The arithmetic mean is useful for calculating the return over one period. For returns over multiple periods, the geometric mean is a better measurement metric as it provides the compounded annual growth rate.
The arithmetic mean is more useful in calculating the expected equity risk premium in a forward-looking context as compared to the geometric mean. The arithmetic mean return predicts the arithmetic mean ending wealth at the end of multiple periods whereas the geometric mean accurately predicts the median and modal wealth.
Suppose an investment of $100 can grow by 100 percent or -50 percent in each year over two periods with equal probability. The geometric mean, in this case, would be zero and the arithmetic mean would be equal to 25 percent. The value of the investment at the end of two periods could be either of these four: $400, $100, $100, $25 (assuming 100 percent rise in both year, 100 percent rise and 50 percent fall in two years alternatively, and 50 percent fall in both years). The arithmetic mean of that is equal to $156.25 which is equal to an annual return of 25 percent. (156.25/100)0.5 - 1 = 25 percent.
However, if the returns are 100 percent in the first year and 50 percent in the second year, then the net value of the portfolio will be $100 only and thus a return equal to the geometric mean of zero percent.
The higher is the uncertainty in the cash flows, the greater is the deviation between geometric mean and arithmetic mean.
If there is zero variance of returns, then the arithmetic mean is equal to the geometric mean.
The arithmetic mean is always greater than or equal to the geometric mean.
Geometric mean ≈ Arithmetic mean - 0.5σ2
where σ2 is the variance of returns
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