Geometric vs Arithmetic Returns

"Average 20% one year, lose 20% the next" sounds like it nets out to roughly flat. It doesn't — a 20% gain followed by a 20% loss on the new, larger balance leaves you down 4%, not even. The plain average of the two numbers (0%) and the actual money-weighted outcome (-4%) disagree, and that gap is not a rounding error — it's the structural difference between an arithmetic average and a geometric one, and it shows up in every volatile return series.

Two different averages of the same numbers

The arithmetic mean is the simple average of per-period returns. The geometric mean is the single, constant per-period growth rate that would have compounded to the same final balance:

$$1 + g = \left[\prod_{t=1}^{T} (1 + r_t)\right]^{1/T} .$$

In words: multiply together each period's growth factor $(1 + r_t)$, take the $T$-th root, and subtract 1 to get $g$, the geometric mean return. Because multiplying compounds effects rather than just adding them, any variation in the $r_t$ pulls the geometric mean below the arithmetic mean — and the two are equal only in the special case where every period's return is identical. A useful approximation for small, roughly normal returns is $g \approx a - \sigma^2/2$, where $a$ is the arithmetic mean and $\sigma$ the volatility of returns: the higher the volatility, the bigger the gap, regardless of what the arithmetic average says.

Push the rate up and watch how compound growth pulls away from simple growth over time — the same underlying mechanism that makes a volatile path's realized compounded return diverge from its own period-by-period average.

Worked example 1 — the ±50% sequence

A strategy returns $+50\%$, $-50\%$, $+50\%$ over three years. The arithmetic mean is $(50 - 50 + 50)/3 = 16.7\%$ — looks solid. The actual compounded outcome: $1.5 \times 0.5 \times 1.5 = 1.125$, so the geometric mean is $1.125^{1/3} - 1 \approx 4.03\%$. The arithmetic average overstates the true annualized growth rate by more than four times.

Worked example 2 — same arithmetic mean, different volatility

Two strategies both report a 10% arithmetic average annual return. Strategy A has low volatility, $\sigma = 5\%$; Strategy B is more volatile, $\sigma = 30\%$. Using the approximation $g \approx a - \sigma^2/2$: Strategy A's geometric return is roughly $10\% - (0.05)^2/2 = 10\% - 0.125\% \approx 9.88\%$ — barely different from its arithmetic average. Strategy B's is roughly $10\% - (0.30)^2/2 = 10\% - 4.5\% = 5.5\%$ — a full 4.5 percentage points below its own quoted average return, purely from volatility, with no change to the "average" number a marketing deck would show.

What this means in practice

Any performance figure quoted as an "average annual return" needs a second look at whether it's arithmetic or geometric — arithmetic is the right number for estimating next period's expected return, but geometric is the right number for how much money actually accumulated over the history shown. Comparing two strategies on arithmetic averages alone can rank the more volatile one artificially high, exactly as in worked example 2, when the geometric (realized, compounded) outcome tells the opposite story.