Table of Contents
The geometric mean return is the constant rate per period that, compounded over every period, turns a starting value into the same ending value as a series of actual returns. Because returns compound, it is the standard way to average them across several periods.
Quick answer
Convert each return to a growth factor (1 + r), multiply the factors, take the nth root, and subtract 1:
g = ((1 + r1) × (1 + r2) × ... × (1 + rn))^(1/n) - 1
Write each return as a decimal, so 12% is 0.12 and -18% is -0.18. For five annual returns of 12%, 6%, -18%, 14%, and 9%, the geometric mean return is about 3.88% per year. The arithmetic mean of the same returns is 4.6%.
To check this example, switch the calculator below to returns mode and enter 12, 6, -18, 14, 9.
The geometric mean return formula
In product notation:
g = (Π(1 + r_i))^(1/n) - 1
Here r_i is the return for period i, Π means multiply all the terms, and n is the number of periods. The result is a return per period: annual inputs give an annual rate, and monthly inputs give a monthly rate.
The formula assumes that every period has the same length, that each return already excludes the effect of deposits and withdrawals (as a time-weighted return does), and that no return is below -100%.
Step by step
- Convert each percentage to a decimal.
- Add 1 to each decimal to get a growth factor.
- Multiply the growth factors together.
- Raise the product to the power 1/n.
- Subtract 1 and convert the result back to a percentage.
Worked example: five annual returns
Start with 10,000 in any currency and apply the five annual returns in order.
| Year | Return | Growth factor | Value at year end |
|---|---|---|---|
| Start | 10,000.00 | ||
| 1 | 12% | 1.12 | 11,200.00 |
| 2 | 6% | 1.06 | 11,872.00 |
| 3 | -18% | 0.82 | 9,735.04 |
| 4 | 14% | 1.14 | 11,097.95 |
| 5 | 9% | 1.09 | 12,096.76 |
The product of the growth factors is:
1.12 × 1.06 × 0.82 × 1.14 × 1.09 = 1.2096760704
Take the fifth root and subtract 1:
g = 1.2096760704^(1/5) - 1 ≈ 0.0388045, or about 3.8804% per year.
As a check, 10,000 × 1.0388045^5 ≈ 12,096.76, which is the same ending value as the table. Earning a steady 3.8804% every year would have produced the same result as the actual uneven sequence.
Why the arithmetic mean overstates growth
The arithmetic mean adds the returns and divides by n: (12 + 6 - 18 + 14 + 9) / 5 = 4.6%. Compounding 10,000 at 4.6% for five years gives about 12,521.56, roughly 424.80 more than the investment actually reached.
Gains and losses do not offset evenly. After an 18% loss, the value must rise about 22% just to recover. The geometric mean is never higher than the arithmetic mean, and the two are equal only when every return is identical. The arithmetic mean describes a typical single period. The geometric mean describes compound growth across the whole series. For a fuller comparison, see geometric mean vs arithmetic mean and which average should I use?
Cumulative return vs average return per period
Two different numbers describe the same series:
| Measure | Formula | Example result |
|---|---|---|
| Cumulative return | product of growth factors - 1 | 20.97% over five years |
| Geometric mean return | (product of growth factors)^(1/n) - 1 | 3.88% per year |
The two measures are linked by (1 + g)^n = 1 + cumulative return. Dividing the cumulative return by the number of periods does not give the geometric mean: 20.97% / 5 is about 4.19%, which overstates the compound annual rate. When the period returns are time-weighted, the product minus 1 is the cumulative time-weighted return for the whole span.
Annualizing monthly or quarterly results
If your inputs are monthly or quarterly returns, the geometric mean is a monthly or quarterly rate. Convert it to an annual rate by compounding:
- Monthly:
annual rate = (1 + g)^12 - 1 - Quarterly:
annual rate = (1 + g)^4 - 1 - Annual inputs: no conversion is needed.
| Input periods | Geometric mean per period | Calculation | Annualized rate |
|---|---|---|---|
| Monthly | 0.8% | (1.008)^12 - 1 | about 10.03% |
| Quarterly | 2.0% | (1.02)^4 - 1 | about 8.24% |
| Annual | 3.88% | none | 3.88% |
Do not multiply a monthly rate by 12. That ignores compounding and would give 9.6% instead of 10.03% in the monthly example. If you have a cumulative return over a span that is not a whole number of years, the annualized rate is (1 + cumulative return)^(1/years) - 1. Annualizing a short span assumes the same rate would continue for a full year. This is a calculation convention, not a prediction.
Negative returns and the -100% limit
Negative returns work in the formula as long as they are above -100%. A return of -18% becomes a growth factor of 0.82, which is positive, so the product and its root are well defined.
A return of exactly -100% means a complete loss. Its growth factor is 0, so the product is 0 and the geometric mean return is -100%, no matter how strong the other periods were. Once a value reaches zero, later percentage gains cannot restore it.
Returns below -100%, which can occur with leveraged or short positions, give a negative growth factor. The standard geometric mean is not defined for that case, and this calculator does not support it.
Cash flows, CAGR, and XIRR
The geometric mean of period returns measures how the investment itself performed, provided each period’s return was calculated to exclude deposits and withdrawals.
CAGR uses only a starting value, an ending value, and the number of years: CAGR = (end value / start value)^(1/years) - 1. With no deposits or withdrawals and whole-year periods, CAGR matches the geometric mean of the annual returns. In the example, (12,096.76 / 10,000)^(1/5) - 1 ≈ 3.88%. Try the CAGR calculator or read CAGR vs geometric mean.
XIRR is a money-weighted return. It requires every cash flow with its date, and it is found by solving for the rate that makes the net present value of those dated flows equal zero. Depending on the pattern of cash flows, there may be no solution or more than one. Because XIRR depends on the timing and size of each flow, it cannot be calculated from a CAGR figure alone. See CAGR vs XIRR and converting CAGR to XIRR for the additional information required.
What the geometric mean does not tell you
A single average hides the path. The example series and a steady 3.8804% per year end at the same value, but in the actual series the investment fell from 11,872.00 to 9,735.04 during year 3. The geometric mean says nothing about that drop or about volatility. Without cash flows, the order of returns does not change the ending value. With regular deposits or withdrawals, the order does matter.
When comparing investments, use the geometric mean alongside measures of risk, such as the standard deviation of returns or the largest peak-to-trough decline. All figures on this page are historical or hypothetical examples. A past geometric mean return is not a forecast of future returns.
Sources
- Wolfram MathWorld: Geometric Mean
- OpenStax, Introductory Business Statistics 2e, 2.5 Geometric Mean
- NIST Dataplot Reference Manual: Geometric Mean
To find the geometric mean of other values, use the geometric mean calculator. If you find an error on this page, please contact us.