## Continuously Compounded Interest

If an amount of moneyis invested so that compound interest is accrued at the rate r% per time period, then aftertime periods the amount of money will have grown to an amount

If however, the interest is compounded more regularly, then something a little bit strange happens. Suppose £1000 is invested at 12% per annum. If interested is compounded annually then at the end of a year, the original £1000 will have grown to £1120. If however, it is compounded monthly, then the monthly rate of interest will be 12/12 =1% and after 1 year the original £1000 will have grown to

In fact if the year is divided intotime periods, so that interest is compounded n times a year, the interest per time period isand the amount of money will have grown to

The table below shows the investment after 1 year for various values of n.

10 | 1126.691779 |

100 | 1127.415743 |

1000 | 1127.488731 |

10000 | 1127.495196 |

100000 | 1127.495975 |

Astends to infinity, this expression tends to a limit

We can generalise this reasoning, so that if annual interest ofis compounded continuously on an investment ofat the end of a year the investment will have grown to and at the end ofyears the principal will have grown to