The risk-free bond value with the same maturity and face value as the bond of the company. 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.

Question

Want to see more full solutions like this?

Answer 1

Question

Chapter 25, Problem 21QP

a)

Summary Introduction

To compute: The risk-free bond value with the same maturity and face value as the bond of the company.

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)

Expert Solution

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 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.

Equation to compute the PV of the continuously compounded amount:

PV=Face value×e(Risk-free rate)(Maturity period)

Compute the PV of the continuously compounded amount:

PV=Face value×e(Risk-free rate)(Maturity period)=$35,000×e−0.05(2)=$31,669.31

Hence, the present value is $31,669.31.

b)

Summary Introduction

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)

Expert Solution

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:

d1=[ln(SE)+(r+σ22)t]σt

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:

d1=[ln(SE)+(r+σ22)t]σt=[In($21,800$35,000)+($0.05+0.6022)×(2)](0.60×2)=[−$0.013438091$0.848528137]=−$0.0158

Hence, d₁is -$0.0158.

Formula to calculate d₂:

d2=d1−σt

Calculate d₂:

d2=d1−σt=−$0.0158−(0.60×2)=−$0.0158−$0.848528137=−$0.8644

Hence, d₂is -$0.8644.

Compute N (d₁) and N (d₂):

N(d1) =0.4937

N(d2) =0.1937

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:

C=S×N(d1)−E×e−Rt×N(d2)

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:

C=S×N(d1)−E×e−Rt×N(d2)=$21,800×(0.4937)−($35,000e−0.05(2))(0.1937)=$10,762.66−$6,134.345276=$4,627.87

Hence, the call price is $4,627.87.

Formulae to calculate the put price using the black-scholes model:

P=E×e−Rt+C−S

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:

P=E×e−Rt+C−S=$35,000e−0.05(2)+$4,627.87−21,800=$14,497.18

Hence, the put price is $14,497.18.

c)

Summary Introduction

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)

Expert Solution

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:

Value of the risky bond=Present value of the sum−Put option price

Calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price=$31,669.31−$14,497.18=$17,172.13

Hence, the risky bond value is $17,172.13.

Equation of the present value to compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)

Compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)$17,172.13=$35,000e−R(2)e−R(2)=$17,172.13$35,000

e−R(2)=0.4,906RD=−(12)ln(0.4,906)RD=0.3560 or 35.60%

Hence, the return on debt is 35.60%.

d)

Summary Introduction

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)

Expert Solution

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:

PV=Face value×e(Risk-free rate)(Maturity period)

Compute the PV with five year maturity period:

PV=Face value×e(Risk-free rate)(Maturity period)=$35,000×e−0.05(5)=$27,258.03

Hence, the present value is $27,258.03.

Formula to calculate the d₁:

d1=[ln(SE)+(r+σ22)t]σt

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:

d1=[ln(SE)+(r+σ22)t]σt=[In($21,800$35,000)+($0.05+0.6022)×(5)](0.60×5)=[$0.676561908$1.341640786]=$0.5043

Hence, d₁is $0.5043.

Formula to calculate d₂:

d2=d1−σt

Calculate d₂:

d2=d1−σt=$0.5043−(0.60×5)=$0.5043−$1.341640786=−$0.8374

Hence, d₂ is -$0.8374.

Compute N (d₁) and N (d₂):

N(d1) =0.69297471

N(d2) =0.20118388

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:

Equity=C=S×N(d1)−E×e−Rt×N(d2)

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:

C=S×N(d1)−E×e−Rt×N(d2)=$21,800×(0.69297471)−($35,000e−0.05(5))(0.20118388)=$15,106.84868−$5,483.875715=$9,622.97

Hence, the call price is $9,622.97.

Formula to calculate the put price using the black-scholes model:

P=E×e−Rt+C−S

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:

P=E×e−Rt+C−S=$35,000e−0.05(5)+$9,622.97−21,800=$15,081

Hence, the put price is $15,081.

Formula to calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price

Calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price=$27,258.03−$15,081=$12,177.03

Hence, the risky bond value is $12,177.03.

Equation of the present value to compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)

Compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)$12,177.03=$35,000e−R(5)e−R(5)=$12,177.03$35,000e−R(5)=$0.3479

RD=−(15)ln(0.3479)RD=0.2112 or 21.12%

Hence, the return on debt is 21.12%.

