Chapter 25, Problem 22QP

a)

Summary Introduction

To compute: The risk-free bond value.

Introduction:

Value of equity is the amount that comprises of the firm’s capital structure as equity shares. It is the total contribution of the equity shareholders to the firm. Value of debt is the amount that comprises of the firm’s capital structure as debt. It is the total contribution of the debt-holders to the firm.

Answer to Problem 22QP

The present market of the risk-free bond is $56,375.05.

Explanation of Solution

Given information: Company K has zero coupon bonds with a maturity period of 5 years at $80,000 face value. The present value on the assets of the company is $77,000 and the standard deviation is 34% per year. The risk-free rate is 7% per year.

Formula to calculate PV (Present Value) of risk-free bond:

$\text{PV}=E{e}^{-rt}$

Calculate PV (Present Value) of risk-free bond:

$\begin{array}{c}\text{PV}=E{e}^{-rt}\\ =\$80,000e^{-0.07\left(5\right)}\\ =\$56,375.05\end{array}$

Hence, PV of risk-free bond is $56,375.05.

b)

Summary Introduction

To compute: The put option value on the assets of the firm.

Introduction:

Value of equity is the amount that comprises of the firm’s capital structure as equity shares. It is the total contribution of the equity shareholders to the firm. Value of debt is the amount that comprises of the firm’s capital structure as debt. It is the total contribution of the debt-holders to the firm.

Answer to Problem 22QP

The put option value is $10,979.13.

Explanation of Solution

Formula to calculate the delta of the call option:

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

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 the delta of the call option:

$\begin{array}{c}{d}_1=\frac{\ln \left(\frac{S}{E}\right)+\left(r+\frac{{\sigma }^2}{2}\right)t}{\sigma \sqrt{t}}\\ =\frac{\ln \left(\frac{\$77,000}{\$80,000}\right)+\left(0.07+\frac{{0.34}^2}{2}\right)\times 5}{0.34\times \sqrt{5}}\\ =\frac{-\$0.038221212+\$0.639}{0.76026311234992}\\ =0.7902\end{array}$

Hence, d₁is 0.7902.

$\text{N}\left(d_1\right)=0.7853$

Note: The cumulative frequency distribution value for 0.7902 is 0.7853.

Hence, the delta for the call option is 0.7853.

Formula to calculate the delta of the put option:

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

Calculate the delta of the put option:

$\begin{array}{c}{d}_2=d_1-\sigma \sqrt{t}\\ =0.7902-0.34\times \sqrt{5}\\ =0.7902-76.02631123\\ =0.0300\end{array}$

Hence, d₂ is 0.0300.

$\text{N}\left(d_2\right)=0.5120$

Note: The cumulative frequency distribution value for 0.0300 is 0.5120.

Hence, the delta for the put option is 0.5120.

Formula to calculate the call price or equity 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

klC is the call price

R is the risk-free rate

t is the period of maturity

Calculate the call price or equity:

$\begin{array}{c}C=S\times \text{N}\left(d_1\right)-E\times e^{-Rt}\times \text{N}\left(d_2\right)\\ =\$77,000\times \left(0.7853\right)-\left(\$80,000e^{-0.07\left(5\right)}\right)\left(0.5120\right)\\ =\$60,468.1-\$28,864.02415\\ =\$31,604.07585\end{array}$

Hence, the call price or equity is $31,604.08.

Formula to calculate price of “put-option” using call-put parity (P):

$S+P=E{e}^{-RT}+C$

Calculate the price of “put-option” using call-put parity (P):

$\begin{array}{c}S+P=E{e}^{-RT}+C\\ P=\left(\$80,000e^{-.07\left(5\right)}\right)+\text{\$31,604}\text{.08}–\text{\$77,000 }\\ =\$56,375.04718+\text{\$31,604}\text{.08}–\text{\$77,000 }\\ =\$10,979.12718\end{array}$

Hence, the price of “put-option” is $10,979.13.

c)

Summary Introduction

To compute: The value and yield on the debt of the company.

Introduction:

Value of equity is the amount that comprises of the firm’s capital structure as equity shares. It is the total contribution of the equity shareholders to the firm. Value of debt is the amount that comprises of the firm’s capital structure as debt. It is the total contribution of the debt-holders to the firm.

Answer to Problem 22QP

The value and yield on the debt is $45,395.92 and 11.33% respectively.

Explanation of Solution

Formula to calculate value of debt:

$\text{Value of debt}=\text{Value}\text{?}\text{of}\text{?}\text{risk-free bond}-\text{Value}\text{?}\text{of}\text{?}\text{put-option on assets}$

Calculate value of debt:

$\begin{array}{c}\text{Value of debt}=\text{Value of risk-free bond}-\text{Value of put-option on assets}\\ \text{=\$56,375}\text{.04718}-\text{\$10,979}\text{.12718}\\ =\$45,395.92\end{array}$

Hence, value of debt is $45,395.92.

Formula to calculate yield on debt:

$\text{Value of debt}=E{e}^{-rt}$

Calculate yield on debt:

$\begin{array}{c}\text{Value of debt}=E{e}^{-rt}\\ \$45,395.92=\$80,000e^{-r\left(5\right)}\\ \frac{\$45,395.92}{\$80,000}=e^{-r\left(5\right)}\end{array}$

$\begin{array}{c}r=-\left(\frac{1}{5}\right)\ln \left(0.567449\right)\\ =0.1133\text{ or }11.33\%\end{array}$

Hence, PV yield on debt is 11.33%.

d)

Summary Introduction

To compute: The firm’s debt value after restructuring its assets.

Introduction:

Value of equity is the amount that comprises of the firm’s capital structure as equity shares. It is the total contribution of the equity shareholders to the firm. Value of debt is the amount that comprises of the firm’s capital structure as debt. It is the total contribution of the debt-holders to the firm.

