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Kelly sizing on an asymmetric bet

MediumMarket Making & Betting Games~5m

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

Show hints (2)+
  1. Maximize the expected log of wealth: G(f)=0.4ln⁡(1+3f)+0.6ln⁡(1−f)G(f)=0.4\ln(1+3f)+0.6\ln(1-f) — the even-money 2p−12p-1 shortcut does not apply.
  2. Set G′(f)=0G'(f)=0: 1.21+3f=0.61−f\frac{1.2}{1+3f}=\frac{0.6}{1-f}.

Answer

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0.2 (± 0.005)

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Asked at: Jane Street, Two Sigma

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