Geometric Mean

What it is

The geometric mean is a type of average appropriate for multiplicative processes—most commonly used to summarize rates of return, growth rates, or any series where compounding matters. For n positive values a1, a2, …, an, the geometric mean is the nth root of their product:

Geometric mean = (a1 × a2 × … × an)^(1/n)

For a series of returns R1, R2, …, Rn (expressed as decimals), the geometric mean return is:

μ_geometric = [(1 + R1)(1 + R2)…(1 + Rn)]^(1/n) − 1

Why it matters

• Accounts for compounding — it reflects the actual multiplicative growth over time.
• More appropriate than the arithmetic mean for serially correlated or volatile returns.
• Provides an “apples-to-apples” comparison of investment performance over multiple periods.
• Always ≤ the arithmetic mean (equality only when all values are the same).

When to use it

Use the geometric mean for:
* Time series of percentage returns (CAGR, time-weighted returns).
* Growth rates (population growth, inflation factors, index multipliers).
Avoid it for simple additive contexts (e.g., average of independent measurements that don’t multiply over time).

Calculation: example

Suppose a portfolio returns the following annual rates: 5%, 3%, 6%, 2%, 4%.

1. Convert to growth factors: 1.05, 1.03, 1.06, 1.02, 1.04
2. Multiply them: 1.05 × 1.03 × 1.06 × 1.02 × 1.04 = 1.2161
3. Take the 5th root: 1.2161^(1/5) ≈ 1.0399
4. Subtract 1 to convert back to a return: 0.0399 → 3.99%

So the geometric mean annual return is about 3.99%, slightly below the arithmetic mean of 4%.

Spreadsheet tip

Most spreadsheet programs have a built-in function:
* Excel / Google Sheets: =GEOMEAN(range)
For returns, supply the growth factors (1 + R) or convert returns appropriately.

Handling negative or zero values

• The geometric mean requires all terms to be positive (strictly > 0).
• For returns, this means each 1 + R must be > 0 (i.e., R > −100%).
• If any factor is zero or negative, the geometric mean is not defined in the real numbers. In practice, returns with losses are represented by their growth factors (e.g., −3% → 0.97). If a factor ≤ 0 appears, you must use alternative measures or transform the data.

Geometric mean of two numbers

For two positive numbers a and b:
Geometric mean = √(a × b)

Key takeaways

• The geometric mean captures compounded growth and is the correct average for multiplicative processes.
• It is especially useful for evaluating multi-period investment returns.
• It will generally be lower than the arithmetic mean and requires positive inputs (growth factors) to be meaningful.

Bottom line

Use the geometric mean when you need the true average growth rate over time—particularly for investments or any series where values multiply period to period. It accounts for compounding and gives a more accurate picture of long-term performance than the arithmetic mean.