Optional reading. The maths behind Kelly, why betting more backfires and where it stops working in real life.
Skip it and just play to see for yourself if you prefer.
The idea
When the odds are in your favour Kelly answers one question. What share of your money should you bet each time so it grows fastest over many bets?
A worked example
Take a coin that lands on heads 60% of the time. The simulator starts there, and you can change the odds whenever you like. You bet on heads and a win pays the same amount you bet (called even money). You win 60% of the time and lose 40%. So your edge is 60 - 40 = 20 points.
At even money the Kelly bet is simply your edge: 20% of your money on every flip. With $10,000 that is $2,000 on the first flip. Win and you have $12,000, so the next bet is $2,400. Lose and you have $8,000, so the next bet is $1,600. The bet always stays at 20% of whatever you have.
A bigger edge means a bigger bet. No edge means no bet:
| Chance of winning | Kelly bet (even money) |
|---|
| 51% | 2% |
| 55% | 10% |
| 60% (the starting coin) | 20% |
| 65% | 30% |
| 70% | 40% |
| 50% or less | 0% |
Why not bet more?
Because every loss shrinks the money your next win can grow. Start with $100 and bet 40% each flip. Win and you have $140. Lose the next one and you're down to $84. One win and one loss have cost you 16%. The bigger the bet the bigger that drag.
With a 60% coin betting 20% grows your money fastest, about 2% per flip on average. Bet 30% and growth slows. Bet more than about 39% and your money shrinks over time even though you still win 60% of the flips. The losses now outweigh your edge.
Why many use half Kelly or less
Half Kelly (10% with a 60% coin) keeps about three-quarters of the growth with much smaller swings. Quarter Kelly (5%) gives up more growth for an even smoother ride. Both keep you far away from bust and protect you if you've overestimated your edge, which is easy to do in real life.
The limits of this game
Here you know the exact odds. In real life you never do. You estimate your edge and estimates are usually too hopeful. Getting it wrong is costly. Say you think the coin lands on heads 60% of the time but it is really 55%. You bet 20%, the Kelly bet for 60%. That's double the right bet for 55% and your money stops growing at all. Betting less than full Kelly protects you against that mistake and against risk of ruin.
Kelly also aims for the fastest growth over a very long run, and the ride can be rough. At full Kelly with a 60% coin there is close to a 1-in-2 chance of falling to half your starting money at some point.
Kelly in investing
Investors can ask the same question: how much of my portfolio should one idea get? In the coin game you know the odds. In investing you have to estimate them yourself, and you will often be wrong.
There is a version of the Kelly formula for investing. With realistic numbers for a single stock it often says to bet more than all your money. For example, a stock you expect to beat cash by 6% a year that swings about 20% a year comes out at 150% of your money. That would mean borrowing. It's one reason investors who use Kelly tend to bet a fraction of what it says.
The coin flip is a simple model. Real investing adds risks it doesn't capture:
- You don't know the odds or the payoff. A coin wins or loses a fixed amount. A stock can double or go to zero. Your edge is only an estimate.
- Swings. Some stocks move far more than others. A 10% position in a stock that moves 8% a day shakes your portfolio about as much as a 25-30% position in one that moves 3%. So a jumpier stock needs a much higher expected return to earn the same size.
- Bets that move together. Each coin flip is independent, but stocks are not. In a sell-off most of them fall at once. Five companies in the same industry or trend can act like one big position. High-beta stocks make this worse.
- Living through drawdowns. Watching your portfolio fall 40% feels very different from seeing it on a chart. Size positions so you can hold on without panic selling at the bottom.
- Leverage and forced selling. Borrowed money adds a new way to go bust. A margin call can force you to sell at the worst moment before your edge plays out.
- Real-world friction. Prices can jump overnight on earnings or news. Costs and taxes eat into returns. You may need the money before the long run arrives.
What carries over well: be honest about how sure you can be. Size down for investments that swing a lot. Keep any one bet small enough that being wrong doesn't knock you out of the game.
Play around
Use the simulator to see the risks of different strategies for yourself. Bet more than Kelly, switch strategy mid-game or rig the coin against yourself and watch what happens. Seeing a run go bust is what makes everything above stick.