Geometric vs. Arithmetic Average

In April 3's post, I referred to the geometric average. What in the world is that? Come on…You remember…You learned this in junior high school. Geometric average can best be understood in contrast to its cousin, arithmetic average.

Let’s say you’re talking to an investment advisor and he wants to give you a sense of what you might expect from a particular investment—maybe the stock market generally; or perhaps his own investment performance. He shows you a string of numbers for X years. Naturally, given the variation in returns from year to year, you want to get a sense of the average. The arithmetic average is simple enough: Add up all the annual returns and divide by X.

It’s simple, but it’s misleading. It leaves you with an inflated sense of the potential impact of the investment on your retirement assets. If you compound your investment at the arithmetic average rate, year after year, you’ll end up with a bigger number than you would in real life. That’s because variation in returns from year to year harms the overall long-term result. Up one year, down the next. The more the variation, the more the arithmetic average return overstates long-term performance. You may as well use your hat size.

What you really want to see—to squeeze out the impact of that variation on your projected future—is the geometric average return. Using that average, if you compound your investment at that rate for X years, you’ll get a realistic idea of what your assets would actually look like in real life over that period.

The actual formula for computing geometric return is kinda’ complicated, but spreadsheet programs like Microsoft Excel make it easy by including a function that computes the geometric average of a string of numbers. (In the case of Excel, it’s “Geomean(String of Numbers).” To make it work, you add one to each number in the string; apply the Geomean function, and then subtract one form the result.)

By their nature, the geometric average is always smaller than the arithmetic average (except where all numbers are identical, which never happens in real life). So if anyone quotes you the arithmetic average performance of an investment, beware. Intentionally or not, he is being misleading.

Oh, one more thing. Past average performance is not a predictor of future performance, so either average is misleading in that sense. But at least the geometric average is misleading in only one way, whereas the arithmetic average is doubly so.