Geometric vs. Arithmetic Mean: The Right Average for Investment Returns
October 21, 2026
The arithmetic mean of +50% and −50% is 0%, but an investor who gains 50% and then loses 50% ends with 75% of the starting money, an annualized loss of 13.4%. The geometric mean captures this: it multiplies growth factors and takes the nth root. For growth rates, the geometric mean is the right average because returns compound.
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Convert each return to a growth factor (1 + return), multiply the factors, take the nth root, and subtract 1. For +50% and −50%: 1.5 × 0.5 = 0.75, and the square root of 0.75 is 0.866, so the geometric mean is −13.4% per year.
For three years of +10%, +20%, and −10%: the factors multiply to 1.1 × 1.2 × 0.9 = 1.188, and the cube root is 1.0591, so the geometric average is 5.91% a year. The arithmetic mean is 6.67%, which overstates the result.
| Returns | Arithmetic mean | Geometric mean |
|---|---|---|
| +50%, −50% | 0.00% | −13.40% |
| +10%, +20%, −10% | 6.67% | 5.91% |
| +5%, +5%, +5% | 5.00% | 5.00% |
Why the Gap Exists
The arithmetic mean is always greater than or equal to the geometric mean, with equality only when all returns are identical. The gap grows with volatility: roughly, geometric ≈ arithmetic − variance ÷ 2. This volatility drag means that the more a portfolio swings, the more compound growth falls below the simple average.
That is why two funds with the same average yearly return can leave investors with different final balances.
Which One to Use
Use the geometric mean (also called the compound annual growth rate, CAGR) to describe what happened to an investment over a period. Use the arithmetic mean to estimate the expected return for a single future year, since each year's outcome is an independent draw.
For ordinary averages of quantities that add, such as test scores or daily temperatures, the arithmetic mean is correct. For ratios, rates of change, and multiplicative processes, use the geometric mean.
Frequently Asked Questions
What is the geometric mean?
The nth root of the product of n values. For returns, it is applied to growth factors.
Why is the arithmetic mean misleading for returns?
It ignores compounding, so volatile returns appear better than the actual growth.
Is CAGR the same as geometric mean?
Yes. CAGR is the geometric average growth rate over a period.
Can the geometric mean be higher than the arithmetic mean?
No. It is always less than or equal to the arithmetic mean.