A strategy returns plus 50% and minus 40% in alternating periods. Give the arithmetic and geometric mean returns per period, explain the gap using Jensen's inequality, and state the approximation for the gap in terms of volatility.

A strategy returns plus 50% and minus 40% in alternating periods. Give the arithmetic and geometric mean returns per period, explain the gap using Jensen's inequality, and state the approximation for the gap in terms of volatility.

Approach: Compound two periods to get the geometric mean, compare it with the simple average, then write the gap as the difference between the expectation of a logarithm and the logarithm of an expectation.

The arithmetic mean is +5% and the geometric mean is -5.13% per period. Two periods compound to 1.5 * 0.6 = 0.9, so the geometric mean growth factor is sqrt(0.9) = 0.9487, that is -5.13% per period, while the arithmetic mean of +50% and -40% is +5%. Jensen's inequality explains the sign of the gap: the logarithm is concave, so E[log(1+R)] <= log(E[1+R]), and the geometric mean, which is the exponential of the left side, is at most the arithmetic mean. Expanding log(1+R) = R - R^2/2 + ... and taking expectations gives geometric mean approximately mu - sigma^2/2, the volatility drag. Here mu = 0.05 and the return variance is 0.45^2 = 0.2025, so the estimate is 0.05 - 0.101 = -5.1%, matching the exact figure. The practical consequence is that a strategy quoted on average returns with 45% period volatility loses money while appearing to make 5%.

Follow-up: At what volatility does a strategy with a 10% arithmetic mean have zero median compound growth, and how does that number bound position sizing?

Key concepts: jensen inequality, geometric mean, arithmetic mean, volatility drag.