M&M Propositions and the BPM Ogden, Jen and O Connor, Chapter 2 Bus 3019, Winter 2004
Outline of the Lecture Modigliani and Miller Propositions With Taxes Without Taxes The Binomial Pricing Model 2
An Ideal Capital Market No transaction costs, taxes, brokerage fees, short-selling is unrestricted. All market participants have the same expectations. All market participants are atomistic. The firm s investment program is fixed and known. The firm s financing is fixed. 3
Modigliani and Miller s Propositions On the Irrelevance of the Capital Structure Proposition I: The market value of a firm is constant regardless of the amount of leverage that it uses to finance its assets. Proposition II: The expected return on a firm s equity is an increasing function of the firm s leverage. 4
Analysis of M&M Proposition I The value of a firm is given by the present value of all the cash flows its assets are expected to generate in the future. The value of a firm is equal to the value of its assets. Unlevered Firm: V U = E U Levered Firm: V L = D + E L. 5
Analysis of M&M Proposition I M&M Proposition I states that V U = V L. Why? Consider an all-equity firm with value V U = E U. Suppose there existed a way to finance this firm s assets with debt and equity such that V L = D + E L > V U. 6
Analysis of M&M Proposition I An arbitrageur could buy α shares of the above firm, place them in a trust and sell debt and equity claims against these shares in proportions such that α(d + E L ) > αe U, making then a riskless profit. 7
Analysis of M&M Proposition I Similarly, someone could buy all of the firm s shares for E U and modify the firm s capital stucture to have V L = D + E L > E U and then resell the firm for a riskless profit of V L V U. 8
Analysis of M&M Proposition I In a frictionless market, this arbitrage oppotunity would lead to an increase in the firm s unlevered equity to the point where for any level of D and E L. V U = E U = D + E L = V L 9
Analysis of M&M Proposition II M&M Proposition II states that a firm s expected return on equity increases with leverage. Why? Take the firm s WACC: ( ) D WACC = r D D + E L ( ) EL + r LE. D + E L Note: There are no taxes here. 10
Analysis of M&M Proposition II A firm s WACC can be seen as the expected return on its assets, r A : r A = r D ( Let s factor out r LE. D D + E L ) ( ) EL + r LE. D + E L 11
Analysis of M&M Proposition II r A = r D ( D D + E L ) ( ) EL + r LE D + E L ( ) ( ) EL D r LE = r A r D D + E L D + E L ( ) D + EL D r LE = r A r D E L E L ( ) D D r LE = r A + 1 r D E L E L r LE = r A + D E L (r A r D ). 12
Analysis of M&M Proposition II r LE = r A + D E L (r A r D ) From M&M Proposition I, r A is constant regardless of D/E L. If the firm s assets are risky and investors require a premium for holding risky assets, then r A > r D, since debtholders are the first claimants to the firm s assets. 13
Analysis of M&M Proposition II Is r LE = r A + D E L (r A r D ) sufficient to say that r LE increases with D? 14
Analysis of M&M Proposition II Is r LE = r A + D E L (r A r D ) sufficient to say that r LE increases with D? No. When D/E L increases, r D also increases and thus r A r D decreases. 15
Analysis of M&M Proposition II Incorporating more details in the analysis, it is possible to show that r LE as D E L. The binomial pricing model, for instance, can be used to show this result. 16
The Binomial Pricing Model Consider a firm that can take on two values at time T, i.e. VT u with probability p, V T = VT d with probability 1 p, with V u T > V d T, where u stands for up and d stands for down. 17
The Binomial Pricing Model Let V Current value of the firm s assets D Current value of the firm s debt E Current value of the firm s equity X Payment promised to debtholders at time T r f Risk-free rate of interest 18
The Binomial Pricing Model What are D and E? At time T, the value of the firm s debt is X if V T X, D T = V T if V T < X. 19
The Binomial Pricing Model: Risk-Free Debt If V d T X, then V u T > X and thus debt is free of risk. Risk-free assets are discounted at the risk-free rate, and thus which gives D = X (1 + r f ) T, E = V D = V X (1 + r f ) T. 20
The Binomial Pricing Model: Risk-Free Debt Using continuous discounting, this can be rewritten as D = Xe r f T E = V Xe r f T. 21
The Binomial Pricing Model: Risk-Free Debt Example Consider a firm with present value V = $400, future value V3 u = $650 with probability p = 0.7, V 3 = = $250 with probability 1 p = 0.3. V d 3 in three years, and a pure-discount debt issue that pays X = $200 in three years. The risk-free interest rate is r f = 5%. 22
The Binomial Pricing Model: Risk-Free Debt Example Note that the above values give us a return on assets of ( ) 1/3 pv3 u r A = + (1 p)v 3 d 1 V ( ).7 650 +.3 250 1/3 = 1 400 = 9.83%. 23
The Binomial Pricing Model: Risk-Free Debt Example What is the current value of debt? Debt is risk-free and thus X can be discounted at the risk-free rate to find D: D = X (1.05) 3 = 200 (1.05) 3 = 173. The current value of equity is then E = V D = 400 173 = 227. 24
