# Arithmetic vs. Geometric Means: Empirical

“Arithmetic vs. Geometric Means: Empirical

Evidence and Theoretical Issues”

by Jay B. Abrams, ASA, CPA, MBA

Copyright 1996

There has been a flurry of articles about the relative merits of using the arithmetic mean (AM) versus the geometric mean (GM). The Ibbotson SBBI Yearbook took the first position that the arithmetic mean is the correct mean to use in valuation. Allyn Joyce’s June 1995 BVR article initiated arguments for the GM as the correct mean. The previous articles have centered around Professor Ibbotson’s famous example using a binomial distribution with 50%-50% probabilities of a +30% and -10% return. The debate has been very interesting, but it is off on a tangent, focused on the wrong issue. There are theoretical and empirical reasons why the arithmetic mean is the correct one. We will look at both in this article.

Theoretical Superiority of Arithmetic Mean

Rather than argue about Ibbotson’s much debated above example, I prefer to cite and elucidate another quote from his book:

In general, the geometric mean for any time period is less than or equal to the arithmetic mean. The two means are equal only for a return series that is constant (i.e., the same return in every period). For a non-constant series, the difference between the two is positively related to the variability or standard deviation of the returns. For example, in Table 6-7, the difference between the arithmetic and geometric mean is much larger for risky large company stocks than it is for nearly riskless Treasury bills.1 The GM measures the magnitude of the returns, as the investor starts with one portfolio value and ends with another. It does not measure the variability of the journey, as does the AM. As noted in SBBI, the GM is backward looking. There is no difference in the GM of two stocks (or portfolios), one of which is highly volatile and the other of which is absolutely 1 SBBI-1996, p. 104

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stable, while the AM is forward looking in that it does impound the volatility of the stocks. As Mark Twain said, “Forecasting is difficult—especially into the future.” Had Twain been acquainted with portfolio theory, he obviously would have been an AM advocate. I suppose the GM advocates could cite Paul McCartney on their side for his song Yesterday. I would then have to counter with Steven Spielberg and Back To The Future. Table I contains an illustration of two stock series. The first one is highly volatile, with a standard deviation of returns of 65%, while the second one has a zero standard deviation. It makes no sense intuitively that the GM is the correct one. That would imply that both stocks are equally risky, since they have the same GM. Would anyone really consider stock #2 equally as risky as #1? If so, let’s trade stocks! Every modern model to calculate discount rates recognizes that investors are risk averse and avoid volatility unless they are adequately compensated for undertaking it. It is more consistent to use the mean that fully impounds risk (AM) than the one that has had risk removed from it (GM).

Another aspect of consistency in favor of the AM can be found in my article on the size effect,2 which demonstrates that stock market portfolio returns correlate very well (98% R2) with the standard deviation of returns, i.e., risk. Which mean returns correlate best with the volatility of returns? Obviously the AM. In other words, the dependent variable (AM returns) is consistent with the independent variable (standard deviation of returns) in the regression. The latter is risk, and the former is the fully risk-impounded rate of return. GM is less consistent. Using CAPM leads us to the same conclusions vis-a-vis AM vs. GM. The equation still says return is some function of risk. It is more consistent to use a fully-riskimpounded return than a risk-neutered return. In the next section, we will confirm this empirically using my Log Size Model.

Table II: Empirical Evidence

Table II contains both the geometric and arithmetic...

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