Half-Kelly gives up little growth for much less risk

Question

Many traders bet "half-Kelly", half the fraction the Kelly criterion prescribes.

Using the continuous growth model, exactly how much growth do you give up at half-Kelly, and how much does your strategy volatility fall? Why is this a good trade?

Accepted Answer

Continuous log-growth as a function of fraction f is g(f) = f\mu - 	frac12 f^2\sigma^2, maximized at f^ = \mu/\sigma^2 with peak growth g^ = \frac{\mu^2}{2\sigma^2}. Now bet half of that, f = 	frac12 f^ = 	frac{\mu}{2\sigma^2}: g\!\left(	frac12 f^ight) = 	frac12 f^\mu - 	frac12\left(	frac12 f^ight)^2\sigma^2 = 	frac12\frac{\mu^2}{\sigma^2} - 	frac12\cdot	frac14\frac{\mu^2}{\sigma^2} = \left(	frac12 - 	frac18ight)\frac{\mu^2}{\sigma^2} = 	frac38\frac{\mu^2}{\sigma^2}. Compare to the maximum g^ = 	frac12\,\mu^2/\sigma^2: \frac{g(	frac12 f^)}{g^} = \frac{3/8}{1/2} = \frac34.…

Half-Kelly gives up little growth for much less risk

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