The Kelly fraction for uneven odds

Question

You are offered a repeatable bet: with probability $p = 0.40$ you win 2 times your stake, and with probability $0.60$ you lose your stake. You may bet any fraction of your bankroll each round.

Derive the growth-maximizing bet fraction, and confirm the bet is worth taking.

Accepted Answer

First check there is an edge at all. Expected profit per \1 staked is 0.40 	imes 2 - 0.60 	imes 1 = 0.80 - 0.60 = 0.20 0, so the bet is favorable. Bet a fraction f of your bankroll each round. A win multiplies wealth by (1 + bf) with b = 2; a loss multiplies it by (1 - f). The long-run growth rate of log-wealth is g(f) = p\ln(1 + bf) + (1-p)\ln(1 - f) = 0.4\ln(1 + 2f) + 0.6\ln(1 - f). Differentiate and set to zero: g'(f) = \frac{pb}{1 + bf} - \frac{1-p}{1 - f} = 0 \implies f^ = \frac{bp - (1-p)}{b} = p - \frac{1-p}{b}. Plugging in: f^ = \dfrac{2(0.4) - 0.6}{2} = \dfrac{0.8 - 0.6}{2} = \dfra…

The Kelly fraction for uneven odds

Your answer

More Market Making questions