Continuous Kelly, leverage from a Sharpe ratio

Question

A strategy earns an expected excess return of $\mu = 8\%$ per year with volatility $\sigma = 20\%$ per year, and returns are roughly normal. You can lever it, choosing a multiple $f$ of your capital to deploy.

What leverage maximizes long-run growth, and what growth rate does it achieve?

Accepted Answer

Compounding continuously, leverage f delivers a long-run growth rate g(f) = f\mu - 	frac12 f^2 \sigma^2, the expected return minus the volatility drag. Maximize: g'(f) = \mu - f\sigma^2 = 0, so f^ = \frac{\mu}{\sigma^2} = \frac{0.08}{0.20^2} = \frac{0.08}{0.04} = 2.0. Run the strategy at 2	imes leverage. The optimal growth rate is g(f^) = \frac{\mu^2}{2\sigma^2} = \frac12\left(\frac{\mu}{\sigma}ight)^2 = 	frac12(	ext{Sharpe})^2 = 	frac12(0.4)^2 = 0.08, that is, 8\% per year of compounded growth, and continuous Kelly growth is always half the squared Sharpe ratio. Half Kelly. At f = 1 (…

Continuous Kelly, leverage from a Sharpe ratio

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