Arithmetic vs geometric mean return

Definition in one sentence

The arithmetic mean return adds up period returns and divides by the number of periods, while the geometric mean return is the steady per-period rate that would turn the starting value into the actual ending value; because losses and gains compound, the arithmetic figure is usually higher than the growth you actually got.

A worked example (hypothetical)

Take an account that rises by some percentage in the first month and then falls by the same percentage in the second month.

• Arithmetic mean: the gain and the loss cancel out, so the average monthly return looks like zero.
• Actual result: the loss is applied to a larger balance than the gain was, so the account ends below where it started.
• Geometric mean: it is negative, which matches what happened to the money.

The bigger the swings, the bigger the gap. With small, steady returns the two averages are almost equal. With large ups and downs, the arithmetic figure can show a positive "average investment return" while the balance has shrunk. To test your own sequence of returns, use the tools on Calculators.

Formal definition

For period returns r₁ … rₙ:

• Arithmetic mean: the sum of the rᵢ divided by n.
• Geometric mean: multiply the growth factors (each period's return added to the starting unit), take the n-th root of the product, then subtract the starting unit again.

Key properties:

• The geometric mean is never above the arithmetic mean. The two are equal only when every period return is identical.
• A common approximation is that the geometric mean equals the arithmetic mean minus a term proportional to the variance of returns. This gap is often called volatility drag.
• The arithmetic mean of log returns corresponds to the geometric mean of simple returns.

Edge cases

• A period that loses the entire capital makes the product zero. The geometric mean then shows total loss, whatever the other periods did.
• Leverage makes the swings larger, which widens the gap between the two averages. Leveraged forex and CFD trading can lose money fast. A high arithmetic average of trade results does not protect the account balance. See Risk management.
• Deposits and withdrawals distort both figures. Time-weighted returns remove the effect of cash flows, while money-weighted returns include it.
• Forecasting vs describing: the arithmetic mean is the usual estimate for a single future period. The geometric mean describes the compound growth that was actually realised over many periods.

Where you see these terms

• Fund factsheets and reports: an "annualised" or CAGR figure is a geometric mean. A plain "average annual return" is often arithmetic, so check which one is meant.
• News articles: long-run market claims often quote simple averages without saying so.
• MetaTrader testing reports: "expected payoff" is an arithmetic average per trade. It is not the compound growth of the account, which you see on the equity curve.
• Your own account statement: the change in balance over time reflects geometric growth, net of cash flows.

Related terms

• Annualized return
• CAGR
• Annualized volatility
• Fat tails and tail risk
• Equity curve
• Basis point
• Trading glossary

Frequent questions

Which is larger, the geometric or the arithmetic mean return?

The arithmetic mean is the same or larger. They match only when all period returns are equal.

Is CAGR the same as the geometric mean return?

Yes. CAGR is the geometric mean return expressed per year.

Why do reports sometimes use the arithmetic mean?

It is simpler to calculate and is the standard estimate for one future period. It does not show how money compounded over many periods.

Does volatility always reduce compound growth?

For a given arithmetic average, more variable returns produce a lower geometric mean. That is why two strategies with the same average can end with very different balances.

Can the arithmetic mean be positive while I lost money?

Yes. Large swings can produce a positive simple average while the compounded result is negative.