Saving & investing
Arithmetic vs. geometric return: Definition, Formula and Example
Arithmetic average return is the simple mean of periodic returns; geometric average return is the constant compounded rate matching their cumulative growth.
Scope: General financial concept; examples are hypothetical.
Other names and related labels: Average annual return; Compound average return.
How it works
An arithmetic mean describes an average single-period outcome. The geometric mean describes linked growth over the full sequence, before external cash-flow effects. Volatility can leave the arithmetic mean positive while compounded wealth is unchanged or lower.
Arithmetic mean = Σ returns / number of periods; geometric mean = [Π(1 + return)]^(1 / number of periods) − 1
Use equally long periods. Standard investment growth factors must be nonnegative; a −100% period wipes out the invested amount.
A gain of 20% followed by a loss of 20%
Begin with $1,000 and add no money.
- After the gain: $1,000 × 1.20 = $1,200.
- After the loss: $1,200 × 0.80 = $960.
- Arithmetic mean: (20% − 20%)/2 = 0%; geometric mean: √0.96 − 1 ≈ −2.0204%.
The arithmetic mean is 0%, but wealth falls 4% and the geometric annual mean is about −2.0204%.
Two consecutive annual returns; no external deposits, taxes or fees.
The worked example uses the assumptions above. Related tools may model a different scope; check their descriptions before comparing results.
Averaging percentages can hide a loss
A 20% loss needs a 25% gain to recover, because the recovery starts from a smaller base. Dollar-weighted performance also depends on when contributions occur, so it need not equal the geometric mean.
Explore with a calculator
- Average Return Calculator
Compare arithmetic and geometric means for a sequence of returns.
Sources and checks
Definition and example arithmetic checked . Sources explain the concepts; numerical examples use the assumptions stated above.
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