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 rA. 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 by the end of the third period and then earned a 20 percent return on whatever was left for the final period. The arithmetic average, however, places equal weight on all returns and, therefore, the impact of the large negative return is diluted by 1/N. As the sample size increases, the impact of a single bad return declines asymptotically and it does not matter if that single bad return occurred early or late in the sample. In reality, that is not how money is earned and that is why we use geometric averages. In this example, the investment indeed earned an average –29 percent return in each period. Had you invested a dollar at the beginning of the first period, that dollar would have shrunk to about $0.25 in four periods. This is certainly not an amount implied by the arithmetic mean.
We will not prove the following formally, but it is intuitive that, in general, as the variance in the individual periodic returns declines, so does the difference between the arithmetic and geometric means. In the limit, if the four returns in our example were identical, then the arithmetic and geometric means would also be identical. Otherwise, it can be shown that the arithmetic mean is always greater than the geometric mean because the arithmetic mean ignores the correlations across returns over time. The takeaway is that these two measures tend to diverge in value as volatility in returns rises.