20. a. Using the Black-Scholes model to value the equity, we get:
Putting these values into Black-Scholes:
b. The value of the debt is the firm value minus the value of the equity, so:
c. Using the equation for the PV of a continuously compounded lump sum, we get:
d. Using the Black-Scholes model to value the equity, we get:
Putting these values into Black-Scholes:
e. The value of the debt is the firm value minus the value of the equity, so:
Using the equation for the PV of a continuously compounded lump sum, we get:
When the firm accepts the new project, part of the NPV accrues to bondholders. This increases
Challenge
21. a. Using the equation for the PV of a continuously compounded lump sum, we get:
b. Using the Black-Scholes model to value the equity, we get:
Putting these values into Black-Scholes:
And using put-call parity, the price of the put option is:
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:
Using the equation for the PV of a continuously compounded lump sum to find the return on debt,
we get:
d. The value of the debt with five years to maturity at the risk-free rate is:
Using the Black-Scholes model to value the equity, we get:
Putting these values into Black-Scholes:
And using put-call parity, the price of the put option is:
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:
Using the equation for the PV of a continuously compounded lump sum to find the return on debt,
we get:
The value of the debt declines because of the time value of money; that is, it will be longer until
22. a. Using the equation for the PV of a continuously compounded lump sum, we get:
b. Using the Black-Scholes model to value the equity, we get:
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:
Using the equation for the PV of a continuously compounded lump sum to find the return on debt,
we get:
d. Using the equation for the PV of a continuously compounded lump sum, we get:
Using the Black-Scholes model to value the equity, we get:
Putting these values into Black-Scholes:
And using put-call parity, the price of the put option is:
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:
Using the equation for the PV of a continuously compounded lump sum to find the return on debt,
we get:
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: $33,098.68 – 36,760.10 = –$3,661.42
This is an agency problem for bondholders. Management, acting to increase shareholder wealth in
23. a. Going back to the chapter on dividends, the price of the stock will decline by the amount of the
b. Using the Black-Scholes model with dividends, we get:
24. a. Going back to the chapter on dividends, the price of the stock will decline by the amount of the
b. Using put-call parity to find the price of the put option, we get:
25. N(d1) is the probability that “z” is less than or equal to N(d1), so 1 – N(d1) is the probability that “z”
26. From put-call parity:
P = E × e–Rt + C – S
Substituting the Black–Scholes call option formula for C and using the result in the previous
question produces the put option formula:
P = E × e–Rt + C – S
P = E × e–Rt + S × N(d1) – E × e–Rt × N(d2) – S
P = S × (N(d1) – 1) + E × e–Rt × (1 – N(d2))
P = E × e–Rt × N(–d2) – S × N(–d1)
27. Based on Black-Scholes, the call option is worth $50! The reason is that present value of the exercise
price is zero, so the second term disappears. Also, d1 is infinite, so N(d1) is equal to 1. The problem is
28. The delta of the call option is N(d1) and the delta of the put option is N(d1) – 1. Since you are selling