The debt value decreases due to the time value of money; in other words, it would be longer till the shareholders obtain their payment. However, the required return on the debt decreases. Under the present situation, it is not likely that the firm would have the assets to pay off the bondholders. As per the new plan, the firm operates additionally, for five years. Hence, the possibility of raising the assets’ value to exceed or to meet the face value of the debt is greater than the firm that operates for only two additional years.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Students have asked these similar questions

Chapter 12, Problem 3 IBM needs to raise $I billion and is trying to decide between a domestic dollar bond issue and a Eurobond issue. The U.S. bond can be issued at a coupon of 6.75 percent, paid semiannually, with underwriting and other expenses totaling 0.95 percent of the issue size. The Eurobond would cost only 0.55 percent to issue but would bear an annual coupon of 6.88 percent. Both issues would mature in 10 years. a. Assuming all else is equal, which is the least expensive issue for IBM? B. What other factors might IBM want to consider before deciding which bond to issue?

D4) Suppose that there is 30-year coupon bond with par value of $100 and Macaulay duration of 20.56. The coupon rate is unknown. Currently, the bond is traded at $90 and the yield is flat at 20% pa. Yield to maturity is an annualized simple interest rate compounded annually. If the bond yield increases by 50 basis points, what is the approximation of the percentage capital gain or loss? Please choose the correct range for the percentage capital gain/loss, i.e., if it is -3.5%, please select “A value between -3% and -4%” A value between -9% and -10% A value between -8% and -9% None of the other answers are correct. A value between -7% and -8% A value between -10% and -11%

45) Costly Corporation plans a new issue of bonds with a par value of $1000, a maturity of 37 years, and an annual coupon rate of 11.0%. Flotation costs associated with a new debt issue would equal 3.0% of the market value of the bonds. Currently, the appropriate discount rate for bonds of firms similar to Costly is 9.0%. The firm's marginal tax rate is 50%. What will the firm's true cost of debt be for this new bond issue? 11.34% 9.29% 4.65% 5.67% 12.74%

Answer 2

Question

Chapter 25, Problem 21QP

a)

Summary Introduction

To compute: The risk-free bond value with the same maturity and face value as the bond of the company.

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)

Expert Solution

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 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.

Equation to compute the PV of the continuously compounded amount:

PV=Face value×e(Risk-free rate)(Maturity period)

Compute the PV of the continuously compounded amount:

PV=Face value×e(Risk-free rate)(Maturity period)=$35,000×e−0.05(2)=$31,669.31

Hence, the present value is $31,669.31.

b)

Summary Introduction

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)

Expert Solution

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:

d1=[ln(SE)+(r+σ22)t]σt

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:

d1=[ln(SE)+(r+σ22)t]σt=[In($21,800$35,000)+($0.05+0.6022)×(2)](0.60×2)=[−$0.013438091$0.848528137]=−$0.0158

Hence, d₁is -$0.0158.

Formula to calculate d₂:

d2=d1−σt

Calculate d₂:

d2=d1−σt=−$0.0158−(0.60×2)=−$0.0158−$0.848528137=−$0.8644

Hence, d₂is -$0.8644.

Compute N (d₁) and N (d₂):

N(d1) =0.4937

N(d2) =0.1937

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:

C=S×N(d1)−E×e−Rt×N(d2)

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:

C=S×N(d1)−E×e−Rt×N(d2)=$21,800×(0.4937)−($35,000e−0.05(2))(0.1937)=$10,762.66−$6,134.345276=$4,627.87

Hence, the call price is $4,627.87.

Formulae to calculate the put price using the black-scholes model:

P=E×e−Rt+C−S

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:

P=E×e−Rt+C−S=$35,000e−0.05(2)+$4,627.87−21,800=$14,497.18

Hence, the put price is $14,497.18.

c)

Summary Introduction

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)

Expert Solution

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:

Value of the risky bond=Present value of the sum−Put option price

Calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price=$31,669.31−$14,497.18=$17,172.13

Hence, the risky bond value is $17,172.13.

Equation of the present value to compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)

Compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)$17,172.13=$35,000e−R(2)e−R(2)=$17,172.13$35,000

e−R(2)=0.4,906RD=−(12)ln(0.4,906)RD=0.3560 or 35.60%

Hence, the return on debt is 35.60%.

d)

Summary Introduction

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)

Expert Solution

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:

PV=Face value×e(Risk-free rate)(Maturity period)

Compute the PV with five year maturity period:

PV=Face value×e(Risk-free rate)(Maturity period)=$35,000×e−0.05(5)=$27,258.03

Hence, the present value is $27,258.03.

