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Optimal stake fraction

Market makingHard

A bettor believes an event has probability $0.32$ , while a bookmaker offers decimal odds of $3.4$ on it (a winning stake of $1$ returns $3.4$ in total). The bettor stakes a fraction $f$ of the bankroll on a single bet, choosing $f$ to maximize the expected logarithm of the bankroll after the bet. What is $f$ ? Give an exact fraction.

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Hint

Net odds are $b = 2.4$ ; expected log growth is maximized where its derivative in $f$ vanishes.

Solution

The net win per unit staked is $b = 3.4 - 1 = 2.4$ , the win probability is $p = 0.32$ and the loss probability is $q = 0.68$ .
The derivative condition gives $f = (b \cdot p - q) \div b = (0.768 - 0.68) \div 2.4$ .
So $f = 0.088 \div 2.4 = \frac{11}{300}$ .

Answer: $\frac{11}{300}$

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