A bet pays 2 to 1 and wins with probability 0.4. What fraction of the bankroll does full Kelly stake?

A bet pays 2 to 1 and wins with probability 0.4. What fraction of the bankroll does full Kelly stake?

Approach: Maximise the expectation of the log of terminal wealth over the stake, which gives the standard edge over odds formula. Then evaluate the growth rate at that stake.

0.1. Kelly stakes f = (b*p - q)/b with b = 2, p = 0.4 and q = 0.6, so f = (0.8 - 0.6)/2 = 0.1 of the bankroll. The expected log growth is 0.4*ln(1.2) + 0.6*ln(0.9) = 0.4*0.18232 - 0.6*0.10536 = 0.00971, about 0.97% per bet, so ln(2)/0.00971 = 71 bets double the money on average. The edge is 0.4*2 - 0.6 = 0.2 per unit staked, and the growth rate is roughly edge^2/(2*variance) for small stakes, which is why a 20% edge converts into only a 1% growth rate.

Follow-up: How does the Kelly stake change if the payout is quoted net of a 5% commission on winnings?

Key concepts: kelly criterion, expected log growth, bankroll, edge.