Lessons · Risk

The Kelly criterion: how much to bet

Having an edge isn't enough. Bet too much and you go broke anyway; bet too little and you leave growth on the table. The Kelly criterion gives the bet size that grows your bankroll fastest over the long run.

The formula

For a bet that wins with probability p, loses with probability q = 1 − p, and pays b to 1 (you win $b per $1 staked), the Kelly fraction of your bankroll is

f* = (b·p − q) / b

For an even-money bet (b = 1) this simplifies to f* = p − q = 2p − 1. A coin that wins 60% of the time at even money: bet 20% of your bankroll each flip.

If b·p − q ≤ 0, the bet has no edge and Kelly says bet nothing.

Why it works

Repeated bets multiply your wealth. After many bets, what matters is the average of the logarithm of the growth factor per bet. Kelly is the fraction that maximizes

g(f) = p · ln(1 + b·f) + q · ln(1 − f)

Setting the derivative to zero gives f* = (bp − q)/b. Because it maximizes long-run growth, a Kelly bettor eventually overtakes any strategy that bets a different fixed fraction.

Over-betting is worse than under-betting

Growth falls off steeply above the Kelly fraction. At even money with p = 0.6, betting 20% grows wealth about 2% per bet in log terms. Betting 40% grows it at roughly zero, and betting more than that shrinks it, even though every single bet has a positive expected value. That's the lesson the Betting Table game teaches: positive expected value is not the same as positive growth.

Fractional Kelly

In real trading, your p and b are estimates. Because over-betting is so costly, traders often bet half Kelly. It gives up a little growth for much smaller swings, and it protects you if your edge is smaller than you think.

Worked example

A bet pays 2 to 1 and wins 40% of the time. Expected value per $1: 0.4 × 2 − 0.6 = +$0.20. Kelly: f* = (2 × 0.4 − 0.6)/2 = 0.10, so bet 10% of your bankroll.

Practice

Play it: the Betting Table game · Next: Making markets