Chapter 22, Problem 25QP

Equity as an Option and NPV A company has a single zero coupon bond outstanding that matures in 10 years with a face value of $10 million. The current value of the company’s assets is $9.05 million, and the standard deviation of the return on the firm’s assets is 39 percent per year. The risk-free rate is 6 percent per year, compounded continuously.

1. a. What is the current market value of the company’s equity?
2. b. What is the current market value of the company’s debt?
3. c. What is the company's continuously compounded cost of debt?
4. d. The company has a new project available. The project has an NPV of $1,200,000. If the company undertakes the project, what will be the new market value of equity? Assume volatility is unchanged.
5. e. Assuming the company undertakes the new project and does not borrow any additional funds, what is the new continuously compounded cost of debt? What is happening here?

Summary Introduction

To compute: Current market value of the company’s equity.

Option Pricing:

Option pricing helps in determining the correct or fair price in the market. It is the value of one share on the basis of which option is traded. Black-Scholes is one of the pricing methods. Further, equity is also used as an option.

Explanation of Solution

Given,

Stock price is $9,050,000.

Exercise price is $10,000,000.

Risk free rate is 0.06.

Time to expire is 10.

Formula to calculate the value of equity by using Black Scholes model is,

$\text{Value}\text{of}\text{equity}=\text{SN}\left({\text{d}}_{\text{1}}\right)-{\text{Ee}}^{-\text{Rt}}\text{N}\left({\text{d}}_{\text{2}}\right)$

Where,

• S is stock price.
• E is exercise price.
• R is risk free rate.
• T is time to expire.

Substitute $9,050,000 for S, $10,000,000 for E, 0.06for R, and 10for T.

$\begin{array}{c}\text{Value of equity}=\$9,050,000\left(0.8467\right)-\$10,000,000e^{-0.06\times 10}\left(0.4164\right)\\ =\$7,662,635-\$10,000,000\times 0.54881\left(0.4164\right)\\ =\$7,662,635-\$2,285,244.84\\ =\$5,377,390.16\end{array}$

Working Note:

Calculation of ${\text{d}}_{\text{1}}$,

$\begin{array}{c}{\text{d}}_{\text{1}}=\frac{\text{In}\left(\frac{\text{S}}{\text{E}}\right)+\left(\text{R}+{\sigma }^2\times 0.50\right)\text{t}}{\sigma \sqrt{t}}\\ =\frac{\text{In}\left(\frac{\$9,050,000}{\$10,000,000}\right)+\left(0.06+{0.39}^2\times 0.50\right)\times 10}{0.39\sqrt{10}}\\ =\frac{-0.0998+1.3605}{0.39\times 3.16227766}\\ =1.0222\end{array}$

${\text{N(d}}_{\text{1}}\text{)}=0.8467$

Calculation of $d_2$,

$\begin{array}{c}{d}_2=d_1-\sqrt{{\sigma }^{2}t}\\ =1.0222-\sqrt{{0.39}^2\times 10}\\ =1.0222-1.2333\\ =-0.2111\end{array}$

$\text{N}\left({\text{d}}_{\text{2}}\right)=0.4164$

Hence, current market value of the company’s equity is $5,377,390.16.

Summary Introduction

To compute: Current market value of the company’s debt.

Explanation of Solution

Given,

Value of company is $9,050,000.

Value of equity is $5,377,390.16.

Formula to calculate the value of debt is,

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

Substitute $9,050,000 as value of company and $5,377,390.16as value of equity.

$\begin{array}{c}\text{Value}\text{of}\text{debt}=\$9,050,000-\$5,377,390.16\\ =\$3,672,609.84\end{array}$

Hence, current market value of company’s debt is $3,672,609.84.

Summary Introduction

To compute: Company’s continuously compounded cost of debt.

Explanation of Solution

Given,

Value of debt is $3,672,609.84.

Face value is $10,000,000.

Time to expire is 10 years.

Formula to calculate the firm’s continuously cost of debt is,

$\text{Value}\text{of}\text{debt}=\text{Face}\text{value}\times {\text{e}}^{-\text{R}\times \text{t}}$

Where,

• R is firm’s continuously cost of debt.
• t is maturity time.

Substitute $3,672,609.84as value of debt, $10,000,000 as face value and 10 for t.

$\begin{array}{c}\$3,672,609.84=\$10,000,000{\text{e}}^{-\text{R}\times 10}\\ \frac{\$3,672,609.84}{\$10,000,000}={\text{e}}^{-\text{R}\times 10}\\ \text{In}\left(0.367260984\right)=-\text{R}\times 10\end{array}$

Simplify the above equation.

