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Future
Value
The future value of a sum of money
invested at interest rate i for one year is given by:
FV = PV ( 1 + i )
where
FV
= future value
PV = present value
i = annual interest rate
PV = present value
i = annual interest rate
If the resulting principal and interest
are re-invested a second year at the same interest rate, the future value is
given by:
FV = PV ( 1 + i ) ( 1 + i )
In general, the future value of a
sum of money invested for t years with the interest credited and
re-invested at the end of each year is:
FV = PV ( 1 + i ) t
Solving
for Required Interest Rate or Time
Given a present sum of money and a
desired future value, one can determine either the interest rate required to
attain the future value given the time span, or the time required to reach the
future value at a given interest rate. Because solving for the interest rate or
time is slightly more difficult than solving for future value, there are a few
methods for arriving at a solution:
- Iteration - by calculating the future value for different
values of interest rate or time, one gradually can converge on the
solution.
- Financial calculator or spreadsheet - use built-in
functions to instantly calculate the solution.
- Interest rate table - by using a table such as the one
at the end of this page, one quickly can find a value of interest rate or
time that is close to the solution.
- Algebraic solution - mathematically calculating the
exact solution.
Algebraic Solution
Beginning with the future value
equation and given a fixed time period, one can solve for the required interest
rate as follows.
FV = PV ( 1 + i ) t
Dividing each side by PV and
raising each side to the power of 1/t:
(
FV / PV ) 1/t = 1 + i
The required interest rate then is
given by:
i = ( FV / PV ) 1/t
- 1
To solve for the required time to
reach a future value at a specified interest rate, again start with the
equation for future value:
FV = PV ( 1 + i ) t
Taking the logarithm (natural log or
common log) of each side:
log
FV = log [ PV ( 1 + i ) t ]
Relying on the properties of
logarithms, the expression can be rearranged as follows:
log
FV = log PV + t log ( 1 + i
)
Solving for t:
t =
|
|
Interest
Factor Table
The term ( 1 + i ) t
is the future value interest factor and may be calculated for an array of time
periods and interest rates to construct a table as shown below:
Table of Future Value Interest Factors
t \ i
|
1%
|
2%
|
3%
|
4%
|
5%
|
6%
|
7%
|
8%
|
9%
|
10%
|
1
|
1.010
|
1.020
|
1.030
|
1.040
|
1.050
|
1.060
|
1.070
|
1.080
|
1.090
|
1.100
|
2
|
1.020
|
1.040
|
1.061
|
1.082
|
1.103
|
1.124
|
1.145
|
1.166
|
1.188
|
1.210
|
3
|
1.030
|
1.061
|
1.093
|
1.125
|
1.158
|
1.191
|
1.225
|
1.260
|
1.295
|
1.331
|
4
|
1.041
|
1.082
|
1.126
|
1.170
|
1.216
|
1.262
|
1.311
|
1.360
|
1.412
|
1.464
|
5
|
1.051
|
1.104
|
1.159
|
1.217
|
1.276
|
1.338
|
1.403
|
1.469
|
1.539
|
1.611
|
6
|
1.062
|
1.126
|
1.194
|
1.265
|
1.340
|
1.419
|
1.501
|
1.587
|
1.677
|
1.772
|
7
|
1.072
|
1.149
|
1.230
|
1.316
|
1.407
|
1.504
|
1.606
|
1.714
|
1.828
|
1.949
|
8
|
1.083
|
1.172
|
1.267
|
1.369
|
1.477
|
1.594
|
1.718
|
1.851
|
1.993
|
2.144
|
9
|
1.094
|
1.195
|
1.305
|
1.423
|
1.551
|
1.689
|
1.838
|
1.999
|
2.172
|
2.358
|
10
|
1.105
|
1.219
|
1.344
|
1.480
|
1.629
|
1.791
|
1.967
|
2.159
|
2.367
|
2.594
|
11
|
1.116
|
1.243
|
1.384
|
1.539
|
1.710
|
1.898
|
2.105
|
2.332
|
2.580
|
2.853
|
12
|
1.127
|
1.268
|
1.426
|
1.601
|
1.796
|
2.012
|
2.252
|
2.518
|
2.813
|
3.138
|
13
|
1.138
|
1.294
|
1.469
|
1.665
|
1.886
|
2.133
|
2.410
|
2.720
|
3.066
|
3.452
|
14
|
1.149
|
1.319
|
1.513
|
1.732
|
1.980
|
2.261
|
2.579
|
2.937
|
3.342
|
3.797
|
15
|
1.161
|
1.346
|
1.558
|
1.801
|
2.079
|
2.397
|
2.759
|
3.172
|
3.642
|
4.177
|
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