Introduction to Corporate Finance

Ross et al.: Fundamentals

of Corporate Finance, Sixth

Edition, Alternate Edition

VIII. Topics in Corporate

Finance

(^836) 24. Option Valuation © The McGraw−Hill
Companies, 2002
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 1.06184^2
$100 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 con-
tinuously 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 val-
ues directly. In particular, the future value of $1 for tperiods at a continuously com-
pounded rate of Rper period is simply:
Future value $1 eRt
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
Future value $100 e.06(2)
$100 2.71828.12
$100 1.1275
$112.75
which is exactly what we had before.
Similarly, we can calculate the present value of $1 to be received in tperiods at a
continuously compounded rate of Rper period as follows:
Present value $1 e
Rt
So, if we want the present value of $15,000 to be received in five years at 8 percent
compounded continuously, we would calculate:
Present value $15,000 e
.08(5)
$15,000 2.71828
.4
$15,000 .67032
$10,054.80
Looking back at our PCP condition, we wrote:
SPPV(E) C
CHAPTER 24 Option Valuation 811
Continuous Compounding
What is the present value of $500 to be received in six months if the discount rate is 9 per-
cent 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:
Present value $500 e
.09(1/2)
$500 2.71828
.045
$500 .956
$478
EXAMPLE 24.3