Chapter 22, Problem 32QP

Two-State Option Pricing and Corporate Valuation Strudler Real Estate, Inc., a construction firm financed by both debt and equity, is undertaking a new project. If the project is successful, the value of the firm in one year will be $213 million, but if the project is a failure, the firm will be worth only $156 million. The current value of the company is $l85 million, a figure that includes the prospects for the new project. Strudler has outstanding zero coupon bonds due in one year with a face value of $l75 million. Treasury bills that mature in one year have an EAR of 7 percent. The company pays no dividends.

1. a. Use the two-state option pricing model to find the current value of the company’s debt and equity.
2. b. Suppose the company has 500,000 shares of common stock outstanding. What is the price per share of the company’s equity?
3. c. Compare the market value of the company’s debt to the present value of an equal amount of debt that is riskless with one year until maturity. Is the firm’s debt worth more than, less than, or the same as the riskless debt? Does this make sense? What factors might cause these two values to be different?
4. d. Suppose that in place of the preceding project the company’s management decides to undertake a project that is even more risky. The value of the company will either increase to $245 million or decrease to $135 million by the end of the year. Surprisingly, management concludes that the value of the company today will remain at exactly $185 million if this risky project is substituted for the Jess risky one. Use the two-state option pricing model to determine the values of the firm's debt and equity if the company plans on undertaking this new project. Which project do bondholders prefer?

Summary Introduction

To determine: Present value of the company’s debt and equity.

Debt:

Debt is the amount of money borrowed by an individual or a company forming a liability upon their assets.

Equity:

Equity is the amount of funds contributed by the shareholders (owners) of the company for use in the business.

Explanation of Solution

Given,

Current value of the firm is $185 million.

Value of the firm if project is successful is $213 million.

Value of the firm if the project falls is $156 million.

Risk free return on treasury bills is 7 percent.

Compute present value of equity.

Formula to calculate value of equity of the proposed project is:

$\text{PV}=\frac{\text{Expected payoff of the option}}{\left(1+\text{r}\right)}$

Substitute $197,952,000 for expected payoff of the option and 0.07 for r in the above equation,

$\begin{array}{c}\text{PV}=\frac{\$197,945,730}{1+0.07}\\ =\$184,996,009.34\end{array}$

Here, the present value of equity in the proposed project is $184,996,009.34.

Compute present value of debt.

Formula to calculate current value of company:

$\text{Current value of the company}=\text{Value of debt + Value of equity}$

Substitute $185 million for current value of the firm and $184,996,009.34 for value of equity in the above equation,

$\begin{array}{c}185,000,000=\text{Value of debt}+\$184,996,009.34\\ \text{Value of debt}=\$185,000,000-\$184,996,009.34\\ =\$3,990.65\end{array}$

Here, the present value of debt in the proposed project is $3,990.65.

Working note:

Calculate the rate of return of the proposed project:

$\begin{array}{c}{\text{R}}_{\text{R}}=\left(\frac{\text{Value of firm if project is successful}}{\text{Current value of the firm}}-1\right)\\ =\left(\frac{213}{185}-1\right)\\ =\left(1.1514-1\right)\\ =0.1514\end{array}$

Where,

• ${\text{R}}_{\text{R}}$ is the rate of return if the project becomes successful.

Here, the rate of return of the proposed project when the project is successful is 15.14 percent.

Calculate the rate of return of the proposed project:

$\begin{array}{c}{\text{R}}_{\text{F}}=\left(\frac{\text{Value of firm if project fails}}{\text{Current value of the firm}}-1\right)\\ =\left(\frac{156}{185}-1\right)\\ =\left(0.8432-1\right)\\ =-0.1568\end{array}$

Where,

• $R_F$ is the rate of return if the project fails.

Here, the rate of return of the proposed project when the project falls is -15.68 percent.

