Consider an unlevered firm. The EBIT (earnings before interest and taxes) generated by the capital equipment depends upon market conditions as follows:

Recession Normal Expansion

EBIT $7,500 $15,000 $30,000

Probability .25 .50 .25

Currently, the total market value of this unlevered firm is $150,000. The current price per share is $15 and there are 10,000 shares outstanding. The current interest rate charged on risk free bonds or borrowing is 6 percent. (Obviously, the beta on risk free debt is zero.) The corporate tax rate is 35 percent and the personal tax rate on dividends and interest income is 28 percent. (As we discussed, if the personal tax rates for interest and dividend payments are the same, you can ignore personal taxes the corporate tax rate is all that matters.) The expected return on the well-diversified market portfolio is 15 percent.

A. If this firm is priced according to CAPM, what is the beta of this firm? To answer this question, you need to figure out what is the expected after-tax earnings (by taking a weighted average of the potential after-tax earnings in each type of economy i.e., recession, normal, or expansion with the weights equal to the probability of each type of economy) and then figure out what required rate of return on assets (or, equivalently, the equity of the unlevered firm) makes this expected perpetual earnings stream equal the value of the equity. (Since this is an all-equity firm (at this point), the equity value is simply the value of the firm, which is $150,000. So, you need to find the r such that $150,000 = {(Expected EBIT)*(1-T_c)}/r, where T_c is the corporate tax rate.) Once you have the required rate of return, you can figure out the firms beta using CAPM (i.e.,, where is the required return on equity (or the assets), is the return on the market, r_f is the risk-free rate, and E[.] denotes the expectation). To figure out the expected after-tax earnings for a particular state of the economy (e.g., recession, normal, or expansion), take the EBIT for that state subtract the tax due. The tax due is simply .35 times that EBIT. Earnings after tax is then just EBIT minus the tax due. Then take the weighted average of the earnings after tax for each state of the economy, using weights that equal the probability of each state.

B. The firm wants to change its capital structure. In particular, it wants to issue $25,000 in debt and use the proceeds to buy back equity.

i. What are the earnings after tax in normal, recession and expansion economies for this newly levered firm? To figure this out, for each state take EBIT and subtract the interest expense (the interest rate times the amount of debt $25,000). This is taxable income. Then multiple taxable income by the tax rate to get the tax due. To get earnings after tax, subtract the tax due from taxable income.

ii. In each of these economic states, can the firm satisfy its interest payment obligation? That is, can it pay the interest on the debt in each state? If yes, what is the beta of debt and what is the required rate of return on debt? If no, what must be true about the debt beta and required rate of return relative to risk free debt?

iii. What will be the beta of the equity after the capital structure change and how will the required rate of return on equity change given the capital structure change? With corporate taxes, M&M II implies that the beta of equity depends upon the unlevered beta (, calculated in 1 above) and the mix of debt (B) and equity (S) according to

,

where is the corporate tax rate (.35) and is the beta of debt. Given this beta, you can figure out the new required return on equity simply using CAPM. Alternatively, the following expression also gives you the required rate of return on equity with corporate taxes:

.

iv. What is the WACC for the newly levered firm? To figure this out, you first need to figure out the value of the levered firm. The value of the levered firm is the value of the unlevered firm plus the corporate tax rate times the level of debt:

.

The new value of equity is simply Then use the values of S and B to calculate the WACC as

v. What happens to the wealth of the equity holders after this recapitalization? Dont forget to include the money they receive in the recapitalization when they sell shares back to the firm.

vi. What price do they have to pay for the shares they repurchase?

C. The firm thinks that if issuing $25,000 in debt and buying back shares is good, issuing $50,000 in debt and buying back shares must be better. What happens to the wealth of the equity holders if the firm recapitalizes by issuing $50,000 (rather than $25,000) in debt and buys back shares? Be careful to check to see if the debt it issues is still risk free.

D. At what level of debt does the debt become risky? Qualitatively, what will happen to the value of the firm and the wealth of equity holders if the firm issues more than this level of debt?

22 S=U+(SB)(1TC)(UB) 0 B rS=rU+(SB)(1TC)(rUrB) VL=VU+TCB S=VLB WACC=(S+BS)rs+(1TC)(S+BB)rB