Integral Calculus, Compound Interest

Compound Interest

If you earn 6% interest on your money, you might receive 6% at the end of the year, or you might receive 3% twice a year. The latter is slightly preferable, as 1.03×1.03 = 1.0609, or 6.09%. This is a form of compound interest, compounded semi-annually. Four quarterly payments of 1.5% is even better. The limit is continuous interest, as though your money were constantly making more money.

Assume you are paid n times a year, receiving 6%/n each time. At the end of the year your funds are multiplied by (1+6%/n)ⁿ. Take the limit as n approaches infinity. Actually we will take the limit of the log, which is n×log(1+6%/n). This is the product of two quantities; one approaches ∞ and the other approaches 0. Instead of multiplying by n, divide by 1/n, hence we can use L'hopital's rule. After canceling -n², the result is 6% over 1+6%/n. This approaches 6% for large n. This was the log of our expression, hence our assetts are multiplied by E^6% at the end of the year. This is an actual increase of 6.18%. If the bank holds the money for half the year, you will receive E^3%. Many institutions use this formula for continuously compounded interest. You are paid for the precise number of hours that your money is in their hands.