Compound interest is interest which is calculated on the original lump sum plus the interest already accumulated.

If a lump sum (PV) is invested in an account paying a rate of compound interest (i), then after a number of periods (n), it will grow into a future value (FV).

This formula can be rearranged to give:

The compound interest can be found by taking the difference between the future value and the present value.

For example, is a lump sum of 10,000 is invested today in an account paying 8% a year compounded annually, then at the end of two years the future value is given by the future value of a lump sum formula as follows.

PV = 10,000 i = 8% n = 2 FV = PV x (1 + i)^{n}FV = 10,000 x (1 + 8%)^{2}FV = 11,664.00

and the compound interest is given by:

Compound interest = FV - PV Compound interest = 11,664 - 10,000 Compound interest = 1,664.00

This is presented in the diagram below

Period | 0 | 1 | 2 | 3 | |
---|---|---|---|---|---|

Future value | 10,000 ↑ | 11,664.00 ↑ |

### How to Calculate Compound Interest Between Two Periods

This same technique can be applied for any two periods.

Using the same example, if we wanted to know the compound interest earned for years 6,7,8 and 9, we calculate the future value at the end of year 5 (PV at the start of year 6), and the future value at the end of year 9 The difference between the two values must be the compound interest earned between year 6 and year 9.

Future value at year 5 (PV at the start of year 6) is given by:

PV = 10,000 i = 8% n = 5 FV = PV x (1 + i)^{n}FV = 10,000 x (1 + 8%)^{5}FV = 14,693.28

and, the future value at the end of year 9 is given by:

PV = 10,000 i = 8% n = 9 FV = PV x (1 + i)^{n}FV = 10,000 x (1 + 8%)^{9}FV = 19,990.05

and the compound interest earned for years 6, 7, 8, and 9 is given as follows:

FV = FV at the end of year 9 PV = FV at the end of year 5 (PV at the start of year 6) Compound interest = FV - PV Compound interest = 19,990.05 - 14,693.28 Compound interest = 5,296.77

This is presented in the diagram below

Period | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|

Future value | 14,693.28 ↑ | 19,990.05 ↑ |

The compound interest between any two periods can be found by taking the difference between the future values at the start and end of the two periods. The future value can be calculated using the future value of a lump sum formula.