Compound returns: why an average can mislead
Compare arithmetic and geometric averages, then test when the sequence of returns really changes capital.
Returns across periods multiply because each percentage applies to a new balance. A simple average of percentages answers a different question and need not describe account growth.
Two different averages
The arithmetic average adds returns and divides by the number of periods. The geometric average finds the constant per-period return that would produce the same compounded ending value. Use periods of equal length.
Our example excludes fees, taxes, deposits and withdrawals. It explains mechanics, not a promised constant return. A contractual savings rate and fluctuating market returns do not represent identical conditions.
Plus 20% and minus 20% do not cancel
An initial 100 becomes 120 after +20%. Losing 20% of 120 then leaves 96. The arithmetic average is 0%, but capital has lost 4%. The geometric average is √0.96 − 1 ≈ −2.0204% per period.
Reversing those two periods without cash flows gives the same ending value: 100 × 0.8 × 1.2 = 96. With identical return factors, changing only their order does not change terminal wealth.
Our own teaching example; all figures are hypothetical.
100 × 1.2 × 0.8 = 96; √(96 / 100) − 1 ≈ −2.0204%
| Sequence | No withdrawals | Withdraw 10 after each period |
|---|---|---|
| +20%, −20% | 96 | 78 |
| −20%, +20% | 96 | 74 |
When sequence does matter
Now withdraw 10 at the end of each period, after its return. First +20%, then −20% gives (100 × 1.2 − 10) × 0.8 − 10 = 78. The reverse gives (100 × 0.8 − 10) × 1.2 − 10 = 74. Total withdrawals are 20 in both cases, but the balances differ.
When comparing histories, record cash-flow timing as well as percentages. A final value also conceals drawdowns along the way. The next article explains why recovering from a loss requires a larger percentage gain.
Does +10% followed by −10% restore the starting balance?
No: 100 × 1.1 × 0.9 = 99. The loss is 1%, although the arithmetic average is zero.
Sources
Sources checked: 2026-10-20
This material and its translations were prepared with AI assistance. Educational calculations were checked programmatically. Sources explain concepts; examples and files were created for this article.
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Next step
Check how much growth is needed to recover from a drawdown.
Drawdown and recovery: why −20% needs +25% →