# Compound interest

Interest payments can be added to the principal. This means higher interest in the next period for which interest is paid. This compounding of interest can, over a period of time, mean that the total amount of interest paid is far higher than the simple interest.

Suppose £100 is invested in an account that pays 10%, which is paid annually. Assume that interest is paid out of the account. Then the total interest paid over the ten years will be £100.

If the interest is allowed to accumulate in the account, then there will be a total of £259.37 in the account after 10 years - the total interest payment will be £159.37.

The longer the time period the greater the effect of compounding. Suppose the same scenario as above but with a 20 year time period. Then the interest paid, if it is withdrawn as soon as it is paid, will be £200. If it is allowed to accumulate over 20 years the account will contain £672.75. Total interest payments will be £572.75.

The effect of compounding will also be stronger if the frequency of payment is greater. Interest of 10% paid quarterly will compound over the period of an year, so the annual rate of compounded interest will be 10.4%, and the total interest paid over a ten year period will be £280.

The total amount (principal plus interest) arising from compounded interest is:

p × (1 + i)t

where p is the amount of the principal
i is the interest rate per period (typically an year) and
t is the number of periods.

If the interest rate series, the calculation becomes:

p × (1 + i1) × (1 + i2) × (1 + i3) ⋅⋅⋅

where the subscript on each i denotes different periods.

## Continuously compounded interest

As the period over which interest is paid becomes smaller the amount given by the formula above converges to the value given by the formula below. This is continuous compounding which can be useful for some calculations. This amount is:

peit

where p is the principal,
i is the interest rate,
t is the time and
e is the constant.

The time and the interest rate must use consistent periods of time: if i is an annual interest rate, then t must be the time in years.