22.21 Equity as an Option Sunburn Sunscreen has a zero coupon bond issue outstanding with a $15,000 face value that matures in one year. The current market value of the firm’s assets is $15,800. The standard deviation of the return on the firm’s assets is 38 percent per year, and the annual risk-free rate is 5 percent per year, compounded continuously. Based on the Black–Scholes model, what is the market value of the firm’s equity and debt?

22.22. 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.

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

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

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

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

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b. Although the NPV of project B is higher, the equity value with
project A is higher. While NPV represents the increase in the
value of the assets of the firm, in this case, the increase in the value
of the firm’s assets resulting from project B is mostly allocated
to the debtholders, resulting in a smaller increase in the value
of the equity. Stockholders would, therefore, prefer project A
even though it has a lower NPV.

c. Yes. If the same group of investors have equal stakes
in the firm as bondholders and stock-holders, then total firm
value matters and project B should be chosen, since it
increases the value of the firmto $17,400 instead of $17,000.
d. Stockholders may have an incentive to take on riskier,
less profitable projects if the firm is leveraged; the higher
the firm’s debt load, all else the same, the greater is this incentive.

22.24. Mergers and Equity as an Option Suppose Sunburn Sunscreen and Frostbite Thermalwear in the previous problems have decided to merge. Because the two companies have seasonal sales, the combined firm’s return on assets will have a standard deviation of 29 percent per year.

1. What is the combined value of equity in the two existing companies? The value of debt?

2. What is the value of the new firm’s equity? The value of debt?

3. What was the gain or loss for shareholders? For bondholders?

4. What happened to shareholder value here?

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22.25. 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 $25 million. The current value of the company’s assets is $21 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. What is the current market value of the company’s equity?

2. What is the current market value of the company’s debt?

3. What is the company’s continuously compounded cost of debt?

4. 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. 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?

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