Formula to calculate the d₁:

d1=[ln(SE)+(r+σ22)t]σt

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:

d1=[ln(SE)+(r+σ22)t]σt=[In($21,800$35,000)+($0.05+0.6022)×(5)](0.60×5)=[$0.676561908$1.341640786]=$0.5043

Hence, d₁is $0.5043.

Formula to calculate d₂:

d2=d1−σt

Calculate d₂:

d2=d1−σt=$0.5043−(0.60×5)=$0.5043−$1.341640786=−$0.8374

Hence, d₂ is -$0.8374.

Compute N (d₁) and N (d₂):

N(d1) =0.69297471

N(d2) =0.20118388

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:

Equity=C=S×N(d1)−E×e−Rt×N(d2)

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:

C=S×N(d1)−E×e−Rt×N(d2)=$21,800×(0.69297471)−($35,000e−0.05(5))(0.20118388)=$15,106.84868−$5,483.875715=$9,622.97

Hence, the call price is $9,622.97.

Formula to calculate the put price using the black-scholes model:

P=E×e−Rt+C−S

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:

P=E×e−Rt+C−S=$35,000e−0.05(5)+$9,622.97−21,800=$15,081

Hence, the put price is $15,081.

Formula to calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price

Calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price=$27,258.03−$15,081=$12,177.03

Hence, the risky bond value is $12,177.03.

Equation of the present value to compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)

Compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)$12,177.03=$35,000e−R(5)e−R(5)=$12,177.03$35,000e−R(5)=$0.3479

RD=−(15)ln(0.3479)RD=0.2112 or 21.12%

Hence, the return on debt is 21.12%.

The debt value decreases due to the time value of money; in other words, it would be longer till the shareholders obtain their payment. However, the required return on the debt decreases. Under the present situation, it is not likely that the firm would have the assets to pay off the bondholders. As per the new plan, the firm operates additionally, for five years. Hence, the possibility of raising the assets’ value to exceed or to meet the face value of the debt is greater than the firm that operates for only two additional years.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 3

Question

Chapter 25, Problem 21QP

a)

Summary Introduction

To compute: The risk-free bond value with the same maturity and face value as the bond of the company.

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)

Expert Solution

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 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.

Equation to compute the PV of the continuously compounded amount:

PV=Face value×e(Risk-free rate)(Maturity period)

Compute the PV of the continuously compounded amount:

PV=Face value×e(Risk-free rate)(Maturity period)=$35,000×e−0.05(2)=$31,669.31

Hence, the present value is $31,669.31.

b)

Summary Introduction

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)

Expert Solution

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:

d1=[ln(SE)+(r+σ22)t]σt

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:

d1=[ln(SE)+(r+σ22)t]σt=[In($21,800$35,000)+($0.05+0.6022)×(2)](0.60×2)=[−$0.013438091$0.848528137]=−$0.0158

Hence, d₁is -$0.0158.

Formula to calculate d₂:

d2=d1−σt

Calculate d₂:

d2=d1−σt=−$0.0158−(0.60×2)=−$0.0158−$0.848528137=−$0.8644

Hence, d₂is -$0.8644.

Compute N (d₁) and N (d₂):

N(d1) =0.4937

N(d2) =0.1937

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:

C=S×N(d1)−E×e−Rt×N(d2)

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:

C=S×N(d1)−E×e−Rt×N(d2)=$21,800×(0.4937)−($35,000e−0.05(2))(0.1937)=$10,762.66−$6,134.345276=$4,627.87

Hence, the call price is $4,627.87.

Formulae to calculate the put price using the black-scholes model:

P=E×e−Rt+C−S

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:

P=E×e−Rt+C−S=$35,000e−0.05(2)+$4,627.87−21,800=$14,497.18

Hence, the put price is $14,497.18.

c)

Summary Introduction

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)

Expert Solution

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:

Value of the risky bond=Present value of the sum−Put option price

Calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price=$31,669.31−$14,497.18=$17,172.13

Hence, the risky bond value is $17,172.13.

Equation of the present value to compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)

Compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)$17,172.13=$35,000e−R(2)e−R(2)=$17,172.13$35,000

e−R(2)=0.4,906RD=−(12)ln(0.4,906)RD=0.3560 or 35.60%

Hence, the return on debt is 35.60%.

d)

Summary Introduction

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)

Expert Solution

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:

PV=Face value×e(Risk-free rate)(Maturity period)

Compute the PV with five year maturity period:

PV=Face value×e(Risk-free rate)(Maturity period)=$35,000×e−0.05(5)=$27,258.03

Hence, the present value is $27,258.03.

