The Modigliani–Miller theorem. (Modigliani and Miller, 1958.) Consider the analysis of the effects of uncertainty about discount factors in Section 9.7. Suppose, however, that the firm finances its investment using a mix of equity and risk-free debt. Specifically, consider the financing of the marginal unit of capital. The firm issues quantity b of bonds; each bond pays 1 unit of output with certainty at time t + τ for all τ ≥ 0. Equity holders are the residual claimant; thus they receive

 

 

(a) Let P(t) denote the value of a unit of debt at t, and V(t) the value of the equity in the marginal unit of capital. Find expressions analogous to (9.35) for P(t) and V(t).

 

(b) How, if at all, does the division of financing between bonds and equity affect the market value of the claims on the unit of capital, P(t)b + V(t)? Explain intuitively.

 

(c) More generally, suppose the firm finances the investment by issuing n financial instruments. Let di(t + τ) denote the payoff to instrument i at time t + τ; the payoffs satisfy d1(t + τ) +···+ dn(t + τ) = π (K(t + τ)), but are otherwise unrestricted. How, if at all, does the total value of the n assets depend on how the total payoff is divided among the assets?

 

(d) Return to the case of debt and equity finance. Suppose, however, that the firm’s profits are taxed at rate θ, and that interest payments are tax-deductible. Thus the payoff to bond holders is the same as before, but the payoff to equity holders at time t + τ is (1 − θ)[π (K(t + τ)) − b]. Does the result in part (b) still hold? Explain.