# CHAPTER 18. Capital Budgeting and Valuation with Leverage. Chapter Synopsis

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1 CHAPTER 18 Capital Budgeting and Valuation with everage Chapter Synopsis 18.1 Overview of Key Concepts There are three discounted cash flow valuation methods: the weighted average cost of capital (WACC) method, the adjusted present value (APV) method, and the flow-to-equity (FTE) method. Despite differences in implementation, all methods can be used to determine the same correct project or firm value. In the next three sections, three assumptions will be made: 1. The project has average risk, so the project s cost of capital can be assessed based on the risk of the firm. 2. The firm s debt-equity ratio is constant, so its weighted average cost of capital will not fluctuate due to leverage changes. 3. Corporate taxes are the only market imperfection. Imperfections such as costs of financial distress, agency costs and benefits of debt, and issuance costs are not significant at the level of debt chosen. These assumptions will be relaxed in the final sections of the chapter The Weighted Average Cost of Capital Method The WACC method calculates the total value of the equity and debt by discounting unlevered free cash flows at the WACC, rwacc. The key steps in the WACC valuation method are: 1. Determine the free cash flows of the firm or project. 2. Compute the after-tax WACC as

2 210 Berk/DeMarzo Corporate Finance, Second Edition E D rwacc = re + rd(1 τ C) E + D E + D where E is the market value of equity, r E is equity cost of capital, D is the market value of debt (net of cash), r D is the debt cost of capital, and τ C is the marginal corporate tax rate. 3. Compute the value of the firm or project, including the tax benefit of leverage, by discounting the free cash flows such that V 0 T FCFt = (1 + r ) t= 0 wacc The intuition of the WACC method is that the firm s weighted average cost of capital represents the average return the firm must pay to its investors (both debt and equity holders) on an after-tax basis. Thus, an acceptable project should generate an expected return of at least the firm s weighted average cost of capital. The WACC method incorporates the tax savings associated with using debt in the r wacc calculation, which uses the after-tax cost of debt The Adjusted Present Value Method In the adjusted present value (APV) method, the total value of an investment is determined by first calculating its unlevered value and then adding the value of the interest tax shield and deducting any costs that arise from other market imperfections. The key steps in the APV valuation method are as follows: 1. Determine the free cash flows of the investment. 2. Determine the investment s value without leverage( V U ) by discounting its free cash flows at the unlevered cost of capital, ru. With a constant debt-equity ratio, E D ru = Pretax WACC = re + rd E+ D E+ D 3. Determine the present value of the interest tax shield. Given the expected debt on date t, Dt, the interest tax shield at time t is τ C Dt. If a constant debt-equity ratio is maintained, ru is the appropriate discount rate to use top discount the tax shields. 4. Add the present value of the interest tax shield to the unlevered value and subtract the present value of any financial distress, agency, and issuance costs: U V = APV = V + PV(interest tax shield) PV(Financial distress, agency, and Issuance costs) The APV method may be easier to apply than the WACC method when the firm does not maintain a constant debt-equity ratio and the level of debt in the future is known. However, the APV method is generally more complicated to apply than the WACC method when a firm uses a constant leverage ratio because you need to know the debt level to compute the interest tax shield, but with a constant debt-equity ratio you need to know the project s value to compute the debt level. As a result, implementing the APV approach with a constant debt-equity ratio requires solving for the project s debt and value simultaneously.