Formula to calculate the d₁:

d1=[ln(SE)+(r+σ22)t]σt

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:

d1=[ln(SE)+(r+σ22)t]σt=[In($21,800$35,000)+($0.05+0.6022)×(5)](0.60×5)=[$0.676561908$1.341640786]=$0.5043

Hence, d₁is $0.5043.

Formula to calculate d₂:

d2=d1−σt

Calculate d₂:

d2=d1−σt=$0.5043−(0.60×5)=$0.5043−$1.341640786=−$0.8374

Hence, d₂ is -$0.8374.

Compute N (d₁) and N (d₂):

N(d1) =0.69297471

N(d2) =0.20118388

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:

Equity=C=S×N(d1)−E×e−Rt×N(d2)

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:

C=S×N(d1)−E×e−Rt×N(d2)=$21,800×(0.69297471)−($35,000e−0.05(5))(0.20118388)=$15,106.84868−$5,483.875715=$9,622.97

Hence, the call price is $9,622.97.

Formula to calculate the put price using the black-scholes model:

P=E×e−Rt+C−S

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:

P=E×e−Rt+C−S=$35,000e−0.05(5)+$9,622.97−21,800=$15,081

Hence, the put price is $15,081.

Formula to calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price

Calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price=$27,258.03−$15,081=$12,177.03

Hence, the risky bond value is $12,177.03.

Equation of the present value to compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)

Compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)$12,177.03=$35,000e−R(5)e−R(5)=$12,177.03$35,000e−R(5)=$0.3479

RD=−(15)ln(0.3479)RD=0.2112 or 21.12%

Hence, the return on debt is 21.12%.

The debt value decreases due to the time value of money; in other words, it would be longer till the shareholders obtain their payment. However, the required return on the debt decreases. Under the present situation, it is not likely that the firm would have the assets to pay off the bondholders. As per the new plan, the firm operates additionally, for five years. Hence, the possibility of raising the assets’ value to exceed or to meet the face value of the debt is greater than the firm that operates for only two additional years.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 4

Question

Chapter 25, Problem 21QP

a)

Summary Introduction

To compute: The risk-free bond value with the same maturity and face value as the bond of the company.

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)

Expert Solution

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 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.

Equation to compute the PV of the continuously compounded amount:

PV=Face value×e(Risk-free rate)(Maturity period)

Compute the PV of the continuously compounded amount:

PV=Face value×e(Risk-free rate)(Maturity period)=$35,000×e−0.05(2)=$31,669.31

Hence, the present value is $31,669.31.

b)

Summary Introduction

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)

Expert Solution

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:

d1=[ln(SE)+(r+σ22)t]σt

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:

d1=[ln(SE)+(r+σ22)t]σt=[In($21,800$35,000)+($0.05+0.6022)×(2)](0.60×2)=[−$0.013438091$0.848528137]=−$0.0158

Hence, d₁is -$0.0158.

Formula to calculate d₂:

d2=d1−σt

Calculate d₂:

d2=d1−σt=−$0.0158−(0.60×2)=−$0.0158−$0.848528137=−$0.8644

Hence, d₂is -$0.8644.

Compute N (d₁) and N (d₂):

N(d1) =0.4937

N(d2) =0.1937

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:

C=S×N(d1)−E×e−Rt×N(d2)

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:

C=S×N(d1)−E×e−Rt×N(d2)=$21,800×(0.4937)−($35,000e−0.05(2))(0.1937)=$10,762.66−$6,134.345276=$4,627.87

Hence, the call price is $4,627.87.

Formulae to calculate the put price using the black-scholes model:

P=E×e−Rt+C−S

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:

P=E×e−Rt+C−S=$35,000e−0.05(2)+$4,627.87−21,800=$14,497.18

Hence, the put price is $14,497.18.

c)

Summary Introduction

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)

Expert Solution

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:

Value of the risky bond=Present value of the sum−Put option price

Calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price=$31,669.31−$14,497.18=$17,172.13

Hence, the risky bond value is $17,172.13.

Equation of the present value to compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)

Compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)$17,172.13=$35,000e−R(2)e−R(2)=$17,172.13$35,000

e−R(2)=0.4,906RD=−(12)ln(0.4,906)RD=0.3560 or 35.60%

Hence, the return on debt is 35.60%.

d)

Summary Introduction

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)

Expert Solution

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:

PV=Face value×e(Risk-free rate)(Maturity period)

Compute the PV with five year maturity period:

PV=Face value×e(Risk-free rate)(Maturity period)=$35,000×e−0.05(5)=$27,258.03

Hence, the present value is $27,258.03.

