# According to Modigliani-Miller Proposition II with corporate taxes, the value of levered equity is:

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1 Homework 2 1. A project has a NPV, assuming all equity financing, of \$1.5 million. To finance the project, debt is issued with associated flotation costs of \$60,000. The flotation costs can be amortized over the project's 5 year life. The debt of \$10 million is issued at 10% interest, with principal repaid in a lump sum at the end of the fifth year. If the firm's tax rate is 34%, calculate the project's APV.15 points NPV of all equity financed project = \$1.5 million Amortization per period =\$60,000/5=\$12,000. Tax shield of amortization per period= \$12,000 (0.34) PV of flotation costs including deductibility of expenses is: -\$60,000 + \$12,000(.34) (1/ / / ) = -\$44,534 Tax shield of interest per period 10,000,000 (0.10) (0.34)=340,000 PV(TS interest )=\$ 340,000 (1/ / / )=\$ 1,288,868 (or, you can compute the NPV of the loan: NPV LOAN = amount borrowed PV(after tax interest payment)- PV (principal) NPV LOAN = \$10,000,000 - \$10,000,000 (0.10) (1-0.34) (1/ / / ) - \$10,000,000/ = \$1,288,868 ) Therefore, APV = \$1,500,000 + \$1,288,868 - \$44,534 = \$2,744, Imagine Inc. just issued has a D/E ratio of 0.4. The required return on the company s unlevered equity is 16% and the pretax cost of the firm s debt is 7%. Sales revenue for the company is expected to remain stable indefinitely at last year s level of \$ 40 million. Variable costs amount to 50% of sales. The tax rate is 35% and the company distributes all its earnings as dividends at the end of each year. ( total 20 points) a. If the company were financed entirely by equity, how much would it be worth? (5 p) If the company were financed entirely by equity, the value of the firm would be equal to the present value of its unlevered after-tax earnings, discounted at its unlevered cost of capital. First, we need to find the company s unlevered cash flows, which are: Sales \$ 40,000,000

2 Variable costs 20,000,000 EBT \$ 20,000,000 Tax 7,000,000 Net income \$ 13,000,000 So, the value of the unlevered company is: V U = \$13,000,000 /.16 V U = \$81,250, b. What is the required return on the firm s levered equity? (5p) According to Modigliani-Miller Proposition II with corporate taxes, the value of levered equity is: R S = R 0 + (B/S)(R 0 R B )(1 t C ) R S =.16 + (.4)(.16.07)(1.35) R S =.1834 or 18.34% c. Use the weighted average cost of capital method to calculate the value of the company. What is the value of the company s equity? What is the value of the company s debt? (5p) In a world with corporate taxes, a firm s weighted average cost of capital equals: R WACC = [B / (B + S)](1 t C )R B + [S / (B + S)]R S So we need the debt-value and equity-value ratios for the company. The debt-equity ratio for the company is: B/S = 0.4 Substituting this in the debt-value ratio, we get: B/V =.4 / (.4 + 1) B/V =.29 And the equity-value ratio is one minus the debt-value ratio, or: S/V = 1.29 S/V =.71 So, using the capital structure weights, the company s WACC is: R WACC = [B / (B + S)](1 t C )R B + [S / (B + S)]R S R WACC =.29(1.38)(.07) +.71(.1834) R WACC =.1440 or 14.40% We can use the weighted average cost of capital to discount the firm s unlevered after tax earnings to value the company. Doing so, we find: V L = \$13,000,000 /.1440 V L = \$ 90,277,777.78

3 Now we can use the debt-value ratio and equity-value ratio to find the value of debt and equity, which are: B = V L (Debt-value) B = \$ 90,277, (.29) B = \$ 25,793, S = V L - B S= 90,277, ,793, S= \$ 64,484, d. Use the flow to equity method to calculate the value of the company s equity. (5p) In order to value a firm s equity using the flow-to-equity approach, we can discount the cash flows available to equity holders at the cost of the firm s levered equity. First, we need to calculate the levered cash flows available to shareholders, which are: Sales \$ 40,000,000 Variable costs 20,000,000 EBIT \$ 20,000,000 Interest 1,805,556 EBT \$ 18,194,444 Tax 6,368,056 Net income \$ 11,826,389 So, the value of equity with the flow-to-equity method is: S = Cash flows available to equity holders / R S S = \$11,826,389/.1834 S = \$ 64,484, Leveraged buyout( total 40 p) A leveraged buyout (LBO) is the acquisition by a small group of equity investors of a public or private company. Generally, an LBO is financed primarily with debt. The new shareholders service the heavy interest and principal payments with cash from operations and/or asset sales. The shareholders generally hope to reverse the LBO within three to seven years by way of a public offering or sale of the company to another firm. A buyout is therefore likely to be successful only if the firm generates enough cash to serve the debt in the early years, and if the company is attractive to other buyers as the buyout matures. In a leveraged buyout, the equity investors are expected to pay off outstanding principal according to a specific timetable. The owners know that the firm s debt-equity ratio will fall and can forecast the dollar amount of debt needed to finance future operations. Sunny food Inc. (SFI) is a young company operating in the online game development. Although in the growing stage, the company s cash flows are relatively steady. The

