Chapter 25, Problem 20QP

a)

Summary Introduction

To compute: The present market value of the equity 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 20QP

The present market value of the equity is $7,584,629.086.

Explanation of Solution

Given information:

A firm has outstanding single zero coupon bonds with a maturity period of 5 years at $20 million face value. The present value on the assets of the company is $18.1 million and the standard deviation is 41% per year. The risk-free rate is 6% per year.

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:

$\begin{array}{c}{d}_1=\frac{\ln \left(\frac{S}{E}\right)+\left(r+\frac{{\sigma }^2}{2}\right)t}{\sigma \sqrt{t}}\\ =\frac{\ln \left(\frac{\$18,100,000}{\$20,000,000}\right)+\left(0.06+\frac{{0.41}^2}{2}\right)\times 5}{0.41\times \sqrt{5}}\\ =\frac{-\$0.099820335+\$0.72025}{0.91678787}\\ =0.6767\end{array}$

Hence, d₁is 0.6767.

$\text{N}\left(d_1\right)=0.75070184$

Note: The cumulative frequency distribution value for 0.6767 is 0.75070184.

Hence, the delta for the call option is $0.75070184.

Formula to calculate the delta of the put option:

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

Calculate the delta of the put option:

$\begin{array}{c}{d}_2=d_1-\sigma \sqrt{t}\\ =0.6767-0.41\times \sqrt{5}\\ =0.6767-0.91678787\\ =-0.2400\end{array}$

Hence, d₂ is -0.2400.

$\text{N}\left(d_2\right)=0.40516513$

Note: The cumulative frequency distribution value for -0.2400 is 0.40516513.

Hence, the delta for the put option is $0.40516513.

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:

$\begin{array}{c}C=S\times \text{N}\left(d_1\right)-E\times e^{-Rt}\times \text{N}\left(d_2\right)\\ =\$18,100,000\times \left(0.75070184\right)-\left(\$20,000,000e^{-0.06\left(5\right)}\right)\left(0.40516513\right)\\ =\$13,587,703.3-\$6,003,074.214\\ =\$7,584,629.086\end{array}$

Hence, the call price or equity is $7,584,629.086.

b)

Summary Introduction

To compute: The present value 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 20QP

The present value on the debt of the company is $10,515,370.91.

Explanation of Solution

Given information:

A firm has outstanding single zero coupon bonds with a maturity period of 5 years at $20 million face value. The present value on the assets of the company is $18.1 million and the standard deviation is 41% per year. The risk-free rate is 6% per year.

Formula to calculate the value of debt:

Value of debt is the value of the firm minus the value of equity.

$\text{Value of debt}=\text{Value of firm}-\text{Value of equity}$

Calculate the value of debt:

It is given that the value of firm is $18,100,000 and value of equity is $7,584,629.086.

$\begin{array}{c}\text{Value of debt}=\text{Value of firm}-\text{Value of equity}\\ \text{=}\$18,100,000-\$7,584,629.086\\ =\$10,515,370.91\end{array}$

Hence, value of debt is $10,515,370.91.

c)

Summary Introduction

To compute: The cost of debt.

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 20QP

The cost of debt is 12.86%.

Explanation of Solution

Given information:

A firm has outstanding single zero coupon bonds with a maturity period of 5 years at $20 million face value. The present value on the assets of the company is $18.1 million and the standard deviation is 41% per year. The risk-free rate is 6% per year.

Compute the debt value:

$\begin{array}{c}\text{Value of debt}=E{e}^{-rt}\\ \$10,515,370.91=\$20,000,000e^{-r(5)}\\ \frac{\$10,515,370.91}{\$20,000,000}=e^{-r(5)}\\ 0.525768545=e^{-r(1)}\end{array}$

$\begin{array}{c}r=-\left(\frac{1}{5}\right)\ln \left(0.525768545\right)\\ =0.1286\text{ or }12.86\%\end{array}$

Hence, the “cost of debt” is 12.86%.

d)

Summary Introduction

To compute: The new market value of the equity.