Formula to calculate the d₁:

d1=[ln(SE)+(r+σ22)t]σt

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:

d1=[ln(SE)+(r+σ22)t]σt=[In($21,800$35,000)+($0.05+0.6022)×(5)](0.60×5)=[$0.676561908$1.341640786]=$0.5043

Hence, d₁is $0.5043.

Formula to calculate d₂:

d2=d1−σt

Calculate d₂:

d2=d1−σt=$0.5043−(0.60×5)=$0.5043−$1.341640786=−$0.8374

Hence, d₂ is -$0.8374.

Compute N (d₁) and N (d₂):

N(d1) =0.69297471

N(d2) =0.20118388

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:

Equity=C=S×N(d1)−E×e−Rt×N(d2)

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:

C=S×N(d1)−E×e−Rt×N(d2)=$21,800×(0.69297471)−($35,000e−0.05(5))(0.20118388)=$15,106.84868−$5,483.875715=$9,622.97

Hence, the call price is $9,622.97.

Formula to calculate the put price using the black-scholes model:

P=E×e−Rt+C−S

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:

P=E×e−Rt+C−S=$35,000e−0.05(5)+$9,622.97−21,800=$15,081

Hence, the put price is $15,081.

Formula to calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price

Calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price=$27,258.03−$15,081=$12,177.03

Hence, the risky bond value is $12,177.03.

Equation of the present value to compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)

Compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)$12,177.03=$35,000e−R(5)e−R(5)=$12,177.03$35,000e−R(5)=$0.3479

RD=−(15)ln(0.3479)RD=0.2112 or 21.12%

Hence, the return on debt is 21.12%.

The debt value decreases due to the time value of money; in other words, it would be longer till the shareholders obtain their payment. However, the required return on the debt decreases. Under the present situation, it is not likely that the firm would have the assets to pay off the bondholders. As per the new plan, the firm operates additionally, for five years. Hence, the possibility of raising the assets’ value to exceed or to meet the face value of the debt is greater than the firm that operates for only two additional years.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 5

Chapter 25, Problem 21QP

a)

Summary Introduction

To compute: The risk-free bond value with the same maturity and face value as the bond of the company.

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)

Expert Solution

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 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.

Equation to compute the PV of the continuously compounded amount:

PV=Face value×e(Risk-free rate)(Maturity period)

Compute the PV of the continuously compounded amount:

PV=Face value×e(Risk-free rate)(Maturity period)=$35,000×e−0.05(2)=$31,669.31

Hence, the present value is $31,669.31.

b)

Summary Introduction

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)

Expert Solution

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:

d1=[ln(SE)+(r+σ22)t]σt

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:

d1=[ln(SE)+(r+σ22)t]σt=[In($21,800$35,000)+($0.05+0.6022)×(2)](0.60×2)=[−$0.013438091$0.848528137]=−$0.0158

Hence, d₁is -$0.0158.

Formula to calculate d₂:

d2=d1−σt

Calculate d₂:

d2=d1−σt=−$0.0158−(0.60×2)=−$0.0158−$0.848528137=−$0.8644

Hence, d₂is -$0.8644.

Compute N (d₁) and N (d₂):

N(d1) =0.4937

N(d2) =0.1937

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:

C=S×N(d1)−E×e−Rt×N(d2)

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:

C=S×N(d1)−E×e−Rt×N(d2)=$21,800×(0.4937)−($35,000e−0.05(2))(0.1937)=$10,762.66−$6,134.345276=$4,627.87

Hence, the call price is $4,627.87.

Formulae to calculate the put price using the black-scholes model:

P=E×e−Rt+C−S

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:

P=E×e−Rt+C−S=$35,000e−0.05(2)+$4,627.87−21,800=$14,497.18

Hence, the put price is $14,497.18.

c)

Summary Introduction

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)

Expert Solution

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:

Value of the risky bond=Present value of the sum−Put option price

Calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price=$31,669.31−$14,497.18=$17,172.13

Hence, the risky bond value is $17,172.13.

Equation of the present value to compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)

Compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)$17,172.13=$35,000e−R(2)e−R(2)=$17,172.13$35,000

e−R(2)=0.4,906RD=−(12)ln(0.4,906)RD=0.3560 or 35.60%

Hence, the return on debt is 35.60%.

d)

Summary Introduction

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)

Expert Solution

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:

PV=Face value×e(Risk-free rate)(Maturity period)

Compute the PV with five year maturity period:

PV=Face value×e(Risk-free rate)(Maturity period)=$35,000×e−0.05(5)=$27,258.03

Hence, the present value is $27,258.03.

