Geometric Mean Return
Compound annual growth rate of returns
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Geometric Mean Return is built for compound annual growth rate of returns — fast, free, and private. Fill in Annual Returns (comma sep %) on Geometric Mean Return and read your answer immediately. Geometric Mean Return puts the headline number first. Geometric Mean Return shines when speed matters. You can disconnect once Geometric Mean Return has loaded; the arithmetic runs locally. Geometric Mean Return sits in the Finance section next to related tools and geometric mean return calculator cagr. Geometric Mean Return goes from question to answer fast. Geometric Mean Return is free, instant and local.
What the defaults give
The six returns multiply to 1.410028, so the geometric mean is 5.89% a year. The arithmetic average of the same numbers is 6.17%.
Both describe the same six years; only the geometric figure is the rate the money actually compounded at.
The formula
Geometric mean = (Π (1 + rᵢ))^(1/n) − 1
Arithmetic mean = Σ rᵢ ÷ n
Each return is turned into a growth factor, the factors are multiplied, and the nth root turns the total growth back into a per-year rate. This is the compound annual growth rate of the series, and it is what an investor earned per year on money left invested throughout.
Worked example
Returns 10%, -5%, 15%, 8%, -3%, 12%
Product = 1.10 × 0.95 × 1.15 × 1.08 × 0.97 × 1.12 = 1.410028
Geometric mean = 1.410028^(1/6) − 1 = 5.89%
Arithmetic mean = 37 ÷ 6 = 6.17%
BRL 100 grows to BRL 141.00 over the six years. At a steady 5.89% it would do exactly the same; at 6.17% compounded it would reach BRL 143.20, which is more than the actual series produced.
Why the arithmetic average is too high
Percentage losses hurt more than equal percentage gains help, because the loss is taken on a smaller base. A 50% loss needs a 100% gain to get back to where you started, and no arithmetic average captures that asymmetry. The gap between the two means is roughly half the variance of the returns, which is why a volatile series shows a wide gap and a smooth one shows almost none.
This is why funds quote annualised (geometric) returns. A fund that returns +50% then -50% has an arithmetic mean of 0% and a geometric mean of -13.4%, and the account balance agrees with the second one.
Using it properly
Order does not matter for the geometric mean: the same six numbers in any sequence give 5.89%. That is a feature, because the calculation is describing compound growth, not a path. What does matter is that every period is included — dropping the worst year flatters the result enormously, which is exactly why the figure should come from a full statement rather than a remembered best run.
Enter returns as percentages separated by commas, and keep the sign on losses.
Frequently Asked Questions
Why is the geometric mean lower than the average?
Because losses are taken on a smaller base and need a larger gain to recover. The gap grows with volatility; here it is 5.89% against 6.17%.
What does 5.89% mean in money terms?
BRL 100 becomes BRL 141.00 over the six years. A steady 5.89% a year produces the same balance as those six uneven returns.
Does the order of the returns matter?
No. Multiplication is commutative, so any sequence of the same six numbers gives the same geometric mean. Only the set of returns matters.
Is the geometric mean the same as CAGR?
Yes, for a series that starts and ends at the same points. Both express total growth as a constant annual rate.
Should I drop a bad year to see the trend better?
No. Removing the worst period raises the result sharply and misrepresents what the money did. Use the complete series, including the losses.
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