interest rate divided by 12 or Monthly interest rate 12 12 Monthly interest rate 01 Now we can set the present value of the lease payments equal to the cost of the equipment or 4000 The lease payments are in the form of an annuity due so PVAdue 1 r Cl 7 11 r2 r 4000 1 01 Cl 7 11 01 01 C 18643 First we will calculate the present value if the college eXpenses for each child The eXpenses are an annuity so the present value of the college eXpenses is PVA C1711 rt r PVA 230001711 065quot 065 PVA 7879337 This is the cost of each child s college eXpenses one year before they enter college So the cost of the oldest child s college eXpenses today will be Pv FVl r Pv 78793371 065 PV 3262835 And the cost of the youngest child s college eXpenses today will be Pv FVl r Pv 78793371 065 PV 2876709 Therefore the total cost today of your children s college eXpenses is Cost today 3262835 2876709 Cost today 6139544 This is the present value of your annual savings which are an annuity So the amount you must save each year will be PVA C1711 r r 6139544 Clt171106515 065 C 652958 CHAPTER 10 RISK AND RETURN THE CAPITAL ASSETPRICING MODEL CAPM Solutions to Questions and Problems NOTE All end of chapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answerfor each problem isfound without rounding during any step in the problem Basic 1 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 7040 11022 5220 The portfolio weight for each stock is WeightA 70405220 5364 WeightB 110225220 4636 3 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 5011 3017 2014 1340 or 1340 9 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 is Boom ERp 07 15 333 1833 or 1833 Bust ERp 13 03 7063 0333 or 333 To nd the eXpected return of the portfolio we multiply the return in each state of the economy by the probability of that state occurring and then sum Doing this we nd ERp 701833 300333 1383 or 1383 This portfolio does not have an equal weight in each asset We still need to nd the return of the portfolio in each state of the economy To do this we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy Doing so we get Boom ERp2007 2015 6033 72420 or 2420 Bust ERp 72013 2003 60706 7 70040 or 7040 And the expected return of the portfolio is ERp 702420 307004 1682 or 1682 To calculate the standard deviation we rst need to calculate the variance To nd the variance we nd the squared deviations from the eXpected return We then multiply each possible squared deviation by its probability and then add all of these up The result is the variance So the variance and standard deviation the portfolio is of 7 702420 7 16822 3070040 7 16822 7 012708 c 7 012708 2 7 1127 or 1127 This portfolio does not have an equal weight in each asset We rst need to nd the return of the portfolio in each state of the economy To do this we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy Doing so we get Boom ERp 7 303 4045 3033 7 3690 or 3690 Good ERp 7 3012 4010 3015 7 1210 or 1210 Poor ERp 7 3001 40i15 30705 7 70720 or 7720 Bust ERp 7 30706 40730 30709 7 71650 or 71650 And the eXpected return of the portfolio is ERp 303690 401210 2570720 0571650 1329 or 1329 To calculate the standard deviation we rst need to calculate the variance To nd the variance we nd the squared deviations from the eXpected return We then multiply each possible squared deviation by its probability and then add all of these up The result is the variance So the variance and standard deviation the portfolio is of 303690 7 13292 401210 7 13292 25 70720 7 13292 0571650 7 13292 of 03171 c 7 03171 2 7 1781 or 1781 12 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 GK 0 3p 10 130131913Bx Solving for the beta of Stock X we get 3X7110 Here we need to nd the riskfree rate using the CAPM Substituting the values given and solving for the riskfree rate we nd ERi 17 Rf 117 Rf19 17 Rf 209 719Rf Rf 0433 or 433 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 7 055 075150 7 1675 or 1675 It is given in the problem that the eXpected return of Stock Y is 17 percent but according to the CAPM the return of the stock based on its level of risk the expected return should be 1675 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 1675 percent For Stock Z we nd ERZ 7 055 075080 7 1150 or 1150 The return given for Stock Z is 105 percent but according to the CAPM the eXpected return of the stock should be 1150 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 1150 percent We can also answer this question using the rewardtorisk ratio All assets must have the same rewardtorisk ratio that is every asset must have the same ratio of the asset risk premium to its beta This follows from the linearity of the SML in Figure 1111 The reward torisk ratio is the risk premium of the asset divided by its 3 This is also know as the Treynor ratio or Treynor indeX We are given the market risk premium and we know the 3 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 17 7 055 150 0767 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 rewardtorisk ratio For Stock Z we nd Rewardtorisk ratio Z 105 7 055 80 0625 N p A 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 rewardtorisk ratio For