A firm is considering its capital structure. Currently the fifirm is unlevered and unlevered firm value equals VU . The fifirm is considering issuing debt and and using the proceeds of the debt issue, P, to repurchase shares. Let D represent the value of debt. If the firm defaults, the firms debtholders will incur some expenses related to fifirm bankruptcy. For example they might have to pay lawyers and court costs, the value of these expenses equals BC(D), i.e., the value of these expenses is a function of the firms level of debt, D. Because debt is tax deductible, debt will generate tax shields worth TS(D).

Because the firm is unlevered, the current owners of the firm are the shareholders. If the firm issues debt with value D, the wealth of the shareholders will equal the value of the levered firm, VL(D), less the value of the new debtholders claim, D, plus the proceeds from the the debt issue. The value of the levered firm equals the value of the unlevered firm plus the value of the the tax shields.

If capital market are perfect (other than taxes) and the capital market is competitive, show that the wealth of current owners will be maximized if the firm maximizes TS(D) )-BC(D)