a)
To compute: The risk-free
Introduction:
Black-Scholes is the model of pricing that is used to determine the theoretical value or fair price for a put option or call option, depending on six variables like type of option, volatility, strike price, underlying stock price, risk-free rate, and time.
a)
Answer to Problem 21QP
The value of the risk-free bond is $31,669.31.
Explanation of Solution
Given information:
Company Z has a zero coupon bond issue, which matures in 2 years with a face value of $35,000. The
Equation to compute the PV of the continuously compounded amount:
Compute the PV of the continuously compounded amount:
Hence, the present value is $31,669.31.
b)
To calculate: The price that the bondholders will pay for the put option on the assets of the firm.
Introduction:
Black-Scholes is the model of pricing that is used to determine the theoretical value or fair price for a put option or call option, depending on six variables like type of option, volatility, strike price, underlying stock price, risk-free rate, and time.
b)
Answer to Problem 21QP
The bondholders must pay $14,479.18 for the put option.
Explanation of Solution
Given information:
Company Z has a zero coupon bond issue, which matures in 2 years with a face value of $35,000. The present value on the assets of the company is $21,800 and the standard deviation of the return on assets is 60% per year. The risk-free rate is 5% per year.
Formula to calculate the d1:
Where,
S is the stock price
E is the exercise price
r is the risk-free rate
σ is the standard deviation
t is the period of maturity
Calculate d1:
Hence, d1is -$0.0158.
Formula to calculate d2:
Calculate d2:
Hence, d2is -$0.8644.
Compute N (d1) and N (d2):
Note: The cumulative frequency distribution value for -0.0158 is 0.4937 and for -$0.8644 is 0.1937.
Formula to calculate the call price using the black-scholes model:
Where,
S is the stock price
E is the exercise price
C is the call price
R is the risk-free rate
t is the period of maturity
Calculate the call price:
Hence, the call price is $4,627.87.
Formulae to calculate the put price using the black-scholes model:
Where,
S is the stock price
E is the exercise price
C is the call price
P is the put price
R is the risk-free rate
t is the period of maturity
Calculate the put price:
Hence, the put price is $14,497.18.
c)
To calculate: The value on the debt of the firm and the yield on the debt.
Introduction:
Black-Scholes is the model of pricing that is used to determine the theoretical value or fair price for a put option or call option, depending on six variables like type of option, volatility, strike price, underlying stock price, risk-free rate, and time.
c)
Answer to Problem 21QP
The value and yield on the debt are $17,172.13 and 35.60% respectively.
Explanation of Solution
Given information:
Company Z has a zero coupon bond issue, which matures in 2 years with a face value of $35,000. The present value on the assets of the company is $21,800 and the standard deviation of the return on assets is 60% per year. The risk-free rate is 5% per year.
Formula to calculate the value of the risky bond:
Calculate the value of the risky bond:
Hence, the risky bond value is $17,172.13.
Equation of the present value to compute the return on debt:
Compute the return on debt:
Hence, the return on debt is 35.60%.
d)
To calculate: The value and the yield on the debt of the firm as per the proposed plan.
Introduction:
Black-Scholes is the model of pricing that is used to determine the theoretical value or fair price for a put option or call option, depending on six variables like type of option, volatility, strike price, underlying stock price, risk-free rate, and time.
d)
Answer to Problem 21QP
The value and the yield on the debt is $12,177.03 and 21.12% respectively.
Explanation of Solution
Given information: The management has proposed a plan where the firm will repay the same face value of the debt amount. However, the repayment will not happen for 5 years.
Equation to compute the PV:
Compute the PV with five year maturity period:
Hence, the present value is $27,258.03.
Formula to calculate the d1:
Where,
S is the stock price
E is the exercise price
r is the risk-free rate
σ is the standard deviation
t is the period of maturity
Calculate d1:
Hence, d1is $0.5043.
Formula to calculate d2:
Calculate d2:
Hence, d2 is -$0.8374.
Compute N (d1) and N (d2):
Note: The cumulative frequency distribution value for 0.5043 is 0.69297471 and for -0.8374 is 0.20118388.
Formula to calculate the call price using the black-scholes model:
Where,
S is the stock price
E is the exercise price
C is the call price
R is the risk-free rate
t is the period of maturity
Calculate the call price:
Hence, the call price is $9,622.97.
Formula to calculate the put price using the black-scholes model:
Where,
S is the stock price
E is the exercise price
C is the call price
P is the put price
R is the risk-free rate
t is the period of maturity
Calculate the put price:
Hence, the put price is $15,081.
Formula to calculate the value of the risky bond:
Calculate the value of the risky bond:
Hence, the risky bond value is $12,177.03.
Equation of the present value to compute the return on debt:
Compute the return on debt:
Hence, the return on debt is 21.12%.
The debt value decreases due to the
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Chapter 25 Solutions
Fundamentals of Corporate Finance
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