Calculate risk neutral probability of an increase in the value of the project:

$\begin{array}{c}r=\left(P_R\times R_R\right)+\left(P_F\times R_F\right)\\ 0.07=\left(P_R\times 0.1514\right)+\left(\left(1-P_R\right)\times -0.1568\right)\\ =0.1514P_R+(-0.1568+0.1568P_R)\\ 0.07+0.1568=0.1514P_R+0.1568P_R\end{array}$

Simplify above equation to calculate P_R.

$\begin{array}{c}0.07+0.1568=0.1514P_R+0.1568P_R\\ 0.2268=0.3082P_R\\ P_{R=}\frac{0.2268}{0.3082}\\ =0.73589\end{array}$

Where,

• $\text{r}$ is the risk free return on treasury bills.
• ${\text{P}}_{\text{R}}$ is the probability of return when the project is successful.
• ${\text{R}}_{\text{R}}$ is the rate of return when project is successful.
• ${\text{P}}_{\text{F}}$ is the probability of return when the project falls.
• ${\text{R}}_{\text{F}}$ is the rate of return when project falls.

Here, the risk neutral probability $P_R$ when the project is successful is 0.73588 or 73.589 percent.

Calculate risk neutral probability of decrease in the value of the project:

$\begin{array}{c}{P}_F=\left(1-P_R\right)\\ =\left(1-0.73589\right)\\ =0.26411\end{array}$

Here, the risk neutral probability $P_F$ when the project fails is 0.26411 or 26.411 percent.

Calculate expected payoff:

$\begin{array}{c}\text{Expected payoff}=\left[\begin{array}{c}\left(P_R\times \text{Value of call option after 1 year}\right)\\ +\left(P_F\times \text{Value of call option after 1 year}\right)\end{array}\right]\\ =\left(0.73589\times \$213,000,000\right)+\left(0.26411\times 156,000,000\right)\\ =\$156,744,570+\$41,201,160\\ =\$197,945,730\end{array}$

Here, the expected payoff of the option available at expiration is $197,945,730.

Hence the current value of company’s debt is $3,990.65 and value of equity is $184,996,009.35.

Summary Introduction

To determine: Price per share of the company’s equity.

Explanation of Solution

Given,

Total equity value is $184,996,009.35.

Number of shares outstanding is 500,000.

Formula to calculate price per share is:

$\text{Price per share}=\frac{\text{Total equity value}}{\text{Number of shares outstanding}}$

Substitute $184,996,009.35 for total equity value and 500,000 for number of shares outstanding in the above equation,

$\begin{array}{c}\text{Price per share}=\frac{\$184,996,009.35}{500,000}\\ =\$370\end{array}$

Here, the price per share is $370.

Hence, the present per share price of the company is $370.

Summary Introduction

To determine: Value of riskless debt.

Explanation of Solution

Given,

Value of debt is $3,990.65.

Risk free return on treasury bills is 7 percent.

Formula to calculate value of risk free return is:

$\text{Value of riskless debt}=\frac{\text{Value of debt}}{\left(1+r\right)}$

Substitute $3,990.65 for value of debt and 0.07 for $r$ in the above equation,

$\begin{array}{c}\text{Value of riskless debt =}\frac{\$3,990.65}{1+0.07}\\ =\$3,729.58\end{array}$

Here, the value of riskless debt is $3,729.58.

Hence, the value of riskless debt is $3,729.58 and the reason for change in value is the possibility that company might not repay its debt in full and thus the amount of debt is less than the amount of riskless debt.

Summary Introduction

To determine: Value of the firm’s debt and equity after the project and preference of bond holders among the two projects.

Explanation of Solution

Given,

Current value of the firm is $185 million.

Value of the firm if project is successful is $245 million.

Value of the firm if the project falls is $135 million.

Risk free return on treasury bills is 7 percent.

