MM Proposition I

The value of the levered firm, V^L, must be equal to the value of the unlevered firm, V^U.

The firm's market value and average cost of capital are completely independent of the capital structure that the firm chooses. That is:

V^L= V^U

where

V^Uis the value of an unlevered or all equity firm
V^Lis the value of a levered firm (a firm which has some debt in its capital structure).

Click here for the proof

Proof:

Without loss of generality, we first make the following simplifying assumptions:

The levered firm is financed with two types of securities, stocks or equity(E^L) and bonds or debt (D^L)
Earnings before interest is X per year in perpetuity for both firms, where X is random
The bonds of the levered firm are riskless and perpetual. That is D^L= C / r_d where C is the coupon payment and r_d is the bondholders required return. Note that this also implies that C = D^Lr_d.
All earnings after interest payments are paid out as dividends. The rate of return demanded by shareholders of the unlevered firm is r_e^U and the rate of return demanded by shareholders of the levered firm is r_e^L.
Individuals can borrow at the same rate of interest as the firm.

Under these conditions the value of the levered firm can be written as

where r_*^L is the average cost of capital of the levered firm and is given by:

Of course, for an unlevered firm V^L= E^L and r_*^U = r_e^U.

Consider two firms that are identical in all respects except capital structure. Firm U has no debt in its capital structure, while Firm L has some debt. Since the two firms are identical in all respects, their earnings before interest payments are equal, that is,

X_U = X_L= X.

MM Proposition I (without tax) can be stated as V^L= V^U. To prove that this is the case, we will assume this is not the case, and show that costless arbitrage profits can be earned. If the market does not have costless arbitrage opportunities, then MM Proposition I must hold.

To begin, suppose that we observed that V^L< V^U in the marketplace. To profit from such an opportunity, buy a proportion of the levered firm's stock, buy a proportion of the levered firm's bonds, and sell a proportion of the unlevered firm's stock. The portfolio transactions are:

Position	Time 0	Time 1
Buy a of levered firm’s stock	-a E ^L	a (X ^L - C)
Buy a of levered firm’s bonds	-a D ^L	-a C
Sell a of unlevered firm’s stock	aV ^U	-a X ^L
Total	a (V ^U- V ^L)	0

Note that the portfolio is certain to have a terminal income of zero and yet the initial value is positive. This implies a costless arbitrage profit of V^U- V^Lmay be earned. If such an opportunity arose, arbitrage activity would begin, driving the levered firm share price up and the unlevered share price down until the market realized an equilibrium where V^L= V^U.

To complete the proof, we need to consider the case where V^L> V^U. If such an opportunity appeared, investors would consider forming a portfolio just the opposite of that described above. They would borrow an amount of money a D^L, sell a proportion of the levered firm's shares, and buy a proportion of the unlevered firm's shares. The arbitrage transactions are:

Position	Time 0	Time 1
Sell a of levered firm’s stock	a E ^L	-a Ca
Sell a of levered firm’s bonds	a D ^L	(X ^L - C)-
Buy a of unlevered firm’s stock	-aV ^U	a X ^L
Total	a (V ^L- V ^U)	0

Again, an arbitrage profit can be earned. The price of the levered firm's stock would fall and the price of the unlevered firm's stock would rise as a result of the arbitrage activity. Market equilibrium would occur where V^L= V^U.