Geometric Mean Return
Geometric Mean Return
Compounded average periodic return. Multiplies (1 + r) across all periods, takes the n-th root, and subtracts 1. The honest "actual realized growth" of the series.
When to use: Use whenever you care about cumulative wealth — long-horizon investing, compound growth, or comparing buy-and-hold strategies. Geometric mean is always ≤ arithmetic mean; the gap widens with volatility.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| GMR | Geometric Mean Return | Geometric average periodic return as a decimal | % |
| Returns | Periodic Returns | Sequence of periodic returns as decimals (e.g. 0.05 for 5%) | % |
Real-Life Examples
Example 1: Three-Year Stock Return
Annual returns: 50%, −30%, 20%.
Given
Step-by-Step
Geometric mean of 8.01% per year. Compare to the arithmetic mean of (50 − 30 + 20)/3 = 13.33% — the gap is large because of the −30% drawdown. The geometric mean is the figure to use for "what did I actually earn?" decisions.
Frequently Asked Questions
For returns no lower than −100%, the geometric mean is no greater than the arithmetic mean, with equality when all returns are equal. A −50% return followed by +100% has zero compound growth but a +25% arithmetic average.
An exact −100% return makes cumulative wealth zero, so the geometric mean is −100%. Returns below −100% are outside this unlevered wealth-compounding model.