# 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.

There are two types of annuities. 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.

## Formula

The future value of an ordinary annuity can be computed using the following formula:

 FV of Ordinary Annuity = R × (1 + i)n − 1 i

Since cash flows in an annuity due accumulate interest for one additional period, the future value of annuity due can be determined by growing the future value of an ordinary annuity for one additional period:

FV of Annuity Due = FV of Ordinary Annuity × (1 + i)

 FV of Annuity Due = R × (1 + i)n − 1 × (1 + i) i

In the above formulas,
i is the periodic interest rate which equals annual percentage rate divided by periods per year;
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 20X1 in an investment account of 9% annual interest rate. Calculate the future value of the annuity on Dec 31, 20X1. Compounding is done on monthly basis.

Solution

```We have,
Periodic Payment       R  = \$700
Number of Periods      n  = 12
Interest Rate          i  = 9%/12 = 0.75%
Future Value          PV  = \$700 × {(1+0.75%)^12-1}/0.75%
= \$700 × {1.0075^12-1}/0.0075
≈ \$700 × (1.0938069-1)/0.0075
≈ \$700 × 0.0938069/0.0075
≈ \$700 × 12.5076
≈ \$8,755```

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 Also calculate the total interest earned on the deposits if the whole amount is withdrawn on the last day of 12th month.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 Periods      n  = 12
Interest Rate          i  = 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```