3 Berk/DeMarzo Corporate Finance, Second Edition The Flow-to-Equity Method In the flow-to-equity (FTE) valuation method, the free cash flows available to equity holders, after taking into account all payments to and from debt holders, are discounted using the levered equity cost of capital. The key steps in the flow-to-equity method for valuing a levered investment are as follows: 1. Determine the free cash flow to equity (FCFE) of the investment as After-tax interest expense FCFE = FCF (1 τ ) (Interest payments) + (Net borrowing) C 2. Determine the equity cost of capital, re. 3. Compute the equity value, E, by discounting the free cash flow to equity using the equity cost of capital. The FTE method has the same disadvantage associated with the APV approach: You need to know the debt level to compute the interest tax shield, but with a constant debt-equity ratio you need to know the project s value to compute the debt level Project-Based Costs of Capital The methods discussed above assumed that both the risk and the leverage of the project under consideration matched those characteristics for the firm as a whole. However, specific projects often have different risk and leverage than the average investment made by the firm. The cost of capital should reflect the risk of owning assets used in the project. The cost of capital for a specific industry can be estimated by studying single-division firms in that industry. If the unlevered cost of capital for the industry happens to be known, then r E can be inferred as: E D D ru = re + rd re = ru + ( ru rd) E + D E + D E Once the equity cost of capital is known, the division s WACC can be determined. The same r estimate can be calculated directly using wacc D r = wacc ru τ CrD E+ D If r U is not known, then the CAPM can be used to determine r E. First, unlever the comparison firm s beta as: β = [ E/( E+ D)] β + [ D/( E+ D )] β. U E D The unlevered beta can be used in the CAPM to estimate the unlevered cost of capital to use in the APV method. For the WACC and FTE methods, the unlevered beta can be relevered to reflect the firm s capital structure as: β = β + [ D/ E ]( β β ). E U U D The levered beta, β E, can be used in the CAPM to determine the industry equity cost of capital, and then the division s WACC can be determined.

4 212 Berk/DeMarzo Corporate Finance, Second Edition To determine the cost of capital for a project, the amount of debt associated with the project must be determined based on the incremental financing needed if the firm takes on the project. The incremental financing of a project need not correspond to the financing that is directly tied to the project. For example, a project financed with a mortgage for 90% of its value by a firm with a policy to maintain a 40% debt-to-value ratio will reduce debt elsewhere in the firm. In that case, the appropriate debt-to-value ratio to use when evaluating the project is 40%, not 90%. A firm s leverage should be evaluated based on its debt net of any excess cash. When a firm s dividend payouts and share repurchases are set in advance and will not be affected by a project s free cash flow, new projects are effectively 100% debt financed. Any cash requirement of the project will be funded using the firm s cash or borrowing, and any cash that the project produces will be used to repay debt or increase the firm s cash APV with Other everage Policies When the firm has a constant interest coverage ratio, it keeps its interest payments to a target fraction of its FCF. Using the APV method, the value of a project for a firm in which k equals incremental interest payments as a target fraction of the project s free cash flow is given by constant interest coverage model: V = APV = V + PV(interest tax shield) = V + τ k V = (1 + τ k) V U U U U C C When the level of debt fluctuates with the value of the project, the project s unlevered cost of capital should be used to discount the tax shields. However, when a firm has a fixed debt schedule, the interest tax shields can be discounted using the debt cost of capital, r D, since the amount of the debt will not fluctuate, and thus the tax shield is less risky than the project Other Effects of Financing If there are fees associated with securing a loan or issuing securities, they should be deducted when computing a project s value. If management believes that the equity will sell at a price that is less than its true value, this mispricing is a cost of the project for the existing shareholders and it should be deducted when determining a project s value. When the probability of financial distress is high, the estimated future free cash flows should be reduced by the expected costs associated with financial distress and agency problems. The probability of financial distress also raises the unlevered cost of capital. Since financial distress is more likely to occur when economic times are bad, the costs of distress cause the value of the firm to fall further in a market downturn. Financial distress costs, therefore, tend to increase the sensitivity of the firm s value to market risk, raising the unlevered cost of capital for highly levered firms Advanced Topics in Capital Budgeting When the firm readjusts is debt annually, based on its expected future free cash flows, the discount rate used for the interest tax shield is rd for the first period, because it is fixed, and then ru from then on. The constant interest coverage model in this case becomes:

