Average Return Calculator (Arithmetic vs Geometric)

How it works

Arithmetic average = (r₁ + r₂ + … + rₙ) ÷ n · geometric average (CAGR) = [(1 + r₁) × (1 + r₂) × … × (1 + rₙ)]^(1/n) − 1 · volatility drag = arithmetic − geometric ≈ σ² ÷ 2 · σ = sample standard deviation of the returns

The arithmetic average adds up the yearly returns and divides by the number of years; the geometric average is the constant yearly return that would turn the starting amount into the same ending value. Because losses hurt more than equal gains help (a 50% loss needs a 100% gain to recover), the geometric average is always at or below the arithmetic one, and the gap — the volatility drag — grows with volatility. Use the geometric average (CAGR) to describe what an investor actually earned and the arithmetic average as the best estimate of a single future year’s expected return.

Worked example

Yearly returns of 12%, −8%, 15%, 6% and −3% have an arithmetic average of 22 ÷ 5 = 4.40%. But 10,000 grows to 10,000 × 1.12 × 0.92 × 1.15 × 1.06 × 0.97 = 12,183.76, a total return of 21.84%, so the geometric average is 1.2184^(1/5) − 1 = 4.03% a year. The 0.37-point gap is the volatility drag; the standard deviation of the returns is 9.76%.

Sources: Wikipedia — Rate of return (arithmetic and geometric average) · Wikipedia — Geometric mean

Frequently asked questions