Finance > Future Value
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
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 |
Finance > Future Value
