Geometric vs arithmetic mean return: when each measure is the right one to use

An investor whose portfolio returns +50% in year one and −50% in year two has lost a quarter of their starting capital, not broken even. The arithmetic mean of those two returns is zero; the geometric mean is approximately −13%. They are different averages of the same data, and they answer different questions about an investment.

What each mean is

The arithmetic mean is the simple average of a set of returns: add them up and divide by the count. For periods r₁, r₂, …, rₙ, the arithmetic mean is (r₁ + r₂ + … + rₙ) / n. It is the figure most people intuitively expect when asked for the average of a series.

The geometric mean is the constant single-period return that would have produced the same compounded outcome as the actual sequence. For periods r₁, r₂, …, rₙ, the geometric mean is the n-th root of the product of (1 + r₁) × (1 + r₂) × … × (1 + rₙ), minus one. Compounded annual growth rate (CAGR) is the geometric mean expressed in annual terms.

For any non-trivial return series with non-zero variance, the geometric mean is strictly less than the arithmetic mean. The two coincide only when every period's return is identical—an unrealistic case. The gap between the two grows with the variance of the return series and is, in fact, approximately equal to half the variance for small returns: geometric mean ≈ arithmetic mean − ½σ².

How they differ in practice

The +50% / −50% example is extreme but illuminates the structural point. Starting capital of USD 100 grows to USD 150 after the first year and falls to USD 75 after the second. The arithmetic mean of the period returns is (50 − 50) / 2 = 0%. The geometric mean is the rate that takes USD 100 to USD 75 over two periods: (75/100)^(1/2) − 1 = −13.4%. The geometric mean tells the investor what they actually experienced; the arithmetic mean does not.

Less dramatic examples produce less dramatic gaps but follow the same logic. A return series of +20%, −10%, +15%, +5% has an arithmetic mean of 7.5% and a geometric mean of approximately 7.0%. The 0.5-percentage-point gap reflects the volatility drag: the series's variance reduces the compounded outcome below what a constant 7.5% return would have produced.

The relationship has a practical implication. Two portfolios with the same arithmetic mean return but different volatilities will produce different terminal wealth: the lower-volatility portfolio compounds more efficiently. This is the volatility tax, and it is why long-term wealth creation typically rewards risk control as much as it rewards higher expected returns.

What the evidence shows

Bodie, Kane, and Marcus (2014) document the gap empirically across major asset classes. For US equities over 1926–2010, the arithmetic mean annual return was approximately 11.9% and the geometric mean was approximately 9.9%—a 2-percentage-point gap that reflects the standard deviation of approximately 20%. For long-term Treasuries over the same period, the arithmetic mean was 5.9% and the geometric mean 5.4%, a smaller gap reflecting the lower volatility of fixed income.

The implication for forecasting is direct. The arithmetic mean is the appropriate input to a single-period expected-return calculation: if an investor wants to know what they should expect to earn in the next year, the arithmetic mean of historical returns is a better estimate than the geometric mean. The geometric mean is the appropriate measure of what an investor has actually earned over a multi-period sequence: if an investor wants to know what their long-run rate of wealth accumulation has been, the geometric mean is the right answer.

Confusing the two leads to systematic forecasting errors. Compounding the arithmetic mean over a long horizon overstates the realised compounded outcome by a factor that depends on the volatility of the series; using the geometric mean as a single-period expected return understates the per-period expectation.

Limitations and trade-offs

Both measures assume the return series is comparable across periods. If the periods themselves are unequal in length—monthly returns mixed with annual returns, for instance—the simple arithmetic and geometric formulas no longer apply without conversion. Annualising matters: the geometric mean of monthly returns must be raised to the 12th power to be compared with an annual figure.

Both measures also assume the return distribution is stable enough that summarising it with a single number is meaningful. For series with extreme skewness or kurtosis—single very large positive or negative observations—even the geometric mean can mislead, because it remains dominated by individual extreme observations. In those cases, the median and quantile statistics complement the mean rather than replace it.

Neither measure tells the investor anything about the dispersion around the average. A 7% geometric mean can come from a series of stable 7% returns or from a series ranging from −30% to +50% that happens to compound to the same outcome. The dispersion is a separate question, addressed by volatility, downside volatility, and drawdown statistics.

Geometric vs arithmetic mean in pfolio

pfolio reports CAGR (the geometric mean of annualised returns) as the headline performance figure for any asset or portfolio, and arithmetic mean return alongside it. Both are computed from the same underlying return series, with the choice of close or adjusted close price input configurable via advanced settings. The metrics are visible in pfolio Insights; the calculation methodology is documented in the metrics we use.

Related articles

• CAGR explained: what compound annual growth rate means for your portfolio
• Cumulative return: how to measure total portfolio growth over time
• Mean return explained: how to measure average investment performance
• Volatility in investing: how to measure and manage portfolio risk
• Log returns vs simple returns: when each convention is appropriate
• Annualisation in investing: how monthly and daily figures are scaled to annual rates