Question

In year​ 1, AMC will earn ​$ before interest and taxes. The market expects these earnings to grow at a rate of per year. The firm will make no net investments​ (i.e., capital expenditures will equal​ depreciation) or changes to net working capital. Assume that the corporate tax rate equals ​%. Right​ now, the firm has ​$ in​ risk-free debt. It plans to keep a constant ratio of debt to equity every​ year, so that on average the debt will also grow by ​% per year. Suppose the​ risk-free rate equals ​%, and the expected return on the market equals ​%. The asset beta for this industry is .

a. If AMC were an​ all-equity (unlevered)​ firm, what would its market value​ be?

b. Assuming the debt is fairly​ priced, what is the amount of interest AMC will pay next​ year? If​ AMC's debt is expected to grow by ​% per​ year, at what rate are its interest payments expected to​ grow?

c. Even though​ AMC's debt is riskless​ (the firm will not​ default), the future growth of​ AMC's debt is​ uncertain, so the exact amount of the future interest payments is risky. Assuming the future interest payments have the same beta as​ AMC's assets, what is the present value of​ AMC's interest tax​ shield?

d. Using the APV​ method, what is​ AMC's total market​ value, VL​? What is the market value of​ AMC's equity?

e. What is​ AMC's WACC? ​(Hint​: Work backward from the FCF and VL​.)

f. Using the​ WACC, what is the expected return for AMC​ equity?

g. Show that the following holds for​ AMC: .                                                      h. Assuming that the proceeds from any increases in debt are paid out to equity​ holders, what cash flows do the equity holders expect to receive in one​ year? At what rate are those cash flows expected to​ grow? Use that information plus your answer to part ​(f​) to derive the market value of equity using the FTE method.

Answer 1

a.         AMC has unlevered FCF of $2,000 ×0.6 = $1,200.

From the CAPM, AMC's unlevered cost of capital is 5% + 1.11 × (11% − 5%) =11.66%.

Discounting the FCF as a growing perpetuity tells us that the value of the firm, assuming growth of 3%, is:

            V(All Equity) = $1,200 / 0.1166 - 0.03 = $13,857.

b.         Since the debt is risk-free, the interest rate paid on it must equal the risk-free rate of 5% (or else there would be an arbitrage opportunity). The firm has $5,000 of debt next year. The interest payment will be 5% of that, or $250. If the debt grows by 3% per year, so will the interest payments.

c.         The expected value of next year's tax shield will be $250 × 40% = $100 and it will grow (with the growth of the debt) at a rate of 3%. But the exact amount of the tax shield is uncertain, since AMC may add new debt or repay some debt during the year, depending on their cash flows. This makes the actual amount of the tax shield risky (even though the debt itself is not). Since the beta of the tax shield due to debt is 1.11, the appropriate discount rate is 5% + 1.11 (11% - 5%) = 11.67%. We can now use the growing perpetuity formula and conclude that

PV(Interest Tax Shields) = $100 / 0.1166 - 0.03 =  $1,155

d.         The APV tells us that the value of a firm with debt equals the sum of the value of an all-equity firm and the tax shield. From previous work (parts (a) and (c)), we get:

V(AMC) = $13,857 + $1,155 = $15,012.

The market value of the equity is therefore V - D = $15,012 - $5000 = $10,012.

e.         Next year's FCF is      $2,000 ×0.6 = $1,200. It is expected to grow at 3%, so the WACC must satisfy:                                                           

            V(AMC) =$1,200 / rwacc - 0.03 = $15,000.

Solving for the WACC, we get WACC = 11 %.

f.          By definition, rwacc = E/V ×  rE + D/V × rD × (1 - τC).

The return on the debt is 5%; the value of the debt is $5,000, the value of the firm is $15,000 and therefore the value of the equity is $15,000 - $5,000 = $10,000. Plugging into the above expression, we get:

            11% = $10,000/$15,000 × rE + $5,000/$15,000 × 5% ×(1-0.4) è rE = 15%

g.         From the CAPM, βE must satisfy 15% = 5% + βE (11% −5%), so we conclude βE =1.66. The relationship holds since ($10,000/$15,000) × 1.66 = 1.11, and the beta of the debt equals 0.

h.         The debt is expected to increase to $5,000 × (1 + 0.03) = $5,150, so the equity holders will get $150 due to the increase in debt. These proceeds will increase by 3% annually. (The second-year debt will be $5,000× (1 + 0.03)2 = $5,304.5, with an increase in debt of $154.5, 3% higher than the $150 proceeds of year 1.) The expected FCF to equity at the end of the first year is therefore EBIT - Interest - Taxes + Debt proceeds, or FCFE = (2000 - 250) × (1 - .40) + 150 = $1200.

This cash flow is expected to grow at 3% per year. Thus, another way to compute the value of equity is to discount these cash flows directly at the MCR for the equity of 15% (from (f)):

E = FCFE / rE - g = 1200 / 15% - 3% = 10,000.

This is the same value we computed in (d), using the APV.