a portfolio that is equally invested in largecompany stocks and longterm bonds Return 124 582 91 For a portfolio that is equally invested in small stocks and Treasury bills Return 175 382 1065 a The eXpected return of the portfolio is the sum of the weight of each asset times the eXpected return of each asset so ERP WAERA wBERB ERP 4015 6025 ERp 2100 or 2100 The variance of a portfolio of two assets can be eXpressed as of wioi w of ZwAwBvoBpAYB op 402402 602652 24060406550 of 24010 N U1 0 the standard deviation is o 24010 2 4900 or 4900 The eXpected return of the portfolio is the sum of the weight of each asset times the eXpected return of each asset so ERP WAERA WBERB ERp 4015 6025 ERp 2100 or 2100 The variance of a portfolio of two assets can be eXpressed as of wioi w1236123 2wAwBvoBpAYB op 402402 602652 240604065750 of 11530 N U1 0 the standard deviation is o 11530 2 3396 or 3396 As Stock A and Stock B become less correlated or more negatively correlated the standard deviation of the portfolio decreases CORPORATE FINANCE AN INTRODUCTORY COURSE DISCUSSION NOTES MODULE 201 OPTIONS AND OPTION PRICING AN INTRODUCTION I The life story of a call option A review of vocabulary An option contract is a nancial contract giving the owner of the option the ght not the obligation to buy call option or sell put option a prespecif1ed asset the underlying asset on or before a prespecif1ed date the expiration date for a prespecif1ed price the exercise price We can clear up a lot of the usual misunderstandings of those thinking about options for the first time if we discuss a usual history of a 11 option contract Due to the introductory nature of this note I will concentrate on call options Suppose you believe that the price of Cisco Systems stock is going to rise over the next several months It is now November 27m just to provide a time frame One way to act on this belief is to simply buy Cisco s stock and enjoy the capital gain This may not be the most effective way for several reasons that will become clear The second way is to purchase a call option on Cisco stock You decide to follow this second strategy You purchase a call option contract on 100 shares the typical size of option contracts2 of Cisco stock with an exercise price of 15 that matures on April 15111 the expiration date Yesterday Cisco closed at 1445 the current stock price and your option contract cost you 170 per share for a total cost to the contract of 17000 This 170 is the current price of the option It is what you pay the seller in order to obtain the right to buy a share of Cisco stock from himher for an additional payment of 15 the exercise or strike price at anytime exchange traded options in the US are American options a European option would be exercisable only on April 15th over the next several months So just before the Thanksgiving holiday you are 170 poorer but you are now the proud owner of a very valuable right Several months have passed and it is now April 15m You are older not much wiser but in possession of some very important information that you did not have before What is that information You now know what actually happened to the price of Cisco stock Let s suppose that Cisco stock is now trading at 2255 per share whenever the stock price is above the exercise price of a call option that option is referred to as being in the money Being the owner of a call option you have a certain right implying you have a choice to make at this time The choice is to exercise the option or not What do you do Looking back at the last paragraph ownership of the option gives you the right to 1 This lecture module is designed to complement Chapter 22 in Ross Wester eld and Jaffe 2 Typical Option contracts on equity are for 100 shares Prices are quoted per share and we will usually talk in terms of an Option on one share of equity as that is usually more understandable buy one share of Cisco stock for 15 Given that the option is in the money this is a valuable right to own More simply here is something that everyone else has to pay 2255 for and you can pay 15 for it That is something good You should exercise the option and pay the seller 15 and get a share of Cisco stock You don t even have to want to own Cisco stock for this to be a good thing Remember the stock market is a market so if you exercise the option and get the stock you can immediately sell it for 2255 Buy low 15 and sell high 2255 the only really well known tenet in nance At this point in time your ownership of the option is then worth 755 This 755 is your payoff at expiration but is of course not your total return Remember you paid 170 for the option back in November Note however the decision to exercise is made independently of the price you paid to purchase the call Any time the call option is in the money to any extent ignoring transactions costs you exercise the option Suppose instead that Cisco stock is trading at 12 per share out of the money What do you do Exercising the option seems like a silly thing to do You would immediately lose 3 if you did Are you stuck with the loss No remember that ownership of the call option conveys a right to purchase