# Future Value of an Annuity

The future value of an annuity is the value of its periodic payments each enhanced at a specific rate of interest for given number of periods to reflect the time value of money. In other words, future value of an annuity is equal to the sum of face value of periodic annuity payments and the total compound interest earned on all periodic payments till the future value point.

## Formula

There are two types of annuity. The one in which payments occur at the end of each period is called ordinary annuity and the other in which payments occur at the beginning of each period is called annuity due. Both types have different formulas for future value calculation:

FV of Ordinary Annuity = R × | (1 + i)^{n} − 1 |

i |

FV of Annuity Due = R × | (1 + i)^{n} − 1 | × (1 + i) |

i |

In the above formulas,

**i** is the interest rate per compounding period;

**n** are the number of compounding periods; and

**R** is the fixed periodic payment.

## Examples

**Example 1:** Mr A deposited $700 at the end of each month of calendar year 2010 in an investment account of 9% annual interest rate. Calculate the future value of the annuity on Dec 31, 2011. Compounding is done on monthly basis.

__Solution__

We have, Periodic Payment

R= $700 Number of Periodsn= 12 Interest Ratei= 9%/12 = 0.75% Future ValuePV= $700 × {(1+0.75%)^12-1}/1% = $700 × {1.0075^12-1}/0.01 ≈ $700 × (1.0938069-1)/0.01 ≈ $700 × 0.0938069/0.01 ≈ $700 × 9.38069 ≈ $6,566.48

**Example 2:** Calculate the future value of 12 monthly deposits of $1,000 if each payment is made on the first day of the month and the interest rate per month is 1.1%. Also calculate the total interest earned on the deposits if the whole amount is withdrawn on the last day of 12th month.

__Solution__

Periodic Payment

R= $1,000 Number of Periodsn= 12 Interest Ratei= 1.1% Future Value = $1,000 × {(1+1.1%)^12-1}/1.1% × (1+1.1%) = $1,000 × {1.011^12-1}/0.011 × (1+0.011) = $1,000 × (1.140286-1)/0.011 × 1.011 ≈ $1,000 × 0.140286/0.011 × 1.011 ≈ $1,000 × 12.75329059 × 1.011 ≈ $12,893.58 Interest Earned ≈ $12,893.58 - $1,000 × 12 ≈ $893.58

Written by Irfanullah Jan