Geometric vs arithmetic mean return: 7.49%, not 8.33%

Quantitative Methods·August 28, 2026·4 min read

A portfolio returns 20%, then −10%, then 15%. A candidate who averages the three numbers reports 8.33% a year, but the portfolio actually compounded at 7.49%, because only the geometric mean return reproduces the ending value the portfolio really reached.

The two means answer different questions

Both are averages of the same three numbers, and both are correct arithmetic. They are not interchangeable, because they are built for different questions.

Arithmetic mean return = (R_1 + R_2 + ... + R_n) / n

Geometric mean return = [(1 + R_1)(1 + R_2)...(1 + R_n)]^(1/n) − 1

The arithmetic mean adds returns, so it treats each period as a fresh, independent draw. That is what makes it the right average when the question is about one period: an equally weighted average of what happened in a typical year. The geometric mean multiplies growth factors instead, so it carries the ending value of each period into the next. When the question is what a sum of money actually grew to over several periods, multiplying is the operation the money performed.

Adding percentages is the intuitive move, and it is the wrong one for multi-period growth. That is the whole trap. Nothing in the wording of an average return question signals which operation to reach for, so the phrase to hunt for is the one about compounding or annual growth over the full horizon.

Working the three-year example

Start with 100. The three annual returns are 20%, −10%, and 15%.

Arithmetic mean = (0.20 − 0.10 + 0.15) / 3 = 0.25 / 3 = 8.33%

Growth factors: 1.20 × 0.90 × 1.15 = 1.242

Geometric mean = 1.242^(1/3) − 1 = 7.49%

Check both against the account balance. The portfolio ends at 100 × 1.242 = 124.20. Compounding the geometric mean gives 100 × 1.0749^3 = 124.20, the same figure. Compounding the arithmetic mean gives 100 × 1.0833^3 = 127.14, which is 2.94 more money than the portfolio ever held.

That 127.14 is not a rounding artifact. It is the distractor, and it is what an answer choice built on 8.33% is quietly asserting.

The gap is the volatility, and it never runs the other way

The geometric mean is never above the arithmetic mean. The two are equal only when every period return is identical, which is another way of saying the returns have no dispersion at all. Any variation at all pushes them apart, and more variation pushes them further apart.

The gap has a size you can estimate. Take the three deviations from 8.33%: 11.67%, −18.33%, and 6.67%. Square them, average the three squares, and the variance is 0.01722, a standard deviation of 13.12%.

Geometric mean ≈ arithmetic mean − variance / 2 = 8.33% − 0.86% = 7.47%

Against the exact 7.49%, that approximation is close enough to sanity-check an answer choice in a few seconds. Push the volatility up and the effect stops being subtle. A portfolio that gains 50% and then loses 50% has an arithmetic mean of exactly zero, while 100 becomes 150 and then 75. The geometric mean is (1.50 × 0.50)^(1/2) − 1 = −13.40%. A zero average return and a quarter of the money gone, from the same two numbers.

Quick reference

One habit removes most of the risk here. Once you have an average return, compound it over the stated horizon and compare the result against the ending value the question gives you. The arithmetic mean will overshoot whenever returns varied, and that mismatch is visible before you commit to an answer choice.

The same split shows up in money-weighted and time-weighted returns: one set of returns, two defensible averages, and a question that has already decided which one it wants.

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