not an obligation to purchase3 If the option is out of the money you simply wad it up in disgust and throw it away giving you a payoff at expiration of zero If Cisco were trading at 15 per share or at the money it would be just worth zero If the call is out of the money you would choose not to exercise there is a cheaper way to buy the underlying asset than by exercising the call option There are lots of prices running around in that story and lots of dates but it is really quite simple when you get used to it II Call Options American call options are the most common types of options An American call option gives the owner the right to purchase the underlying asset for a xed price on or before the expiration date of the contract To think about option contracts it helps to work backwards Let s think about the value of a call option at expiration of the contract Continuing with the above example we saw that if Cisco was trading at 2255 per share we will often denote the stock price as S as shorthand on April 15111 options commonly expire at the end of the month I was fixated on this date for some reason the value of the call at expiration was 755 This is just the difference between the prevailing stock price S and the exercise price E of the call provided that the option is in the money at the expiration date T remember it is a right not an obligation If instead the stock price were 35 per share at the expiration of the call its value would be S 7 E 35 7 15 20 As long as the call expires in the money its value at expiration increases dollar for dollar with increases in the price of the underlying stock 3 The only way this right or option can have value is however if the seller of the option contract the short side of the option has an obligation to do what the owner the long side of the contract desires Thus if the option ends in the money and the owner decides to exercise the call the seller is obligated to sell the underlying asset for the exercise price If the option expires out of the money the value of the option at expiration is zero This is true not matter what the price of the underlying stock is as long as it is out of the money Thus whether Cisco is trading for 1499 or it is trading for 499 on April 15111 the value of the call at expiration is zero This gives the famous hockey stick diagram for the value of a call option at expiration as a function of the stock price Remember a call option is a derivative security This means it derives its value solely from the value of the underlying asset Thus at expiration the only thing determining the value of the call is the price of the underlying stock compared to the exercise price Value of call C at expiration in 450 Exercise Stock Price S at Price E expiration in Symbolically we can write the value of the call CT4 at exercise as a function of the stock price S and the exercise price E as CT maxS 7 E 0 This figure shows us that the call option has limited liability once you purchase it you can never face a situation where you will lose further value The worst you do at expiration is zero and the value of the call increases dollar for dollar with the stock price when the stock price is above the exercise price of the option 111 Put Options An American put option gives the owner the right to sell the underlying asset for a fixed price on or before the expiration date of the put Again let s describe the value of a put option at expiration 4 We add the subscript T to highlight that it is the value of the call at the expiration date and not prior to the expiration date of the option Suppose instead of a call option in our Cisco example we were talking about a put option with the same parameters ie the same expiration date and exercise price If you own the put option you have the right to sell a share of Cisco for 15 on or before the expiration date The value of this right at expiration is very different from a call s value Suppose that Cisco s stock price is indeed 35 on April 15m What is the put worth What is the right to sell for 15 something that can be sold or bought for 35 Zero This is a put option that is out of the money For a put option at expiration when the stock price is greater than the strike price the put is out of the money and has no value When the stock price is less than the exercise price the put is in the money and has value Suppose Cisco s stock price is instead 5 on April 15m Now the put allows you to sell a share of Cisco stock for 15 instead of the 5 market price This is a 10 value Do you need to own Cisco stock to take advantage of this value No buy a share for 5 and exercise the put selling the share for 15 realizing a gain of 10 on the transaction Buy low and sell high The lower is the stock price the higher is the value of the put at expiration The value of the put at expiration P can be written P maxE 7 S 0 If the exercise price is higher than the stock price E 7 S is positive and higher than 0 so the value of the put is the higher value E 7 S If the exercise price is lower than the stock price E 7 S is negative and 0 is the higher of the two values and thus the value of the put Graphically Value