Formula to calculate the d₁:

d1=[ln(SE)+(r+σ22)t]σt

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:

d1=[ln(SE)+(r+σ22)t]σt=[In($21,800$35,000)+($0.05+0.6022)×(5)](0.60×5)=[$0.676561908$1.341640786]=$0.5043

Hence, d₁is $0.5043.

Formula to calculate d₂:

d2=d1−σt

Calculate d₂:

d2=d1−σt=$0.5043−(0.60×5)=$0.5043−$1.341640786=−$0.8374

Hence, d₂ is -$0.8374.

Compute N (d₁) and N (d₂):

N(d1) =0.69297471

N(d2) =0.20118388

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:

Equity=C=S×N(d1)−E×e−Rt×N(d2)

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:

C=S×N(d1)−E×e−Rt×N(d2)=$21,800×(0.69297471)−($35,000e−0.05(5))(0.20118388)=$15,106.84868−$5,483.875715=$9,622.97

Hence, the call price is $9,622.97.

Formula to calculate the put price using the black-scholes model:

P=E×e−Rt+C−S

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:

P=E×e−Rt+C−S=$35,000e−0.05(5)+$9,622.97−21,800=$15,081

Hence, the put price is $15,081.

Formula to calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price

Calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price=$27,258.03−$15,081=$12,177.03

Hence, the risky bond value is $12,177.03.

Equation of the present value to compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)

Compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)$12,177.03=$35,000e−R(5)e−R(5)=$12,177.03$35,000e−R(5)=$0.3479

RD=−(15)ln(0.3479)RD=0.2112 or 21.12%

Hence, the return on debt is 21.12%.

The debt value decreases due to the time value of money; in other words, it would be longer till the shareholders obtain their payment. However, the required return on the debt decreases. Under the present situation, it is not likely that the firm would have the assets to pay off the bondholders. As per the new plan, the firm operates additionally, for five years. Hence, the possibility of raising the assets’ value to exceed or to meet the face value of the debt is greater than the firm that operates for only two additional years.

Answer 6

Chapter 25, Problem 21QP

a)

Summary Introduction

To compute: The risk-free bond value with the same maturity and face value as the bond of the company.

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)

Expert Solution

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 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.

Equation to compute the PV of the continuously compounded amount:

PV=Face value×e(Risk-free rate)(Maturity period)

Compute the PV of the continuously compounded amount:

PV=Face value×e(Risk-free rate)(Maturity period)=$35,000×e−0.05(2)=$31,669.31

Hence, the present value is $31,669.31.

Answer 7

Chapter 25, Problem 21QP

a)

Summary Introduction

To compute: The risk-free bond value with the same maturity and face value as the bond of the company.

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.

Answer 8

a)

Expert Solution

Answer to Problem 21QP

The value of the risk-free bond is $31,669.31.

Answer 9

a)

Expert Solution

Answer 10

a)

Expert Solution

Answer 11

Expert Solution

Answer 12

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.

Equation to compute the PV of the continuously compounded amount:

PV=Face value×e(Risk-free rate)(Maturity period)

Compute the PV of the continuously compounded amount:

PV=Face value×e(Risk-free rate)(Maturity period)=$35,000×e−0.05(2)=$31,669.31

Hence, the present value is $31,669.31.

Answer 13

b)

Summary Introduction

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)

Expert Solution

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:

d1=[ln(SE)+(r+σ22)t]σt

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:

d1=[ln(SE)+(r+σ22)t]σt=[In($21,800$35,000)+($0.05+0.6022)×(2)](0.60×2)=[−$0.013438091$0.848528137]=−$0.0158

Hence, d₁is -$0.0158.

Formula to calculate d₂:

d2=d1−σt

Calculate d₂:

d2=d1−σt=−$0.0158−(0.60×2)=−$0.0158−$0.848528137=−$0.8644

Hence, d₂is -$0.8644.

Compute N (d₁) and N (d₂):

N(d1) =0.4937

N(d2) =0.1937

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:

C=S×N(d1)−E×e−Rt×N(d2)

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:

C=S×N(d1)−E×e−Rt×N(d2)=$21,800×(0.4937)−($35,000e−0.05(2))(0.1937)=$10,762.66−$6,134.345276=$4,627.87

Hence, the call price is $4,627.87.

