Which average do you want?
While a comparison of arithmetic and geometric averages may sound like a lesson for a statistics textbook, the topic is central to some major investment questions.
Consider, for example, the question of how much of your stock portfolio should be in shares of small companies. History suggests your portfolio will be more volatile with small stocks. But how much extra return can you expect for accepting the higher risk?
Based on arithmetic average, small stocks gained 17.1% per year from 1926 to 2007, far better than the 12.3% average of large stocks, according to Morningstar. In other words, if you add up the percentage changes for all 82 years from 1926 to 2007 and divide by 82, the average is 17.1% for small stocks and 12.3% for large stocks.
Based on geometric average, small stocks gained 12.5% per year, versus 10.4% for large stocks. That means money invested in 1926 and allowed to grow until 2007 would have increased at an annual compound rate of 12.5% with small stocks and 10.4% with large stocks.
So, have small stocks outperformed large stocks by 4.8% per year or 2.1% per year? Which average is appropriate? The answer depends on how you plan to invest. The geometric average looks backward, measuring the change in wealth over time. If you plan to reinvest proceeds and let your portfolio compound over time, use the geometric average.
The arithmetic average represents a typical performance in a single year. If you plan to hold a portfolio for only a single year, or not reinvest any proceeds, use the arithmetic average. For an investor who puts $100,000 in small stocks every year, regardless of prior returns or the size of his or her portfolio, the arithmetic average is the best measure of expected return.
Most investors let their portfolios compound over time, and geometric average more closely approximates the growth likely over the long haul for such investors. Also, geometric average does a better job of accounting for risk. Only when period returns are identical will geometric and arithmetic return be equal. Otherwise the geometric return will always be lower than the arithmetic return, with the gap between the two widening as period returns become more volatile.
When considering what factors to include in our Quadrix® stock-rating system, we consider both geometric and arithmetic averages. One measurement we use is the average return of the top one-fifth of stocks in an index based on a factor. Using this methodology with S&P 1500 stocks for rolling 12-month periods since 1994, the table below shows the 20 most effective scores in Quadrix based on geometric average return.
Quadrix scores that lead to volatile returns, such as long-term expected profit growth and price/book value ratio, have relatively large differences between arithmetic and geometric averages. The differences are small for scores that produce steadier results, like the Quadrix Overall score.
For the most part, however, factors that work well based on arithmetic average tend to work based on geometric average. In fact, all but one of the 20 most effective scores based on geometric average are also among the top 20 based on arithmetic average. The Overall score, the most important score in Quadrix, ranks No. 4 among all scores on geometric average and No. 5 on arithmetic average.