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 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:

$\text{PV}=\text{Face value}\times e^{\left(\text{Risk-free rate}\right)}{}^{\left(\text{Maturity period}\right)}$

Compute the PV of the continuously compounded amount:

$\begin{array}{c}\text{PV}=\text{Face value}\times e^{\left(\text{Risk-free rate}\right)}{}^{\left(\text{Maturity period}\right)}\\ =\$35,000\times e^{-0.05\left(2\right)}\\ =\$31,669.31\end{array}$

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.

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:

$d_1=\frac{\left[\text{ln}\left(\frac{S}{E}\right)+\left(r+\frac{{\sigma }^2}{2}\right)t\right]}{\sigma \sqrt{t}}$

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:

$\begin{array}{c}{d}_1=\frac{\left[\text{ln}\left(\frac{S}{E}\right)+\left(r+\frac{{\sigma }^2}{2}\right)t\right]}{\sigma \sqrt{t}}\\ =\frac{\left[\text{In}\left(\frac{\$21,800}{\$35,000}\right)+\left(\$0.05+\frac{{0.60}^2}{2}\right)\times \left(2\right)\right]}{\left(0.60\times \sqrt{2}\right)}\\ =\left[\frac{-\$0.013438091}{\$0.848528137}\right]\\ =-\$0.0158\end{array}$

Hence, d₁is -$0.0158.

Formula to calculate d₂:

$d_2=d_1-\sigma \sqrt{t}$

Calculate d₂:

$\begin{array}{c}{d}_2=d_1-\sigma \sqrt{t}\\ =-\$0.0158-\left(0.60\times \sqrt{2}\right)\\ =-\$0.0158-\$0.848528137\\ =-\$0.8644\end{array}$

Hence, d₂is -$0.8644.

Compute N (d₁) and N (d₂):

$N\left(d_1\right)$=0.4937

$N\left(d_2\right)$=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:

$C=S\times \text{N}\left(d_1\right)-E\times e^{-Rt}\times \text{N}\left(d_2\right)$

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:

$\begin{array}{c}C=S\times \text{N}\left(d_1\right)-E\times e^{-Rt}\times \text{N}\left(d_2\right)\\ =\$21,800\times \left(0.4937\right)-\left(\$35,000e^{-0.05\left(2\right)}\right)\left(0.1937\right)\\ =\$10,762.66-\$6,134.345276\\ =\$4,627.87\end{array}$

Hence, the call price is $4,627.87.

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

$P=E\times e^{-Rt}+C-S$

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:

$\begin{array}{c}P=E\times e^{-Rt}+C-S\\ =\$35,000e^{-0.05\left(2\right)}+\$4,627.87-21,800\\ =\$14,497.18\end{array}$

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.

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:

$\text{Value of the risky bond}=\text{Present value of the sum}-\text{Put option price}$

Calculate the value of the risky bond:

$\begin{array}{c}\text{Value of the risky bond}=\text{Present value of the sum}-\text{Put option price}\\ =\$31,669.31-\$14,497.18\\ =\$17,172.13\end{array}$

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

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

$\text{Value of the risky bond}=\text{Face value of the bond}{e}^{-R\left(\text{Maturity period}\right)}$

Compute the return on debt:

$\begin{array}{c}\text{Value of the risky bond}=\text{Face value of the bond}{e}^{-R\left(\text{Maturity period}\right)}\\ \$17,172.13=\$35,000e^{-R\left(2\right)}\\ e^{-R\left(2\right)}=\frac{\$17,172.13}{\$35,000}\end{array}$

$\begin{array}{c}{e}^{-R\left(2\right)}=0.4,906\\ R_D=-\left(\frac{1}{2}\right)\text{ln}\left(0.4,906\right)\\ R_D=0.3560\text{ or 35}\text{.60\%}\end{array}$

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.

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:

$\text{PV}=\text{Face value}\times e^{\left(\text{Risk-free rate}\right)}{}^{\left(\text{Maturity period}\right)}$

Compute the PV with five year maturity period:

$\begin{array}{c}\text{PV}=\text{Face value}\times e^{\left(\text{Risk-free rate}\right)}{}^{\left(\text{Maturity period}\right)}\\ =\$35,000\times e^{-0.05\left(5\right)}\\ =\$27,258.03\end{array}$

Hence, the present value is $27,258.03.

Formula to calculate the d₁:

$d_1=\frac{\left[\text{ln}\left(\frac{S}{E}\right)+\left(r+\frac{{\sigma }^2}{2}\right)t\right]}{\sigma \sqrt{t}}$

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:

$\begin{array}{c}{d}_1=\frac{\left[\text{ln}\left(\frac{S}{E}\right)+\left(r+\frac{{\sigma }^2}{2}\right)t\right]}{\sigma \sqrt{t}}\\ =\frac{\left[\text{In}\left(\frac{\$21,800}{\$35,000}\right)+\left(\$0.05+\frac{{0.60}^2}{2}\right)\times \left(5\right)\right]}{\left(0.60\times \sqrt{5}\right)}\\ =\left[\frac{\$0.676561908}{\$1.341640786}\right]\\ =\$0.5043\end{array}$

Hence, d₁is $0.5043.

Formula to calculate d₂:

$d_2=d_1-\sigma \sqrt{t}$

Calculate d₂:

$\begin{array}{c}{d}_2=d_1-\sigma \sqrt{t}\\ =\$0.5043-\left(0.60\times \sqrt{5}\right)\\ =\$0.5043-\$1.341640786\\ =-\$0.8374\end{array}$

Hence, d₂ is -$0.8374.

Compute N (d₁) and N (d₂):

$N\left(d_1\right)$=0.69297471

$N\left(d_2\right)$=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:

$\text{Equity}=C=S\times \text{N}\left(d_1\right)-E\times e^{-Rt}\times \text{N}\left(d_2\right)$

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:

$\begin{array}{c}C=S\times \text{N}\left(d_1\right)-E\times e^{-Rt}\times \text{N}\left(d_2\right)\\ =\$21,800\times \left(0.69297471\right)-\left(\$35,000e^{-0.05\left(5\right)}\right)\left(0.20118388\right)\\ =\$15,106.84868-\$5,483.875715\\ =\$9,622.97\end{array}$

Hence, the call price is $9,622.97.

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

$P=E\times e^{-Rt}+C-S$

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:

$\begin{array}{c}P=E\times e^{-Rt}+C-S\\ =\$35,000e^{-0.05\left(5\right)}+\$9,622.97-21,800\\ =\$15,081\end{array}$

Hence, the put price is $15,081.

Formula to calculate the value of the risky bond:

$\text{Value of the risky bond}=\text{Present value of the sum}-\text{Put option price}$

Calculate the value of the risky bond:

$\begin{array}{c}\text{Value of the risky bond}=\text{Present value of the sum}-\text{Put option price}\\ =\$27,258.03-\$15,081\\ =\$12,177.03\end{array}$

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

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

$\text{Value of the risky bond}=\text{Face value of the bond}{e}^{-R\left(\text{Maturity period}\right)}$

Compute the return on debt:

$\begin{array}{c}\text{Value of the risky bond}=\text{Face value of the bond}{e}^{-R\left(\text{Maturity period}\right)}\\ \$12,177.03=\$35,000e^{-R\left(5\right)}\\ e^{-R\left(5\right)}=\frac{\$12,177.03}{\$35,000}\\ e^{-R\left(5\right)}=\$0.3479\end{array}$

$\begin{array}{c}{R}_D=-\left(\frac{1}{5}\right)\text{ln}\left(0.3479\right)\\ R_D=0.2112\text{ or }21.12\%\end{array}$

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.