Arithmetic vs. Geometric Means: Empirical

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“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
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Regression #1: Returns as a Function of Risk
The arithmetic mean outperforms3 the geometric mean in this regression, with Adjusted R2 of 97.78% versus 81.93% and t-statistic of 19.9 versus 6.5. Additionally, the constant of
2 “A Breakthrough in Calculating Reliable Discount Rates,” Valuation, August, 1994. Additionally, Grabowski and King subsequently published articles on the same topic in Business Valuation Review, June 1995, p. 69-74 and September 1996, p.
3 In other words, AM does a better job of explaining risk than GM.
5.24% for the arithmetic mean makes economic sense, i.e., it matches the 70-year average Treasury Bond rate—the best proxy for the risk-free rate—while the constant for the geometric mean matches nothing in particular; and how could it? The geometric mean is stripped of some of the information relating to volatility, so therefore it will obviously do an inferior job of forecasting returns based on volatility.
Let’s see how it does in explaining returns as a function of log size.
Regression #2: Returns as a Function of Log Size
Again, the arithmetic mean outperforms the geometric mean. Its adjusted R2 is 91.43% compared to 84.48% for the geometric mean. The absolute value of Its t-statistic is 9.2,
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