Formula to calculate value of equity of the proposed project is:

$\text{PV}=\frac{\text{Expected payoff of the option}}{\left(1+\text{r}\right)}$

Substitute $197,945,730 for expected payoff of the option and 0.07 for r in the above equation,

$\begin{array}{c}\text{PV}=\frac{\$197,953,000}{1+0.07}\\ =\$185,002,803.74\end{array}$

Here, the present value of equity in the proposed project is $185,002,803.74.

Formula to calculate value of debt in the proposed project is:

$\text{Current value of the company}=\text{Value of debt + value of equity}$

Substitute $185 million for current value of the firm and $185,002,803.74 for value of equity in the above equation,

$\begin{array}{c}\$185,000,000=\text{Value of debt}+\$185,002,803.74\\ \text{Value of debt}=\$185,000,000-\$185,002,803.74\\ =-\$2,803.74\end{array}$

Here, the present value of debt in the proposed project is $-\$2,803.74$.

Bond holders of the company would prefer the conservative project because,

• Generally, the risk project increases the value of equity of the firm.
• Highly risky project tends to decrease the value of company’s debt.

Working notes:

Calculate the rate of return of the proposed project:

$\begin{array}{c}{R}_R=\left(\frac{\text{Value of firm if project is successful}}{\text{Current value of the firm}}-1\right)\\ =\left(\frac{245}{185}-1\right)\\ =\left(1.3243-1\right)\\ =0.3243\end{array}$

Where,

• $R_R$ is the rate of return if the project becomes successful.

Here, the rate of return of the proposed project when the project is successful is 32.43 percent.

Calculate the rate of return of the proposed project:

$\begin{array}{c}{R}_F=\left(\frac{\text{Value of firm if project fails}}{\text{Current value of the firm}}-1\right)\\ =\left(\frac{135}{185}-1\right)\\ =\left(0.7297-1\right)\\ =-0.2703\end{array}$

Where,

• $R_F$ is the rate of return if the project fails.

Here, the rate of return of the proposed project when the project falls is -27.03 percent.

Calculate risk neutral probability of an increase in the value of the project:

$\begin{array}{c}r=\left(P_R\times R_R\right)+\left(P_F\times R_F\right)\\ 0.07=\left(P_R\times 0.3243\right)+\left(\left(1-P_R\right)\times -0.2703\right)\\ 0.07=0.3243P_R+(-0.2703+0.2703P_R)\\ 0.07+0.2703=0.3243P_R+0.2703P_R\end{array}$

Simplify above equation to calculate P_R.

$\begin{array}{c}0.07+0.2703=0.3243P_R+0.2703P_R\\ P_R=\frac{0.3403}{0.5946}\\ P_R=0.5723\end{array}$

Where,

• $r$ is the risk free return on treasury bills.
• $P_R$ is the probability of return when the project is successful.
• $R_R$ is the rate of return when project is successful.
• $P_F$ is the probability of return when the project falls.
• $R_F$ is the rate of return when project falls.

Here, the risk neutral probability $P_R$ when the project is successful is 0.5723 or 57.23 percent.

Calculate risk neutral probability of decrease in the value of the project:

$\begin{array}{c}{P}_F=\left(1-P_R\right)\\ =\left(1-0.5723\right)\\ =0.4277\end{array}$

Here, the risk neutral probability $P_F$ when the project fails is 0.4277 or 42.77 percent.

Calculate expected payoff:

$\begin{array}{c}\text{Expected payoff}=\left[\begin{array}{c}\left(P_R\times \text{Value of call option after 1 year}\right)\\ +\left(P_F\times \text{Value of call option after 1 year}\right)\end{array}\right]\\ =\left(0.5723\times \$245,000,000\right)+\left(0.4277\times \$135,000,000\right)\\ =\$140,213,500+\$57,739,500\\ =\$197,953,000\end{array}$

Here, the expected payoff of the option available at expiration is $197,593,000.

Hence the current value of company’s debt is $-\$2,803.74$ and value of equity is $185,002,803.74. Thus, the bond holders should prefer the conservative project as it is beneficial for them.