$\begin{array}{c}-1.0017=-\text{R}\times 10\\ \text{R}=0.10017\\ \text{or}\\ 10.017\%\end{array}$

Hence, the company’s continuously cost of debt is 10.02%.

Summary Introduction

To compute: New market value of equity.

Explanation of Solution

Given,

Exercise price is 10,000,000.

Risk free rate is 0.06.

Time to expire is 10 years.

Calculated stock price:

The stock price is $10,250,000.

Formula to calculate the value of equity by using Black Scholes model is,

$\text{Value}\text{of}\text{equity}=\text{SN}\left({\text{d}}_{\text{1}}\right)-{\text{Ee}}^{-\text{Rt}}\text{N}\left({\text{d}}_{\text{2}}\right)$

Where,

• S is stock price.
• E is exercise price.
• R is risk free rate.
• T is time to expire.

Substitute $10,250,000for S, 10,000,000 for E, 0.06 for R, and 10 for T.

$\begin{array}{c}\text{Value of equity}= \left(\$10,250,000\left(0.8693\right)\right)-\left(\$10,000,000e^{-0.06\times 10}\left(0.4562\right)\right)\\ =\$8,810,325-\left(\$10,000,000\times 0.54885\left(0.4562\right)\right)\\ =\$8,810,325-\$2,503,853.7\\ =\$6,306,471.3\end{array}$

Working Note:

Calculation of ${\text{d}}_{\text{1}}$,

$\begin{array}{c}{\text{d}}_{\text{1}}=\frac{\text{In}\left(\frac{\text{S}}{\text{E}}\right)+\left(\text{R}+{\sigma }^2\times 0.50\right)\text{t}}{\sigma \sqrt{\text{t}}}\\ =\frac{\text{In}\left(\frac{\$10,250,000}{\$10,000,000}\right)+\left(0.06+{0.39}^2\times 0.50\right)\times 10}{0.39\sqrt{\times 10}}\\ =\frac{0.02469+1.3605}{1.2333}\\ =1.1232\end{array}$

${\text{N(d}}_{\text{1}}\text{)}=0.8693$

Calculation of $d_2$,

$\begin{array}{c}{\text{d}}_{\text{2}}={\text{d}}_{\text{1}}-\sqrt{{\text{σ}}^{\text{2}}\text{t}}\\ =1.1232-\sqrt{{0.39}^2\times 10}\\ =1.1232-1.2333\\ =-0.1101\end{array}$

${\text{N(d}}_{\text{2}}\text{)}=0.4562$

Calculate the revised stock price,

$\begin{array}{c}\text{Revised}\text{stock}\text{price}=\text{Current}\text{market}\text{value}\text{of}\text{assests}+\text{NPV}\\ \text{=\$9,050,000+\$1,200,000}\\ \text{=\$10,250,000}\end{array}$

Hence, new market value of equity is $6,406,471.3.

Summary Introduction

To compute: The new continuously compounded cost of debt.

Explanation of Solution

Given,

Value of company is $9,050,000.

Value of equity is $6,406,471.3.

Time to expire is 10 years.

Calculated value:

Value of equity is $6,406,471.3.

Value of the stock is $10,250,000.

Formula to calculate the value of debt is,

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

Substitute $10,250,000as value of company and $6,406,471.3 as value of equity.

$\begin{array}{c}\text{Value}\text{of}\text{debt}=\$10,250,000-\$6,406,471.3\\ =\$3,843,528.7\end{array}$

Formula to calculate the new continuously cost of debt is,

$\text{Value}\text{of}\text{debt}=\text{Face}\text{value}\times {\text{e}}^{-\text{R×t}}$

Where,

• R is firm’s continuously cost of debt.
• t is maturity time.

Substitute $3,843,528.7 as value of debt, $10,000,000 as face value and 10 for t.

$\begin{array}{c}\$10,000,000=\$3,843,528.7{\text{e}}^{-\text{R}\times \text{10}}\\ 2.6018={\text{e}}^{-\text{R}\times \text{10}}\\ \text{In}\left(2.6018\right)=-R\times 10\end{array}$

Simplify the above equation.

$\begin{array}{c}\text{0}\text{.9562}=\text{R}\times \text{10}\\ \text{R}=\text{0}\text{.09562}\\ \text{or}\\ \text{9}\text{.56\%}\end{array}$

Hence, the company’s new continuously cost of debt is 9.56%.