Prof. Thomas J. Chemmanur
MF891: Doctoral Seminar in Corporate Finance
Problem Set – I (A)
1.
M-M Proposition I (Irrelevance of capital Structure)
Think of two firms, identical in every way except in their capital structure: the first is
financed by 100% equity (unlevered), while the other is financed part by equity and part by
riskless debt (levered). Assume that CAPM assumptions hold, but that these firms have no
systematic risks (beta = 0). Also other M-M- I assumptions hold: no market imperfections of any
kind. Let the distribution of their operating income per year be as follows (assume perpetual cash
flows).
Y
Probability
1,500,000
0.5
1,100,000
0.5
Levered firm has issued 5,000,000 in debt with 10% interest (again debt is a perpetuity).
a) Find the value of levered and unlevered firm, Vl and Vu respectively.
2.
M-M Proposition II
Assume an all equity firm with the following probability distribution of operating income,
Y, and with 100,000 shares outstanding. Let CAPM assumptions hold, and asset beta of the firm
A=1. Risk-free rate, rf=10%, and market return, rM=12.5%.
Y
Probability
Earnings per share(eps)
100,000
0.25
1.00
250,000
0.50
2.50
400,000
0.25
4.00
a) Find the expected earnings per share, expected return on equity, value of this unlevered firm,
Vu, and price per share.
Now assume that the firm changes its capital structure to 50% debt and 50% equity. They
do that by issuing debt at 10% perpetual rate, and buying back equity with it.
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b) Find the expected earnings per share, expected return on equity, value of this levered firm, Vl,
and price per share.
3.
Extensions on M-M. The M-M theorem assumes that the firm has only two classes of
securities, perpetual debt and equity. Suppose that the firm has issued a third class of securities –
preferred stock – and that y% of preferred dividends may be written off as an expense (0y1).
Assume that the firm has preferred stock of face value FP, with a preferred dividend rate of rp.
Preferred dividends and debt are riskless (use riskfree rate, r). Assume that only corporate tax, C,
exists. No personal taxes. Also other assumptions of M-M hold. Rate of return of an unlevered
company in risk-class k is U.
a) What is the appropriate expression for the value of the levered firm?
b) What is the appropriate expression for the after-tax cost of capital, L?
4.
Given a world with corporate taxes, C, a personal tax rate paid on bonds, what would be
the effect of a decrease in the corporate tax rate on
a) the aggregate amount of debt in the economy, and
b) the optimal capital structure of firms?
Plot a graph of the equilibrium in the debt market in this case.
5.
Miller Equilibrium. For this question assume that there are three classes of investors
who pay different taxes on bond income; PB1=0%, PB2=15%,PB3=30%. And corporate tax rate is
C=20%. All other assumptions of the original Miller Model hold. As before, the equilibrium
return on municipals is ro and interest income on municipals is not taxable. Describe the Miller
equilibrium in this case; both in words and by graph. Comment on the optimal capital structure of
firms in this case.
6.
Miller equilibrium with tax on equity income. For this question assume again that PB
is a continuum (tax rate on interest income for investors vary continuously, starting at zero, i.e.,
there are some tax-free investors). To begin with, assume a world with (tax-free) return on
municipal bonds, r0 = 4%, corporate tax rate C = 40%, and a personal tax rate on income from
equity (only one rate for all investors), PS = 10 %. In such a setting, give:
a) The equilibrium in the bond market (demonstrate graphically), and
b) The aggregate amount of debt in the economy, and
c) the optimal capital structure of firms?
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d) If now the government hikes up the tax rate on equity income to 15%, how will the equilibrium
change (demonstrate graphically)?
7.
Modigliani-Miller Proposition I with risky debt issued:
The derivation (proof) we did in class of the Modigliani-Miller propositions I (irrelevance of
capital structure for firm value) was done in a setting with no bankruptcy: i.e., we made use of the
assumption (as in the original Modigliani-Miller paper) all debt issued by firms is risk-free debt
(no probability of bankruptcy). Could you now derive the Modigliani-Miller Proposition I under
the assumption that the debt issued by firms may be risky or risk-free (i.e., there is some
possibility that the firm may go bankrupt)? All other assumptions remain the same as in the
original Modigliani-Miller paper: i.e., we assume perfect capital markets.
Hint: You may get the answer to the above by reading Chapter 2.1 of the text book by Matos.
You may also need to read chapter 1 of that book (“Valuation”) to fully understand Chapter 2.1.
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