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Unformatted text preview: Finance 100 Problem Set Capital Structure (Alternative Solutions) Note: Where appropriate, the “final answer” for each problem is given in bold italics for those not interested in the discussion of the solution. I. Formulas This section contains the formulas you might need for this homework set: 1. The Weighted Average Cost of Capital (WACC): r A = r E ( E V ) + r D ( D V ) (1) where r A is the return on assets, r E is the return on equity, r D is the return on debt, E is the value of equity, D is the value of debt and V is the total value of the firm ( D + E ). 2. The Weighted Average of Debt and Equity Betas: β A = β E ( E V ) + β D ( D V ) (2) where β A is the beta on assets, βE is the equity beta, βD is the debt beta, E is the value of equity, D is the value of debt and V is the total value of the firm ( D + E ). 3. The CAPM: E ( r i ) = r f + β i ( E ( r M ) r f ) (3) 1 where E ( r i ) is the expected return on any asset i , r f is the riskfree return, E ( r M ) is the expected return on the market and β i is the asset’s beta, also equal to: β i = Cov ( r i ,r M ) V ar ( r M ) (4) Often we drop the expectations and simply used realized rates of return. 4. Present Value of an Annuity Formula: A = a (1 + i ) + a (1 + i ) 2 + ... + a (1 + i ) N 1 + a (1 + i ) N = a · 1 (1 + i ) N i (5) where a is the amount of the annuity payment, i is the periodic interest rate and N is the total number of periods. II. Problems 1. The CAPM (equation (3)) implies (on average): r E = r f + β E ( r M r f ) r D = r f + β D ( r M r f ) Substituting in for the unknowns in both equations yields: r E = 0 . 10 + 1 . 5(0 . 18 . 10) = 0 . 22 . 12 = 0 . 10 + β D (0 . 18 . 10) = ⇒ β D = 0 . 25 Since D/E = 1, we have E/V = D/V = 0 . 5. Thus, the WACC implies r A = r E ( E V ) + r D ( D V ) = 0 . 22 × . 5 + 0 . 12 × . 5 = 0 . 17 Similarly, the equation (2) implies β A = β E ( E V ) + β D ( D V ) = 1 . 5 × . 5 + 0 . 25 × . 5 = 0 . 875 2 2. 2.a The debt beta is zero since the return to debt is equal to the risk free return. The asset beta follows from equation (2): β A = β E ( E V ) + β D ( D V ) = 0 . 8 × . 90 + 0 × . 10 = 0 . 72 The cost of capital follows from the CAPM relation: E ( r A ) = r f + β A ( E ( r M ) r f ) = 0 . 05 + 0 . 72(0 . 13 . 05) = 0 . 1076 The risk premium of the stock market is the difference between the return on the market and the riskfree asset, or 13% 5% = 8% . 2.b The company’s equity is worth $360 million and is 90% of the total value of the company implying the total value of the company is $ 400 million. The total amount of new debt outstanding is 60%, or 0 . 6 × $400 = $240 million . Hence the company has to issue $ 200 million of debt to pay a dividend of $ 200 million....
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 Spring '09
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