of put P at expiration in Exercise Price E 450 Exercise Stock Price S at Price E expiration in The put option has positive value at expiration on exactly that set of stock prices for which the call option has no value at expiration This is why they are so powerful in combination Note that ownership of the stock is just a 450 line from the origin IV The Short Side It is important to recall that the seller short side of the option contract incurs an obligation If the owner of the option wishes to exercise it the seller is obligated to comply The seller s value at expiration is then just the negative of the owner s value We can see this most easily in pictures Value of Value of Seller s Sell a call seller s Sell a put Position at Position at Expiration Expiration Share price at Exercise P 39 expiration Exercise nce Price Share price at expiration Note that the seller s value at expiration will be negative when the option is in the money This is simply because options are a side bet between the buyer and the seller of the option about what will happen to the stock price relative to the exercise price a bet where the buyer can never lose any money at expiration Thus when the buyer owner wins the seller loses V Put Call Parity A very useful skill to have is to be able to evaluate the payoff to portfolios of options and other instruments and to use these payoffs to evaluate current prices The text has a good discussion of this We will illustrate this process by deriving a very important relation called putcall parity Let s consider the value of a portfolio of two options written on the same underlying stock One option is a put option with a strike price of 20 and one year to expiration and the other is a call option with the same strike price and expiration date What is the payoff at expiration of a portfolio that is long the call and short the put What is the portfolio payoff as a function of the stock price at expiration It centers a round the exercise price When the stock price is below the exercise price the call has no value but the put has value Since we are short the put we own the negative of this value think of the first part of the sell a put diagram The portfolio value looks just like a short position in a put when the stock price is below 20 When the stock price is above the 20 exercise price the put has no value so our short position doesn t cost us anything However when the stock price is above 20 the call we own is valuable Our portfolio for stock prices above 20 looks like a long position in a call Portfolio long C Long a call P Port a call and 511011 a put 20 S 20 S 20 S Note that this looks suspiciously like but for one thing the description I gave of an ownership position in the stock itself Consider a different portfolio one that is long the stock and is short a risk free bond with face value 20 and one year to maturity This short position is the same as borrowing at the risk free rate so that in one year you are required to pay back 20 Note that the two components of this portfolio can be thought to depend only on what happens to the price of the stock The stock itself clearly has a value in one year dependent on what happens to the stock price The bond is risk free so doesn t change no matter what and is trivially a function of the stock price in one year Draw the picture Practice drawing others Portfolio long S Long the stock E Short a bond Port the stock and short a bond I 20 S S 20 S 20 The short position in the bond just serves to shift the stock position down by 20 at each point See any similarity between the payouts of the two portfolios at the end of the year Because the final payouts on the two portfolios will be the same no matter what happens their current prices must be the same We used C P and S for the current prices of the call put and the stock respectively The current value of the bond that pays off the exercise price of the put and the call in one year is just E 1r Our pictures tell us that the following must be true C 7 P S 7 E1r This is the absence of arbitrage This is more commonly written S P C E 1r or the current price of the stock plus the price of a put must equal the price of a call plus the discounted value of the common exercise price for the put and the call VI Reading the Option Quotes This is a copy of a small section of the WSJ options quotes pages from 112702 Let s see what we can learn from it 39 ycALLQEPuIL omowsmxa EXP VOL VQL39 LAST Besthy 2250 Dec 1044 410 712 085 2562 2250 Jan 1868 490 120 135 2562 25 Dec 6789 230 1127 165 2562 25 Jan 33232 285 225 39 2562 2750 Dec 6197 120 911 285 2562 2750 Jan 5874 170 75 3 2562 30 Dec 1481 050 The first thing to notice is that we are focusing on the traded options for BestBuy Best buy has 7 calls and 7 puts trading on the market the most active is Chicago Reading from left to right we see that the underlying stock is BestBuy and BestBuy closed yesterday 112602 at 2562 The first line denotes the options with the lowest strike price here 2250 and the earliest expiration here the end of December The call saw 1044 contracts each for 100 shares trade and the closing price on the call