What is the geometric mean return formula?

RG = [(1 + R1)(1 + R2)…(1 + Rn)]1/n − 1, with each return written as a decimal.

Why is the geometric mean lower than the arithmetic mean?

Losses hurt more than equal gains help, because a gain is applied to a smaller base after a loss. The more volatile the returns, the bigger the gap.

When should I use the geometric mean?

To describe past compound performance over several periods. Use the arithmetic mean as an estimate of the expected return for a single future period.

Can I enter negative returns?

Yes, any return above −100%. A return of −100% or less would wipe out the investment.

Geometric Mean Return Formula

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Geometric Mean Return Calculator

The geometric mean return is the constant per-period return that would produce the same final wealth as the actual sequence of returns. You convert each return to a growth factor (1 + R), multiply them together, take the n-th root and subtract 1. It is the right average to use when you want to describe what an investment actually earned over time.

Formula

R_G = [(1 + R₁)(1 + R₂) ⋯ (1 + R_n)]^1/n − 1

Where

R_G: geometric mean (compound average) return per period
R_i: return in period i, as a decimal
n: number of periods

Calculator

Returns per period (%)

One return per period, in percent. Losses are negative.

Result and step-by-step solution

Geometric mean return

3.5716%

Arithmetic mean: 4.2%. The geometric mean is what the investment actually compounded at: 19.18% in total.

Convert each return to a growth factor
(1 + 0.1) × (1 + −0.05) × (1 + 0.2) × (1 + 0.08) × (1 + −0.12) = 1.1918016
Take the n-th root
1.1918016^1/5 = 1.03571626
Subtract 1
R_G = 3.5716%
Compare with the arithmetic mean
Σ R_in = 0.215 = 4.2% (always ≥ the geometric mean)

How to use the formula

The arithmetic mean, the simple average of the returns, is always at least as large and often noticeably larger. The classic illustration: a 50% gain followed by a 50% loss has an arithmetic mean of 0%, yet 100 becomes 150 and then 75, a real loss. The geometric mean, −13.4% a period, tells the truth. The gap grows with volatility.

In the default example the returns are 10%, −5%, 20%, 8% and −12%. The arithmetic mean is 4.20%, but the geometric mean is 3.57%, and the investment grew by 19.18% in total. Fund fact sheets report multi-year performance as a geometric (annualized) return. For the simple average and the standard deviation of the same returns see the arithmetic mean return page; with only a start and end value, use CAGR.

Frequently asked questions

What is the geometric mean return formula?: R_G = [(1 + R₁)(1 + R₂)…(1 + R_n)]^1/n − 1, with each return written as a decimal.
Why is the geometric mean lower than the arithmetic mean?: Losses hurt more than equal gains help, because a gain is applied to a smaller base after a loss. The more volatile the returns, the bigger the gap.
When should I use the geometric mean?: To describe past compound performance over several periods. Use the arithmetic mean as an estimate of the expected return for a single future period.
Can I enter negative returns?: Yes, any return above −100%. A return of −100% or less would wipe out the investment.

Last reviewed: September 26, 2026. Calculations run in your browser and were checked against numpy-financial, SciPy and published spreadsheet examples.