4 company is currently all equity financed. The company s financial analyst, M. Weng, thinks that the company would be better off if its debt to value ratio is kept at 25%. Moreover, he thinks that the company s price to earnings ratio is low enough to attract acquisition of richer capital. As a result, the company is a good candidate for a leveraged buyout. After several contact with his partner, Brenne Capital, Weng provide projects of the cash flows for the company as follows (in million yuan): Sales 2, , , , , Costs Depreciation EBT 1, , , , , Capital expenditures Asset sales 1, At the end of five years, Weng estimates that the growth rate in cash flows will be 5% per year. The capital expenditures are for new projects and the replacement of equipment that wears out. After a thorough analysis on the company s financials, Brenne Capital finds that in five years, the company can be resold or go public. The acquisition will involve a considerable amount of debt for each of the next five years if the LBO is undertaken. The interest payments in each year are estimated by: Interest payments 1,500 1,600 1,600 1,500 1,500 The company currently has a required return on assets of 12%. The yield to maturity of debt used for acquisition is estimated to be 10% for the next five years. When the debt is refinanced in five years, the new yield to maturity is estimated to be 7%. SFI has currently 200 million shares of stock outstanding that sell for \$80 per share. The corporate tax rate is 35%. a. What is the value of SFI if it remains all equity financed? (10 points) The value of the all-equity financed firm, can be calculated by two steps Step 1: Calculating the present value of unlevered cash flows for the first five years. Step 2: Calculating the present value of the unlevered cash flows beyond the first five years. Step 1: Calculating the present value of unlevered cash flows for the first five years. The income statement presented does not include interest, so it is the projected unlevered cash flows of the company. To find the cash flows each year, we find the operating cash flow by adding depreciation back to net income. Next, we subtract any capital expenditures, changes in net working capital, and add the asset sales. So, the unlevered cash flows can be computed as CFu= Net income Capital expenditure + asset sales CFu= EBT tax capital expenditures + asset sales

5 Hence the unlevered CF for each year will be: Sales 2, , , , , Costs Depreciation EBT 1, , , , , Tax Net income , Capital expenditures Asset sales 1, Unlevered cash flows 1, , , , , Since these are unlevered cash flows, we need to discount at the unlevered cost of equity. Because the company currently has no debt, the cost of unlevered equity is equal to the required return on assets (Ro=12%). So, using this discount rate, we find the present value of the unlevered cash flows for the next five years will be: PV = 1,850/ , / , / ,124.19/ ,259.90/ PV = 5, Step 2: Calculating the present value of the unlevered cash flows beyond the first five years. The assumption given is that the cash flows will grow at 5% into perpetuity. Again, we discount these cash flows at the unlevered return on equity. So, the value of these cash flows in Year 5 will be: Unlevered CF value in Year 5 = [1, (1 +.05)] / (.12.05) Unlevered CF value in Year 5 = 18, The value today of this terminal value is: PV = 18, / PV = 10, Hence, the value of the unlevered firm is Vu= PV (CF in 1 st five years)+ PV (CF after the 1 st five years) Vu= 5, , Vu= 15, b. After LBO, if debt is reduced and maintained at 25 % of the value of the firm, what will be the weighted average cost of capital for SFI in five years? (5p)

6 25% can be seen as the target debt-value ratio for the company after LBO. Note that after 5 years, the new debt cost becomes 7% and the cost of unlevered equity capital is just R0= 12%. The levered cost of equity can be found with to Modigliani-Miller Proposition II with corporate taxes: R S = R 0 + (B/S)(R 0 R B )(1 t C ) R S =.12+ (.25/0.75)( )(1.35) R S = 13.08% Now, we can calculate the WACC for the company beyond Year 5. The WACC at this point will be: R WACC = [B / (B + S)](1 t C )R B + [S / (B + S)]R S R WACC = [.25](1.35)(.07) + [0.75](.1281) R WACC = 10.95% c. After LBO, what is the value of SFI in five years, providing that the debt to value ratio is 25%? (5p) It is appropriate to use the WACC method to calculate a terminal value for the firm at the target capital structure. The terminal value of the levered company, which will be: V L = CF 5 (1+g)/( R WACC - g) V L = [ (1 +.05)] / ( ) V L = 22, d. After LBO, what is the value of interest tax shield in five years, providing that the debt to value ratio is 25%?(5p) We must calculate the value of tax shields associated with debt used to finance the operations of the company after the first five years. The assumption given in the case is that debt will be reduced and D/E maintained at 25 percent from that date forward. Under this assumption it is appropriate to use the WACC method to calculate a terminal value for the firm at the target capital structure. This in turn can be decomposed into an all-equity value and a value from tax shields. Note that we need to use the interest rate on the debt beyond Year 5 in these calculations. Using Modigliani-Miller s valuation of a levered firm: V L = V U + tcb we can value the interest tax shield as: 22, = 18, Interest tax shield Interest tax shield in five years= 3, e. What is the adjusted present value of SFI if it is undertaken by LBO? (10p)