was 410 The put was more thinly traded at 712 contracts and closed at 085 The dots for the last put indicate the put with a strike price of 30 expiring at the end of December did not trade on 112602 There is actually a lot illustrated here but 1 will refer back to much of it later Note now that for the December expirations the call options decline in value and the put options increase in value as the strike price rises Make sense Also the January contracts are always more valuable than the December contracts Why See below VII Pricing Call Options We will approach this in stages in an attempt to keep consistent with the textbook Bounding the call value Think about an American call option how low can its current price be and still be rational It turns out that at any point from the initiation of the contract till expiration the call s price cannot fall below the value of the call if it were immediately exercised For example if the call is in the money this value is the difference between the current stock price and the strike price If the call is out of the money its exercised value would be zero hence the quotes around the term Why do 1 think this is a lower bound on the price ofthe call Suppose the stock price is 30 and the exercise price is 20 with 2 months before expiration of the call Why must the price of the call at that time be more than 10 Suppose its price was 8 Then do the following buy the call immediately exercise it and sell the stock This strategy costs you 8 and gets you 10 for an arbitrage pro t of 2 Note also that the price of the call can never be less than zero This must be true because it can never cost you anything and it might provide some value at expiration An easily established upper bound on the price of the call is the price of the stock Remember the call represents a way to purchase the stock for the exercise price If the price of the call were higher than the price of the stock it would represent a truly expensive to purchase the stock As a function of the current stock price the current price of the call can be represented by a curve that falls within these bounds Current price of the call Upper bound stock price all price as a function of the stock price Lower bound stock price minus E Current price of the stock What we can notice is that as the stock price gets very high the current price of the call approaches the stock price minus the exercise price We can also notice that the slope of the call price function how much the call price changes for small changes in the stock price is greater the higher is the stock price and that this slope is higher in the positive direction than the negative direction These points are important when considering executive stock options for understanding the incentive effects of option compensation This is however a bit imprecise when we are thinking about trading options Let s try and do better Factors determining the price of an American call There are 5 basic factors that in uence the price of an American call option The factors and their in uences on the call price C are listed below 0 The exercise price E all else equal the higher the exercise price the lower is the price of the call This is because the exercise price is what you have to pay for the stock when you exercise the option Call value at expiration is S 7 E This is lower the higher is E when the call is in the money and with a higher E it is less likely the call will be in the money o The expiration date T all else equal the longer the time to expiration the more valuable is the call Think of it this way If I am choosing between a 6 month American call and an 8 month American call with the same exercise price I would always choose the 8 month call Why Worst comes to worst I can treat the 8 month call just as I would the 6 month call and ignore the extra 2 months Thus the price of the 8 month call must be at least as high as the 6 month call However given that this is an option and there is no obligation on the owner s part the extra 2 months have positive value The stock price S all else equal because the call is an instrument that allows you to buy the stock the higher the stock price the more valuable is this instrument look at the picture just above This just says that the curve is always going up as we go to the right The risk free rate rf all else equal the higher is the risk free rate the more valuable is the call This is true because the call lets you buy the stock for a xed price Thus part of the value of the call is the fact that you payout a xed amount E sometime in the future to get the stock In determining the current price of the call one component must be the present value of this payout The higher is the discount rate the lower is the present value of the payout and so the higher is the value of the call The volatility of the price of the underlying stock 622 This is sometimes tricky intuition till you remember that the call is an option and ownership of it conveys a right The riskier is that stock the higher is the probability that the stock price will be very far from what we expect Increased probability that the stock price will