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

A4 3 You bought one of BB Co.’s 10% coupon bonds one year ago for $1100. These bonds make annual payments, have a face value of $1000 each, and mature seven years from now. Suppose you decide to sell your bonds today, when the required return on the bonds is 8%. If the inflation rate was 3% over the past year, what would be your total real return on investment according to the Exact Fisher Formula?

[1C] Direction: Solve the problem. Show your neat and complete solutions. An 8% corporate bond with face value of P100,000 matures in 5 years. The yield to maturity is 7% and the coupon is paid annually. Find the current price of the bond?

6 MLK Bank has an asset portfolio that consists of $100 million of 30-year, 8 percent annual coupon, $1,000 bonds that sell at par. a-1. What will be the bonds' new prices if market yields change immediately by ± 0.10 percent? a-2. What will be the new prices if market yields change immediately by ± 2.00 percent? b-1. The duration of these bonds is 12.1608 years. What are the predicted bond prices in each of the four cases using the duration rule? b-2. What is the amount of error between the duration prediction and the actual market values? Complete this question by entering your answers in the tabs below. Required A1 Required A2 Required B1 Required B2 What will be the bonds' new prices if market yields change immediately by ± 0.10 percent? (Do not round intermediate calculations. Enter all answers as positive numbers. Round your answers to 2 decimal places. (e.g., 32.16)) At + 0.10% At - 0.10% Bonds' New Price $ 988.85 1,011.36 < Required A1 Required A2

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.