We have seen that MM1 with taxes and bankruptcy costs is written

S + B = S0 + T S P VBC ,

where S is the levered firm's value of equity, B is the levered firm's value of debt, S0 is the unlevered firm's value, T S is the levered firm's tax shields, and P VBC is the levered firm's present value of expected bankruptcy costs.

The cost of levered equity is rS, the cost of debt is rB, and the cost of unlevered equity is r0.

Assume that expected bankruptcy costs are discounted at the unlevered cost of equity.

Cash-flows are not necessarily perpetual!

1) Assuming that the levered firm's corporate tax rate is zero, provide a formula for the cost of levered equity rS. The formula has to depend on: S0, S, B, P VBC , rB, and r0 only.

2) Assume that: 1) the corporate tax rate is zero, 2) firms stop their business in one year, 3) earnings before interest and taxes in one year are expected to be worth EBIT, 4) the probability that the levered firm defaults in one year is p, 5) bankruptcy costs are worth BC, 6) face value of debt is F, and 7) coupon rate is also rB. Provide a formula for the cost of levered equity rS. The formula has to depend on: S, EBIT, p, BC, F, rB, and r0 only.

3) Assuming that firms never default, provide a formula for the cost of levered equity rS. The formula has to depend on: S0, S, B, T S, rB, and r0 only. Hint: Use the fact that the expected return on a portfolio P invested in J securities is: rP = x1r1 + x2r2+, . . . , +xJ rJ , where r1, r2, r3, . . . are the expected returns of each security and x1, x2, x3, . . . the weights/fractions of the portfolio invested in each security.

4) Assume that: 1) firms never default, 2) firms stop their business in one year, 3) earnings before interest and taxes in one year are expected to be worth EBIT, 4) the corporate tax rate is Tc, 5) face value of debt is F, and 6) coupon rate is also rB. Provide a formula for the cost of levered equity rS. The formula has to depend on: S, EBIT, Tc, F, rB, and r0 only.

5) Assuming that both expected bankruptcy costs and expected tax savings are strictly positive, provide a formula for the cost of levered equity rS. The formula has to depend on: S0, S, B, T S, P VBC , rB, and r0 only. Hint: Use the fact that the expected return on a portfolio P invested in J securities is: rP = x1r1 + x2r2+, . . . , +xJ rJ , where r1, r2, r3, . . . are the expected returns of each security and x1, x2, x3, . . . the weights/fractions of the portfolio invested in each security.

6) Assume that: 1) firms stop their business in one year, 2) earnings before interest and taxes in one year are expected to be worth EBIT, 3) the corporate tax rate is Tc, 4) the cash-flow to debtholders in one year is expected to be IP + F, where IP is the expected interest payment, 5) the probability that the levered firm defaults in one year is p, and 6) bankruptcy costs are worth BC. Provide a formula for the cost of levered equity rS. The formula has to depend on: S, EBIT, Tc, IP, F, p, BC, rB, and r0 only.