Formulae to calculate the put price using the black-scholes model:

P=E×e−Rt+C−S

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:

P=E×e−Rt+C−S=$35,000e−0.05(2)+$4,627.87−21,800=$14,497.18

Hence, the put price is $14,497.18.

Answer 14

b)

Summary Introduction

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.

Answer 15

b)

Expert Solution

Answer to Problem 21QP

The bondholders must pay $14,479.18 for the put option.

Answer 16

b)

Expert Solution

Answer 17

b)

Expert Solution

Answer 18

Expert Solution

Answer 19

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:

d1=[ln(SE)+(r+σ22)t]σt

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:

d1=[ln(SE)+(r+σ22)t]σt=[In($21,800$35,000)+($0.05+0.6022)×(2)](0.60×2)=[−$0.013438091$0.848528137]=−$0.0158

Hence, d₁is -$0.0158.

Formula to calculate d₂:

d2=d1−σt

Calculate d₂:

d2=d1−σt=−$0.0158−(0.60×2)=−$0.0158−$0.848528137=−$0.8644

Hence, d₂is -$0.8644.

Compute N (d₁) and N (d₂):

N(d1) =0.4937

N(d2) =0.1937

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:

C=S×N(d1)−E×e−Rt×N(d2)

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:

C=S×N(d1)−E×e−Rt×N(d2)=$21,800×(0.4937)−($35,000e−0.05(2))(0.1937)=$10,762.66−$6,134.345276=$4,627.87

Hence, the call price is $4,627.87.

Formulae to calculate the put price using the black-scholes model:

P=E×e−Rt+C−S

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:

P=E×e−Rt+C−S=$35,000e−0.05(2)+$4,627.87−21,800=$14,497.18

Hence, the put price is $14,497.18.

Answer 20

c)

Summary Introduction

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)

Expert Solution

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:

Value of the risky bond=Present value of the sum−Put option price

Calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price=$31,669.31−$14,497.18=$17,172.13

Hence, the risky bond value is $17,172.13.

Equation of the present value to compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)

Compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)$17,172.13=$35,000e−R(2)e−R(2)=$17,172.13$35,000

e−R(2)=0.4,906RD=−(12)ln(0.4,906)RD=0.3560 or 35.60%

Hence, the return on debt is 35.60%.

Answer 21

c)

Summary Introduction

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.

Answer 22

c)

Expert Solution

Answer to Problem 21QP

The value and yield on the debt are $17,172.13 and 35.60% respectively.

Answer 23

c)

Expert Solution

Answer 24

c)

Expert Solution

Answer 25

Expert Solution

Answer 26

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:

Value of the risky bond=Present value of the sum−Put option price

Calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price=$31,669.31−$14,497.18=$17,172.13

Hence, the risky bond value is $17,172.13.

Equation of the present value to compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)

Compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)$17,172.13=$35,000e−R(2)e−R(2)=$17,172.13$35,000

e−R(2)=0.4,906RD=−(12)ln(0.4,906)RD=0.3560 or 35.60%

Hence, the return on debt is 35.60%.

Answer 27

d)

Summary Introduction

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)

Expert Solution

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:

PV=Face value×e(Risk-free rate)(Maturity period)

Compute the PV with five year maturity period:

PV=Face value×e(Risk-free rate)(Maturity period)=$35,000×e−0.05(5)=$27,258.03

Hence, the present value is $27,258.03.

Formula to calculate the d₁:

d1=[ln(SE)+(r+σ22)t]σt

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:

d1=[ln(SE)+(r+σ22)t]σt=[In($21,800$35,000)+($0.05+0.6022)×(5)](0.60×5)=[$0.676561908$1.341640786]=$0.5043

Hence, d₁is $0.5043.

Formula to calculate d₂:

d2=d1−σt

Calculate d₂:

d2=d1−σt=$0.5043−(0.60×5)=$0.5043−$1.341640786=−$0.8374

Hence, d₂ is -$0.8374.

Compute N (d₁) and N (d₂):

N(d1) =0.69297471

N(d2) =0.20118388

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:

Equity=C=S×N(d1)−E×e−Rt×N(d2)

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:

C=S×N(d1)−E×e−Rt×N(d2)=$21,800×(0.69297471)−($35,000e−0.05(5))(0.20118388)=$15,106.84868−$5,483.875715=$9,622.97

Hence, the call price is $9,622.97.

Formula to calculate the put price using the black-scholes model:

P=E×e−Rt+C−S

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:

P=E×e−Rt+C−S=$35,000e−0.05(5)+$9,622.97−21,800=$15,081

Hence, the put price is $15,081.

