Corporate_Finance_9th_edition_Solutions_Manual_FINAL0

The number of shares that should be shorted sell is

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Unformatted text preview: alues into the Black-Scholes model, we find the equity value is: 1 ) = –.0115 1 ) = .5185 1 ) = .4136 1 ) = .7536 446 Equity = $27,200(.6979) – ($25,000e–.05(1))(.4954) = $7,202.84 The value of the debt is the firm value minus the value of the equity, so: D = $27,200 – 7,202.84 = $19,997.16 The return on the company’s debt is: $19,997.16 = $25,000e–R(1) .79989 = e–R RD = –ln(.79989) = 22.33% 24. a. The combined value of equity and debt of the two firms is: Equity = $3,111.31 + 7,202.84 = $10,314.15 Debt = $12,688.69 + 19,997.16 = $32,685.85 b. For the new firm, the combined market value of assets is $43,000, and the combined face value of debt is $40,000. Using Black-Scholes to find the value of equity for the new firm, we find: d1 = [ln($43,000/$40,000) + (.05 + .292/2) × 1] / (.29 × d2 = .5668 – (.29 × N(d1) = .7146 N(d2) = .6090 Putting these values into the Black-Scholes model, we find the equity value is: E = $43,000(.7146) – ($40,000e–.05(1))(.6090) = $7,553.51 The value of the debt is the firm value minus the value of the equity, so: D = $40,000 – 7,553.31 = $35,446.49 c. The change in the value of the firm’s equity is: Equity value change = $7,553.51 – 10,314.15 = –$2,760.64 The change in the value of the firm’s debt is: Debt = $35,446.49 – 32,685.85 = $2,760.64 d. In a purely financial merger, when the standard deviation of the assets declines, the value of the equity declines as well. The shareholders will lose exactly the amount the bondholders gain. The bondholders gain as a result of the coinsurance effect. That is, it is less likely that the new company will default on the debt. 1 ) = .2768 1 ) = .5668 447 25. a. Using Black-Scholes model to value the equity, we get: d1 = [ln($21,000,000/$25,000,000) + (.06 + .392/2) × 10] / (.39 × d2 = .9618 – (.39 × N(d1) = .8319 N(d2) = .3930 Putting these values into Black-Scholes: E = $21,000,000(.8319) – ($25,000,000e–.06(10))(.3930) = $12,078,243.48 b. The value of the debt is the firm value minus the value of the equity, so: D = $21,000,000 – 12,078,243.48 = $8,921,756.52 c. Using the equation for the PV of a continuously compounded lump sum, we get: $8,921,756.52 = $25,000,000e–R(10) .35687 = e–R10 RD = –(1/10)ln(.35687) = 10.30% d. Using Black-Scholes model to value the equity, we get: d1 = [ln($22,200,000/$25,000,000) + (.06 + .392/2) × 10] / (.39 × d2 = 1.0068 – (.39 × N(d1) = .8430 N(d2) = .4104 Putting these values into Black-Scholes: E = $22,200,000(.8430) – ($25,000,000e–.06(10))(.4104) = $13,083,301.04 e. The value of the debt is the firm value minus the value of the equity, so: D = $22,200,000 – 13,083,301.04 = $9,116,698.96 Using the equation for the PV of a continuously compounded lump sum, we get: $9,116,698.96 = $25,000,000e–R(10) .36467 = e–R10 RD = –(1/10)ln(.36467) = 10.09% 10 ) = –.2265 10 ) = 1.0068 10 ) = –.2715 10 ) = .9618 448 When the firm accepts the new project, part of the NPV accrues to bondholders. This increases the present value of the bond, thus reducing the return on the bond. Additionally, the new project makes the firm safer in the sense that it increases the value of assets, thus increasing the probability the call will end in-the-money and the bondholders will receive their payment. 26. a. In order to solve a problem using the two-state option model, we first need to draw a stock price tree containing both the current stock price and the stock’s possible values at the time of the option’s expiration. Next, we can draw a similar tree for the option, designating what its value will be at expiration given either of the 2 possible stock price movements. Price of stock Today 1 year $98 $85 $70 ? $0 =Max(0, $70 – 85) Today Call option price with a strike of $85 1 year $13 =Max(0, $98 – 85) The stock price today is $85. It will either increase to $98 or decrease to $70 in one year. If the stock price rises to $98, the call will be exercised for $85 and a payoff of $13 will be received at expiration. If the stock price falls to $70, the option will not be exercised, and the payoff at expiration will be zero. If the stock price rises, its return over the period is 22.50 percent [= ($98/$80) – 1]. If the stock price falls, its return over the period is –12.50 percent [= ($70/$80) – 1]. We can use the following expression to determine the risk-neutral probability of a rise in the price of the stock: Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) Risk-free rate = (ProbabilityRise)(ReturnRise) + (1 – ProbabilityRise)(ReturnFall) .025 = (ProbabilityRise)(0.2250) + (1 – ProbabilityRise)(–0.1250) ProbabilityRise = .4286 or 42.86% This means the risk neutral probability of a stock price decrease is: ProbabilityFall = 1 – ProbabilityRise ProbabilityFall = 1 – .4286 ProbabilityFall = .5714 or 57.14% Using these risk-neutral probabilities, we can now determine the expected payoff of the call option at expiration. The expected payoff at expiration is:...
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