Kelly sizing on an asymmetric bet

MediumMarket Making & Betting Games~5m

A repeated bet wins with probability $p = 0.4$ . A win pays +3 dollars per $1 staked (net); a loss forfeits the entire $1 stake. You reinvest a fixed fraction $f$ of your bankroll each time. What fraction $f$ maximizes long-run growth? Give a decimal.

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Maximize the expected log of wealth: $G(f)=0.4\ln(1+3f)+0.6\ln(1-f)$ - the even-money $2p-1$ shortcut does not apply.
Set $G'(f)=0$ : $\frac{1.2}{1+3f}=\frac{0.6}{1-f}$ .

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0.2 (± 0.005)

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Kelly sizing on an asymmetric bet

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