Arithmetic Mean Return
Simple average of periodic returns. Sum of returns divided by the number of periods.
When to use: Use as the unbiased estimator of expected return for a single future period — the right input to risk models like CAPM and to volatility/Sharpe calculations. Do NOT use to describe realized cumulative growth (use Geometric Mean Return for that).
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| AMR | Arithmetic Mean Return | Simple 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 Returns
Annual returns: 50%, −30%, 20%.
Given
Step-by-Step
Arithmetic mean of 13.33%. This is the right input to forward-looking models (CAPM expected return, Sharpe denominator inputs) but overstates realized growth — geometric mean is 8.01%.
Frequently Asked Questions
Because arithmetic mean is the unbiased estimator of next-period's expected return — the right number for forward-looking risk models. Geometric mean is the biased downward estimator that captures realized compounding, but underestimates expected future returns.
Approximately: AMR − GMR ≈ σ²/2 (Jensen's inequality). At 20% volatility, expect about a 2pp gap; at 40% volatility, about 8pp. This is sometimes called "variance drag" or "volatility tax."
Forward-looking risk and expected-return models (CAPM, MPT optimization, Sharpe ratios) use arithmetic mean. Backward-looking realized performance disclosures use geometric mean. Confusing the two is one of the most common errors in performance reporting.