Corporate_Finance_9th_edition_Solutions_Manual_FINAL0

The price of a call option will decrease when the

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Unformatted text preview: holes: E = $19,000(.6188) – ($40,000e–.05(5))(.1493) = $7,105.26 And using put-call parity, the price of the put option is: Put = $40,000e–.05(5) + $7,105.26 – $19,000 = $19,257.29 The value of a risky bond is the value of a risk-free bond minus the value of a put option on the firm’s equity, so: Value of risky bond = $31,152.03 – 19,257.29 = $11,894.74 Using the equation for the PV of a continuously compounded lump sum to find the return on debt, we get: Return on debt: $11,894.74 = $40,000e–R(5) .29737 = e–R5 RD = –(1/5)ln(.29737) = 24.26% The value of the debt declines because of the time value of money, i.e., it will be longer until shareholders receive their payment. However, the required return on the debt declines. Under the current situation, it is not likely the company will have the assets to pay off bondholders. Under the new plan where the company operates for five more years, the probability of increasing the value of assets to meet or exceed the face value of debt is higher than if the company only operates for two more years. 31. a. Using the equation for the PV of a continuously compounded lump sum, we get: PV = $50,000 × e–.06(5) = $37,040.91 5 ) = .3023 5 ) = –1.0394 457 b. Using Black-Scholes model to value the equity, we get: d1 = [ln($46,000/$50,000) + (.06 + .502/2) × 5] / (.50 × d2 = .7528 – (.50 × N(d1) = .7742 N(d2) = .3575 Putting these values into Black-Scholes: E = $46,000(.7742) – ($50,000e–.06(5))(.3575) = $22,372.93 And using put-call parity, the price of the put option is: Put = $50,000e–.06(5) + 22,372.93 – 46,000 = $13,413.84 c. The value of a risky bond is the value of a risk-free bond minus the value of a put option on the firm’s equity, so: Value of risky bond = $37,040.91 – 13,413.84 = $23,627.07 Using the equation for the PV of a continuously compounded lump sum to find the return on debt, we get: Return on debt: $23,627.07 = $50,000e–R(5) .4725 = e–R(5) RD = –(1/5)ln(.4725) = 14.99% d. Using the equation for the PV of a continuously compounded lump sum, we get: PV = $50,000 × e–.06(5) = $37,040.91 Using Black-Scholes model to value the equity, we get: d1 = [ln($46,000/$50,000) + (.06 + .602/2) × 5] / (.60 × d2 = .8323 – (.60 × N(d1) = .7974 N(d2) = .3052 Putting these values into Black-Scholes: E = $46,000(.7974) – ($50,000e–.06(5))(.3052) = $25,372.50 5 ) = .7528 5 ) = –.3653 5 ) = .8323 5 ) = –.5094 458 And using put-call parity, the price of the put option is: Put = $50,000e–.06(5) + 25,372.50 – 46,000 = $16,413.41 The value of a risky bond is the value of a risk-free bond minus the value of a put option on the firm’s equity, so: Value of risky bond = $37,040.91 – 16,413.41 = $20,627.50 Using the equation for the PV of a continuously compounded lump sum to find the return on debt, we get: Return on debt: $20,627.50 = $50,000e–R(5) .41255 = e–R(5) RD = –(1/5)ln(.41255) = 17.71% The value of the debt declines. Since the standard deviation of the company’s assets increases, the value of the put option on the face value of the bond increases, which decreases the bond’s current value. e. From c and d, bondholders lose: $20,627.50 – 23,627.07 = –$2,999.57 From c and d, stockholders gain: $25,372.50 – 22,372.93 = $2,999.57 This is an agency problem for bondholders. Management, acting to increase shareholder wealth in this manner, will reduce bondholder wealth by the exact amount by which shareholder wealth is increased. 32. a. Since the equityholders of a firm financed partially with debt can be thought of as holding a call option on the assets of the firm with a strike price equal to the debt’s face value and a time to expiration equal to the debt’s time to maturity, the value of the company’s equity equals a call option with a strike price of $320 million and 1 year until expiration. In order to value this option using the two-state option model, first draw a tree containing both the current value of the firm and the firm’s possible values at the time of the option’s expiration. Next, draw a similar tree for the option, designating what its value will be at expiration given either of the 2 possible changes in the firm’s value. The value of the company today is $300 million. It will either increase to $380 million or decrease to $210 million in one year as a result of its new project. If the firm’s value increases to $380 million, the equityholders will exercise their call option, and they will receive a payoff of $60 million at expiration. However, if the firm’s value decreases to $210 million, the equityholders will not exercise their call option, and they will receive no payoff at expiration. 459 Value of company (in millions) Today 1 year $380 $300 $210 Equityholders’ call option price with a strike of $320 (in millions) Today 1 year $60 ? $0 =Max(0, $210 – 320) =Max(0, $380 – 320) If the project is successful and the company’s value rises, the percentage incre...
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