functions

Geometric Mean Rate Of Return Calculator | Geometric return

How the Geometric Mean Rate Of Return Calculator works

Under the hood, this calculator applies Geometric mean = [(1+r₁)(1+r₂)…(1+rₙ)]^(1/n) − 1, so the geometric mean rate of return calculator result you see is genuine math, not a lookup table.

The reference table below covers the most common geometric mean rate of return calculator cases at a glance, the methodology section breaks the calculation into verifiable steps, and the FAQ tackles the edge cases.

Quick reference

Method	What it answers
Arithmetic mean	Simple average of yearly returns
Geometric mean	Compounded average actually earned
CAGR	Geometric return from start & end value
Why it differs	Volatility drags compounded returns
Use	Compare investments over time

info Geometric Mean Rate Of Return Calculator

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info Educational

Includes the formula and step-by-step explanation so you understand the math, not just the answer.

Interpretation guide

Return pattern	Geometric vs arithmetic
Steady returns	The two are nearly equal
Volatile returns	Geometric is noticeably lower
A big loss year	Geometric mean falls sharply
Reported "average"	Use geometric for honest results

lightbulb Five volatile years: +12, −8, +25, +4, +10

A $10,000 portfolio posts annual returns of +12%, −8%, +25%, +4%, +10% over five years. The arithmetic average is (12 − 8 + 25 + 4 + 10) ÷ 5 = 8.6% — but that ignores compounding.

Result: Multiply the growth factors: 1.12 × 0.92 × 1.25 × 1.04 × 1.10 = 1.473472, so $10,000 → $14,734.72. Geometric mean = 1.473472^(1/5) − 1 ≈ 0.0807, i.e. 8.07% per year.

What this means: 8.07% is the single constant rate that truly reproduces the ending balance: $10,000 × 1.0807^5 = $14,734.72. The 8.6% arithmetic figure would have predicted $15,070 — overstating growth by $335. Volatility always drags the geometric mean below the arithmetic one.

Formula & methodology

Formula: Geometric mean = [(1+r₁)(1+r₂)…(1+rₙ)]^(1/n) − 1

How the geometric mean return is calculated

The tool applies Geometric mean = [(1+r₁)(1+r₂)…(1+rₙ)]^(1/n) − 1, compounding each period's return and taking the nth root — the constant annual rate that reproduces the actual ending value of the investment.

Example: returns of +50% and −50% have an arithmetic mean of 0% but a geometric mean of √(1.5 × 0.5) − 1 ≈ −13.4%, matching the real loss: $100 → $150 → $75.

Reading the result: the geometric mean is always at or below the arithmetic mean, and the gap widens with volatility — it is the correct average for compounding returns, portfolio performance, and any series where each period builds on the previous one. The arithmetic mean overstates growth whenever returns vary.

Authoritative source: Wolfram MathWorld

Frequently asked questions

What is the geometric mean rate of return?

It is the average growth rate that compounds over multiple periods: Geometric mean = [(1+r₁)(1+r₂)…(1+rₙ)]^(1/n) − 1. Unlike the arithmetic average, it accounts for compounding, so it answers the question 'what constant annual return would produce the same final value?' It is the correct measure for comparing investment performance over time.

How is geometric mean different from arithmetic mean?

The arithmetic mean simply averages the yearly returns, while the geometric mean multiplies the growth factors and takes a root. The geometric mean is always lower when returns vary and is the correct measure for multi-year performance.

Is geometric mean the same as CAGR?

The Geometric Mean Rate Of Return Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Calculates geometric mean rate of return using Geometric mean = [(1+r₁)(1+r₂)…(1+rₙ)]^(1/n) − 1, in real time in your browser. There is no limit on usage, and it works on any device.

Why is the geometric mean lower than the arithmetic mean?

The geometric mean multiplies the period growth factors and takes the nth root, which penalizes volatility: a +50% year followed by a −50% year has an arithmetic average of 0% but a geometric mean of about −13.4%, matching the actual loss of value. Because real returns are volatile, the geometric mean is always at or below the arithmetic mean.

When should I use the geometric mean?

Use the geometric mean whenever returns compound over time — multi-year investment performance, fund comparisons, and any series where each period builds on the previous one. It gives the constant annual rate that reproduces the actual ending value. Use the arithmetic mean only for non-compounding contexts, like average monthly sales that do not build on each other.

Why does the simple average overstate my investment growth?

The arithmetic mean ignores compounding — it pretends each year's return applies to the original balance. The geometric mean multiplies the actual growth factors (1.12 × 0.92 × …) and takes the nth root, giving the one constant rate that reproduces your real ending balance. Whenever returns vary year to year, only the geometric figure is honest.

How are losing years handled in the formula?

A −8% year enters as a growth factor of 0.92, not as a negative number to average. That keeps the compounding chain intact: +50% then −50% is 1.5 × 0.5 = 0.75, a real 25% loss, and the geometric mean reports −13.4% — while the arithmetic mean wrongly claims 0%.

Is this the same thing as CAGR?

Yes — CAGR (compound annual growth rate) is exactly the geometric mean of the year-by-year growth factors, just under a different name. If you know only the starting value, ending value, and number of years, CAGR = (end/start)^(1/n) − 1 gives the same answer.

What inputs break the geometric mean?

A year worse than −100% is impossible — you can't lose more than everything. A −100% year makes the product zero and the geometric mean −100%, which is correct: the portfolio is gone and no later gain can revive it. The formula needs at least one period and all factors positive.

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