The Kelly criterion
Bet too little and you grow slowly. Bet too much and you go broke. Find the fraction in between.
Win this many dollars for every dollar staked. Lose the stake otherwise.
200 bettors staking 50% each time, on a log scale where each step up is ten times more money. Dashed: the starting amount. From 0.000001× to 533.0M×.
Kelly says bet 20.0% of your wealth, for a typical growth of 2.03% per bet. You're betting 50%, beyond the 39% line, so even with the odds in your favour your typical wealth shrinks (-3.34% per bet).
Growth rate for every bet size
Long-run growth per bet, g(f). It peaks at Kelly and falls back to zero at about twice Kelly.
Same luck, four strategies
Median wealth over time, log scale. Every strategy faces the same coin flips.
| Strategy | Bet size | Median result | Ended down | Lost 90% at some point |
|---|---|---|---|---|
| Half Kelly | 10.0% | 91× | 1% | 0% |
| Kelly | 20.0% | 420× | 8% | 10% |
| Double Kelly | 40.0% | 0.48× | 51% | 75% |
| Your fraction | 50.0% | 0.00004× | 87% | 92% |
The advanced version: Kelly for a stock market
For an investment with continuous returns, the Kelly fraction becomes excess return divided by variance. Above 100% means borrowing to invest, which is why real investors who use Kelly almost always bet a fraction of it: the inputs are estimates, and overestimating the edge pushes you past the cliff.
Try this
- Set a 60% chance at even money. Kelly says 20%. Now bet 40% and watch the median go nowhere.
- Press new luck a few times with full Kelly. How often does it fall 90% along the way? Is that a ride you could stomach?
- Drop the chance to 50% at even money. What does Kelly tell you to bet when there is no edge?
The math
- the Kelly fraction of wealth to bet
- chances of winning and losing
- net payout per dollar staked
- long-run log growth per bet; for stocks, f* = (μ − r) / σ²