With the concepts of descriptive statistics in hand, we will see why the geometric mean is appropriate for making investment statements about past performance. We will also explore why the
arithmetic mean is appropriate for making investment statements in a forward-looking context.
For reporting historical returns, the geometric mean has considerable appeal because it
is the rate of growth or return we would have had to earn each year to match the actual,
cumulative investment performance. In our simplified Example 3-8, for instance, we purchased
a stock for ¤100 and two years later it was worth ¤100, with an intervening year at ¤200. The
geometric mean of 0 percent is clearly the compound rate of growth during the two years.
Specifically, the ending amount is the beginning amount times (1 + RG)
- The geometric
mean is an excellent measure of past performance.
Example 3-8 illustrated how the arithmetic mean can distort our assessment of historical
performance. In that example, the total performance for the two-year period was unambiguously 0 percent. With a 100 percent return for the first year and −50 percent for the second,
however, the arithmetic mean was 25 percent. As we noted previously, the arithmetic mean is
always greater than or equal to the geometric mean. If we want to estimate the average return
over a one-period horizon, we should use the arithmetic mean because the arithmetic mean
is the average of one-period returns. If we want to estimate the average returns over more
than one period, however, we should use the geometric mean of returns because the geometric
mean captures how the total returns are linked over time.
As a corollary to using the geometric mean for performance reporting, the use of
semilogarithmic rather than arithmetic scales is more appropriate when graphing past
performance.49 In the context of reporting performance, a semilogarithmic graph has an
arithmetic scale on the horizontal axis for time and a logarithmic scale on the vertical
axis for the value of the investment. The vertical axis values are spaced according to the
differences between their logarithms. Suppose we want to represent £1, £10, £100, and
£1,000 as values of an investment on the vertical axis. Note that each successive value
represents a 10-fold increase over the previous value, and each will be equally spaced
on the vertical axis because the difference in their logarithms is roughly 2.30; that is,
ln 10 − ln 1 = ln 100 − ln 10 = ln 1,000 − ln 100 = 2.30. On a semilogarithmic scale, equal
48It is useful to know that we can conduct a Jarque-Bera (JB) statistical test of normality based on
sample size n, sample skewness, and sample excess kurtosis. We can conclude that a distribution is not
normal with no more than a 5 percent chance of being wrong if the quantity JB = n[(S2
K /6) + (K 2
E /24)]
is 6 or greater for a sample with at least 30 observations. In this mutual fund example, we have
only 10 observations and the test described is only correct based on large samples (as a guideline, for
n ≥ 30). Gujarati (2003) provides more details on this test.
49See Campbell (1974) for more information.
128 Quantitative Investment Analysis
movements on the vertical axis reflect equal percentage changes, and growth at a constant
compound rate plots as a straight line. A plot curving upward reflects increasing growth rates
over time. The slopes of a plot at different points may be compared in order to judge relative
growth rates.
In addition to reporting historical performance, financial analysts need to calculate
expected equity risk premiums in a forward-looking context. For this purpose, the arithmetic
mean is appropriate.
We can illustrate the use of the arithmetic mean in a forward-looking context with
an example based on an investment’s future cash flows. In contrasting the geometric and
arithmetic means for discounting future cash flows, the essential issue concerns uncertainty.
Suppose an investor with $100,000 faces an equal chance of a 100 percent return or a −50
percent return, represented on the tree diagram as a 50/50 chance of a 100 percent return or
a −50 percent return per period. With 100 percent return in one period and −50 percent
return in the other, the geometric mean return is √2(0.5) − 1 = 0.
The geometric mean return of 0 percent gives the mode or median of ending wealth
after two periods and thus accurately predicts the modal or median ending wealth of
$100,000 in this example. Nevertheless, the arithmetic mean return better predicts the
arithmetic mean ending wealth. With equal chances of 100 percent or −50 percent returns,
consider the four equally likely outcomes of $400,000, $100,000, $100,000, and $25,000
as if they actually occurred. The arithmetic mean ending wealth would be $156,250 =
($400,000 + $100,000 + $100,000 + $25,000)/4. The actual returns would be 300 percent,
0 percent, 0 percent, and −75 percent for a two-period arithmetic mean return of (300 +
0 + 0 − 75)/4 = 56.25 percent. This arithmetic mean return predicts the arithmetic mean
ending wealth of $100,000 × 1.5625 = $156,250. Noting that 56.25 percent for two periods
is 25 percent per period, we then must discount the expected terminal wealth of $156,250 at
the 25 percent arithmetic mean rate to reflect the uncertainty in the cash flows.
Uncertainty in cash flows or returns causes the arithmetic mean to be larger than
the geometric mean. The more uncertain the returns, the more divergence exists between
the arithmetic and geometric means. The geometric mean return approximately equals the
arithmetic return minus half the variance of return.50 Zero variance or zero uncertainty in
returns would leave the geometric and arithmetic returns approximately equal, but real-world
uncertainty presents an arithmetic mean return larger than the geometric. For example, Dimson
et al. (2002) reported that from 1900 to 2000, U.S. equities had nominal annual returns
with an arithmetic mean of 12 percent and standard deviation of 19.9 percent. They reported
the geometric mean as 10.1 percent. We can see the geometric mean is approximately the
arithmetic mean minus half of the variance of returns: RG ≈ 0.12 − (1/2)(0.1992) = 0.10.
50See Bodie, Kane, and Marcus (2001)
