L1

Kelly Criterion

Junior Quant · Brainteasers & Logic

bet f · WwealthW½ (Heads)½ (Tails)W + 2f·W= W(1 + 2f)W − f·W= W(1 − f)

Payoff = $2 net win on heads, lose stake on tails · find f that maximises long-run growth

Question

You are offered a bet: pay $1 to win $2 if a fair coin lands heads. You start with $10. Using the Kelly criterion, what fraction of your wealth should you bet per round to maximise long-run growth rate? Derive the criterion from first principles.