## Options and the Valuation of Risky Bonds

Let's continue with the case we just examined of a firm with \$12 million in assets and a six-year, zero-coupon bond with a face value of \$10 million. Given the other numbers, we showed that the bonds were worth \$5.484 million. Suppose that the holders of these

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3Delta is used to evaluate the effect of a small change in the underlying asset's value, so it might look like we shouldn't use it to evaluate a shift of \$100 thousand. "Small" is relative, however, and \$100 thousand is small relative to the \$12 million total asset value.

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828 PART EIGHT Topics in Corporate Finance bonds wish to eliminate the risk of default. In other words, the holders want to turn their risky bonds into risk-free bonds. How can they do this?

The answer is that the bondholders can do a protective put along the lines we described earlier in the chapter. In this case, the bondholders want to make sure that their bonds will never be worth less than their face value of \$10 million, so the bondholders need to purchase a put option with a six-year life and a \$10 million face value. The put option is an option to sell the assets of the firm for \$10 million.

Remember that if the assets of the firm are worth more than \$10 million in six years, the shareholders will pay the \$10 million. If the assets are worth less than \$10 million, the stockholders will default, and the bondholders will receive the assets of the firm. At that point, however, the bondholders will exercise their put and sell the assets for \$10 million. Either way, the bondholders get their \$10 million.

So, what we have discovered is that a risk-free bond is the same thing as a combination of a risky bond and a put option on the assets of the firm with a matching maturity and a strike price equal to the face value of the bond:

Value of risky bond + put option = Value of risk-free bond [24.7]

In our example, the face value of the debt is \$10 million, and the risk-free rate is 6 percent, so the value of the bonds if they were risk free is:

Value of risk-free bonds = \$10 million X e- 06(6) = \$6.977 million

If we compare this to the value of the risky bonds, \$5.484 million, we see that the put option is worth \$6.977 - 5.484 = \$1.493 million. Notice that the value of the risk-free bonds is also the present value of the strike price at the risk-free rate.

We can check that this put value is correct. We know the value of the underlying assets is \$12 million, value of the call option (the stock) is \$6.516 million, and the present value of the strike price is \$6.977 million. Using the PCP condition:

P = \$6.977 + 6.516 - \$12 = \$1.493 million which is exactly what we calculated.

We can restate our result here as follows:

Value of risky bond = Value of risk-free bond - put option

This shows us that anything that increases the value of the put option decreases the value of the firm's bonds. With this in mind, we can use the PCP condition to bring together and unite a lot of our discussion in this chapter (and this book!). Using the PCP condition, we can write:

Remember that, in this case, the stock is the underlying asset. Now, if we are thinking of the stock in a firm as being a call option on the assets of the firm, here is how we would interpret this:

Value of assets (S) = Value of stock (C) + (E X e-Rt - P) [24.9]

where E, the strike price, is the face value of the firm's debt. Notice that, as we have just seen, the term in parentheses is the value of the firm's risky bonds, so this expression is really just the balance sheet identity:

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VIII. Topics in Corporate Finance

24. Option Valuation

© The McGraw-Hill Companies, 2002

CHAPTER 24 Option Valuation

Value of assets (S) = Value of stock (C) + Value of bonds (E X e

Thus, the PCP condition and the balance sheet identity say the same thing, but recognizing the nature of the optionlike features of the equity and debt in a leveraged firm leads to a far richer understanding of corporate finance. We illustrate some important examples in the next section.

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