What Is Geometric Mean? Formula and Examples

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The geometric mean is the average you get by multiplying a set of n positive numbers together and taking the nth root of the product. Arithmetic averaging adds values. Geometric averaging multiplies them, so it fits quantities that compound, such as investment returns, growth rates and ratios. In finance, the geometric mean of a series of periodic returns is the single constant rate that would have turned the starting value into the ending value over the same number of periods.

The Geometric Mean Formula

For any set of positive numbers x₁, x₂, … xₙ:

Geometric mean = (x₁ × x₂ × … × xₙ)^(1/n)

The simplest case is two numbers. The geometric mean of a and b is √(ab). For 2 and 8, the product is 16 and its square root is 4. The arithmetic mean of the same pair is 5.

Returns need one adjustment, because a return can be negative or zero and the formula needs positive inputs. Each return is converted into a growth factor first:

Geometric mean return = [(1 + R₁)(1 + R₂)…(1 + Rₙ)]^(1/n) − 1

where R₁ through Rₙ are the periodic returns written as decimals. A 5% gain becomes 1.05, and a 3% loss becomes 0.97. Subtracting 1 at the end converts the averaged growth factor back into a rate.

An equivalent route uses logarithms: take the natural log of each growth factor, average the logs, then raise e to that average and subtract 1. This gives the same answer and is how many statistical tools compute the figure internally. It also avoids overflow when multiplying long series.

Worked Example

Hypothetical portfolio, annual returns over five years: 5%, 3%, 6%, 2%, 4%.

1. Convert to growth factors: 1.05, 1.03, 1.06, 1.02, 1.04.
2. Multiply: 1.05 × 1.03 × 1.06 × 1.02 × 1.04 ≈ 1.2161.
3. Take the fifth root (raise to the power 0.2): 1.2161^0.2 ≈ 1.0399.
4. Subtract 1: 0.0399, or about 3.99% per year.

The arithmetic mean of the same returns is (5 + 3 + 6 + 2 + 4) ÷ 5 = 4.00%. The difference is small here because the returns are steady. A $10,000 starting value growing at 3.99% for five years reaches about $12,161, which matches the product in step 2 exactly. Compounding at 4.00% would slightly overstate the result.

When volatility widens the gap

Hypothetical two-year example: an investment gains 50% in year one and loses 50% in year two.

• Arithmetic mean: (50% − 50%) ÷ 2 = 0%.
• Geometric mean: (1.50 × 0.50)^(1/2) − 1 = 0.75^0.5 − 1 ≈ −13.4% per year.

A $100 investment rises to $150, then falls to $75. The arithmetic mean suggests the investor broke even. The geometric mean reports the loss that actually happened. This shortfall is often called volatility drag. A common rule of thumb approximates it as the geometric mean ≈ arithmetic mean − (variance ÷ 2). The approximation works best when returns are small and fails for swings as large as this one.

Why the Geometric Mean Matters in Finance

Investment returns build on each other. Each period’s gain or loss applies to a balance that already reflects earlier results. The geometric mean is the only simple average that respects this. Several practical uses follow from it:

• Reporting historical performance. Fund and portfolio returns over multiple years are usually annualized geometrically, so the figure represents a rate the investor could actually have compounded.
• Comparing investments. Because it works only with return figures, you can compare two assets over the same period without knowing how much money was invested in each.
• Growth rates. Average annual growth in revenue, earnings, prices or population is a geometric average of the yearly growth factors. Averaging inflation rates across years to find a cumulative price change works the same way.
• Index construction. Some price indexes and ratio-based measures use geometric averaging so that a proportional change in any component carries the same weight.

The longer the horizon and the more volatile the returns, the further the arithmetic mean drifts from reality. That makes the geometric mean the more appropriate summary for long-term results.

How to Calculate the Geometric Mean in Excel or Google Sheets

Both programs include a GEOMEAN function. It accepts only positive values, so enter growth factors rather than percentage returns. With the five factors from the example in cells B2:B6:

=GEOMEAN(B2:B6)-1

This returns about 0.0399. Format the cell as a percentage to display 3.99%. Two equivalent alternatives:

• =PRODUCT(B2:B6)^(1/COUNT(B2:B6))-1
• =EXP(AVERAGE(LN(B2:B6)))-1 (entered as an array formula in older Excel versions)

On a handheld financial calculator without a dedicated function, multiply the growth factors, use the y^x key with an exponent of 1/n, and subtract 1.

Geometric Mean vs. Arithmetic Mean

The two averages answer different questions. The geometric mean describes realized compound growth over a past period. The arithmetic mean is the better estimate of the expected return for one future period drawn from the same distribution, which is why some forecasting and risk models, including those behind the Sharpe ratio, use it. For positive numbers, the geometric mean can never exceed the arithmetic mean. This result is known as the AM-GM inequality.

Geometric Mean vs. CAGR and Time-Weighted Return

The geometric mean of annual returns and the compound annual growth rate (CAGR) are the same number reached two ways. CAGR uses only the starting and ending values: (ending ÷ beginning)^(1/years) − 1. The geometric mean uses each year’s return. If the yearly returns fully explain the change in value, both give an identical result.

Time-weighted return also links period returns geometrically. It strips out the effect of deposits and withdrawals, so it measures the investment rather than the investor’s timing. A money-weighted return (internal rate of return) does the opposite: it reflects when cash moved in and out, and it can differ substantially from the geometric mean. A simple return on investment (ROI) figure shows total gain over a period without annualizing it at all.

Limitations

• Negative and zero values. The geometric mean is not defined for a set of raw numbers that includes negatives, and any zero forces the product to zero. For returns, converting to growth factors solves this: −3% becomes 0.97. A −100% return gives a factor of 0, which correctly produces a geometric mean of −100% because the investment was wiped out.
• Backward-looking. It summarizes a path that already happened. It does not describe the dispersion of returns, so two series with the same geometric mean can carry very different risk.
• Sensitive to the period chosen. Starting or ending the measurement in a crash or a rally can change the result considerably.
• Not an expected-return estimate. Using the historical geometric mean as next year’s expected return will tend to understate the single-period expectation when returns vary.

The Bottom Line

The geometric mean multiplies values and takes the nth root, which makes it the correct average for anything that compounds. For investments, average the growth factors (1 + R), then subtract 1. The result is the constant annual rate that reproduces the actual ending value. It will always be at or below the simple average, and the gap measures how much volatility cost along the way.