Continuous Compounding

It is kind of intuitive that given a APR, say 10%, the more frequently we compound it, the higher the EAR should be. The rationale is, we'd like to have interest come as early as possible so the interest can generate as more interest as possible. If you issue the loan, you would love to charge compounded interest as frequently as possible, right? The question is, how much can it be if your period is infinitely small?

There is good news and bad news. The good news is bankers, back to 17th century, have already figured this issue all out. The bad news, is that there is an upper limit of the EAR however frequently you compound APR.

This limit of interest rate is called continuous compounding interest rate. Its value is given by:

You already know what is APR here. However, what is "e"? "e" here is the natural log base, or roughly 2.71828.

So, 10% compounded infinite number of times per year, or generally referred to as "10% compounded continuously", is really not too high:

It is slightly higher than the EAR of 10% compounding per 6 months, which is 10.25%.