Case I The Debt Is Risk Free

Suppose that in one year the firm's assets will be worth either $1,100 or $1,200. What is the value today of the equity in the firm? The value of the debt? What is the interest rate on the debt?

To answer these questions, we first recognize that the option (the equity in the firm) is certain to finish in the money because the value of the firm's assets ($1,100 or $1,200) will always exceed the face value of the debt. In this case, from our discussion in previous sections, we know that the option value is simply the difference between the value of the underlying asset and the present value of the exercise price (calculated at the risk-free rate). The present value of $1,000 in one year at 12.5 percent is $888.89. The current value of the firm is $950, so the option (the firm's equity) is worth $950 - 888.89 = $61.11.

What we see is that the equity, which is effectively an option to purchase the firm's assets, must be worth $61.11. The debt must therefore actually be worth $888.89. In fact, we really didn't need to know about options to handle this example, because the debt is risk-free. The reason is that the bondholders are certain to receive $1,000. Because the debt is risk-free, the appropriate discount rate (and the interest rate on the debt) is the risk-free rate, and we therefore know immediately that the current value of the debt is $1,000/1.125 = $888.89. The equity is thus worth $950 - 888.89 = $61.11, as we calculated.

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