7 Using APV, the value of SFI is the sum of the value of unlevered firm plus the present value of The interest tax shields for the first five years and that beyond the first five years. The present value of interest tax shields for the first five years is equal to 35%* interest payment. They are computed as Interest payments 1, , , , , Interest tax shield The interest tax shield each year is the interest paid times the tax rate. To find the present value of the interest tax shield, we need to discount these at the pretax cost of debt, so the present value of the interest tax shield for the first five years is: PV = (1,500)(.35) / (1,600)(.35) / (\$1,600)(.35) / (\$1,500)(.35) / (\$1,500)(.35) / PV = 2, While the present value of the interest tax shield beyond the first 5 years is PV =3, / PV =2, To sum up, Value of unlevered CF =10, Value of unlevered CF = 15, And the value of the interest tax shield today is: Value of interest tax shield = , Value of interest tax shield = So, the total value of the company today is: Value of company today =15, Value of company today = f. If Brenne Capital decides to undertake LBO, what is the most they should offer per share? the most the group should offer per share is: Price = /200 Price = According to the US system, a right is attached to each share of stock owned by the existing shareholders (definition by Ross, Jaffe and Westerfield). For example, if two old shares are required to purchase one new share, then the existing shareholders need two

8 rights in order to get one new share. Hence, the number of rights he receives is equal to the number of old shares he owns. Show that with this definition, the value of a right can be derived as Value of a right = Po Pex = (Po Ps)/(N+1) Where Po, Ps, Pex stand for the price for old shares, the subscription price and the ex-right price, respectively, and N is the number of rights needed to buy one new share at the subscription price. (10p) Using P O as the price for old share, and P S as the subscription price, we can express the price per share of the stock ex-rights as: Pe X = [NP O + P S ]/(N + 1) And the equation for the value of a right is: Value of a right = P O Pe X Substituting the ex-rights price equation into the equation for the value of a right and rearranging, we get: Value of a right = P O {[NP O + P S ]/(N + 1)} Value of a right = [(N + 1)P O NP O P S ]/(N+1) Value of a right = [P O P S ]/(N + 1) 5. Bright Life Inc currently has 1 million shares outstanding sold at \$15 per share. The firm is planning to raise \$3 million to finance a new project. What is the ex-right stock price, the value of a right, and the appropriate subscription prices, if (total 15p) a. Two shares of outstanding stock are entitled to purchase one additional share of the new issue. (5p) The number of new shares offered through the rights offering is the existing shares divided by the rights per share, or: New shares = 1,000,000 / 2 New shares = 500,000 And the new price per share after the offering will be: Pex = Pex= Current market value + Proceeds from offer Old shares + Newshares 1,000,000(\$15) + \$3,000,000 1,000, ,000 Pex = \$12

9 The subscription price is the amount raised divided by the number of new shares offered, or: Subscription price = \$3,000,000 / 500,000 Subscription price = \$6 And the value of a right is: Value of a right = (Ex-rights price Subscription price) / Rights needed to buy a share of stock Value of a right = (\$12 6) / 2 Value of a right = \$3 Pex = b. Four shares of outstanding stock are entitled to purchase one additional share of the new issue. (5p) Following the same procedure, the number of new shares offered through the rights offering is: New shares = 1,000,000 / 4 New shares = 250,000 And the new price per share after the offering will be: Current market value + Proceeds from offer Old shares + New shares Pex = 1,000,000(\$15) + \$3,000,000 1,000, ,000 Pex = \$14.40 The subscription price is the amount raised divided by the number of new shares offered, or: Subscription price = \$3,000,000 / 250,000 Subscription price = \$12 And the value of a right is: Value of a right = (Ex-rights price Subscription price) / Rights needed to buy a share of stock Value of a right = (\$ ) / 4 Value of a right = \$0.6 c. How does the stockholders wealth change from a to b? (5p) Since rights issues are constructed so that existing shareholders' proportionate share will remain unchanged, we know that the stockholders wealth should be the same between the

10 two arrangements. However, a numerical example makes this clearer. Assume that an investor holds 4 shares, and will exercise under either a or b. Prior to exercise, the investor's portfolio value is: Current portfolio value = Number of shares Stock price Current portfolio value = 4(\$15) Current portfolio value = \$60 After exercise, the value of the portfolio will be the new number of shares time the ex-rights price, less the subscription price paid. Under a, the investor gets 2 new shares, so portfolio value will be: New portfolio value = 6(\$12) 2(\$6) New portfolio value = \$60 Under b, the investor gets 1 new share, so portfolio value will be: New portfolio value = 5(\$14.40) 1(\$12) New portfolio value = \$60 So, the shareholder's wealth position is unchanged either by the rights issue itself, or the choice of which right's issue the firm chooses.

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