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

We analyse the relationships between return calculation methods, risk and observation periods. We show that the mean of a return set calculated using logarithmic returns is less than the mean calculated using simple returns by an amount related to the variance of the set. This implies that there is not a one-to-one relationship between mean logarithmic and mean simple returns and also that risk and return calculations are not independent as the measure of risk is part of the measure of return. Finally we draw on examples from the extant literature to illustrate that these effects can be very important particularly when dealing with short observation periods.