Growth stacks by multiplication — unwinding a cumulative return into a yearly rate takes a root (the geometric mean), not division.

Experiment

Hands-on experiment

Predict first — a 50% cumulative return over 3 years. The annual average?

Roll both answers for 3 years

Starting with $1,000 — which one lands exactly on $1,500?

Read more — why it exists · insights · common mistakes · formulasExpand ▾

Why

Why does this exist?

'50% in 3 years, so 16.7% a year' feels natural — and is wrong. Compound 16.7% for 3 years and you get 58.9%, because returns multiply; they don't add.

What multiplication built, multiplication must unwind: solve (1+r)³ = 1.5, i.e., the cube root of 1.5 minus 1 = 14.5% per year. That is the Compound Annual Growth Rate.

Fund ads, company earnings, channel growth — wherever 'average annual %' appears, the arithmetic-vs-geometric gap decides your judgment. The bigger the swings, the bigger the gap.

Misconception

Common misconceptions

Cumulative return ÷ years = annual average return.

✓

Growth compounds (multiplies), so division always overestimates. Roll 50%÷3 = 16.7% for three years and you get 58.9% — the true answer is ³√1.5 − 1 = 14.5%.

Alternating +50% and −50% averages 0% — break-even.

✓

1.5 × 0.5 = 0.75 — a 25% loss. The arithmetic mean is 0%, but the geometric mean is √0.75 − 1 ≈ −13.4%/yr. Volatility itself eats returns — the 'volatility tax'.

Formula

Writing it as math

What the verification experiment confirmed, formalized.

Definition of CAGR

The n-th root of the total multiple over n years — asking 'if growth had been steady, what rate was it?'

Compound growth, the original

CAGR is this equation solved backwards for r. Compounding (powers) and CAGR (roots) are a matched pair.

Rule of 72

6% doubles in ~12 years; 9% in ~8. An approximation born from ln 2 ≈ 0.693 — CAGR for mental math.

In Real Life

Where you meet it in real life

Reading fund and ETF ads

'120% cumulative return' means nothing without the period. Over 10 years that's 8.2% CAGR — converting the big headline number to a yearly rate is verification step one.

Company earnings and growth stocks

'Revenue CAGR 25%' means doubling roughly every 3 years (Rule of 72). The growth language of investment reports is CAGR throughout.

National growth rates

3% annual growth doubles GDP in 24 years. Economies that once grew 7% a year doubled every decade.

The volatility tax

An asset alternating +30% and −20% has a +5% arithmetic mean but only ~+2% CAGR. The wilder the swings, the further the geometric mean falls below the arithmetic — why steady returns are underrated.

Math Behind