A contract costs 0.4 dollars and pays 1 dollar if an event happens, otherwise nothing. You believe the event has probability 1/2. What fraction of your bankroll should you spend on contracts to maximise the expected logarithm of wealth?

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A contract costs $0.4$ dollars and pays $1$ dollar if an event happens, otherwise nothing. You believe the event has probability $\frac{1}{2}$ . What fraction of your bankroll should you spend on contracts to maximise the expected logarithm of wealth?

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Hint

Turn the price into net odds per dollar spent.

Solution

Per dollar spent, a success returns a profit of $\frac{1 - 0.4}{0.4} = \frac{3}{2}$ , so $b = \frac{3}{2}$ , $p = \frac{1}{2}$ and $q = \frac{1}{2}$ .
The numerator is $b p - q = \frac{3}{4} - \frac{1}{2} = \frac{1}{4}$ .
Dividing by $b$ gives $\frac{\frac{1}{4}}{\frac{3}{2}} = \frac{1}{6}$ .

Answer: $\frac{1}{6}$

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