Arithmetic Versus Geometric Mean Returns

Take a strategy that returns +50% in year one and −50% in year two. The plain average of those two numbers is 0% — sounds break-even. But $100 growing 50% becomes $150, and $150 falling 50% becomes $75. The strategy lost a quarter of its money over two years, not zero. The plain average — the arithmetic mean — answers a different question than the one most people actually care about, which is: what return, compounded steadily, would have produced the same ending wealth? That second number is the geometric mean, and it is always less than or equal to the arithmetic mean whenever returns vary at all.

The two means, defined in words

The arithmetic mean of $n$ returns $r_1, \dots, r_n$ is just the ordinary average:

$$\bar{r}_A = \frac{1}{n}\sum_{i=1}^n r_i .$$

In plain English: add up the returns and divide by how many there are — the number you'd report if someone asked "what did this strategy return per period, on average?" without thinking further.

The geometric mean is the constant per-period return that compounds to the same ending value as the actual sequence:

$$\bar{r}_G = \left[\prod_{i=1}^n (1+r_i)\right]^{1/n} - 1 .$$

In plain English: multiply together each period's growth factor, take the $n$-th root, and subtract 1 — the single repeated return that would have gotten you from start to finish exactly as the real, varying sequence did.

Why they diverge: volatility eats the difference

The gap between the two means widens with volatility. A useful approximation is:

$$\bar{r}_G \approx \bar{r}_A - \frac{\sigma^2}{2} ,$$

where $\sigma^2$ is the variance of returns. In plain English: the more a strategy's returns bounce around, the further its true compounded growth rate falls below its simple average return — this gap is sometimes called the "volatility drag," and it's not a rounding error, it's a real cost of variance on compounded wealth.

Worked example: the +50%/−50% strategy

Arithmetic mean: $\bar{r}_A = \frac{0.50 + (-0.50)}{2} = 0\%$. Geometric mean: $\bar{r}_G = \sqrt{(1.50)(0.50)} - 1 = \sqrt{0.75} - 1 \approx -13.4\%$. Over two years, compounding −13.4% twice: $(1-0.134)^2 \approx 0.75$, matching the actual $75 ending value from $100. The arithmetic mean of 0% would tell an investor the strategy is a wash; the geometric mean of −13.4% correctly reports that it's actually losing money at a compounded 13.4% per year.

What this means in practice

Whenever a backtest reports a strategy's "average return," check which mean it is: the arithmetic mean is the right number for estimating a single period's expected return (useful for risk models and optimization), while the geometric mean is the right number for describing actual realized, compounded performance over multiple periods. Reporting the arithmetic mean as if it described compounded growth is one of the most common — and most misleadingly flattering — errors in presenting backtest results, especially for volatile strategies where the gap between the two can be large.