5 Berk/DeMarzo Corporate Finance, Second Edition 213 U U 1 + r U U V = APV = V + PV(interest tax shield) = V + τ Ck V. 1 + rd If investors are taxed on the income they receive from holding equity or debt, it will raise the return they require to hold those securities. The WACC method already accounts for personal taxes since the equity and debt cost of capital in the market already reflects the effects of investor taxes. The APV approach must be modified in the presence of personal taxes because it requires an estimate of the unlevered cost of capital. Thus, rd and τ C should be replaced by: Appendix: Foundations and Further Details * (1 τi) (1 τ C)(1 τ e) rd r D and τ * = 1. (1 τ ) (1 τ ) e 18A.1 Deriving the WACC Method For a project financed by both debt and equity, a firm will have to pay investors a total of E(1 + re) + D(1 + rd). Assume that the project generates free cash flows of FCF1 at the end of the year, the interest tax shield of the debt provides a tax savings of τc (interest on debt) τc rd D, and the investment will continue beyond next year with a continuation value of V. 1 Thus, to satisfy investors, the project cash flows must be such that E (1 + r ) + D(1 + r ) = FCF + τ r D+ V Because V = E+ D, we can write the WACC as: 0, E D 1 c D 1 i E D rwacc = re + re ( 1 τ c ). V0 V0 The value of the project today can then be expressed as the present value of next period s free cash flows and continuation value: FCF1 + V1 V0 =. 1 + rwacc For a multi-year project, and assuming the WACC remains constant: FCF1 FCF2 FCF3 V0 = K. 1 + rwacc ( 1+ rwacc ) ( 1+ rwacc ) That is, the value of a levered investment is the present value of its future free cash flows using the weighted average cost of capital. 18A.2 The evered and Unlevered Cost of Capital If an investor holds a portfolio of all of the equity and debt of the firm, then the investor will receive the free cash flows of the firm plus the tax savings from the interest tax shield. Since these are the same cash flows an investor would receive from a portfolio of the unlevered

6 214 Berk/DeMarzo Corporate Finance, Second Edition firm and a separate tax shield security that paid the investor the amount of the tax shield each period, these two portfolios generate the same cash flows, so by the aw of One Price they have the same market values: V = E + D = V U + T. where T is the present value of the interest tax shield. Because these portfolios have equal cash flows, they must also have identical expected returns, which implies ErE + DrD = V U ru + TrT. where rt is the expected return associated with the interest tax shields. The relationship between re, rd, and ru will depend on the expected return rt, which is determined by the risk of the interest tax shield. 18A.3 Solving for everage and Value Simultaneously When we use the APV method, we need to know the debt level to compute the interest tax shield and determine the project s value. But if a firm maintains a constant debt-to-value ratio, we need to know the project s value to determine the debt level. When a firm maintains a constant leverage ratio, to use the APV method we must solve for the debt level and the project value simultaneously. While complicated to do by hand, it can be done in Excel using iteration. The solution is to enter the debt capacity as an Excel formula that sets the debt capacity to be the desired percentage of the project s value. Now, the levered value depends on the debt capacity, and the debt capacity depends on the levered value, but Excel will keep calculating until the values for the debt capacity and levered value of the spreadsheet are consistent Selected Concepts and Key Terms Adjusted Present Value (APV) A valuation method in which the total value of an investment is determined by first calculating its unlevered value and then adding the present value of the interest tax shields and deducting any costs that arise from other market imperfections. Debt Capacity The amount of debt at date t that is required to maintain the firm s target debt-to-value ratio. Flow-to-Equity (FTE) A valuation method in which the free cash flows available to equity holders, after taking into account all payments to and from debt holders, are discounted using the levered equity cost of capital. Free Cash Flow to Equity (FCFE) The free cash flow in a period that remains after subtracting the after-tax value of interest payments made in the period, adding the value of debt that was issued, and subtracting any debt repayments made.

