Effective Interest

Nature isn't the only place where continuous compounding applies. Oddly enough, it is sometimes used by banks!

Back in the early 1970s, the interest rates that banks could pay on regular savings accounts were strictly regulated. Typically, a bank could pay no more than 5% on such accounts. After a time, banking deregulation and double-digit inflation led to this limit being revoked and interest rates rose higher, at least until the 1990s.

Since the amount of interest that they could pay depositors was regulated by the government, one way that banks competed in the 1970s was by offering favorable compounding intervals. If one bank compounded monthly while another compounded daily, the second bank would enjoy a competitive advantage even though both were paying 5% annual interest.

Of course, there was no reason to stop at daily compounding. A bank could, if it chose, compound every hour or every minute or every second. And in fact, many banks took it to the limit and began compounding continuously.

What's the difference between 5% interest, compounded annually, and 5% interest, compounded continuously? One way to get at the answer to this question is to ask a different question. What interest rate, compounded annually, is equivalent to 5% interest compounded continuously. (This rate is called the effective interest rate of the continuously compounded investment.)

We can easily solve this problem with Maple:

solve(compound(P, R, 1) = continuous(P, .05, 1), R);

Note that the effective interest rate is independent of the initial balance.