Formula to calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price

Calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price=$27,258.03−$15,081=$12,177.03

Hence, the risky bond value is $12,177.03.

Equation of the present value to compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)

Compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)$12,177.03=$35,000e−R(5)e−R(5)=$12,177.03$35,000e−R(5)=$0.3479

RD=−(15)ln(0.3479)RD=0.2112 or 21.12%

Hence, the return on debt is 21.12%.

The debt value decreases due to the time value of money; in other words, it would be longer till the shareholders obtain their payment. However, the required return on the debt decreases. Under the present situation, it is not likely that the firm would have the assets to pay off the bondholders. As per the new plan, the firm operates additionally, for five years. Hence, the possibility of raising the assets’ value to exceed or to meet the face value of the debt is greater than the firm that operates for only two additional years.

Answer 28

d)

Summary Introduction

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.

Answer 29

d)

Expert Solution

Answer to Problem 21QP

The value and the yield on the debt is $12,177.03 and 21.12% respectively.

Answer 30

d)

Expert Solution

Answer 31

d)

Expert Solution

Answer 32

Expert Solution

Answer 33

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:

PV=Face value×e(Risk-free rate)(Maturity period)

Compute the PV with five year maturity period:

PV=Face value×e(Risk-free rate)(Maturity period)=$35,000×e−0.05(5)=$27,258.03

Hence, the present value is $27,258.03.

Formula to calculate the d₁:

d1=[ln(SE)+(r+σ22)t]σt

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:

d1=[ln(SE)+(r+σ22)t]σt=[In($21,800$35,000)+($0.05+0.6022)×(5)](0.60×5)=[$0.676561908$1.341640786]=$0.5043

Hence, d₁is $0.5043.

Formula to calculate d₂:

d2=d1−σt

Calculate d₂:

d2=d1−σt=$0.5043−(0.60×5)=$0.5043−$1.341640786=−$0.8374

Hence, d₂ is -$0.8374.

Compute N (d₁) and N (d₂):

N(d1) =0.69297471

N(d2) =0.20118388

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:

Equity=C=S×N(d1)−E×e−Rt×N(d2)

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:

C=S×N(d1)−E×e−Rt×N(d2)=$21,800×(0.69297471)−($35,000e−0.05(5))(0.20118388)=$15,106.84868−$5,483.875715=$9,622.97

Hence, the call price is $9,622.97.

Formula to calculate the put price using the black-scholes model:

P=E×e−Rt+C−S

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:

P=E×e−Rt+C−S=$35,000e−0.05(5)+$9,622.97−21,800=$15,081

Hence, the put price is $15,081.

Formula to calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price

Calculate the value of the risky bond:

Value of the risky bond=Present value of the sum−Put option price=$27,258.03−$15,081=$12,177.03

Hence, the risky bond value is $12,177.03.

Equation of the present value to compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)

Compute the return on debt:

Value of the risky bond=Face value of the bonde−R(Maturity period)$12,177.03=$35,000e−R(5)e−R(5)=$12,177.03$35,000e−R(5)=$0.3479

RD=−(15)ln(0.3479)RD=0.2112 or 21.12%

Hence, the return on debt is 21.12%.

The debt value decreases due to the time value of money; in other words, it would be longer till the shareholders obtain their payment. However, the required return on the debt decreases. Under the present situation, it is not likely that the firm would have the assets to pay off the bondholders. As per the new plan, the firm operates additionally, for five years. Hence, the possibility of raising the assets’ value to exceed or to meet the face value of the debt is greater than the firm that operates for only two additional years.

Answer 34

Ch. 25.1 - Prob. 25.1ACQ Ch. 25.1 - Prob. 25.1BCQ Ch. 25.2 - Prob. 25.2ACQ Ch. 25.2 - Prob. 25.2BCQ Ch. 25.3 - Prob. 25.3ACQ Ch. 25.3 - Prob. 25.3BCQ Ch. 25.4 - Why do we say that the equity in a leveraged firm...Ch. 25.4 - Prob. 25.4BCQ Ch. 25.5 - Prob. 25.5ACQ Ch. 25.5 - Prob. 25.5BCQ

Answer to Problem 21QP

Explanation of Solution

Answer to Problem 21QP

Explanation of Solution

Answer to Problem 21QP

Explanation of Solution

Answer to Problem 21QP

Explanation of Solution

Want to see more full solutions like this?

Chapter 25 Solutions