Geometric Mean Return

The calculation begins by converting each periodic return into a wealth relative, that is, one plus the return expressed as a decimal. A return of eight percent becomes 1.08; a loss of twelve percent becomes 0.88. These relatives are multiplied together to give the total growth factor for the whole horizon, the factor by which one unit of currency has grown. Taking the nth root of that product, where n is the number of periods, strips the compounding back to a single constant per-period rate, and subtracting one restores it to return form.

Because only the starting and ending wealth matter to the product, the ordering of the returns is irrelevant: reversing the sequence produces the same compound rate. This is why the measure is identical to the compound annual growth rate when the periods are years, and why it can be computed directly from beginning and ending values when the intermediate path is unknown.

For any series containing at least two different returns, the geometric mean is strictly below the arithmetic mean; the two coincide only when every periodic return is identical. The gap widens with dispersion, which is why the shortfall is described as volatility drag or variance drain. A useful approximation at Level I is that the geometric mean is roughly the arithmetic mean minus one half of the variance of the returns, so a strategy with high realised volatility surrenders a meaningful part of its average return to compounding asymmetry.

The intuition is arithmetic rather than statistical: a loss requires a proportionally larger gain to recover, since the gain applies to a smaller base. Dispersion around a given average therefore erodes terminal wealth, and only the geometric mean records that erosion.

Fund factsheets, index provider documentation and GIPS-compliant presentations quote annualised returns that are geometric by construction. The time-weighted rate of return chains sub-period holding period returns multiplicatively, then annualises the result by taking the appropriate root, so that external cash flows the manager does not control cannot distort the reported figure. This is the standard for judging manager skill.

The money-weighted return, or internal rate of return, answers a different question: it is the discount rate equating all cash flows to zero and therefore reflects the timing and size of investor contributions. An investor who adds capital just before a strong period will report a money-weighted return above the fund's geometric mean. Candidates should read a quoted "annualised return" as a compound rate unless the source explicitly states otherwise.

Expect to be asked to convert a series of annual returns into a compound annual figure, to back out an annualised rate from beginning and ending portfolio values, or to compare geometric and arithmetic results and identify which describes realised performance. On the calculator, the fastest route is to enter the total growth factor, raise it to the power of one divided by the number of periods, and subtract one.

Arithmetic mean is not wrong, merely answers a different question. It is the unbiased estimate of a single-period expected return drawn at random from the distribution, and so is the appropriate input when forecasting one period ahead or building an expected return for a risk premium estimate. Geometric mean describes what actually happened to invested capital across multiple periods. Note also that a return of negative one hundred percent produces a zero wealth relative, forcing the entire compound rate to negative one hundred percent.