C OST OF CAPITAL OF PROJECTS

Project cost of capital from CAPM  Applying the CAPM to estimate the cost of capital of projects  Recall: the cost of capital is the expected return on a capital market investment opportunity having the same risk as the project  Recall: opportunity cost: this capital market expected return is forgone if I invest in the project  Recall: only relevant risk matters, which is measured by beta in the CAPM  Let’s look at the estimation of the parameters of the CAPM!

Risk-free rate  Look for an asset which has no risk…  As low default risk as possible: government bonds with the best rating  But: such yields are usually fixed (riskless) in nominal terms → inflation risk  Take inflation-indexed (inflation-protected) government bonds  About risk in reinvesting the interest payments: take zero-coupon bonds  What maturity? – Matching that of the project  Typical value: 1-3% p.a., in real terms

Average market risk premium  Risk premium for β = 1, that is, E(r M ) – r f  Taking a global approach – diversification globally is reasonable  Approximation of a global market portfolio: a global stock market index, e.g., MSCI ACWI  Estimation: from historical data  It is assumed, necessarily, that the expectation about the average market risk premium is stable over time  Thus, take the average of the past differences between the returns:  Typical value: 5-6% p.a., in real terms

Project beta  Recall: slope of the security characteristic line, resulting from a linear regression between returns  Problem: market portfolio returns are indeed observable, but there are no data on project returns  Projects are not traded (publicly) → no price data → no return data  Estimation compromise: take something „comparable”: industry betas  Industry betas from stocks of many firms in a given industry  Take the beta of an industry the project activities mostly correspond to (if not so clear, may even take weighted average of several industry betas)  Look at some examples…

Country risk  Risks associated with the country the project is undertaken in (e.g., political)  Question: is it diversifiable? (i.e., completely idiosyncratic)  If intended to be taken into account, in the cost of capital:  where r crp is calculated based on, e.g., credit ratings of government bonds

F INANCIAL CALCULATIONS

Common cash flow patterns (I.)  Annuity:  What is the present value of an annuity of A = 100 lasting for 15 periods, if the discount rate is 12%? (681)  How long should an annuity of A = 50 be in order for its present value to be at least 100, if the discount rate is 18%? (2.7 ~ 3)  Which option would you prefer: $10,000 today or $1,000 each year for 15 years, if the discount rate is 10%? (first > $7,606)  At least how much should the amount A of an annuity lasting for 10 periods be in order for the present value of the annuity to be at least 80, if the discount rate is 15%? (16)

Common cash flow patterns (II.)  Perpetuity:  What is the present value of a perpetuity of A = 100, if the discount rate is 20%? (500)  How much should the annual amount A of a perpetuity be in order for its present value to be at least 250, if the discount rate is 15%? (37.5)  What discount rate makes the present value of a perpetuity of A = 25 equal to 100? (25%)

Common cash flow patterns (III.)  Growing annuity:  Growing perpetuity:

Common cash flow patterns (IV.)  Growing annuity & perpetuity calculations:  Assume that g = 3% and r = 10%  What is the present value if F 1 = 100 and the series lasts for 5 periods? (400)  How much should F 1 be in order for the present value to be 320? (g.a.: 80; g.p.: 22.4)  If F 1 = 100, at least how many periods should the series last for in order for the present value to be larger than 500? (6.55 ~ 7)  Run the same, but for g = 10% (455; 70.4; 5.5 ~ 6)  The pattern is a growing perpetuity. If F 1 = 100 and r = 10%, then what g, or if g = 3%, then what r makes the present value equal to 1250? (g = 2%; r = 11%)

NPV and IRR  We have discussed the formulas and the decision rules…  Issues related to the IRR  Multiple roots  „Double relativity” – ranking mutually exclusive projects with different initial investment and/or life Project Cash flow ($)IRR (%) NPV r=10% F0F0 F1F1 A-10,000+20, ,182 B-20,000+35, ,818 Cash flow ($) IRR (%) NPV r=10% ProjectF0F0 F1F1 F2F2 F3F3 F4F4 F5F5 etc. C-9,0006,0005,0004, ,592 D-9,0001,800 …209,000

