Geometric mean returns or growth are calculated by taking account of compounding.
(Contrasted with the arithmetic mean, which ignores compounding).
Example 1: Positive returns or growth
The geometric mean return calculated from sample returns of 4%, 5% and 6% is given by:
(1.04 x 1.05 x 1.06)(1/3) - 1
Relationship between geometric mean and arithmetic mean
When returns or growth are positive, geometric means are smaller figures than arithmetic means.
In Example 1 above, the arithmetic mean is:
(4% + 5% + 6%) / 3 = 5.0000%
The geometric mean of +4.9968% is a smaller positive number than the arithmetic mean of +5.0000%.
On the other hand, when returns or growth are negative, the geometric mean is a larger negative number - further away from zero - than the arithmetic mean.
Example 2: Negative returns or decline
The geometric mean return calculated from three negative sample returns of -(4)%, -(5)% and -(6)% is given by:
( (1 - 0.04) x (1 - 0.05) x (1 - 0.06) )(1/3) - 1
(0.96 x 0.95 x 0.94)(1/3) - 1
The negative geometric mean of -(5.0035)% is a larger negative number - further away from zero - than the arithmetic mean of -(5.0000)%.
(The arithmetic mean of the negative returns of -(4)%, -(5)% and -(6)% is the three items added together and divided by 3.)