Arithmetic vs. Geometric Means: Empirical
1
“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
2
stable, while the AM is forward looking in that it does
“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
2
stable, while the AM is forward looking in that it does