7 Berk/DeMarzo Corporate Finance, Second Edition 215 Concept Check Questions and Answers What are the three methods we can use to include the value of the tax shield in the capital budgeting decision? The WACC valuation method, the adjusted present value (APV) valuation method, and the flow to equity valuation method In what situation is the risk of a project likely to match that of the overall firm? This would generally be the case only if the firm is in one line of business. For a firm with multiple divisions, the risk of the projects in different divisions may be different Describe the key steps in the WACC valuation method. The key steps in the WACC valuation method are: 1) determining the free cash flow of the investment; 2) compute the weighted average cost of capital; and 3) compute the value of the investments, including the tax benefit of leverage, by discounting the free cash flow of the investment using the WACC How does the WACC method take into account the tax shield? It uses the after-tax debt cost of capital, thus accounting for the fact that interest expense is paid before taxes and results in an interest tax shield Describe the adjusted present value (APV) method. The adjusted present value method is a valuation method in which we determine the levered value of an investment by first calculating its unlevered value, and then adding the value of the interest tax shield and deducting any costs that arise from other market imperfections At what rate should we discount the interest tax shield when a firm maintains a target leverage ratio? When a firm maintains a target leverage ratio, future interest tax shields have similar risk to the project s cash flows, and so should be discounted at the project s unlevered cost of capital Describe the key steps in the flow to equity method for valuing a levered investment. The key steps in the flow to equity method for valuing a levered investment are: 1) determine the free cash flow to equity of the investment; and 2) compute the equity value by discounting the free cash flow to equity using the equity cost of capital Why does the assumption that the firm maintains a constant debt-equity ratio simplify the flow-to-equity calculation? Because we are using the same discount rate for the free cash flow and the tax shield, the cash flows of the project and the tax shield can be combined first and then discounted at the rate ru How do we estimate a project s unlevered cost of capital when the project s risk is different from that of a firm? When a project s risk is different from that of the firm as a whole, we must estimate its cost of capital separately from that of the firm. We estimate the project s unlevered cost of capital by looking at the unlevered cost of capital for other firms with similar market risk as the project.

8 216 Berk/DeMarzo Corporate Finance, Second Edition What is the incremental debt associated with a project? The incremental debt associated with a project is the change in a firm s total debt (net of cash) with the project versus without the project What condition must the firm meet to have a constant interest coverage policy? A firm has a constant interest coverage policy if it sets debt to maintain its interest expenses as a fraction of free cash flow What is the appropriate discount rate for tax shields when the debt schedule is fixed in advance? When debt levels are set according to a fixed schedule, the appropriate discount rate for the predetermined interest tax shields is the debt cost of capital How do we deal with issuance costs and security mispricing costs in our assessment of a project s value? Issuance cost, as well as any costs or gains from mispricing of issued securities, should be included in the assessment of a project s value How would financial distress and agency costs affect a firm s use of leverage? Financial distress costs are likely to lower the expected free cash flow of a project and raise its unlevered cost of capital. Taking these effects into account together with agency costs may limit a firm s use of leverage When a firm has pre-determined tax shields, how do we measure its net debt when calculating its unlevered cost of capital? When debt is set according to a fixed schedule for some period of time, the interest tax shields for the scheduled debt are known, relatively safe cash flows. These safe cash flows will reduce the effect of leverage on the risk of the firm s equity. Thus, when a firm has predetermined tax shields, we should deduct the value of these safe tax shields from the debt when measuring the firm s net debt If the firm s debt-equity ratio changes over time, can the WACC method still be applied? When a firm does not maintain a constant debt-equity ratio for a project, the WACC method becomes more difficult to use, because when the proportion of debt financing changes, the project s equity cost of capital and WACC will not remain constant over time. With a bit of care, however, this method can still be used and, of course, will lead to same result as other methods. Examples with Step-by-Step Solutions Solving Problems Problems may involve calculating the value of a firm or project using the weighted-average cost of capital (WACC) method, the adjusted present value (APV) method, or the flow-to-equity (FTE) method under the assumption that a firm uses a constant leverage ratio. Problem 1 below demonstrates how to use each of the methods and shows that each method leads to the same value. Problems may also require determining the correct project or firm cost of capital. For projects in a different industry, or for non-public firms, the cost of capital for a specific industry can be estimated by studying single-division firms in that industry.

