# DUKE UNIVERSITY Fuqua School of Business. FINANCE CORPORATE FINANCE Problem Set #4 Prof. Simon Gervais Fall 2011 Term 2.

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1 DUK UNIRSITY Fuqua School of Business FINANC CORPORAT FINANC Problem Set #4 Prof. Simon Gervais Fall 2011 Term 2 Questions 1. Suppose the corporate tax rate is 40%. Consider a firm that earns \$1,000 before interest and taxes each year (in perpetuity) with no risk. The firm s capital expenditures equal its depreciation expenses each year, and it will have no changes to its net working capital. The riskfree interest rate is 5%. (a) Suppose the firm has no debt and pays out its net income as a dividend each year. What is the value of the firm s equity? (b) Suppose instead the firm makes interest payments of \$500 per year. What is the value of equity? What is the value of debt? (c) What is the difference between the total value of the firm with leverage and without leverage? (d) The difference in part (c) is equal to what percentage of the value of the debt? 2. Western Lumber Company expects to have a free cash flow of \$4.25 million in the coming year. Free cash flows are expected to grow at a rate of 4% per year thereafter. Western Lumber has an equity cost of capital of 10% and a debt cost of capital of 6%. The corporate tax rate is 35%. If Western Lumber maintains a (constantly rebalanced) debt-equity ratio of 0.50, what is the present value of its interest tax shield? 3. Suppose that the Drazil Susej Corporation (DSC) has an equity cost of capital of 8.5%, a debt cost of capital of 7%, a marginal corporate tax rate of 35%, and a debt-equity ratio of 2.6. Suppose that Goodyear does not plan to rebalance its debt, i.e., its current debt is perpetual. (a) What is DSC s WACC? (b) What is DSC s unlevered cost of capital? 4. Kurz Manufacturing is currently an all-equity firm with 20 million shares outstanding and stock price of \$7.50 per share. Although investors currently expect Kurz to remain an allequity firm, Kurz plans to announce that it will borrow \$50 million and use the funds to repurchase shares (i.e., Kurz s announcement is not anticipated by investors and thus not reflected in the current stock price). Kurz will pay interest only on this debt, and it has no further plans to increase or decrease the amount of debt. Kurz is subject to a 40% corporate tax rate. (a) What is the market value of Kurz s existing assets before the announcement? 1

2 (b) What is the market value of Kurz s assets (including any tax shields) just after the debt is issued, but before the shares are repurchased? (c) What is Kurz s share price just before the share repurchase? How many shares will Kurz repurchase? (d) What are Kurz s market value balance sheet and share price after the share repurchase? 5. Merck s simplified balance sheets (using book and market values) are currently as follows: Balance Sheet (book values in millions) Net working capital 1,473 Long-term debt 5,269 Long-term assets 14,935 quity 11,139 Total assets 16,408 Total liabilities 16,408 Balance Sheet (market values in millions) Net working capital 1,473 Long-term debt 5,269 Market value of long-term assets 51,212 quity 47,416 Total assets 52,685 Total liabilities 52,685 Suppose that Merck decides to move to a 50% book debt-to-value ratio by issuing debt and using the proceeds to repurchase shares. The corporate tax rate is 40%; consider only corporate taxes. Now construct Merck s balance sheet (with market values only) to reflect the new capital structure, making sure to add an item called P(additional tax shields) on the asset side of the balance sheet. Before it changes its capital structure, Merck has 1,248 million shares outstanding. What is the stock price before and after the change? 6. Hula nterprises is considering a new project to produce solar water heaters. The finance manager wishes to find an appropriate risk adjusted discount rate for the project. The(equity) beta of Hot Water, a firm currently producing solar water heaters, is 1.3. Hot Water has a debt to total value ratio of 0.4. The expected return on the market is 14% and the risk-free rate is 8%. Throughout this problem, assume that the debt is risk-free and that it is constant (i.e., the debt is never rebalanced). (a) Suppose the corporate tax rate is 30%. What is the asset (or unlevered) beta for the solar water heater project? (b) If Hula is an equity financed firm, what is the weighted average cost of capital for the project? (c) If Hula has a debt to equity ratio of 2, what is the weighted average cost of capital for the project? (d) Hula s chairman wishes to know why the cost of capital for Hot Water cannot be used directly. xplain why. (e) The finance manager believes that the solar water heater project can only support 30 cents of debt for every dollar of asset value, i.e., the debt capacity is 30 cents for every dollar of asset value. This is lower than the cents to every dollar of asset value (debt to equity ratio of 2) that current projects can support. Current projects have higher collateral value than the assets of the new solar heater project. Hence she 2

