## Continuous Compounding A Refresher Course

Back in Chapter 6, we saw that the effective annual interest rate (EAR) on an investment depends on compounding frequency. We also saw that, in the extreme, compounding can occur every instant, or continuously. So, as a quick refresher, suppose you invest \$100 at a rate of 6 percent per year compounded continuously. How much will you have in one year? How about in two years?

In Chapter 6, we saw that the EAR with continuous compounding is:

where q is the quoted rate (6 percent, or .06, in this case) and e is the number 2.71828 ..., the base of the natural logarithms. Plugging in the numbers, we get:

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### CHAPTER 24 Option Valuation 811

or about 6.2 percent. Notice that most calculators have a key labeled "ex," so doing this calculation is simply a matter of entering .06 and then pressing this key. With an EAR of 6.184 percent, your \$100 investment will grow to \$106.18 in one year. In two years, it will grow to:

Future value = \$100 X 1.061842 = \$100 X 1.1275 = \$112.75

When we move into option valuation, continuous compounding shows up quite a bit, and it helps to have some shortcuts. In our examples here, we first converted the continuously compounded rate to an EAR and then did our calculations. It turns out that we don't need to do the conversion at all. Instead, we can calculate present and future values directly. In particular, the future value of \$1 for t periods at a continuously compounded rate of R per period is simply:

For example, looking back at the problem we just solved, the future value of \$100 in two years at a continuously compounded rate of 6 percent is

which is exactly what we had before.

Similarly, we can calculate the present value of \$1 to be received in t periods at a continuously compounded rate of R per period as follows:

So, if we want the present value of \$15,000 to be received in five years at 8 percent compounded continuously, we would calculate:

= \$15,000 X 2.71828-4 = \$15,000 X .67032 = \$10,054.80

Continuous Compounding

What is the present value of \$500 to be received in six months if the discount rate is 9 percent per year, compounded continuously?

In this case, notice that the number of periods is equal to one-half because six months is half of a year. Thus, the present value is:

Ross et al.: Fundamentals VIII. Topics in Corporate 24. Option Valuation of Corporate Finance, Sixth Finance

Looking back at our PCP condition, we wrote:

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812 PART EIGHT Topics in Corporate Finance

If we assume that R is the continuously compounded risk-free rate per year, then we could write this as:

where t is the time to maturity (in years) on the options.

Finally, suppose we are given an EAR and we need to convert it to a continuously compounded rate. For example, if the risk-free rate is 8 percent per year compounded annually, what's the continuously compounded risk-free rate? Going back to our first formula, we had that:

Now, we need to solve for q, the continuously compounded rate. Plugging in the numbers, we have:

We need to take the natural logarithm (ln) of both sides to solve for q:

or about 7.7 percent. Notice that most calculators have a button labeled "ln," so doing this calculation involves entering 1.08 and then pressing this key.

EXAMPLE 244 I EVe" MOe ^^

-1 Suppose a share of stock sells for \$30. A three-month call option with a \$25 strike sells for

\$7. A three-month put with the same maturity sells for \$1. What's the continuously compounded risk-free rate?

We need to plug the relevant numbers into the PCP condition:

Notice that we used one-fourth for the number of years because three months is a quarter of a year. We now need to solve for R:

\$24 = \$25 x e-R(1/4) .96 = e-R(1/4) ln(.96) = ln(e-R(1/4)) -.0408 = -R(1/4) R = .1632

or about 16.32 percent, which would be a very high risk-free rate!

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