9 Berk/DeMarzo Corporate Finance, Second Edition 217 Other problems may apply the APV for a firm with a fixed debt schedule, a constant interest coverage ratio, or for a firm that adjusts its level of debt annually. Examples 1. You are valuing a firm that you expect will have earnings before interest and taxes (EBIT) of \$26 million every year beginning in one year. Each year, you expect that depreciation will equal capital spending and there will be no new required investments in working capital. The firm has 10 million shares outstanding, uses 50% debt financing, has a debt cost of capital of 6%, a beta of 0.8, and a tax rate of 50%. The risk-free rate is 6%, and the market risk premium is 5%. [A] What is the value per share using the weighted average cost of capital (WACC) method? [B] What is the value per share using the adjusted present value method (APV) method? [C] What is the value per share using the flow-to-equity (FTE) method? Step 1. Determine the firm s unlevered free cash flows. EBIT \$26,000,000 Interest 0 EBT 26,000,000 Taxes 13,000,000 Unlevered Net Income 13,000,000 Free Cash Flow Plus: Depreciation = Capital Spending ess: Capital Spending = Depreciation ess: Increases in Working Capital 0 Free Cash Flow \$13,000,000 Step 2. Determine the after-tax weighted-average cost of capital (WACC). The equity cost of capital can be determined from the CAPM. r = r + β( E[ R ] r )= (0.05) = 10% E f Mkt f The WACC can now be calculated as: E D rwacc = re + rd(1 τ C) = 0.50(0.10) (0.06)(1 0.50) = 6.5% E+ D E+ D Step 3. Determine the firm value, equity value, and value per share using the WACC method. Since the firm s free cash flows are a perpetuity: T FCFt FCF \$13,000,000 V 0 = = = = \$200million t= 1 (1 + r wacc) rwacc The WACC method values the firm as a whole, so the equity value is: E= V D = \$200 million 0.50(\$200 million) = \$100 million And the value per share is \$100 million \$ million = Step 4. Determine the unlevered cost of capital to use for the APV method. The unlevered equity beta, assuming that the debt beta is approximately zero, is:

10 218 Berk/DeMarzo Corporate Finance, Second Edition β = [ E/( E+ D)] β + [ D/( E+ D)] β [ E/( E+ D )] β = 0.50(0.80) = 0.40 U E D E So the unlevered cost of equity, which equals the unlevered cost of capital, is: re = rf + β ( E[ R Mkt ] r f ) = (0.05) = 8% Step 5. Determine the firm value, equity value, and value per share using the APV method. Assuming that the present value of financial distress, agency and issuance costs are zero: V = APV = V U + PV(Interest tax shield) \$13 million \$100(0.06)(0.50) = + = = \$200 million The APV method values the firm as a whole, so the equity value is: E = V D = \$200 million 0.50(\$200 million) = \$100 million. \$100 million And the value per share is = \$ million Step 6. Determine the annual free cash flow to equity (FCFE) for the FTE method. EBIT \$26,000,000 Interest (6% \$100 million) 6,000,000 EBT 20,000,000 Taxes 10,000,000 Unlevered Net Income 10,000,000 Free Cash Flow Plus: Depreciation = Capital Spending ess: Capital Spending = Depreciation ess: Increases in Working Capital 0 Plus: Net Borrowing 0 Free Cash Flow to Equity (FCFE) \$10,000,000 Step 7. Determine the equity value and value per share using the FTE method. The equity value is the present value of the flows to equity, which are a perpetuity, discounted at the levered cost of equity: E FCFE FCFE \$10,000,000 T t = = = = t= 1 (1 + r E) re 0.10 \$100 million The FTE method values the firm as a whole, so the value per share is \$100 million \$ million = 2. Suppose that the firm in problem 1 was expected to have a constant interest coverage ratio in which interest payments were going to be maintained at 50% of free cash flow. What would the value per share be now? Step 1. Determine which technique would be best to use. The easiest method would be to use here is the constant interest coverage APV equation discussed in the chapter.

