*How to calculate the future value of a present sum of money.*

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 |

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