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1 1 1 1 0 0 0 Value of company Bonds Warrants Bonds + warrants Solvay Business School Risky debt A.Farber Introduction Risky debt valuation - Binomial model Professor André Farber Revised May 2003 This worksheet illustrates the valuation of a risky zero-coupon based on a three-step binomial version of Merton's model. An all-equity firm wants to raise some amount of external financing (the proceed of the issue) to finance an investment project. It considers issuing a 3-year zero-coupon. You are asked to analyze the term of the issue using a binomial model with 1 step per year. You are given the following data: Market data: risk-free interest rate , market risk premium Company data: number of shares outstanding, current stock price, standard deviation of returns (volatility), beta asset Zero-coupon: Face value, Maturity (3 years), Issue price You are asked: - to calculate the market values of the equity and of the debt following the issue; - to compute the expected return on equity and the expected return on debt Solvay Business School Risky debt A.Farber Introduction the issue) Solvay Business School Risky debt Page 4/11 Data and questions Risky debt valuation using the binomial model Professor André Farber Basic data: Number shares 500 Price per share 25 Volatility 40% Beta Asset 1.2 Market risk premium 5% Riskless rate 4% Zero-coupon Face value of issue 12,000 Maturity 3 Proceed 10,000 To analyze this issue, use a binomial model with 1 step per year. 1. Build a three-step binomial tree for the value of company We will first calculate the market values of debt and equity. Remember that to achieve this, we apply the risk-neutral valuation principle. We assume that we are in a risk-neutral world. This is done by using risk-neutral probabilities. 2. Calculate the value of the debt and of the equity at maturity 3. Based on the risk-neutral valuation principle, compute the following: - the market value of the debt - the market value of the equity - the market value of the option to default 4. Is the issue properly priced? Who gains or looses from any mispricing? The next questions analyze expected returns on the equity and on the debt. To do so, it is useful to move back to the "real" world and to calculate the "true" probabilities. These true probabilities are obtained in two steps. Step 1: Based on the CAPM, compute the expected return on equity of the unlevered firm (ra) Step 2: Compute the true probability (q) as the solution of: q*u + (1-q) * d = 1+ra 5. Calculate the expected return on the equity of the levered firm. Two differents methods leading to the same result: a) To use the CAPM, proceed as follow: - calculate the delta of equity - calculate the beta of equity for the levered firm Solvay Business School Risky debt Page 5/11 Data and questions - use CAPM equation to get the expected return on equity b) Direct calculation based on true probabilities: - calculate the realized returns in the up and down cases; - calculate the expected return using the true probabilities. 6. Calculate the expected return on the debt of the levered firm. Once again, two different methods lead to the same result: a) View the risky debt as a portfolio composed of risk-free debt and a (short) put option Step 1: calculate the expected return on the put option - calculate the delta of the put (using the put-call parity relationship or based on the binomial tree) -calculate the beta of the put -use the CAPM equation to get the expected return on the put option Step 2: calculate the expected return on the risky debt as a weighted average of returns on the risk-free debt and on the put option b) Direct calculation based on true probabilities: - calculate the realized returns in the up and down cases; - calculate the expected return using the true probabilities. 