The Binomial Pricing Model: Risky Debt Suppose now that debt is not risk-free. That is, suppose that V d T < X < V u T. The value of debt at time T is then X if V T = VT u D T =, VT d if V T = VT d. 25
The Binomial Pricing Model: Risky Debt The value of equity at time T is VT u E T = X if V T = VT u, 0 if V T = VT d. Let E u T = V u T X and let Ed T = 0. 26
The Binomial Pricing Model: Risky Debt How can we find D and E in this case? We have the risk-free discount rate and thus we can find the present value of any risk-free asset. Can we form a risk-free portfolio with V, D and E? 27
The Binomial Pricing Model: Risky Debt A risk-free portfolio is a portfolio that provides the same payoff in each state of the world u and d. How to make a portfolio that pays K, say, whether V T = VT u or V T = VT d? What can K be? What payoff can we guarantee with certainty? 28
The Binomial Pricing Model: Risky Debt Consider a portfolio P, which involves the purchase of all of the firm s assets and the short sale of a fraction δ of the firm s equity. The payoff of portfolio P at time T is then VT u δeu T in state u, VT d δed T in state d. 29
The Binomial Pricing Model: Risky Debt For portfolio P to be risk-free, we need V u T δe u T = V d T δe d T δ = V u T V d T E u T Ed T The present value of portfolio P, V δe, is then given by V δe = V T δe T (1 + r f ) T.. 30
The Binomial Pricing Model: Risky Debt With δ = V u T V d T E u T Ed T, we have V T δe T = V u T δe u T = (1 δ)v u T + δx = V d T δe d T = V d T and thus V δe = V d T (1 + r f ) T. 31
The Binomial Pricing Model: Risky Debt The current market value of equity is then E = 1 ( VT d V δ (1 + r f ) T and the current market value of debt is D = V E. ) 32
The Binomial Pricing Model: Risky Debt Example Consider a firm with present value V = $400, future value V3 u = $650 with probability p = 0.7, V 3 = = $250 with probability 1 p = 0.3. V d 3 in three years, and a pure-discount debt issue that pays X = $400 in three years. The risk-free interest rate is r f = 5%. 33
The Binomial Pricing Model: Risky Debt Example Same firm as before, except that X = 400. Debt is not default-free anymore. What is the current value of debt and equity? First thing to do: find δ in the portfolio V δe such that the latter be risk-free. 34
The Binomial Pricing Model: Risky Debt Example To have V3 u δeu 3 = V 3 d δed 3, we need δ = V u 3 V d 3 E u 3 Ed 3, where E u 3 = max { 0, V u 3 X } = max { 0, 650 400 } = 250 E d 3 = max { 0, V u 3 X } = max { 0, 250 400 } = 0. 35
The Binomial Pricing Model: Risky Debt Example This gives δ = V u 3 V d 3 E u 3 Ed 3 = 650 250 250 0 = 1.6 and thus E = 1 δ ( V V d 3 (1 + r f ) 3 ) = 1 1.6 ( 400 250 ) (1.05) 3 = 184 and D = V E = 400 184 = 216. 36
The Binomial Pricing Model: Risky Debt Example Note that the return required by bondholders in this case is r D = ( ) 400 1/3 1 = 22.8%. 216 37
M&M Propositions with Taxes Consider an unlevered firm, denoted U, that expects constant earnings before interest and taxes, denoted EBIT, forever. Each period, if the corporate tax rate is T c, shareholders receive (1 T c )EBIT and the government receives T c EBIT. 38
M&M Proposition I with taxes Let E U = V U denote the present value of the payment (1 T c )EBIT forever. Let G U denote the present value of the payment T c EBIT forever. Let r 0 denote the rate at which investors discount EBIT, i.e. E U = V U = (1 T c)ebit r 0. 39
M&M Proposition I with taxes Consider a levered firm, Firm L, with the same EBIT as U, but with a perpetual debt issue D with coupon rate i. Interest payments are tax exempt. Shareholders receive (1 T c )(EBIT id) each period forever, bondholders receive id each period forever, and the government receives T c (EBIT id) each period forever. 40
M&M Proposition I with taxes Let Q U = E U + G U and Q L = E L + D + G L. From M&M Proposition I without taxes, we know that Q U = Q L since EBIT is the same for both firms each period. EBIT should therefore be discounted at the same rate for both firm U and firm L, namely r 0. 41
M&M Proposition I with taxes Each period, the total cash flow to shareholders and bondholders of Firm L is (1 T c )(EBIT id) + id = (1 T c )EBIT + T c id. Assuming that T c id is as safe as debt itself, we have V L = (1 T c)ebit r 0 + T cid r D, where r D is bondholders required return. 42
M&M Proposition I with taxes For debt to have been issued at par, we need i = r D and thus V L = (1 T c)ebit r 0 + T cr D D r D = V U + T c D. 43
M&M Proposition II with taxes How does r LE, the return required by the levered firm s shareholders, compare with r 0? Using E L = (1 T c)(ebit r D D) r LE r LE = (1 T c)(ebit r D D) E L and (1 T c )EBIT = r 0 V U = r 0 (V L T c D) = r 0 (E L + (1 T c )D), 44
M&M Proposition II with taxes we find r LE = r 0(E L + (1 T c )D) (1 T c )r D D E L = r 0 + D E L (1 T c )(r 0 r D ). 45
M&M Propositions with taxes and the WACC WACC L = D (1 T c )r D + E L r LE V L V L = D V L (1 T c )r D + E L V L ( r 0 + D E L (1 T c )(r 0 r D ) = D(1 T c)r D + E L r 0 + D(1 T c )(r 0 r D ) V L = (V L D)r 0 + D(1 T c )r 0 V L = V Lr 0 T c Dr 0 V L = r 0 ( 1 T ) cd V U + T c D ) 46
M&M Propositions with taxes and the WACC ( WACC L = r 0 1 T ) cd V U + T c D = r 0 V U V U + T c D 47