Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate EditionVIII. Topics in Corporate
Finance(^852) 24. Option Valuation © The McGraw−Hill
Companies, 2002
Now that we know the value of the equity, we can calculate the value of the debt us-
ing 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:
$5.484 $10 eRD(6)
.5484 eRD(6)
RD1/6 ln(.5484)
.10
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 consider-
ation. For example, suppose the firm undertakes a project with an NPV of $100 thou-
sand, 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 $100 thousand
$84.9 thousand.^3 Why?
The reason is that the firm has made its assets more valuable, which means that de-
fault 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.
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
CHAPTER 24 Option Valuation 827
(^3) Delta 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.
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 cur-
rent 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(d 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 mil-
lion,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.
EXAMPLE 24.9