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Continuous Compounding:
Some Basics
W.L. Silber
Because you may encounter continuously compounded growth rates elsewhere,
and because you will
encounter continuously compounded discount rates when we
examine the BlackScholes option pricing formula, here is a brief introduction to what
happens when something grows at
r
percent per annum, compounded continuously.
We know that as
∞
→
n
(1)
L
71828183
.
2
1
1
=
=
+
e
n
n
In our context, this means that if $1 is invested at 100% interest, continuously
compounded, for one year, it produces $2.71828 at the end of the year.
It is also true that if the interest rate is
r
percent, then $1 produces
r
e
dollars after
1 year.
For example, if
06
.
=
r
we have
0618365
.
1
1
$
06
.
=
⋅
e
After two years, we would have:
127497
.
1
)
2
(
06
.
06
.
06
.
=
=
⋅
e
e
e
More generally, investing
P
at
r
percent, continuously compounded, over
t
years,
produces (grows to) the amount
F
according to the following formula:
(2)
F
Pe
rt
=
For example, $100 invested at 6 percent, continuously compounded, for 5 years
produces
98588
.
134
$
100
$
)
5
(
06
.
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 Spring '08
 Miyakawa

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