Kelly sizing a 60/40 coin at even money

Question

You have a \10{,}000 bankroll and can repeatedly bet on a coin that lands heads with probability 0.6, at even money (win 1 for each 1 staked). You can bet any fraction of your bankroll each flip, as many flips as you like. How much do you bet per flip, and why not more?

Accepted Answer

Because winnings compound multiplicatively, the right objective is to maximize the expected growth rate, equivalently E[\log(	ext{wealth})]. Betting fraction f at even money: g(f) = 0.6\ln(1+f) + 0.4\ln(1-f). Set g'(f) = \frac{0.6}{1+f} - \frac{0.4}{1-f} = 0: 0.6(1-f) = 0.4(1+f) \implies 0.2 = f. Bet f^ = 20\% of bankroll, \2{,}000 on the first flip, and re-size to 20% of the current bankroll every flip. (General even-money Kelly: f^ = p - q = 0.6 - 0.4.) The achieved growth rate is g(0.2) = 0.6\ln 1.2 + 0.4\ln 0.8 \approx 0.6(0.1823) - 0.4(0.2231) \approx 0.0201, about 2.0\% of bankroll per…

Kelly sizing a 60/40 coin at even money

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