Profitability index (PI) (I.)  Case of scarce capacity allocation  General formulation:  Capital rationing:  Formulation may also be for PV instead of NPV  Then: decision criterion: PI > 1  Note that PI is relative with respect to initial investment → problem when comparing projects with different initial investment

Profitability index (PI) (II.)  Example: consider the following projects:  Ranking according to PI: D > B > C > A > E  Assume that the capital constraint is 150 – then D, B, A projects would be undertaken  Because F 0 altogether is = 150 After D and B, insufficient allotment remains for C  E is by no means to be undertaken, because PI E < 1 (NPV E < 0) F0F0 PVPI A-50601,20 B-20301,50 C ,36 D ,63 E-70500,71

Profitability index (PI) (III.)  Run the same, but for a constraint of 200  Then, by following the same logic, again D, B, A projects would be undertaken, total NPV is 150 in this case  But take a more careful look: what if D and C were undertaken? Total F 0 = = 190 < 200 Total NPV = = 170 > 150 ! And we are interested in total NPV, that is to be maximized  Explanation for the issue: PI is relative – not only the specific benefit matters, but the amount of unspent dollars also (cf. backpack problem…)

Annual equivalent (AE) (I.)  For comparing mutually exclusive and recurring projects with unequal lives  Their annual cash flow equivalents are compared  It is assumed that the projects will be undertaken repeatedly with the same conditions, until a common finite cessation point or infinity  A level annual amount is sought the NPV of the annuity of which equals the NPV of the project within the “cycle”  In essence, the cash flow pattern of the project is “flattened” in this approach

Annual equivalent (AE) (II.)  Illustration:  We can choose from two types of air-conditioners to be installed in our office. Type A can be purchased for $10,000 and operated annually for $1,500 with expected life-span of 7 years. Type B can be purchased for $14,000 and operated annually for $1,100 with expected life- span of 10 years. The discount rate is 9%. Which type should we choose, assuming that we would stick with the choice until infinity? (-$3,487 < -$3,281, so B)

F INANCING DECISIONS

Financing decisions  Investment decisions: what to acquire?  Financing decisions: from what kinds of sources?  Thus far silently assumed: financing completely with equity  Question: does it matter if financing not completely with equity?  In other words: is it worth to employ some debt beside equity?  Would shareholders be better off then?

Capital structure  The proportions of equity (E) and debt (D) in financing  D/E ratio – leverage  Unlevered vs. levered firm  Before-tax value (A BT ) and after-tax value (A)  Value of corporate tax (T cE )

MM propositions  Modigliani and Miller  Capital structure does not matter in a perfect world…  Proposition I.: The value of a firm does not depend on its capital structure; that is, the proportion of debt and equity financing is irrelevant to value. The value of a firm is determined by its real assets and not by its capital structure.  Proposition II.: The cost of equity capital (E(r E )) and its relevant risk (β E ) of a firm both increase as the firm’s leverage (i.e., debt-equity ratio) increases.  It would be good because calculations could be done assuming completely equity financing

Perfect, in terms of the CAPM (I.)

Perfect, in terms of the CAPM (II.)  The weighted average cost of capital (A = D + E):

Imperfections  We look at two cases of imperfections:  Tax savings Interest on debt is (corporate) tax-deductible More debt → more interest → less tax to be paid This saving goes to shareholders – an argument pro debt financing  Efficiency losses (costs of financial distress) Higher leverage increases costs, decreases efficiency E.g., customers, suppliers, employees, monitoring An argument contra debt financing  But the combined effect can be regarded negligible…

Effect of tax savings

Effect of efficiency losses

Combined effect

Combined effect – implications  If negligible: we are back to the MM propositions  Even in an imperfect world, capital structure choice may be considered irrelevant  For simplicity, it is then valid to do calculations assuming completely equity financing