Section 4.6. Compound Interest
The
compound interest formula
is given by
•
A
n
(
t
) =
P
(1 +
r
n
)
nt
where
P
= Principal (Present Value)
A
n
(
t
) = Amount (Future Value)
r
= annual rate of interest (in decimal form)
n
= number of compounds per year
t
= number of years
Example.
$1000 is invested at a 6% annual rate.
Find the amount in 2 years if the interest is
compounded (a) yearly, (b) quarterly, (c) monthly, (d) daily.
Solution.
LetA
=
A
n
(
t
) =
P
(1 +
r
n
)
nt
= 1000(1 +
.
06
n
)
2
n
. Then
(a)
A
= 1000(1 +
.
06
1
)
2
= 1
,
123
.
60
(
n
= 1)
(b)
A
= 1000(1 +
.
06
4
)
8
= 1
,
126
.
49
(
n
= 4)
(c)
A
= 1000(1 +
.
06
12
)
24
= 1
,
127
.
16
(
n
= 12)
(d)
A
= 1000(1 +
.
06
365
)
730
= 1
,
127
.
49
(
n
= 365)
Note. In the above example,
A
increases as
n
increases. What happens to
A
as
n
→ ∞
? The answer
to this question is given by the following.
•
A
n
(
t
) =
P
(1 +
r
n
)
nt
→
Pe
rt
as
n
→ ∞
When
A
is given by
A
=
Pe
rt
, the interest is said to be
compounded continuously
.
Note: The continuous compounding formula shows how the number
e
≈
2
.
718
· · ·
occurs naturally.
Example.
Find the amount,
A
, after two years if $1,000 is invested at a
nominal rate
of 6%
compounded continuously.
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 Fall '11
 Kutter
 Algebra, Annual rate, continuous compounding formula

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