Geometric Mean

When values compound, such as investment returns over several years, the arithmetic mean can overstate the average. The geometric mean accounts for compounding.

For returns, each percentage return is converted into a growth factor by adding 1. For example, a 10% gain becomes 1.10 and a 5% loss becomes 0.95. These factors are multiplied together, the appropriate root is taken and 1 is subtracted.

Suppose an investment returns +10%, +20% and −5% over three years. The arithmetic mean is about 8.3%. The geometric mean is about 7.8%. An investment growing at exactly 7.8% each year would end up with the same final value as the actual investment.

The difference becomes dramatic with large swings. If an investment gains 50% and then loses 50%, the arithmetic mean return is 0%, but 100 becomes 150 and then 75. The geometric mean return is about −13.4% a year, which correctly reflects the loss.

The geometric mean is the same idea as the compound annual growth rate. It is always less than or equal to the arithmetic mean, and the gap grows as returns become more volatile.