Arithmetic returns are not symmetrical, and this is a problem. The positive and negative percent of ordinary returns of an equal magnitude do not cancel each other out, result in a net change. For example, if the price of a security increased from 100 to 125 one day, then the next day fell back to 100, it would be easy to calculate the return on the first day as 25% and the loss on the second day as 20%, that being 25 as a percentage of 125. However, if these two days' returns are just added up, they do not explain the outcome that we know that we ended up where we started at 100. Instead, they give a total of plus 5%. In other words, arithmetic returns are not time additive, making them inappropriate for time series data like equity prices. Logarithmic returns solve that issue. They're also called continuously compounded returns because they assume returns are compounded continuously rather than in steps, as is the case for arithmetic returns. The continuously compounded return as an asset climbs from 100 to 125 is calculated as the natural logarithm of 125 divided by 100, or LN on a calculator, open bracket, 125, over 100.

This would result in 22.32%. The continuously compounded return as the asset falls from 125 to 100 is calculated as the natural logarithm of 100 divided by 125, or LN of 100 divided by 125. This would result in minus 22.32%. Adding these two together gives us the result of 0%, which solves the problem caused by arithmetic approaches.