Average Return Calculator
Compare the arithmetic average of a series of annual returns against the geometric average (CAGR) — they are rarely the same number, and the gap matters.
How to use this calculator
- 1List each year's return as a percentage, one per line or separated by commas.
- 2Add a starting dollar amount to see the concrete difference in ending value.
How the calculation works
Arithmetic mean = Σr / n. Geometric mean = [Π(1 + rᵢ)]^(1/n) − 1- Σr / n
- The simple average of the return percentages
- Π(1 + rᵢ)
- The product of each year's growth factor — how a dollar actually compounded
Only the geometric mean answers "what constant annual return would have produced the same ending value" — which is what most people actually mean by "average return." The arithmetic mean answers a different, less useful question.
The gap between the two grows with volatility: a wild sequence of +50%/-50% has an arithmetic average of 0% but actually loses 25% of the original value ($100 → $150 → $75), a geometric average of about -13.4%.
Worked example
Returns of +20%, -10%, +15%, -5%, +25%
- 1.Arithmetic mean: (20 − 10 + 15 − 5 + 25) / 5 = 45 / 5 = 9%.
- 2.Compound growth factor: 1.20 × 0.90 × 1.15 × 0.95 × 1.25 = 1.474875.
- 3.Geometric mean: 1.474875^(1/5) − 1 ≈ 8.08%.
- 4.$10,000 actually grew to $14,748.75, not the $15,386.24 the arithmetic average would suggest.
Result: Geometric 8.08% vs. arithmetic 9.00%
Two different questions, two different averages
The arithmetic mean answers "what is the simple average of these yearly percentages" — add them up, divide by the count. The geometric mean answers a completely different question: "what single constant annual rate, compounded every year, would have turned the starting amount into the actual ending amount." People asking about investment performance almost always mean the second question, even when the figure quoted to them is the first.
Why compounding breaks the simple average
Gains and losses are not symmetric once they compound, because a loss shrinks the base a later gain has to work from. A 50% loss requires a 100% gain just to get back to even — not another 50% — because the 50% gain is calculated on the smaller, already-reduced balance. The arithmetic mean ignores this entirely, treating every year's percentage as equally weighted regardless of order or the size of the balance it applied to, which is exactly why it systematically overstates the real compound result whenever returns vary from year to year.
Where this gap shows up in the real world
The distinction is not an academic technicality — it changes real-world comparisons in several common situations.
- Investment performance marketing — a fund reporting an "average annual return" without specifying which average can make historical performance look better than what an investor who actually held the fund the entire period experienced — which is why regulators generally require standardized, geometric-based reporting for advertised fund returns.
- Backtested trading strategies — a strategy report built on a simple average of period returns can look far more attractive than the same strategy's true compound result, especially if the strategy has occasional large losses.
- Comparing a volatile stock to a steadier index — two investments with the same arithmetic average return over a period rarely end up worth the same amount — the more volatile one almost always compounds to less, because it has more, and larger, drawdowns to recover from along the way.
- Retirement withdrawal planning — the order returns arrive in matters enormously once withdrawals begin — a market downturn early in retirement does more lasting damage than the same downturn later, a phenomenon usually called sequence-of-returns risk, which an average-return figure alone says nothing about.
A rule of thumb for the size of the gap
The gap between the two averages grows with how much the individual yearly returns vary — and it grows roughly with the square of that variation, not in direct proportion to it. Practically, that means doubling the year-to-year swings in a set of returns does not just double the gap between arithmetic and geometric averages — it roughly quadruples it. A steady, low-volatility set of returns will show the two averages sitting close together; a wild, high-volatility set will show them pulling noticeably apart, exactly as the earlier +50%/−50% example in the formula above illustrates.
What this assumes, and where it stops
Assumptions
- Returns compound annually with no withdrawals or additional contributions between periods.
Limitations
- Does not account for fees, taxes, or the effect of contributions and withdrawals made during the period, all of which change the real-world result.
Common questions
Which average should a fund actually report?
Regulators generally require the geometric average (as "annualized return" or CAGR) precisely because the arithmetic average can make historical performance look better than what an investor who held the whole period actually experienced. If you see "average annual return" without specification, it is worth checking which one is meant.
Why is the geometric mean always lower?
Losses and gains are not symmetric once you compound them — a 50% loss needs a 100% gain just to break even, not another 50%. The arithmetic mean treats every percentage as equally weighted regardless of order or compounding, so it systematically overstates the real result whenever returns vary.
Formula and content last reviewed on .
Results are estimates for information only, not professional advice.
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