In contrast to discrete compounding, continuous compounding means that the returns are compounded continuously. The frequency of compounding is so large that it reaches infinity. These are also called log returns.

Suppose the rate of return is 10% per annum. The effective annual rate on a continuously compounded basis will be:

Effective Annual Rate = er – 1

\=e^0.10 – 1

\=10.517%

This means that if 10% was continuously compounded, the effective annual rate will be 10.517%.

We can also perform the reverse calculations. If a portfolio earned 10.517% in one year, then what would be the equivalent continuously compounded rate?

It will be ln(1+r) = ln (1.10517) = 10%

Let’s take another example. An investment of $100 grows to $130 in one year. What is the continuously compounded rate of return?

We can calculate it in two ways. First calculate the return earned, i.e., (130 -100)/100 = 30%. 30% is the holding period return.

The continuously compounded rate of return = ln(1.30) = 26.23%

Alternatively you can simply calculate ln(130/100) = 26.23%

An important property of continuously compounding rates is that they are additive.