be very high clearly improves the value of the call What about the associated increased probability that the stock price will be very very low It s an option If the stock price is below the exercise price of the option the option has zero value and it doesn t matter how low it goes the option value is still zero One way to think of this that is often helpful is that the call option benefits from the upside risk and is protected from the downside risk so risk is good Binomial option pricing model Lets think about the price of a one year call option written on a stock with a very simply probability distribution for what its price will be one year from now This distribution will seem silly but it makes the analysis easy is educational and can be extended to something very useful The binomial model for a stock price says that over a given period of time we ll say a year for ease of intuition the stock price can move from its current level S only either up to say S 5 or down to S v For example suppose that the current price of ABC stock is 50 Further suppose that one year from now its price will be either 70 with probability p or 20 with probability 1 p The risk free rate is 10 What we want to be able to do with this information is to find the current price of a call option that expires in one year and has a strike price of 50 the call is written at the money The way we will nd the price of the call C is to follow the same approach we used in establishing the putcall parity relationship If we can nd of portfolio of things for which we know the current price the underlying stock and borrowing and we can make the payoff on the portfolio exactly equal to the payoff on the call option regardless of what happens then we can price the call Remember since the call is a derivative security the regardless of what happens is easy to determine The only thing that will change the payoff on the call option is if the price of the underlying stock changes What is the payoff at expiration of this call option ie what do we have to replicate If the stock price rises to 70 then the value of the call is 20 70 50 If the value of the stock falls to 20 then the call expires out of the money is not exercised and its value will be 0 Thus we need a portfolio worth 20 if the stock price rises and 0 if the stock price falls Break this into easier pieces We need a portfolio that is perfectly correlated with the stock price and has a total change of 20 20 0 across the possibilities Well the stock itself is certainly perfectly correlated with itself but it has a total change from good state to bad state of 50 70 20 There is too much risk in one share of stock to match the risk of the call option But if we use a little thought and calculate what is called the delta or the hedge ratio of the call option we can see that 20 50 2 539115 of a share of stock will have the same risk as the call option and is perfectly correlated with its payoff If the stock price rises to 70 then the value of 2 5th of a share is 28 If the stock price falls to 20 then the value of 25thofa share is 8 The risk ofthis partial share of stock is indeed 20 28 8 The 25th ofa share right now costs us 20 25 50 We are on our way but we have now a portfolio that has a payoff of 28 if the stock price rises and 8 if the stock price falls not exactly what we are looking for Notice however that all is not lost Each of these payoffs is exactly 8 larger than we want So if we arrange to also have to payout 8 regardless of what happens to the stock price we will have the portfolio payoff we are looking for That s easy to do borrow money now so that the required principal and interest payment equals 8 in a year This loan is the second part of our portfolio The appropriate loan brings in 727 8l 10 now and requires us to payout 8 in a year Thus we now have a portfolio that is long 2 539115 of a share of the underlying stock and a loan that currently costs 1273 20 727 The portfolio has a payoff of 20 28 8 if the stock price rises and 0 8 8 if the stock price falls Since this portfolio exactly replicates the payoff of the call option the current cost of the portfolio must be the current price of the call or there would be a simple arbitrage Let s write down the simple formula we implicitly used to find this We formed a portfolio of a part of a share of stock and a loan The portion of a share we used was the Delta of the call option The amount of the loan was just the discounted value of what we need to repay to replicate the call payoff We can write the call price as Current price of the call Current price of the stock gtlt Delta 7 Amount Borrowed 1273 50 X 25 7 727 We can even think of the amount borrowed in a more illuminating way The 727 is the discounted value of the payout we want to make of 8 To look ahead we could also of course represent the 8 as a fraction of the exercise price of the call Thus we can write this last term in the above formula as a fraction of the discounted value of the exercise price on the call Price ofthe call Price ofthe stock gtlt Delta 7 Elrf gtlt lK Where I am just using UK as a way to denote the idea that we can represent