Average Return Calculator
Calculate the arithmetic mean and geometric mean (CAGR-style) of a series of annual percentage returns.
How It's Calculated
Formula
\text{Arithmetic Mean} = \frac{\sum r_i}{n}\\ \text{Geometric Mean} = \left[\prod_{i=1}^{n} (1 + r_i)\right]^{1/n} - 1When you have a series of annual percentage returns, there are two common ways to average them, and they answer different questions. The arithmetic mean simply adds up every return and divides by how many there are — it's the average you'd naturally compute, but it doesn't account for how compounding actually works over time. The geometric mean (often called CAGR — compound annual growth rate — in an investment context) instead multiplies together each year's growth factor (1 + return), takes the nth root, and subtracts 1: this correctly reflects the single constant annual rate that would produce the same overall multi-year growth. The geometric mean is always less than or equal to the arithmetic mean whenever returns vary from year to year (a gap known as "volatility drag"), and the two are equal only when every year's return is identical. For evaluating actual investment performance over multiple years, the geometric mean is the more accurate figure — the arithmetic mean can overstate performance whenever returns are volatile.
Worked Examples
Three-year series: +10%, +20%, -10%
- Arithmetic mean: (10 + 20 − 10) / 3 = 20 / 3 = 6.67%
- Growth factors: 1.10 × 1.20 × 0.90 = 1.188
- Geometric mean: 1.188^(1/3) − 1 = 1.05910... − 1 = 5.91%
- The geometric mean (5.91%) is lower than the arithmetic mean (6.67%) because of volatility drag
High-volatility pair: +50%, -50%
- Arithmetic mean: (50 − 50) / 2 = 0%
- Growth factors: 1.50 × 0.50 = 0.75
- Geometric mean: 0.75^(1/2) − 1 = 0.86603 − 1 = -13.40%
- Despite an arithmetic average of 0%, the actual compounded result is a loss — this is the classic case for why geometric mean matters
Frequently Asked Questions
Why is the geometric mean usually lower than the arithmetic mean?
Because compounding is multiplicative, not additive. A loss followed by an equal-sized percentage gain (or vice versa) doesn't cancel out to zero once you compound — losing 50% then gaining 50% leaves you down 25% overall, not back to even. The geometric mean captures this correctly; the arithmetic mean does not, which is why it tends to overstate the real, compounded result whenever returns vary.
When are the arithmetic and geometric means the same?
Only when every return in the series is identical — with zero volatility, there's no compounding distortion for the geometric mean to correct for, so both calculations converge on the same figure.
Which figure should I use to evaluate investment performance?
The geometric mean (CAGR). It answers "what single constant annual rate would have produced this same overall result," which is what you actually experienced. The arithmetic mean is a simpler statistical average but can be misleading for judging real compounded performance.
Why can't I enter a return of -100% or lower?
A -100% return means a total loss — the investment's value hits zero, and 1 + (-100%) = 0. A geometric mean formula built on multiplying growth factors together breaks down at that point (multiplying by zero, or by a negative number below -100%, produces a result with no meaningful real-valued nth root), so those inputs are rejected rather than producing a misleading number.