3. A Reconsideration of the Modigliani-Miller Propositions

E.g., Brealey et al. (2013), Brigham and Ehrhardt (2010), Copeland et al. (2004), Levy and Sarnat (1994), Ross et al. (2012), Stiglitz (1969), among others.

A middleman (person D) can also do the arbitrage and get one sheep: Find a person (E) who is willing to pay six sheep for B and C’s cow, and buy A’s cow with five sheep for B and C, and then do the exchanges among A, B and C, and E simultaneously.

Modigliani and Miller (1958) use the following method to prove their first proposition. Consider two firms generate the same perpetual stream of cash flow, \( \tilde{X} \), in each year and differ only in their capital structure. The market value of the unlevered firm is \( V_{U} \). The market value of the levered firm is \( V_{L} \equiv S_{L} + B \), where \( S_{L} \) is the market value of equity, and \( B \) is the market value of riskless debt. One strategy an investor can take is to buy 15 % of the shares of the levered firm. That is, he invests \( 0.15S_{L} \) in the beginning and at the end of each year obtains payoffs \( 0.15(\tilde{X} - Interest) \). Another strategy is to buy 15 % of the shares of the unlevered firm, and also borrow \( 0.15B \) from a bank on his own account on the same terms as the firm. That is, the investor invests \( 0.15(V_{U} - B) \) in the beginning and at the end of each year obtains payoffs \( 0.15(\tilde{X} - Interest) \). Since both the strategies produce exactly the same results: \( 0.15(\tilde{X} - Interest) \), the initial costs of the two strategies must be the same, i.e., \( 0.15S_{L} = 0.15(V_{U} - B) \) or \( V_{U} = V_{L} \equiv S_{L} + B \). This kind of proof hinges on the assumption that individuals and corporations can borrow at the same rate (see also Brealey et al. 2003, p. 468; Ross et al. 2010, p. 495).

When all the investors of the levered firm cooperate to do the arbitrage (or there are middlemen), it is no need to sell short and assume equal access (i.e., the types of securities that can be issued by firms can also be issued by investors on personal account, see Fama 1978). If there is only one firm and investors value levered and unlevered firm differently, then the firm can simply change its debt-equity ratio (i.e., a costless window dressing) to benefit its investors.

Equation (3.3) can be written as:where \( \tilde{r}_{WACC} = \tilde{X}/(S_{L} + B) \). Suppose debt is riskless, i.e., \( \tilde{r}_{B} \equiv r_{B} \). Then the variance of the rate of return on equity is:

Since the Modigliani-Miller first proposition holds, changes in the debt-equity ratio \( (B/S_{L} ) \) will not affect the firm’s value \( (S_{L} + B) \), and the probability density function of \( \tilde{r}_{WACC} \) (and \( Var(\tilde{r}_{WACC} ) \)) does not change. Thus, when the debt-equity ratio increases, the variance of the rate of return on equity will also increase.

See also Proposition 5.1 in Chap. 5.