Geometric Mean Return

Builds onArithmetic Mean Return · Holding Period Return (HPR) — if this page feels steep, start there.

R_G = \left[\prod_{i=1}^{n}(1+R_i)\right]^{1/n} - 1

$R_G$: the geometric (compound) mean return
$\prod_{i=1}^{n}$: the big pi is like sigma but for MULTIPLYING: multiply all the terms together
$(1+R_i)$: each period's 'wealth relative': your money multiplied by this each period (a 5% gain is ×1.05)
$[\cdots]^{1/n}$: raising to the power 1/n is taking the n-th root — it undoes n periods of compounding
$-\,1$: subtract the 1 to turn the growth factor back into a return

Why it must be true

Returns don't add across time — they compound. Lose 50% then gain 50% and you're down 25%, though the arithmetic average says zero. The geometric mean is the single constant rate that, compounded $n$ times, reproduces your actual ending wealth. It is the honest answer to "what did I earn per year?"

Work in wealth relatives $(1+R)$ : they multiply cleanly, the negative-return trap disappears, and taking the $n$ -th root undoes the compounding. $R_G$ never exceeds the arithmetic mean, and the gap grows with volatility — volatility drag made visible.

The derivation

Let ending wealth per dollar be the product of wealth relatives:

W_n = (1+R_1)(1+R_2)\cdots(1+R_n)

Define $R_G$ as the constant rate producing the same wealth: $(1+R_G)^n = W_n$ . Take the $n$ -th root:

1 + R_G = \left[\prod_{i=1}^n (1+R_i)\right]^{1/n} \quad\Rightarrow\quad R_G = \left[\prod (1+R_i)\right]^{1/n} - 1

When to reach for it

The constant annual rate that reproduces a realized multi-period outcome — track records, CAGR, 'annualized since inception'.

Listen for

“compound annual growth rate”“annualized return over the period”“what constant rate gives the same ending value”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

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Volatility drag: geometric ≈ arithmetic − σ²/2. Compute the easy arithmetic mean first, then shade it down — the geometric answer is always at or below it.
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Returns close together → the two means nearly coincide; returns spread wide → big gap. −50% then +50% has arithmetic 0% but geometric ≈ −13.4%.

Traps in applying it

✗Multiplying raw returns instead of wealth relatives (1 + R) — negative periods break the naive product.
✗Forgetting to subtract the 1 after taking the root.
✗Feeding it into a one-period-ahead forecast, where the arithmetic mean is the right expectation.

Limits & criticisms

It is backward-looking by construction — a description of the path taken, not an expectation of the next step. It is undefined once any period loses 100% or more, and by collapsing the path into one number it hides the volatility and drawdowns an investor actually lived through.

Where it came from

Geometric progressions go back to Greek mathematics, but finance rediscovered the geometric mean the hard way: performance claims based on arithmetic averages systematically overstated what investors actually earned, a gap (the volatility drag) that grows with risk. Index designers and fund analysts adopted compound-growth reporting to close it.

Today every CAGR in a fund factsheet, every "annualized return since inception," is a geometric mean — and the CFA curriculum tests the arithmetic-vs-geometric distinction relentlessly because marketing documents still blur it.

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