4by12

Continuous Compound Interest

Long time no blogging! A lot has happened since my last post. I got married, was out on a honeymoon, left my job and returned to school to finish my degree. But now I’m back, so read on and enjoy!

We were in the office a while back, and we started talking about compound interest. You can calculate compound interest over different time intervals: Yearly, monthly, weekly, and so on. What will happen if we take the limit as the interval approaches zero, i.e. use continuous time? How much interest will we get?

Let’s see. If M(t) is the money you have at time t, how much money will you have at t+dt for some interval dt? That’s M(t) plus an additional amount that’s proportional to M(t) — the interest. Let’s assume this additional amount is also proportional to dt (this seems like a reasonable assumption to make). So we get:

For some constant which will determine the rate at which interest is accumulated. Rearranging, we get:

And the limit as dt approaches zero is:

So this tells us continuous compound interest grows exponentially in time, which isn’t too surprising when you think about it.

After working this out at the office, that final equation was left on the whiteboard. A couple of days later, our product manager walked in and saw it. He immediately recognized it. Apparently, this formula was taught in his MBA program as the maximum amount of interest one can get…