Kelly on a 3-to-1 long shot

Question

You can repeatedly bet on an event with win probability $p = 0.4$ . A win pays net odds $b = 3$ (you win $\$ 3 $for every$ $1$ staked); a loss forfeits the stake. You may bet any fraction of your bankroll.

What fraction should you bet, and what long-run growth rate does it achieve?

Accepted Answer

Maximize expected log growth of betting fraction f: g(f) = p\,\ln(1 + bf) + q\,\ln(1 - f), \qquad q = 1 - p = 0.6. The general Kelly optimum is f^ = \frac{pb - q}{b} = p - \frac{q}{b}. Plug in p = 0.4,\ q = 0.6,\ b = 3: f^ = \frac{0.4 \cdot 3 - 0.6}{3} = \frac{1.2 - 0.6}{3} = \frac{0.6}{3} = 0.20. Bet 20\% of bankroll. The numerator pb - q = 1.2 - 0.6 = 0.6 is your edge per dollar staked; dividing by the odds b = 3 scales that edge into the safe fraction. Growth rate. At f^ = 0.20: g(0.20) = 0.4\ln(1 + 0.6) + 0.6\ln(0.8) = 0.4(0.4700) + 0.6(-0.2231) = 0.1880 - 0.1339 = 0.0541, about 5.4\% per…

Kelly on a 3-to-1 long shot

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