Chapter 22, Problem 22QP

Equity as an Option and NPV Suppose the firm in the previous problem is considering two mutually exclusive investments. Project A has an NPV of $1,200, and Project B has an NPV of $1,600. As a result of taking Project A, the standard deviation of the return on the firm’s assets will increase to 55 percent per year. If Project B is taken, the standard deviation will fall to 34 percent per year.

a. What is the value of the firm’s equity and debt if Project A is undertaken? If Project B is undertaken?

b. Which project would the stockholders prefer? Can you reconcile your answer with the NPV rule?

c. Suppose the stockholders and bondholders are, in fact, the same group of investors. Would this affect your answer to (b)?

d. What does this problem suggest to you about stockholder incentives?

Summary Introduction

To compute: Value of the firm’s equity and debt under project A and project B.

Option Pricing:

Option pricing helps in determining 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

Project A

Given,

Stock price is $\$21,700+\$1,200=\$22,900$

Exercise price is 20,000.

Risk free rate is 0.05.

Time to expire is 1 year.

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 $22,900 for S, $20,000 for E, 0.05 for R, and 1 for T.

$\begin{array}{c}\text{Value of equity}=\left(\$22,900\left(0.8291\right)\right)-\left(\$20,000e^{-0.05\times 1}\left(0.5239\right)\right)\\ =\$18,986.39-\left(\$20,000\times 0.9512294\left(0.5239\right)\right)\\ =\$18,986.76-\$9,966.98\\ =\$9,019.78\end{array}$

Formula to calculate the value of debt is,

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

Substitute $22,900 as value of firm and $9,019.78 as value of equity.

$\begin{array}{c}\text{Value}\text{of}\text{debt}=\$22,900-\$9,019.78\\ =\$13,880.22\end{array}$

Project B

Given,

Stock price is $21,700.

Exercise price is 20,000.

Risk free rate is 0.05.

Time to expire is 1 year.

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

Where,

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

Substitute $21,700 for S, $20,000 for E, 0.05 for R, and 1 for T.

$\begin{array}{c}\text{Value of equity}=\left(\$21,700\left(0.7088\right)\right)-\left(\$20,000e^{-0.05\times 1}\left(0.5832\right)\right)\\ =\$15,380.96-\left(\$20,000\times 0.9512294\left(0.5832\right)\right)\\ =\$4,285.82\end{array}$

Formula to calculate the value of debt is,

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

Substitute $21,700 as value of firm and $4,285.82 as value of equity.

$\begin{array}{c}\text{Value}\text{of}\text{debt}=\$21,700-\$4,285.82\\ =\$17,414.18\end{array}$

Working Note:

Formula to calculate $d_1$ is,

${\text{d}}_{\text{1}}=\frac{\text{In}\left(\frac{\text{S}}{\text{E}}\right)+\left(\text{R}+\frac{{\text{σ}}^{\text{2}}}{\text{2}}\right)\text{t}}{\sqrt{{\text{σ}}^{\text{2}}\text{t}}}$

Calculation of $d_1$ for Project A,

$\begin{array}{c}{d}_1=\frac{\text{In}\left(\frac{\$22,900}{\$20,000}\right)+\left(0.05+\frac{{0.55}^2}{2}\right)\times 1}{\sqrt{{0.55}^2\times 1}}\\ =\frac{0.1354+0.20125}{0.55}\\ =0.6121\end{array}$

From normal distribution table ${\text{N(d}}_{\text{1}}\text{)}=0.8291$

Calculation of $d_1$ for Project B,

$\begin{array}{c}{d}_1=\frac{\text{In}\left(\frac{\$21,700}{\$20,000}\right)+\left(0.05+\frac{{0.34}^2}{2}\right)\times 1}{\sqrt{{0.34}^2\times 1}}\\ =\frac{0.08158+0.1078}{0.34}\\ =0.557\end{array}$

From normal distribution table ${\text{N(d}}_1\text{)}=0.7088$

Formula to calculate $d_2$ is,

$d_2=d_1-\sqrt{{\sigma }^{2}t}$

Calculation of $d_2$ for Project A,

$\begin{array}{c}{d}_2=0.6121-\sqrt{{0.55}^2\times 1}\\ =0.6121-0.55\\ =0.0621\end{array}$

From normal distribution table ${\text{N(d}}_2\text{)}=0.5239$

Calculation of $d_2$ for Project B,

$\begin{array}{c}{d}_2=0.557-\sqrt{{0.34}^2\times 1}\\ =0.557-0.34\\ =0.217\end{array}$

From normal distribution table ${\text{N(d}}_{\text{2}}\text{)}=0.5832$

Hence, for Project A the value of firm’s equity is $9,019.78, value of firm’s debt is$13,880.22 and for Project B the value of firm’s equity is $4,285.82 and value of firm’s debt is $17,414.18.

Summary Introduction

To identify: Project that would be preferred by stockholders.

Answer to Problem 22QP

• Here, equity’s value is higher in Project A than Project B.
• Project A does not create more bondholders.

Explanation of Solution

• If Project A is considered, it has increased the firm’s assets to$1,200.
• If Project B is considered, it has increased the firm’s assets to$1,600.
• NPV rules say Project B should be accepted, but value of equity is more in the case of Project A rather than Project B, which shows that Project A has less of bondholders.
• Thus, Project A is more attractive.

Hence, the stockholders prefer Project A.

Summary Introduction

To identify: Project that would be preferred by stockholders if both stockholders and bondholders are same.

Answer to Problem 22QP

As Project B adds more value to the firm, this would be a good option.

Explanation of Solution

• If stockholders and bondholder would be the same, in that case their interest would also be the same and they can get benefits equally.
• Since Project A increases the firm’s assets to$1,200 and Project B increases the firm’s assets to$1,600.
• Thus, Project B is more attractive.

Hence, the stockholders prefer Project B.

Summary Introduction

To explain: The effect on stockholders incentives.

Answer to Problem 22QP

In case of leveraged firm, stockholders would definitely prefer those projects, which would increase value of equity.

Explanation of Solution

• Reason for opting equity source is that in the case of debt source, risk is borne by the bondholders and benefits are limited to their debt value, which is not happening in the case of equity sources.
• All benefits after paying the debt, goes to the stockholders pocket.
• Thus, the stockholders incentive would relate to the project that adds more value to the equity.

Hence, stockholder’s incentives are more related with the project that contains equity.