Continuous Kelly, sizing a strategy by its Sharpe

Question

A strategy earns an expected excess return $\mu$ per period with variance $\sigma^2$ (roughly normal returns). You choose a leverage $f$ : hold $f$ dollars of the strategy per dollar of capital.

What leverage maximizes long-run growth, and what growth rate does it earn? Evaluate for $\mu = 8\%$ and $\sigma = 16\%$ per year.

Accepted Answer

Scaling the strategy by f gives per-period mean f\mu and variance f^2\sigma^2. Expected log growth is approximately g(f) \approx f\mu - 	frac{1}{2}f^2\sigma^2. This is the continuous-time Kelly objective: expected return penalized by half its variance (the drag from compounding volatility). Differentiate and set to zero: g'(f) = \mu - f\sigma^2 = 0 \;\implies\; \boxed{f^ = \frac{\mu}{\sigma^2}.} The optimal growth rate is g(f^) = \frac{\mu^2}{\sigma^2} - \frac{1}{2}\frac{\mu^2}{\sigma^2} = \frac{\mu^2}{2\sigma^2} = \frac{1}{2}\left(\frac{\mu}{\sigma}ight)^2 = 	frac{1}{2}\,	ext{SR}^2, exa…

Continuous Kelly, sizing a strategy by its Sharpe

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