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1
Chapter Five
1
2
TIME VALUE OF MONEY WITH AN ANNUITY
3
4
FUTURE VALUE OF AN ANNUITY
5
6
Let
n
equal deposits of an amount
A
are made at the beginning of each year.
7
What would be the future value of this annuity if the annual rate of interest is
k
?
8
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The mathematical formula can be developed as follows:
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11
Amount deposited
A
A
A
A
A
A
12
13
Beginning of year
1
2
3
4
(n  1)
n
14
15
16
By expression (4.1), at point
n
the future value of the first deposit of
A
dollars =
17
A
(1 +
k
)
n
1
. The future value of the second deposit of
A
dollars =
A
(1 +
k
)
n
2.
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19
The future value of the (
n
 1)
th
deposit of
A
dollars =
A
(1 +
k
), and the future
20
value of the
n
th
deposit of A dollars =
A
. Therefore, the future value of the annuity
21
will be:
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1
2
2
1
1
1
1
1
1
1
1
n
n
n
k
k
k
A
A
k
A
k
A
k
A
FVA
24
25
Note that the terms in the brackets are in geometric series with common ratio of
26
(1+
k
).
By our discussion in Chapter 1, this series can be simplified to:
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k
k
A
k
k
A
FVA
n
n
1
1
1
1
1
1
(5.1)
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Note that in these computations the future value is obtained right after the last
31
payment. Also, it does not matter whether the payments are made at the beginning
32
or at the end of each year. In expression (5.1), there are
n
payments, while the
33
number of the compounding period is only (
n
 1) no matter when they start.
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35
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36
37
The future value of an annuity (
FVA
) of
n
deposits of $
A
with an annual
38
rate of interest
k
is given by
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k
k
A
FVA
n
1
1
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Example 5.1
:
What is the future value of an annuity of an annual $15 deposit:
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(a) at the beginning of each year for three years?
46
(b) at the end of each year for three years?
47
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Assume that the annual interest rate is 15 percent.
49
Answer
:
Since the numbers of payments are small, the value of the annuity can be
50
obtained either directly or by using formula (5.1).
51
The following diagram will help solve the problem. We are assuming a
52
deposit is made at the beginning of each year, and the value is calculated at the
53
beginning of the third year.
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55
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Amount deposited
15
15
15
57
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Beginning of year
0
1
2
3
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By calculating the results directly, we have the future value at year 3
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= 15(l + .15)
2
+ 15(l + .15) + 15
63
= 19.8375 + 17.25 + 15
64
= $52.0875
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Alternatively, using expression (5.1):
A
= 15,
k
= .15 and
n
= 3.
Therefore,
67
68
0875
.
52
$
15
.
1
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This note was uploaded on 11/06/2011 for the course FIN 3414 taught by Professor Chaiyuthpadungsaksawasdi during the Fall '10 term at FIU.
 Fall '10
 CHAIYUTHPADUNGSAKSAWASDI
 Finance, Time Value Of Money, Annuity, Future Value, Interest

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