The Kelly criterion: how much to bet when the odds are in your favour

Finding a bet with positive expected value is half the problem. The other half is how much to stake. Bet too little and you waste the edge; bet too much and a run of losses ruins you. The Kelly criterion gives the stake that maximises long-run growth.

The formula

For a bet that pays b to 1 with win probability p and loss probability q = 1 − p:

f = (bp − q) / b

where f is the fraction of your bankroll to stake.

For an even-money bet (b = 1) this becomes f = p − q.

Examples

• Even money, 60% to win: f = 0.6 − 0.4 = 20%.
• Even money, 55% to win: f = 10%.
• Pays 2 to 1, 40% to win: f = (2 × 0.4 − 0.6) / 2 = 10%.
• Pays 3 to 1, 25% to win: f = (0.75 − 0.75) / 3 = 0. No edge, no bet.

If the formula gives a negative number, do not bet.

Why it works

Kelly maximises the expected logarithm of wealth, which is the same as maximising the long-run growth rate. A bet that risks everything has the highest expected value in one round and certain ruin over many.

Why traders bet less

• You do not know p exactly. Overestimating your edge with full Kelly means overbetting, which is costly.
• Swings are large. Full Kelly produces deep drawdowns.
• Half Kelly gives about three quarters of the growth with far smaller swings, so it is a common practical choice.

The coin game

A well-known experiment gives people a coin that lands heads 60% of the time and lets them bet on it repeatedly. Many go bust or bet erratically. The Kelly answer is to stake 20% of the current bankroll on heads every time.

In interviews

• "You have a 60% coin and 100. How much do you bet?" Around 20, and less if unsure of the 60%.
• "Why not bet everything?" Because one loss ends the game.
• "What if you can only play ten times?" Kelly is still a good guide; with few rounds and a cap on winnings, you would also consider the payout limit.

Related: expected value questions.