For each dollar staked, a bet returns a profit of 1 dollar with probability 1/2, and with probability 1/2 loses only half the stake (a loss of 1/2 dollar). You stake a fraction f of your bankroll each time, with 0 < f < 1, to maximise the expected logarithm of wealth. What is f?

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For each dollar staked, a bet returns a profit of $1$ dollar with probability $\frac{1}{2}$ , and with probability $\frac{1}{2}$ loses only half the stake (a loss of $\frac{1}{2}$ dollar). You stake a fraction $f$ of your bankroll each time, with $0 < f < 1$ , to maximise the expected logarithm of wealth. What is $f$ ?

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Hint

Differentiate the expected log growth and set it to zero.

Solution

Wealth is multiplied by $1 + f$ or by $1 - \frac{f}{2}$ , each with probability $\frac{1}{2}$ .
The derivative of the expected log growth is $\frac{1}{2 ( 1 + f )} - \frac{\frac{1}{4}}{1 - \frac{f}{2}} = \frac{1}{2 + 2 f} - \frac{1}{4 - 2 f}$ .
Setting it to zero gives $2 + 2 f = 4 - 2 f$ , so $f = \frac{1}{2}$ .

Answer: $\frac{1}{2}$

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