Calculating and Comparing Security Returns is Harder than you Think : A Comparison between Logarithmic and Simple Returns

We show that there are some troubling differences between mean returns calculated using logarithmic returns and those calculated using simple returns. The mean of a set of returns calculated using logarithmic returns is less than the mean calculated using simple returns by an amount related to the variance of the set of returns. This implies that there is not a one-to-one relationship between mean logarithmic and mean simple returns. In addition, when logarithmic returns are used, ceteris paribus, higher variance will automatically reduce expected returns. We illustrate when these effects are important drawing on examples from the extant literature