Investment Theory and Risk Management, + Website

Geometric and Arithmetic Averages

If things weren't already complicated enough, we now see that there are two distinct averages—geometric as well as arithmetic. It is important to understand the difference. If you want to know what an asset actually returned, then geometrically link the N gross returns over the relevant time. And, upon doing that, if you then want to know what the average return (geometric) was for each period in the return series, then take the Nth root and subtract one. Using a trailing series of the past 12 monthly returns as an example, we get:

The annual return is r_A. The geometric average of the monthly returns is

Clearly, this is different from the arithmetic average:

The difference is not subtle. For example, suppose we observe a sequence of four returns {0.9, 0.1, –0.9, 0.2}. The arithmetic average is 0.075 (7.5 percent), while the geometric average is –29 percent! Why the large discrepancy? If you had a dollar invested over these four periods, the return you would have received would have been affected to a greater degree (in a negative way) by the third period's negative 90 percent return, that is, you would have lost 90 percent of your accumulated investment ...