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 reinvested 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 reinvested 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 builtin 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