Chapter 25, Problem 17QP

Equity as an Option and NPV [LO4] Suppose the firm in the previous problem is considering two mutually exclusive investments. Project A has an NPV of $1,900, and Project B has an NPV of $2,800. As the result of taking Project A, the standard deviation of the return on the firm’s assets will increase to 46 percent per year. If Project B is taken, the standard deviation will fall to 29 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: The value of company’s debt and equity for both the projects

Introduction:

Value of equity:

The value of equity is the amount comprised in the firm’s capital structure as equity shares. It is the total contribution of the equity shareholders to the firm.

Value of debt:

The value of debt is the amount comprised in the firm’s capital structure as debt. It is the total contribution of the debt holders to the firm.

Answer to Problem 17QP

The values of the company’s equity and debt for Project A are $6,416.10 and $14,583.90 and for Project B are $6,274.76 and $15,625.24.

Explanation of Solution

Given information:

Company SS is planning two mutually exclusive projects. The NPV of Project A is $1,900 and for Project B, it is $2,800. The return on the standard deviation on the assets of the firm will raise to 46% per year. If Project B is taken, the standard deviation will decrease to 29% per year.

Explanation:

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 for Project A:

$\begin{array}{c}{d}_1=\frac{\ln \left(\frac{S}{E}\right)+\left(r+\frac{{\sigma }^2}{2}\right)t}{\sigma \sqrt{t}}\\ =\frac{\ln \left(\frac{\$21,000}{\$17,000}\right)+\left(0.06+\frac{{0.46}^2}{2}\right)\times 1}{0.46\times \sqrt{1}}\\ =\frac{0.21131+0.1658}{0.46}\\ =0.8198\end{array}$

Hence, d₁is $0.8198.

$\text{N}\left(d_1\right)=0.79383493$

Note: The cumulative frequency distribution value for 0.8198 is 0.79383493.

Hence, the delta for the call option is $0.7938.

Formula to calculate the delta of the put option:

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

Calculate the delta of the put option for Project A:

$\begin{array}{c}{d}_2=d_1-\sigma \sqrt{t}\\ =0.8198-0.46\times \sqrt{1}\\ =0.8198-0.46\\ =0.3598\end{array}$

Hence, d₂ is $0.3598.

$\text{N}\left(d_2\right)=0.64050165$

Note: The cumulative frequency distribution value for 0.3598 is 0.64050165.

Hence, the delta for the put option is $0.6405.

Formula to calculate the 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 equity for project A:

$\begin{array}{c}\text{Equity}=C=S\times \text{N}\left(d_1\right)-E\times e^{-Rt}\times \text{N}\left(d_2\right)\\ =\$21,000\times N(0.8198)-N(0.3598)\times \left(\$17,000e^{-0.06\left(1\right)}\right)\\ =\text{ \$21,000}\left(0.79383493\right)–0.64050165\times \$16,009.99707\end{array}$

$\begin{array}{c}=\$16,670.53353-\$10,254.42954\\ =\$6,416.10\end{array}$

Hence, the equity for Project A is $6,416.10.

Formula to calculate the debt amount:

$\text{Debt}=\text{Asset value}-\text{Equity}$

Calculate the debt amount for Project A:

$\begin{array}{c}\text{Debt}=\text{Asset value}-\text{Equity}\\ =\$21,000-\$6,416.10399\\ =\$14,583.90\end{array}$

Hence, the debt amount for Project A is $14,583.90.

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 for Project B:

$\begin{array}{c}{d}_1=\frac{\ln \left(\frac{S}{E}\right)+\left(r+\frac{{\sigma }^2}{2}\right)t}{\sigma \sqrt{t}}\\ =\frac{\ln \left(\frac{\$21,900}{\$17,000}\right)+\left(0.06+\frac{{0.29}^2}{2}\right)\times 1}{0.29\times \sqrt{1}}\\ =\frac{0.25327+0.10205}{0.29}\\ =1.2253\end{array}$

Hence, d₁is $1.2253.

$\text{N}\left(d_1\right)=0.88976890$

Note: The cumulative frequency distribution value for 1.2253 is 0.88976890.

Hence, the delta for the call option is $0.8898.