3 is not sure that the debt to equity ratio of 2 used in the weighted average cost of capital calculation is valid. What is the appropriate capital structure to use? What is the weighted average cost of capital that you will arrive at with this capital structure? 7. Company XYZ is currently financed with 40% debt (a 40% debt-to-value ratio). The average current yield on government securities is 10%. The expected return on the S&P500 portfolio is 20%. The corporate tax rate is 20% and XYZ s stock has a beta of 1.6. XYZ can borrow at 200 basis points above the prevailing government rate. (a) What is XYZ s weighted average cost of capital under the current capital structure? (b) What is XYZ s cost of capital if it decides to pursueapolicy of always using 100% equity financing instead? (i) First, solve this under the assumption that XYZ s debt will never be rebalanced. (ii) Now, solve this under the assumption that XYZ s debt will be constantly rebalanced. 8. If it were unlevered, the overall firm beta for Wild Widgets Inc. (WWI) would be 0.9. WWI has a target debt/equity ratio of 1/2 and plans to constantly rebalance its debt in order to maintain it. The expected return on the market is 16%, and Treasury bills are currently selling to yield 8%. WWI one-year bonds(with a face value of \$1,000) carry an annual coupon of 7% and are selling for \$ The corporate tax rate is 34%. (a) What is WWI s cost of debt? (b) What is WWI s weighted average cost of capital? (c) What is WWI s cost of equity? 3

4 Solutions 1. Because the firm s cash flows are riskless, we have r = r D = 5%. (a) If the firm has no debt and pays out its net income (of \$600 = \$1,000(1 0.40)) as a dividend each year, the firm s equity is worth U = \$ = \$12,000. (b) If the firm makes (riskless) interest payments of \$500 per year, then the equity-holders will receive an annual dividend of \$300 = (\$1,000 \$500)(1 0.40). Their equity is then worth L = \$ = \$6,000. The bondholders receive \$500 every year. Their debt is worth (c) The levered value of the firm is D L = \$ = \$10,000. L = L +D L = \$6,000+\$10,000 = \$16,000. The difference between the levered value and the unlevered value is L U = \$16,000 \$12,000 = \$4,000. (d) This difference is the present value of the interest tax shield, which is a fraction t c of the debt D L : P(interest tax shield) = t c D L = (0.40)(\$10,000) = \$4, We can calculate the value of Western Lumber s interest tax shield by comparing its value with and without leverage. The expected return on Western Lumber s unlevered assets is given by r A = ( ) D r D + ( ) ( ) ( ) r = (0.06) + (0.10) = 8.67% If Western Lumber was all-equity financed, then its value would be U = \$4.25M = \$91.07M. To calculate Western Lumber s levered value, we first need to calculate its weighted-average cost of capital: WACC = D D + (1 t c)r D + D + r = (1 0.35)(0.06) + 1 (0.10) = 7.97%

5 The levered value of Western Lumber is therefore L = \$4.25M = \$107.14M. This levered value exceeds the unlevered value by the present value of the interest tax shield, that is, by P(interest tax shield) = L U = \$107.14M \$91.07M = \$16.07M. 3. We have r = 8.5%, r D = 7%, t c = 0.35, and D/ = 2.6 (which corresponds to D/ = = 0.722). (a) DSC s weighted average cost of capital is WACC = D (1 t c)r D + r = (0.722)(1 0.35)(0.07) +(0.278)(0.085) = 5.65%. (b) We know that the weighted average cost of capital satisfies ( ) D WACC = 1 t c r A 5.65% = [ 1 (0.35)(0.722) ] r A. This implies that r A = 7.56%. 4. (a) Because Kurz Manufacturing is initially all-equity financed, its value is the value of its equity: U = U = 20 million \$7.50 = \$150 million. (b) Right after the debt is issued, Kurz s value increases by the amount of debt it raised (\$50 million), and by the tax shield that this debt creates(0.40 \$50 million = \$20 million). That is, its value increases by \$70 million. (c) Before the debt is repurchased, the total value of the firm is \$150 million+\$70 million = \$220 million. Since the debt is worth \$50 million (i.e., the debtholders get what they pay for), the equity must be worth \$220 million \$50 million = \$170 million. This implies that Kurz s share price is \$170 million 20 million = \$8.50. This in turn implies that the \$50 million raised through the debt issue will allow Kurz to buy back \$50 million \$8.50 = million shares. (d) After the share repurchase, the total market value of Kurz Manufacturing is \$220 million \$50 million = \$170 million. On the asset side of the balance sheet, Kurz s unlevered assets are worth \$150 million, as before, and the debt tax shield is \$20 million. On the liability side of the balance sheet, the debt is worth \$50 million, and the equity is worth \$120 million (20 million million = million shares trading at \$8.50 each). 5. The long-term debt is increased from \$5,269 to 50% of book = 50% \$16,408 = \$8,204, 5

