When should I use geometric mean instead of arithmetic mean for investment returns?

This is one of the most frequently tested distinctions in CFA Level I quant. The short answer: use arithmetic mean for forecasting a single future period's return, and geometric mean for measuring actual compound growth over multiple periods.

The Core Difference

Arithmetic Mean = (R1 + R2 + ... + Rn) / n

Geometric Mean = [(1 + R1)(1 + R2)...(1 + Rn)]^(1/n) - 1

Why They Diverge

The arithmetic mean ignores compounding effects. The geometric mean captures the actual wealth path because it accounts for the fact that gains and losses compound on different bases.

Example — Thornfield Capital Fund:

Year	Return
1	+40%
2	-30%
3	+20%

Arithmetic mean = (40 - 30 + 20) / 3 = 10.00%

Geometric mean = [(1.40)(0.70)(1.20)]^(1/3) - 1 = [1.176]^(1/3) - 1 = 5.53%

If you invested $100,000:

• After Year 1: $140,000
• After Year 2: $98,000
• After Year 3: $117,600

Your actual compound annual growth rate is 5.53%, not 10%. The arithmetic mean overstates the true growth because it ignores the asymmetry of compounding — a 30% loss requires a 42.9% gain to recover, not 30%.

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The Inequality

Arithmetic mean >= Geometric mean, always. The gap widens as volatility increases. This is called volatility drag — higher return dispersion means a bigger penalty to compound growth.

Rule of thumb: Geometric mean is approximately equal to Arithmetic mean minus half the variance.

G ≈ A - σ²/2

When to Use Each

Scenario	Use
Forecasting next year's expected return	Arithmetic
Reporting historical fund performance	Geometric
Comparing managers over multi-year track records	Geometric
Input for mean-variance optimization	Arithmetic

Exam tip: If a vignette gives you annual returns and asks "what was the compound annual growth rate," that is always geometric mean. If it asks "what return would you expect next year," that is arithmetic mean.

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