A fund earns a 10% arithmetic expected return with 40% volatility and borrows at zero. At what leverage does the expected compound growth rate fall to zero, and what is the unlevered geometric return?

A fund earns a 10% arithmetic expected return with 40% volatility and borrows at zero. At what leverage does the expected compound growth rate fall to zero, and what is the unlevered geometric return?

Approach: Expected log growth at leverage f is f*mu minus half of f^2*sigma^2. Solve for the root and evaluate the unlevered case.

1.25. Expected log growth at leverage f is f*mu - f^2*sigma^2/2 = 0.10*f - 0.08*f^2, which is zero at f = 0.10/0.08 = 1.25 and is maximised at f* = 0.625. Unlevered the geometric return is mu - sigma^2/2 = 0.10 - 0.08 = 2% a year against a 10% arithmetic mean, so 8 points a year are lost to variance drag. At f = 1.25 the arithmetic return is 12.5% and the drag is 0.5*(1.25*0.4)^2 = 12.5%, so they cancel and the median path is flat while the mean keeps rising, the gap being carried by an ever thinner right tail. Any leverage above 1.25 compounds to zero with probability one even though the expected wealth grows.

Follow-up: At what leverage is the median 10 year outcome maximised if the borrowing rate is 3% rather than zero?

Key concepts: variance drag, geometric return, log growth, leverage.