6 that is it is increased by \$8,204 \$5,269 = \$2,935. This implies that the debt tax shield is increasedby0.40 \$2,935 = \$1,174, andthemarketvalueofthefirmisnow\$52,685+\$1,174 = \$53,859. The new balance sheet (with market values) will look as follows: Balance Sheet (market values in millions) Net working capital 1,473 Long-term debt 8,204 Mkt value of long-term assets 51,212 quity 45,655 P(additional tax shields) 1,174 Total assets 53,859 Total liabilities 53,859 Before the change, the equity is worth \$47,416, and 1,248 shares are outstanding, so that the stock price is P = 47,416 1,248 = Upon the firm s announcement of its plans to repurchase shares, the firm s value should go up by the extra debt tax shield that this will generate. That is, the equity goes up to \$47,416+\$1,174 = \$48,590, and each share is then worth P = 48,590 1,248 = The amount raised from the new debt, \$2,935, is therefore used to repurchase 2, = shares. 6. (a) When the tax rate is 30%, the asset beta (unlevered beta) of the solar heater project is β 1.3 β A = 1+(1 t c ) D = 1+(1 0.30) 2 = Notice that the debt-to-equity ratio is = 2 3 when the debt-to-value ratio is 0.4. (b) If Hula is an all-equity financed firm, the weighted average cost of capital for the project is just the unlevered cost of equity. The unlevered cost of equity is given by the CAPM: r A = r unlevered = r f +β A (r m r f ) = 0.08+(0.8863)( ) = 13.32%. (c) The levered beta corresponding to a debt-to-equity ratio of 2 is [ β levered = 1+(1 t c ) D ] β A = [1+(1 0.30)(2)](0.8863) = Hence the return on equity corresponding to the leverage of 2 is given by the CAPM: r levered = 0.08+(2.13)( ) = 20.8%. Thus the weighted average cost of capital for the project is ( ) D p ( ) WACC = (1 t c )r p p D + r p = 0.667(1 0.30)(0.08) (0.208) = 10.7%. 6

7 (d) Hot Water may have a different capital structure and thus its weighted average cost of capital is not applicable. ven though the business risk is the same, the difference in capital structure implies a different weighted average cost of capital. (e) You are explicitly told that the new project has a lower debt capacity. Thus it can only support 30 cents of debt for every dollar of asset value. xisting assets can support cents. Thus the long run debt capacity of the new project is different. Hence we need to account for this while doing our capital budgeting. The appropriate leverage adjustment is the debt to value ratio of 0.3. First, the levered beta corresponding to the debt to total value ratio of 0.3 is calculated as follows: β levered = [ 1+(1 t c ) D ] β A = [ 1+(1 0.30) The corresponding return on equity, as given by the CAPM, is r = ( ) = 14.9%, and the weighted average cost of capital is then ( ) D p ( ) WACC = (1 t c )r p p D + r p ] (0.8863) = = 0.3(1 0.30)(0.08) + 0.7(0.149) = 12.1%. Hence 12.1% rather than 10.7% is the right answer. 7. (a) With a debt-to-value ratio of 40%, we have D = 40% 60% compute the cost of equity: = 2/3. We use the CAPM to r = r f +β (r m r f ) = ( ) = 26%. Noticing that r D = 10%+2% = 12%, we have ( ) ( ) D WACC = (1 t c )r D + r = 0.40(1 0.20)(0.12) +0.60(0.26) = 19.44%. (b) The risk premium on XYZ s debt is 0.02 = β D ( ), for a β D of 0.2. (i) When the debt is permanent, we can unlever the equity beta to get an (unlevered) asset beta as follows: Using CAPM, we find β A = β + ( D ) (1 tc )β D 1+ ( D ) (1 tc ) = (1 0.20)(0.2) (1 0.20) = r A = r f +β A (r m r f ) = ( ) = 21.13%. 7

8 (ii) When the debt is constantly rebalanced, we can unlever the equity beta to get an (unlevered) asset beta as follows: ( ) ( ) D β A = β D + β = (0.40)(0.2) +(0.60)(1.6) = Using CAPM, we find r A = r f +β A (r m r f ) = ( ) = 20.40%. 8. From the data we have r m = 0.16, r f = 0.08, D/ = 1/3, and / = 2/3. Also the beta of an otherwise identical but unlevered firm is 0.9. This means that r A = r f +(r m r f )β A = 0.08+( )(0.9) = 15.2%. (a) WWI one-year coupon bonds have a face value of \$1,000. One year from now, they will pay the face value and the 7% coupon, i.e. 1,000(1+0.07) = 1,070. Since WWI s bonds are now selling for \$972.72, we can deduct the value for WWI s (pre-tax) cost of debt: r D = 1,070 1 = 10% (b) The weighted average cost of capital for a levered firm can be calculated as follows: WACC L = r A D t cr D = (0.34)(0.10) = %. 3 (c) Since WACC L = r D (1 t c ) D +r, we have = 0.10(1 0.34) 1 3 +r 2 3, which implies r = 17.80%. 8

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