Kelly for a continuous edge, f = μ/σ²

Question

A strategy has an expected excess return of $\mu = 8\%$ per year with volatility $\sigma = 20\%$ per year, and returns are roughly Gaussian.

What leverage (fraction of bankroll) does the Kelly criterion prescribe, and what long-run growth rate does it achieve?

Accepted Answer

For a continuously-compounded, near-Gaussian return, log-growth as a function of the invested fraction f is g(f) = f\mu - 	frac12 f^2\sigma^2. The first term is the edge you earn per unit of exposure; the second is the "variance drag" that compounding imposes. 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. Kelly wants 2× leverage, put twice your bankroll into the strategy. Plug back in for the growth rate: g(f^) = \frac{\mu^2}{2\sigma^2} = \frac{0.08^2}{2 	imes 0.04} = \frac{0.0064}{0.08} = 0.08 = 8\%	ext{/yr}. Two sanit…

Kelly for a continuous edge, f = μ/σ²

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