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 20QP

The new market value of the equity is $9,201,195.626.

Explanation of Solution

Given information:

A firm has outstanding single zero coupon bonds with a maturity period of 5 years at $20 million face value. The present value on the assets of the company is $18.1 million and the standard deviation is 41% per year. The risk-free rate is 6% per year.

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:

$\begin{array}{c}{d}_1=\frac{\ln \left(\frac{S}{E}\right)+\left(r+\frac{{\sigma }^2}{2}\right)t}{\sigma \sqrt{t}}\\ =\frac{\ln \left(\frac{\$20,200,000}{\$20,000,000}\right)+\left(0.06+\frac{{0.41}^2}{2}\right)\times 5}{0.41\times \sqrt{5}}\\ =\frac{0.009950330853+\$0.72025}{0.91678787}\\ =0.7965\end{array}$

Hence, d₁is 0.7965.

$\text{N}\left(d_1\right)=0.78712926$

Note: The cumulative frequency distribution value for 0.7965 is 0.78712926.

Hence, the delta for the call option is $0.78712926.

Formula to calculate the delta of the put option:

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

Calculate the delta of the put option:

$\begin{array}{c}{d}_2=d_1-\sigma \sqrt{t}\\ =0.7965-0.41\times \sqrt{5}\\ =0.6767-0.91678787\\ =-0.1203\end{array}$

Hence, d₂ is -0.1203.

$\text{N}\left(d_2\right)=0.45212275$

Note: The cumulative frequency distribution value for -0.1203 is 0.45212275.

Hence, the delta for the put option is $0.45212275.

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:

$\begin{array}{c}\text{Equity}=C=S\times \text{N}\left(d_1\right)-E\times e^{-Rt}\times \text{N}\left(d_2\right)\\ =\$20,200,000\times \left(0.78712926\right)-\left(\$20,000,000e^{-0.06\left(5\right)}\right)\left(0.45212275\right)\\ =\$15,900,011.05-\$6,698,815.424\\ =\$9,201,195.626\end{array}$

Hence, the call price or equity is $9,201,195.626.

e)

Summary Introduction

To compute: The new debt cost.

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 20QP

The new cost of debt is 11.96%.

Explanation of Solution

Given information:

A firm has outstanding single zero coupon bonds with a maturity period of 5 years at $20 million face value. The present value on the assets of the company is $18.1 million and the standard deviation is 41% per year. The risk-free rate is 6% per year.

Formula to calculate value of debt:

Value of debt is value of firm minus value of equity.

$\text{Value of debt}=\text{Value of firm}-\text{Value of equity}$

Calculate the value of debt:

$\begin{array}{c}\text{Value of debt}=\text{Value of firm}-\text{Value of equity}\\ \text{=\$20,200,000}-\text{\$9,201,195}\text{.626}\\ =\$10,998,804.37\end{array}$

Hence, value of debt is $10,998,804.37.

Formula to calculate “cost of debt:”

Substitute the “value of debt” formula below to calculate the “cost of debt." “Value of debt” is calculated by multiplying the exercise price and the exponential powered minus an interest rate and time period.

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

Calculate the “cost of debt” with the new project:

$\begin{array}{c}\text{Value of debt}=K{e}^{-rt}\\ \$10,998,804.37=\$20,000,000e^{-r(5)}\\ \frac{\$10,998,804.37}{\$20,000,000}=e^{-r(5)}\\ 0.549940218=e^{-r(5)}\end{array}$

$\begin{array}{c}r=-\left(\frac{1}{5}\right)\ln \left(0.549940218\right)\\ =0.1196\text{ or11}\text{.96\%}\end{array}$

Hence, the “cost of debt” is 11.96%.

The impact from the acceptance of a new project on equity value:

The consideration of a new project increases the value of total assets. At same time, it increases the present value of bonds and equity. It makes the company secure and safe (due to an increase of total value of assets). However, the acceptance of a new project reduces the return on bondholders from 12.86% to 11.96%. This reduction of return is only due to the increase of bond value.