A market neutral book has an 8% expected excess return and 16% volatility. What leverage does full Kelly imply, and why would a desk run a quarter of it?

A market neutral book has an 8% expected excess return and 16% volatility. What leverage does full Kelly imply, and why would a desk run a quarter of it?

Approach: Full Kelly leverage for a continuous return stream is the expected excess return over the variance. Then compare the growth lost against the estimation error in the numerator.

3.125. Full Kelly leverage is f* = mu/sigma^2 = 0.08/0.0256 = 3.125, which runs the book at 3.125*16% = 50% volatility for a log growth of mu^2/(2*sigma^2) = 12.5% a year. Quarter Kelly runs at 12.5% volatility and keeps 2*0.25 - 0.0625 = 43.75% of the growth. That trade is worth taking because f* is a ratio of two estimated numbers and mu is the badly estimated one. A Sharpe ratio of 0.5 measured over five years has a standard error near 1/sqrt(5) = 0.45, so the point estimate of f* can be wrong by a factor of two, and 2*f* has zero growth while 0.5*f* still has 75%. Fractional Kelly buys insurance against the side of that error that ends the fund.

Follow-up: If mu is known only up to a normal prior, what is the certainty equivalent optimal leverage?

Key concepts: kelly leverage, fractional kelly, sharpe ratio, estimation error.