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:

1. Iteration - by calculating the future value for different values of interest rate or time, one gradually can converge on the solution.

2. Financial calculator or spreadsheet - use built-in functions to instantly calculate the solution.

3. 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.

4. 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 =	log ( FV / PV ) log ( 1 + i )

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