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Geometric mean from linked return factors

For n equal periods with positive growth factors, the geometric mean is the nth root of their product minus one.

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Match the endpoint under a constant rate

A constant per-period rate g over n intervals should satisfy (1+g)^n = Π(1+rt). Taking the nth root produces an endpoint-equivalent rate. It is a historical summary of the supplied path, not a statement that each actual interval earned g.

Convert a factor to a rate

The product is a growth factor. Take its root first, then subtract one. Taking a root of the cumulative percentage directly mixes a rate with a factor.

g=[∏t=1n(1+rt)]1/n−1g=\left[\prod_{t=1}^{n}(1+r_t)\right]^{1/n}-1

Periods must have equal duration for this periodic mean interpretation.

Use the covered horizon

If n equal subperiods represent m periods per year, the covered horizon is n/m years. Annual growth is the product raised to m/n. A periodic geometric mean equals an annual rate only when the periods themselves are years.

Keep zeros and negative bases explicit

For nonnegative wealth, a terminal complete loss gives zero endpoint growth and a −100% simple equivalent rate. Its logarithm is undefined. A return below −100% falls outside this no-borrowing wealth model and needs a different interpretation.

Further references