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
Because losses hurt more than gains help in a multiplicative process. A 50% loss requires a 100% gain to break even — the arithmetic mean treats them as cancelling, but the geometric mean accounts for the asymmetry of compounding.
When volatility is high. The variance drag is approximately σ²/2: at 20% volatility, geometric is roughly 2 percentage points below arithmetic; at 40% volatility (typical of single stocks), the gap is closer to 8 points.
For historical performance: geometric (it's what you actually earned). For expected future returns when computing variance / Sharpe: arithmetic (it's the unbiased estimator). The two are NOT interchangeable — using the wrong one is a common mistake.