Continuous Compound Interest

The formula for calculating the amount to which an investment grows is given by

where

is the principal amount invested

the amount – principal plus interest - afteryears

is the interest rate for the period in question – 7% would be writteni

is the number of periods.

Something strange happens as the number of periods is increased, and the interest rate per period correspondingly decreased.

If £1000 is invested for 1 years at a fixed rate of 6% per annum the amount after 1 year is

If the amount is paid quarterly four times a year then the interest rate per quarter is 1.5%. The number of periods is now 4. The amount after four years is

Extra interest is earned just by compounding more often even though the interest rate has not gone up.

If the interest is compounded n times a year, then the interest rate per period is 0.06 over n so the amount at the end of the year is

£1000*(1 +{0.06 over n})^n

If we let n tend to infinity- compounding interest every tiny fraction of a nanosecond, we can use the identityto write for the amount after one year,

We can generalise this to any interest rate r and any number of yearsIfis invested for years at a rate of interestcompounded continuously, the amount in the account after n years is

This is continuous compound interest, and always earns more interest than when interest is added after each time period longer than zero.