Which average do you want?

3/2/2009


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.

ANNUAL RETURNS
Asset Class
Arithmetic
Average
(%)
Geometric
Average
(%)
Small Stocks
17.1
12.5
Large Stocks
12.3
10.4
Long-Term Corp. Bonds
6.2
5.9
Long-Term Gov’t Bonds
5.8
5.5
Int.-Term Gov’t Bonds
5.5
5.3
U.S. Treasury Bills
3.8
3.7
Inflation
3.1
3.0
Source: Morningstar (1926 to 2007)

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.

Stock selection
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.

MOST EFFECTIVE SCORES IN QUADRIX
Based on the geometric average return of the top one-fifth of S&P 1500 stocks since 1994, the 20 Quadrix scores listed below have done the best job of picking winners. For example, the top one-fifth of S&P 1500 stocks based on EPS revisions for the current quarter have delivered a geometric average return of 16.0% in rolling 12-month periods since 1994.
Score Used To Select Top One-Fifth
Of S&P 1500 Stocks
Arithmetic
Average
(%)
Geometric
Average
(%)
Difference
(%)
EPS Revisions, Current Qtr.
17.8
16.0
1.8
EPS Revisions, Next Qtr.
16.5
14.9
1.6
Enterprise Value/EBITDA Ratio
16.5
14.6
1.9
Overall Score
15.9
14.4
1.6
Long-Term Expected Profit Growth
17.3
14.2
3.1
Price/Free Cash Flow Ratio
15.8
13.7
2.1
Price/Cash Flow Ratio
14.7
13.0
1.7
Price/Earnings Ratio
14.7
12.7
2.0
Value Score
14.5
12.4
2.1
Six-Month Total Return
14.3
12.2
2.1
Price/Book Value Ratio
14.8
12.1
2.7
Five-Year Net-Income Growth
14.0
12.1
1.9
Tangible Book Value — 1-Year Chg.
14.0
12.1
1.9
Performance Score
13.8
11.9
1.9
Cash Conversion Trend
13.3
11.8
1.5
Three-Month Total Return
13.8
11.8
2.0
Five-Year Sales Growth
13.8
11.8
2.0
Gross Profit Margin
13.6
11.8
1.8
Five-Year Cash-Flow Growth
13.8
11.8
2.0
Operating Profit Margin
13.1
11.7
1.3
All stocks in S&P 1500 Index
12.1
10.6
1.5

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.


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