Geometric Mean

What Is a Geometric Mean?

The geometric mean is a way of finding the typical value of a set of numbers when those numbers are meant to be multiplied together rather than added. To calculate it, you multiply all the values, then take the nth root of that product, where n is how many values there are. For two numbers, it is the square root of their product.

For a set of values x1 through xn, the formula is:

GM = (x1 × x2 × ... × xn)^(1/n)

Why It Matters for Investors

Returns compound over time, so a plain arithmetic mean of yearly returns can give a misleading picture. Consider an investment that gains 10% in one year and loses 10% the next. The arithmetic mean is 0% per year, which suggests the investment broke even. But $100 rises to $110, then falls to $99, so it actually lost 1% overall. The geometric mean captures this correctly. Using growth factors of 1.10 and 0.90, the geometric mean is the square root of 0.99, about 0.995, which corresponds to a return of roughly −0.5% per year. That is the annualized return consistent with the actual result.

For investing, the formula is usually written in terms of returns:

Average return = [(1 + r1) × (1 + r2) × ... × (1 + rn)]^(1/n) − 1

This is the same idea behind the compound annual growth rate (CAGR), which describes how a value grew each year on average over a period.

Key Properties

• For positive numbers, the geometric mean is always less than or equal to the arithmetic mean. The two are equal only when all the values are identical.
• The larger the spread between yearly results, the bigger the gap between the two averages. This gap is one reason investors pay attention to volatility.
• It only works with positive values, so return calculations use growth factors (1 plus the return) rather than raw percentages.

The geometric mean is a useful way to summarize performance, but it describes a steady rate that would produce the same final result. It does not describe the typical experience in any single year. Reading it alongside total return and standard deviation gives a fuller picture.