Arithmetic vs geometric return

Take +50% then -50%. The arithmetic mean is 0%, but 100 becomes 150 then 75, a geometric return of -13.4% per period. Nothing was stolen; compounding simply punishes variance.

A useful approximation is geometric = arithmetic - variance/2. A strategy averaging 12% a year with 30% volatility compounds at roughly 12 - 0.09/2 x 100 = 7.5%. Cut volatility to 15% with the same average and you compound at about 10.9%, which is why volatility-targeting can raise realised growth without raising expected return.

When someone quotes a strategy's average monthly return, ask for the compounded figure. The difference between the two is a direct measure of how rough the ride was.

See it drawn

Original diagrams for the ideas on this page. Illustrative, not real market data.

Compounding against a flat return. Two accounts start at $10,000 and earn 10% a year for fifteen years. Leaving the gains in means each year earns on a larger balance, so the curve bends away from the straight line and ends $16,772 higher.