Arithmetic and Geometric Returns: Two Different Averages

Choose the question you want to answer

For n period returns, the arithmetic mean is their sum divided by n. The geometric mean is the nth root of the product of their growth factors, minus one. A return of +10% has growth factor 1.10; a return of −10% has factor 0.90. Multiply factors to reconstruct a reinvested historical path with no external cash flows.

PerformanceAnalytics documents geometric chaining and arithmetic annualization choices. A report must identify which convention it uses rather than calling both numbers an unqualified average. Neither a historical arithmetic mean nor a geometric mean alone is a guaranteed future growth rate.

See why equal gains and losses do not cancel

Start with 10,000 monetary units. A +20% period produces 12,000. A following −20% period leaves 9,600. The arithmetic mean of +20% and −20% is 0%, but the compound growth factor is 1.20 × 0.80 = 0.96, giving a total return of −4%.

The per-period geometric mean is the square root of 0.96 minus one, approximately −2.02041%. Applying that constant rate twice reproduces 9,600. This is an original arithmetic example; it does not assume that markets follow a particular distribution or that the historical sequence will repeat.

Keep periods and scale aligned

For a second example, returns of +5%, +5% and +5% have both arithmetic and geometric means of 5% per period. The three-period cumulative return is 1.05 cubed minus one, or 15.7625%, not 15%. When all period returns match, the two averages coincide, but cumulative growth still compounds.

A monthly geometric mean should not be labeled an annual rate without an explicit transformation. Likewise, irregular holding-period returns should not simply be treated as equally timed monthly observations. If you compare daily source-account returns with weekly follower-account returns, first construct a common reporting frequency and timestamp convention.

Separate return accounting from deposits

An account that rises from 10,000 to 15,000 after a 5,000 deposit has not necessarily earned 50%. Raw equity changes combine investment outcomes and cash flows. Compute a consistent return series that accounts for deposits and withdrawals before averaging or compounding it. Preserve those cash flows in the underlying ledger so the adjustment remains reviewable.

The ordinary real-valued geometric formula also needs valid growth factors. A total loss produces a factor of zero and a terminal value of zero; returns below −100% create a domain problem for this simple wealth-compounding interpretation. Negative starting equity needs a different analysis. Report sample length, costs and the worst path decline alongside the mean, because equal endpoint growth can conceal very different periods of risk and recovery.