The Kelly criterion

Flat staking is simple, but it ignores a valuable piece of information: some bets have more value than others. The Kelly criterion (John L. Kelly Jr., 1956) tells you what fraction of your bankroll to stake to maximise the long-term growth of your capital.

The formula

$f^*$ is the fraction of your current bankroll to stake. If $f^* \le 0$, you don't bet.

The intuition

Kelly maximises the logarithmic growth of capital, i.e. the average compounded growth rate. It has two remarkable properties:

• In the long run, it grows the bankroll faster than any other proportional staking strategy.
• Since it stakes a fraction of the current bankroll, it never leads to total ruin in theory: when you lose, your stakes automatically shrink.

Why full Kelly is dangerous

In practice, almost nobody stakes full Kelly:

1. Extreme volatility. With full Kelly, there's a 50% chance of seeing your bankroll halved at some point, and about a 10% chance of it falling to a tenth of its starting level.
2. Your $p$ is an estimate. If you overestimate your probability, you stake too much. And staking twice Kelly brings expected growth down to almost zero; beyond that, it turns negative.
3. Asymmetry. Staking a little less than Kelly costs little; staking a little more costs a lot.

The solution: fractional Kelly

You stake a fraction of $f^*$, typically ¼ or ½ Kelly.

½ Kelly keeps about three quarters of the growth with half the volatility. It's an excellent compromise.