Formula to calculate the delta of the put option:

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

Calculate the delta of the put option for Project B:

$\begin{array}{c}{d}_2=d_1-\sigma \sqrt{t}\\ =1.2253-0.29\times \sqrt{1}\\ =1.2253-0.29\\ =0.9353\end{array}$

Hence, d₂ is $0.9353.

$\text{N}\left(d_2\right)=0.82518314$

Note: The cumulative frequency distribution value for 0.9353 is 0.82518314.

Hence, the delta for the put option is $0.8252.

Formulae to calculate the 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 equity for project B:

$\begin{array}{c}\text{Equity}=C=S\times \text{N}\left(d_1\right)-E\times e^{-Rt}\times \text{N}\left(d_2\right)\\ =\$21,900\times N(1.2253)-N(0.9353)\times \left(\$17,000e^{-0.06\left(1\right)}\right)\\ =\text{ \$21,900}\left(0.88976890\right)–0.82518314\times \$16,009.99707\\ =\$19,485.93891-\$13,211.17965\end{array}$

$=\$6,274.76$

Hence, the equity for Project B is $6,274.76.

Formula to calculate the debt amount:

$\text{Debt}=\text{Asset value}-\text{Equity}$

Calculate the debt amount for Project B:

$\begin{array}{c}\text{Debt}=\text{Asset value}-\text{Equity}\\ =\$21,900-\$6,274.76\\ =\$15,625.24\end{array}$

Hence, the debt amount for Project B is $15,625.24.

Summary Introduction

To discuss: The project that the stockholders prefer, using the rule of NPV.

Introduction:

Value of equity:

The value of equity is the amount comprised in the firm’s capital structure as equity shares. It is the total contribution of the equity shareholders to the firm.

Value of debt:

The value of debt is the amount comprised in the firm’s capital structure as debt. It is the total contribution of the debt holders to the firm.

Explanation of Solution

Given information:

Company SS is planning two mutually exclusive projects. The NPV of Project A is $1,900 and for Project B, it is $2,800. The return on the standard deviation on the assets of the firm will raise to 46% per year. If Project B is taken, the standard deviation will decrease to 29% per year.

Explanation:

As per the rule of NPV (Net Present Value), project-B is more beneficial (higher value) than Project A. The selection of Project B would increase the value of firm rather than when selecting Project A. However, the firm is highly levered.

Hence, the selection of Project B is highly beneficial to debt-holders. This project selection creates only a small increase in the equity’s value. Therefore, the stockholders will select Project A, even if the NPV is lower.

Summary Introduction

To discuss: The selection of the project if the stocks and bondholders are the same investors.

Introduction:

Value of equity:

The value of equity is the amount comprised in the firm’s capital structure as equity shares. It is the total contribution of the equity shareholders to the firm.

Value of debt:

The value of debt is the amount comprised in the firm’s capital structure as debt. It is the total contribution of the debt holders to the firm.

Explanation of Solution

Given information:

Company SS is planning two mutually exclusive projects. The NPV of Project A is $1,900 and for Project B, it is $2,800. The return on the standard deviation on the assets of the firm will raise to 46% per year. If Project B is taken, the standard deviation will decrease to 29% per year.

Explanation:

If the stockholders and bondholders are the same investors, then Project B must be selected. It increases the value of the firm from 21,000 to 21,900. This value addition increases the value of debt as well as equity kept by the same investors. Therefore, investors do not feel the leakage of earnings.

Summary Introduction

To discuss: The impact of leverage on a stockholder’s incentive.

Introduction:

Value of equity:

The value of equity is the amount comprised in the firm’s capital structure as equity shares. It is the total contribution of the equity shareholders to the firm.

Value of debt:

The value of debt is the amount comprised in the firm’s capital structure as debt. It is the total contribution of the debt holders to the firm.

Explanation of Solution

Given information:

Company SS is planning two mutually exclusive projects. The NPV of Project A is $1,900 and for Project B, it is $2,800. The return on the standard deviation on the assets of the firm will raise to 46% per year. If Project B is taken, the standard deviation will decrease to 29% per year.

Explanation:

Stockholders are the “risk takers” of the company. The highly levered capital structure increases the risk of equity and dilutes the earnings of equity. Therefore, stockholders receive less incentive for taking a higher risk from highly levered firms.