11 Berk/DeMarzo Corporate Finance, Second Edition 219 Step 2. The APV is now: U U U U V = APV = V + PV(interest tax shield) = V + τck V = (1 + τck) V \$13,000,000 = ( ) = \$ Step 3. Determine the value per share now. The APV method values the firm as a whole, so the equity value is: E = V D Interest expense 0.30(13,000,000) The amount of debt, D, is now = = \$65 million rd 0.06 Assuming that the firm currently has this much debt: E = V D = \$ million \$65 million = \$ million. And the value per share is \$ million = \$ million 3. Suppose that firm in problem 1 adjusts its debt annually based on its expected future free cash flows such that its interest payments are maintained at 50% of free cash flow. What would the value per share be now? Step 1. Determine which technique would be best to use. The easiest method would be to use now is the adjustment to the APV equation discussed in the chapter which values a firm assuming that it adjusts its debt annually based on its expected future free cash flows. Step 2. The APV is now: U U 1 + r U U V = APV = V + PV(interest tax shield) = V + τ Ck V 1 + rd 1.08 = (0.30) = \$ Step 3. Determine the value per share now. The APV method values the firm as a whole, so the equity value is: E = V D Interest expense 0.30(13,000,000) The amount of debt, D, is now = = \$65 million rd 0.06 Assuming that the firm currently has this much debt: E = V D = \$ million \$65 million = \$ million. \$ million And the value per share is = \$ million The value is slightly higher than the value in problem 2 because the discount rate used for the interest tax shield is rd for the first period because it is fixed, and then ru from then on.

12 220 Berk/DeMarzo Corporate Finance, Second Edition Questions and Problems 1. You are evaluating a project that costs \$5 million. The project is expected to generate free cash flow of \$1 million the first year, and this free cash flow is expected to be the same every year. The levered equity cost of capital of 17%, the firm has a debt cost of capital of 5%, and a tax rate of 40%. Your firm maintains a debt-equity ratio of 1.0. [A] What is the NPV of the project including any tax shields from leverage? [B] [C] How much debt will the firm initially take on as a result of the project? How much of the project s value is attributable to the present value of interest tax shields? 2. As in problem 1, you are evaluating a project that costs \$5 million. The project is expected to generate free cash flow of \$1 million the first year, and this free cash flow is expected to be the same every year. The levered equity cost of capital of 17%, the firm has a debt cost of capital of 5%, and a tax rate of 40%. Your firm maintains a debt-equity ratio of 1.0. [A] [B] What is the free cash flow to equity for this project? What is its NPV computed using the FTE method? How does it compare with the NPV based on the WACC method? 3. In year 1, your firm is expected to report \$10 million in earnings before interest and taxes (EBIT). The market expects that EBIT will grow at a rate of 3% per year forever. The firm will never make new capital investments and will not change net working capital. The firm currently has \$50 million in risk-free debt, and it plans to keep a constant ratio of debt to equity every year, so that the debt will also grow by 3% per year. The corporate tax rate equals 40%, the risk-free rate is 5%, the expected return on the market is 10%, and the asset beta for this industry is Using the APV method, what is your firm s total market value? What is the value of the equity? 4. Nordstrom is spinning off its casual clothing segment, and you are trying to determine the value of the spun-off entity. You estimate that the division s free cash flow in the coming year to be \$100 million, and you expect the free cash flows will grow by 3% per year in subsequent years. Because the spin-off isn t publicly traded yet, you do not have an accurate assessment of the division s equity beta. However, you do have beta data for Gap, Inc., a firm it closely resembles. Gap has an equity beta of 1.1, a debt beta of 0.10, and a debt-equity ratio of Nordstrom has a beta of 2.2 and a debt-equity ratio of 1.0, and the spun-off firm will maintain this leverage ratio and has been told by its investment bankers that the debt cost of capital will be 6%. The corporate tax rate is 40%, the risk-free rate is 5%, and the expected return on the market portfolio is 10%. [A] Estimate the division s WACC. [B] Estimate the spun off unit s share value based on the 100 million that will be outstanding after the spinoff. 5. As in problem 1, you are evaluating a project that costs \$5 million. The project is expected to generate free cash flow of \$1 million the first year, and this free cash flow is expected to be the same every year. The levered equity cost of capital of 17%, the firm has a debt cost of capital of 5%, and a tax rate of 40%. Your firm maintains a debt-equity ratio of 1.0. [A] If the project was funded exclusively by obtaining a \$5 million bank loan, what WACC should be used to evaluate the project? [B] Suppose that the firm intends to maintain a debt-equity ratio of 1.0 and the debt associated with this project will be fixed at 50% of its value. What is the value of the tax shield now?