7. The final step in our analysis is to calculate the weighted average cost of capital. This a way to check whether previous results are correct. MM 1958 applies in this model: the WACC is independent of the capital structure. Solvay Business School Risky debt Page 6/11 Data and questions Solvay Business School Risky debt Page 7/11 Data and questions binomial tree) Solvay Business School Risky debt Page 8/11 Solution Risky debt valuation DATA SOLUTION (summary - see below for details) Number shares 500 Initial value of equity 12,500 Price per share 25.00 Value of company 22,500 Volatility 40.00% Equity 12,605 Beta Asset 1.20 Debt 9,895 Market risk premium 5% Yield to maturity 6.64% Riskless rate 4.00% Zero-coupon Interest rate on issue 6.27% Face value of issue 12,000 Expected return on equity 13.86% Maturity 3 Expected return on debt 5.08% Proceed 10,000 WACC 10.00% Parameters of binomial model (1 step per year) u= 1.4918 d= 0.6703 pneutral = 0.450 Time 0 1 2 3 Risk neutral probabilities 1. VALUE OF COMPANY # paths Proba/path Probability 74,703 1 9.11% 9.11% 50,075 33,566 33,566 3 11.14% 33.41% 22,500 22,500 15,082 15,082 3 13.61% 40.84% 10,110 6,777 1 16.64% 16.64% Values at maturity VALUE OF ZERO-COUPON BOND Riskless debt - Put = Risky debt Equity 12,000 12,000 0 12,000 62,703 11,538 11,095 12,000 12,000 0 12,000 21,566 9,895 11,538 9,634 12,000 12,000 0 12,000 3,082 8,776 6,777 12,000 5,223 6,777 0 Risk-neutral expected future values 12,000 869 11,131 14,178 Discount factor (using the risk-free interest rate) 0.889 0.889 0.889 0.889 3. Market values 10,668 773 9,895 12,605 2fab2ba1-cdb5-49cf-a152-becffa58048d.xls 4/14/2011 Solvay Business School Risky debt Page 9/11 Solution Notes: Market value of the company 22,500 Value before issue 12,500 + New issue 10,000 Given the term of the issue of the zero coupon Market value of equity 12,605 Market value of debt 9,895 Yield to maturity (this is not an expected return) 6.64% The value of the option to bankrupt is the value of the put option Market value of riskless debt 10,668 Market value of risky debt 9,895 Value of the put option 773 4. Note: the bond issue in overvalued Issue price 10,000 Market value 9,895 Difference 105 The mispricing has an impact on the market value of equity: Market value of equity after issue 12,605 = Initial market value of equity 12,500 Difference 105 5. Expected return on equity Beta asset 1.20 Risk premium on the market portfolio 5.00% Expected return on unlevered firm (CAPM) 10.00% Proba of up movement (true proba) q 0.523 > risk-neutral p = .450 At time 1: V E D Up 33,566 22,471 11,095 Down 15,082 5,448 9,634 Delta of equity (viewed as a call option) 0.92 Reminder : delta = (Cu-Cd)/(uS-dS) Omega of equity = delta *V/E 1.64 Beta equity = Omega * Beta asset 1.97 Expected return on equity 13.86% using CAPM Other calculation leading to same result: Return on equity if up 78.28% Return on equity if down -56.78% Expected return on equity 13.86% Note: the expected return on equity will vary over time as the value of the company changes. 2fab2ba1-cdb5-49cf-a152-becffa58048d.xls 4/14/2011 Solvay Business School Risky debt Page 10/11 Solution 6. Expected return on debt At time 1: Riskless V debt Put Up 33,566 11,095 0 Down 15,082 11,095 1,461 Delta put -0.08 =delta call - 1 or (Pu-Pd)/(uS-dS) Omega of the put option = Delta*V/Put -2.30 Beta of the put option = Omega * Beta asset -2.76 Expected return on the put option -9.81% Check Return on put if up -100.00% per period Return on put if down 89.09% per period Expected return -9.81% Beta debt 0.22 Expected return on the risky debt 5.08% Details of calculation Market Fraction Expected Beta Value invested return Riskless debt 10,668 1.078 4.00% 0.00 Put -773 -0.078 -9.81% -2.76 Risky debt 9,895 1.000 5.08% 0.22 Direct calculation. Current value of risky debt 9,895 At time 1: V Debt Return Up 33,566 11,095 12.12% Down 15,082 9,634 -2.64% Expected return on the risky debt 5.08% Note: the expected return on the risky debt varies over time 7. Weighted average cost of capital Market Expected value return Company 22,500 100.00% Equity 12,605 13.86% 56.02% Debt 9,895 5.08% 43.98% Cost of debt Weighted average cost of capital 10.00% Reminder : by CAPM using beta asset 10.00% This had to be expected since we are in a MM world 2fab2ba1-cdb5-49cf-a152-becffa58048d.xls 4/14/2011 Solvay Business School Risky debt Page 11/11 Solution Here we illustrate the evolution over time of the beta equity, cost of equity, beta debt, cost of debt 0 1 2 Beta equity 1.56 1.79 1.97 2.46 2.58 3.38 Expected return on equity 11.80% 12.96% 13.86% 16.32% 16.91% 20.88% Beta debt 0.00 0.00 0.22 0.00 0.42 0.87 Expected return on risky debt 4.00% 4.00% 5.08% 4.00% 6.09% 8.35% 2fab2ba1-cdb5-49cf-a152-becffa58048d.xls 4/14/2011

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