Answer to Problem 22QP

The debt value of the firm is 13.43%.

Explanation of Solution

Formula to calculate PV (Present Value) of risk-free bond:

$\text{PV}=E{e}^{-rt}$

Calculate PV (Present Value) of risk-free bond:

$\begin{array}{c}\text{PV}=E{e}^{-rt}\\ =\$80,000e^{-0.07\left(5\right)}\\ =\$56,375.05\end{array}$

Hence, PV of risk-free bond is $56,375.05.

Formula to calculate the delta of the call option:

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

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 the delta of the call option:

$\begin{array}{c}{d}_1=\frac{\ln \left(\frac{S}{E}\right)+\left(r+\frac{{\sigma }^2}{2}\right)t}{\sigma \sqrt{t}}\\ =\frac{\ln \left(\frac{\$77,000}{\$80,000}\right)+\left(0.07+\frac{{0.43}^2}{2}\right)\times 5}{0.43\times \sqrt{5}}\\ =\frac{-0.038221212+0.81225}{0.96150923}\\ =0.8050\end{array}$

Hence, d₁is 0.8050.

$\text{N}\left(d_1\right)=0.78959016$

Note: The cumulative frequency distribution value for 0.8050 is 0.78959016.

Hence, the delta for the call option is 0.7896.

Formula to calculate the delta of the put option:

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

Calculate the delta of the put option:

$\begin{array}{c}{d}_2=d_1-\sigma \sqrt{t}\\ =0.8050-0.43\times \sqrt{5}\\ =0.8050-0.96150923\\ =-0.1565\end{array}$

Hence, d₂ is -0.1565.

$\text{N}\left(d_2\right)=0.43781946$

Note: The cumulative frequency distribution value for -0.1565 is 0.43781946.

Hence, the delta for the put option is 0.4378.

Formula to calculate the call price or equity 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 or equity:

$\begin{array}{c}\text{Equity}=C=S\times \text{N}\left(d_1\right)-E\times e^{-Rt}\times \text{N}\left(d_2\right)\\ =\$77,000\times \left(0.78959016\right)-\left(\$80,000e^{-0.07\left(5\right)}\right)\left(0.43781946\right)\\ =\$60,798.44232-\$24,682.09271\\ =\$36,116.35\end{array}$

Hence, the call price or equity is $36,116.35.

Formula to calculate price of “put-option” using call-put parity (P):

$S+P=E{e}^{-RT}+C$

Calculate the price of “put-option” using call-put parity (P):

$\begin{array}{c}S+P=E{e}^{-RT}+C\\ P=\left(\$80,000e^{-.07\left(5\right)}\right)+\text{\$36,116}\text{.35}–\text{\$77,000 }\\ =\$56,375.04718+\text{\$36,116}\text{.35}–\text{\$77,000 }\\ =\$15,491.40\end{array}$

Hence, the price of “put-option” is $15,491.40.

Formula to compute the value of risky bond:

$\text{Value of the risky bond}=\text{PV}-\text{Put price}$

Compute the value of risky bond:

$\begin{array}{c}\text{Value of the risky bond}=\text{PV}-\text{Put price}\\ =\$56,375.04718-\$15,491.40\\ =\$40,883.65\end{array}$

Hence, the value of the risky bond is $40,883.65.

Formula to compute the value of debt:

$\begin{array}{c}\text{Value of risky bond}=E{e}^{-RT}\\ \$40,883.65=\left(\$80,000e^{-R\left(5\right)}\right)\\ e^{-R\left(5\right)}=0.511045625\end{array}$

$\begin{array}{c}{R}_D=-\left(\frac{1}{5}\right)\text{ln}\left(0.5110\right)\\ =0.1343\text{ or }13.43\%\end{array}$

Hence, the value of debt is 13.43%.

The impact of time period on the bondholder’s yield:

The yields on debt show 11.33% with 34% standard deviation and 13.43% with 43% standard deviation. Here, the increase of “standard deviation” increases the value of debt. This is the reason for increasing the yield on debt from $11.33% to $13.43%. However, the risk to earn the return also increases proportionately.

e)

Summary Introduction

To discuss: The impact of bondholders and shareholders, if the firm restructures its assets and how this creates an agency problem.

Introduction:

Value of equity is the amount that comprises of the firm’s capital structure as equity shares. It is the total contribution of the equity shareholders to the firm. Value of debt is the amount that comprises of the firm’s capital structure as debt. It is the total contribution of the debt-holders to the firm.

Explanation of Solution

Formula to calculate gain or loss of shareholders and bondholders:

$\text{Equity value change}=\text{New firm's value of equity}-\text{Combined value of equity}$

$\text{Debt value change}=\text{New firm's value of debt}-\text{Value of debt}$

Calculate the gain or loss of shareholders:

$\begin{array}{c}\text{Equity value change}=\text{New firm's value of equity}-\text{Value of equity}\\ \text{=\$36,116}\text{.35}-\$31,604.08\\ =\$4,512.27\end{array}$

Hence, the gain of shareholders is $4,512.27.

Calculate the gain or loss of bondholders:

$\begin{array}{c}\text{Debt value change}=\text{New firm's value of debt}-\text{Value of debt}\\ \text{=\$40,883}\text{.65}-\$45,395.92\\ =-\$4,512.27\end{array}$

Hence, the loss of bondholders is -$4,512.27.

The impact of a firm’s reconstruction to shareholders and bondholders:

Reconstruction is favorable to shareholders. However, it creates an agency problem to bondholders. The company management is acting favorably to shareholders. They increased the wealth of shareholders by diluting the absolute amount of wealth from the bondholders. Therefore, this reconstruction is helpful to shareholders and difficult for bondholders.