13 Berk/DeMarzo Corporate Finance, Second Edition 221 Solutions to Questions and Problems 1. [A] E D rwacc = re + rd(1 τ C) = 0.50(0.17) (0.05)(1 0.40) = 10% E + D E + D T FCFt FCF \$1 million NPV = = \$5 million + = 5 million + = \$5 million t=1 (1 + rwacc) rwacc 0.10 [B] The firm is financed with 50% debt, so D = 0.50(10) = \$2.5 million. [C] \$2.5(0.05)(0.40) PV(Interest tax shield) = = \$0.50 million 0.10 After-tax interest expense [A] FCFE = FCF (1 τ ) (Interest payments) + (Net borrowing) C =\$1 million (1.40)[5 0.05] = \$0.85 million T FCFEt FCFE \$0.85 million [B] E = = = = \$5 million t= 1 (1 + re) re 0.17 So, the NPV is \$5 million + 2(\$5 million) = \$5 million the same as when using the WACC method. 3. r = r + β ( E[ R ] r ) = (0.05) = 8% E V f Mkt f U = APV = V + PV(interest tax shield) = \$10 million \$50 million(0.05)(0.40) = + = = \$220 million E = = \$170 million 4. [A] Unlever the Gap beta: β = [ E /( E+ D)]β + [ D /( E+ D)]β =[1 / (1 +.2)]1.1 + [.2 / (1 +.2)]0.10 = [B] U E D Relever the beta to reflect the division s debt-equity ratio of 1.0. β = β + [ D / E](β β ) = ( )=1.77 E U U D Calculate the levered cost of equity: r = r + β ( E[ R ] r ) = (0.05) = 13.8% E f Mkt f Calculate the WACC: E D rwacc = re + rd(1 τ C) = 0.50(0.138) (0.06)(1 0.40) = 8.7% E + D E + D T FCFt FCF \$100 million V0 = = = = \$1.75 billion t= 1 (1 + rwacc) rwacc g The WACC method values the firm as a whole, so the equity value is: E = V D = \$1.75 billion 0.50(\$1.75 billion) = \$875 million. And the value per share is \$875 million \$ million =

14 222 Berk/DeMarzo Corporate Finance, Second Edition 5. [A] The loan will reduce debt elsewhere in the firm, so the appropriate debt-to-value ratio to use when evaluating the project is 50% of the project s value, not 100%. E D So rwacc = re + rd(1 τ C) = 0.50(0.17) (0.05)(1 0.40) = 10% E + D E + D [B] When a firm has a fixed debt schedule, the interest tax shields can be discounted using the debt cost of capital, r D, since the amount of the debt will not fluctuate and thus the tax shield is less risky than the project. PV(Interest tax shield) = = \$5(0.05)(0.40) \$2.0 million 0.05

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