A strategy returns +50% or -40% with equal probability each year, independently across years. Compute the expected arithmetic return, the expected log return, and the median wealth multiple after 10 years, and say which figure you quote to an investor.

A strategy returns +50% or -40% with equal probability each year, independently across years. Compute the expected arithmetic return, the expected log return, and the median wealth multiple after 10 years, and say which figure you quote to an investor.

Approach: Average the return and the log return separately, then find the median path by taking the most likely count of up years.

0.59049. The arithmetic mean return is 0.5*0.50 + 0.5*(-0.40) = +0.05, so expected wealth grows 5% a year and E[W_10] = 1.05^10 = 1.629. Growth of an actual path is driven by the log return, whose mean is 0.5*log(1.5) + 0.5*log(0.6) = -0.0527, so the geometric mean is e^{-0.0527} - 1 = -5.13% a year. The median outcome after 10 years is 5 up years and 5 down years, giving (1.5*0.6)^5 = 0.9^5 = 0.59049, a 41% loss. Jensen's inequality is why the two disagree: log is concave, so E[log(1 + r)] < log(1 + E[r]), and with a spread of 0.45 either side the gap is approximately sigma^2/2 = 0.101. That gap is volatility drag. Quote the geometric figure, because the arithmetic mean is carried by a few extreme paths while the typical path loses money.

Follow-up: What fraction of full Kelly sizing would you have to apply to this bet to make the median path grow, and what does that fraction do to the arithmetic mean?

Key concepts: Jensen's inequality, volatility drag, geometric mean, arithmetic mean.