## Valuing the Equity in a Leveraged Firm

Consider a firm that has a single zero-coupon bond issue outstanding with a face value of $10 million. It matures in six years. The firms' assets have a current market value of $12 million. The volatility (standard deviation) of the return on the firm's assets is 40 percent per year. The continuously compounded risk-free rate is 6 percent. What is the current market value of the firm's equity? Its debt? What is its continuously compounded cost of debt?

What this case amounts to is that the stockholders have the right, but not the obligation, to pay $10 million in six years. If they do, they get the assets of the firm. If they don't, they default and get nothing. So, the equity in the firm is a call option with a strike price of $10 million.

Using the Black-Scholes formula in this case can be a little confusing because now we are solving for the stock price. So, the symbol "C" is the value of the stock and the symbol "S" is the value of the firm's assets. With this in mind, we can value the equity of the firm by plugging the numbers into the Black-Scholes OPM with S = $12 million and E = $10 million. When we do so, we get $6.516 million as the value of the equity, with a delta of .849.

© The McGraw-Hill Companies, 2002 Edition, Alternate Edition

### CHAPTER 24 Option Valuation 827

Now that we know the value of the equity, we can calculate the value of the debt using the standard balance sheet identity. The firm's assets are worth $12 million and the equity is worth $6.516 million, so the debt is worth $12 million — $6.516 million = $5.484 million.

To calculate the firm's continuously compounded cost of debt, we observe that the present value is $5.484 million and the future value in six years is the $10 million face value. We need to solve for a continuously compounded rate, RD, as follows:

So, the firm's cost of debt is 10 percent, compared to a risk-free rate of 6 percent. The extra 4 percent is the default risk premium, i.e., the extra compensation the bondholders demand because of the risk that the firm will default and bondholders will receive assets worth less than $10 million.

We also have that the delta of the option here is .849. How do we interpret this? In the context of valuing equity as a call option, the delta tells us what happens to the value of the equity when the value of the firm's assets changes. This is an important consideration. For example, suppose the firm undertakes a project with an NPV of $100 thousand, meaning that the value of the firm's assets will rise by $100 thousand. We now see that the value of the stock will rise (approximately) by only .849 X $100 thousand = $84.9 thousand.3 Why?

The reason is that the firm has made its assets more valuable, which means that default is less likely to occur in the future. As a result, the bonds gain value, too. How much do they gain? The answer is $100 - 84.9 = $15.1, in other words, whatever value the stockholders don't get.

### Equity as a Call Option

Consider a firm that has a single zero-coupon bond issue outstanding with a face value of $40 million. It matures in five years. The risk-free rate is 4 percent. The firm's assets have a current market value of $35 million, and the firm's equity is worth $15 million. If the firm takes a project with a $200 thousand NPV, approximately how much will the stockholders gain?

To answer this question, we need to know the delta, so we need to calculate N(o'1). To do this, we need to know the relevant standard deviation, which we don't have. We do have the value of the option ($15 million), though, so we can calculate the ISD. If we use C = $15 million, S = $35 million, and E = $40 million along with the risk-free rate of 4 percent and time to expiration of five years, we get that the ISD is 48.2 percent. With this value, the delta is .725, so, if $200,000 in value is created, the stockholders will get 72.5 percent of it, or $145,000.

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