the amount we want to payback as a fraction of the exercise price We ll see why we would want to do this in a bit The Black Scholes model Fisher Black and Myron Scholes developed this simple intuition of a replicating portfolio in a much richer model They assumed that the stock price followed a Brownian motion They then went on to show that despite the fact that this means the stock price can wander all over and imposes a normal distribution over the possible future price realizations the same simple intuition can be used The nice thing is that unlike the binomial model this is in fact a very useful way to portray the movement of the stock price over time The way to think about the Brownian motion is that we are just taking the binomial model of the stock price movement and making the period very very small At each instant in time we assume that the stock price can go up or down by a fixed amount these days a penny Over a year there are lots and lots of instants of time and there are lots and lots of combinations of first step is up next step is down next step is down Think of a decision tree spreading out at each successive instant How can we make it all work We can set up the replicating portfolio at the beginning or now and we can determine its current price but this portfolio will only perfectly replicate the call over the first instant of time After the price moves either up or down we have the wrong portfolio going forward What we have to do then is rebalance our portfolio If the first step is for the stock price to go down the call is less in the money than it was at the start This will mean that over the next instant there is less variability in the payoff of the call This lowers the delta and reduces the amount we want to borrow To accomplish the required rebalancing we can sell some stock and repay some of our loan with the proceeds Conversely if the stock price goes up the call is more in the money This raises the delta and makes us need to payback more money to replicate the call over the next instant To accomplish this from our beginning portfolio we can borrow more money and buy more stock The beauty of the Brownian motion is that each of the required rebalancings is self nancing In other words if we have to buy more stock and borrow more to replicate the call over the next instant because the stock price just went up the amount we need to borrow is just enough to purchase the added shares of stock at the new higher price to achieve a replicating portfolio for the call over the next instant The same is true if the stock price falls Thus what we have to do is to set up an initial portfolio that will replicate the call payolT over the first instant in time and then while we have to continually rebalance this portfolio to make sure we replicate the call at each instant all the successive rebalancings are self financing What this means is that the only out of pocket expense we face is the cost of the initial portfolio This portfolio along with the self financing rebalancing strategy will perfectly replicate the call at each instant in time This means that the cost of the initial portfolio must equal the price of the call How do we find the cost of this initial portfolio I ll spare you the stochastic calculus required to develop the equation but the formula is reasonably simple CSXNd1 Exe xNd2 where d1 lnSEr1202t02t d2 2 d1 lazt This is actually a relatively simple formula Remember however that its development did spawn an industry that is among the most active on the financial markets and won the Nobel prize in economics for Scholes Black had passed away so unfortunately could not share in this richly deserved honor What does it say You can find the call price C if you know the current stock price S the exercise price E the risk free rate r the time to maturity in years or fraction of years t and the variance of the stock price 62 Note that except for the variance of the stock price all these variables can be read off the call contract or out of the Wall Street Journal The variance we have to estimate from stock price data that is readily available The Nd is just notation for the cumulative distribution function for the normal distribution and we can read this off a table in the text or have a spreadsheet calculate it for us Also look at the first line of the formula It says the current call price is just equal to the current stock price times some number Ndl is the delta in the BlackScholes formulation minus the discounted value e39rt is used instead of l1r of the exercise price times some other fraction Ndz is just what I had used UK to indicate Thus it is exactly the same as the answer from the binomial model This is not surprising given our discussion of the BlackScholes model being just an extension of the binomial model but it is comforting to see that that mess can actually be understood The textbook has several examples of the use of this formula and several problems for you to practice on Those of you planning to go on in finance if there are any left after my course are well advised to try a few

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