The Kelly Criterion: how much to bet, and what over-betting costs you

Most trading talk is about direction. Will it go up. Is the signal any good. What is the edge.

Almost none of it is about size. But you can be right about direction and still end with nothing. Size is where compounding either works for you or eats you.

The Kelly Criterion answers that second question. It gives you one number: the share of your money to put at risk on each bet. The formula is short. The interesting part is what happens when you ignore it and bet more.

The question a good signal never answers

Say you find a genuinely good bet. It wins 60% of the time. It pays even money, so you risk a dollar and either win a dollar or lose it. On average you make 20 cents per dollar staked. Casinos are built on far less.

Now the real question. You have $1,000 and you can play this coin 600 times. How much do you put on each flip?

Everybody knows “all of it” is wrong. Very few people can say why. The usual answer is “you might lose everything”, which is not quite right. If you only ever stake a share of what is left, you never actually reach zero. So what goes wrong instead?

Why the average lies

Here is the trap, stripped down.

Flip a fair coin. Heads, your money grows 50%. Tails, it falls 40%. Bet your whole stack every time.

The average flip gains 5%. Half the time you get +50%, half the time you get −40%. So the average bankroll after 100 flips is $1,000 × 1.05¹⁰⁰, or about $131,000.

Now follow one real life instead of the average. Fifty heads and fifty tails, in any order, leaves you with:

Five dollars. From a bet that makes money on average, played the most ordinary way possible.

Both numbers are true. The average gets pulled up by a tiny number of futures where you catch an absurd run of heads and end up with billions. The middle outcome, the one you should plan for, shrinks by about 5.1% per flip. Nearly everyone gets that second line.

The reason is simple. Returns multiply. Averages add.

A 50% gain followed by a 40% loss is not +10%. It is 1.5 × 0.6 = 0.9, so you are down 10%. Order does not matter, so no amount of clever sequencing saves you. What matters is the product of all your returns.

The natural way to handle a product is to take its logarithm. That turns it back into a sum, and sums behave. That one move is the whole of Kelly. Everything below is bookkeeping.

The mathematics

Setting it up

Let

• W₀ be your starting money,
• p be the chance of winning, and q = 1 − p the chance of losing,
• b be the gain per dollar staked when you win,
• a be the loss per dollar staked when you lose (usually a = 1, meaning a loss costs you the whole stake),
• f be the share of your current money you risk on every bet.

Current is the important word. You re-stake a share of whatever you have now, so the bet shrinks after losses and grows after wins.

One bet multiplies your money by (1 + bf) if you win, or (1 − af) if you lose.

Play n bets and win k of them. Since each bet just multiplies, the order does not matter and you end up with:

Turning the product into a rate

We do not care about W₀ itself, and we do not want a number that keeps growing with n. We want the rate: how fast money compounds per bet.

So take logs and divide by n.

Now use the one fact that makes this work. The share of bets you win, k/n, settles down to p as n grows. That is the law of large numbers.

So the left side, which is the real compound growth rate of your real money, settles down to a fixed number that depends only on f:

This step is worth a pause, because people skip it. g(f) is not a taste or a preference. Nobody assumed you like logarithms or that you are cautious by nature. g(f) is the growth rate your money will actually end up at if you keep betting this share. Picking f to make g(f) as big as possible is not choosing an attitude to risk. It is choosing the outcome.

Finding the peak

Differentiate g with respect to f:

Set it to zero and cross multiply:

Expand both sides:

pb − abpf = qa + abqf

Move the f terms to one side:

pb − qa = abf(p + q)

And p + q = 1, so that bracket disappears:

That is the Kelly Criterion. When a = 1, which is the usual case, it becomes the version people quote:

Read the top line out loud. pb − qa is your edge, the profit you expect per dollar staked. The bottom line, ab, is the size of the swing. Kelly is edge divided by odds. That is all it is.

Try it on the 60/40 coin. p = 0.6, q = 0.4, a = b = 1, so f* = 0.6 − 0.4, which is 20% of your money. Not all of it. Not a token amount. Twenty percent, and that already feels aggressive for a bet this good.

Checking that it really is a peak

Differentiate once more:

Both terms are negative everywhere that matters, so the curve bends downward the whole way. There is exactly one peak and f* is it.

Two more things fall out for free.

g(0) = 0. Bet nothing, grow nothing. Obvious, but it anchors the curve.

g′(0) = pb − qa, which is the edge. If the edge is zero or negative, the curve is already sloping down at f = 0. Then f* is not positive and the honest answer is to bet nothing. Kelly refuses to size a bad bet.

At the far end, as f gets close to 1/a, the term ln(1 − af) runs off to minus infinity and growth collapses. So the curve starts at zero, rises, turns over, and comes back down through zero.

The shape of the answer

The formula gives you one number. The curve is the part that should change how you trade.

Look at where the curve crosses zero on the way down. For this coin it happens near 39%, which is close to twice the Kelly fraction. That is the general rule when the edge is modest.

Stake exactly that much and you can place a bet with a 20 cent edge six hundred times and finish where you started. Stake more and growth is negative. You own a profitable bet and your money still shrinks.

That is the sentence to carry around. Betting too much does not just lower your return. Past a point it flips the sign. The edge never went away. The sizing ate it.

The reason is not mysterious. Bigger bets mean bigger falls, and a fall costs more than the matching rise returns. Lose 50% and you need +100% to get level. Lose 80% and you need +400%. Swings are not just uncomfortable. They come straight out of compound growth, and they grow with the square of your bet size while the benefit only grows in a straight line.

That last sentence is worth proving, so here it is.

Half the bet, three quarters of the growth

For a small edge you can expand the log and keep two terms. Write R for the return per dollar staked, μ for its average and σ² for its variance. Then:

There it is in plain sight. The reward term is linear in f. The penalty term is squared in f. Maximise it and you get f* ≈ μ/σ², with a peak growth of μ²/2σ².

Now bet some multiple c of the Kelly fraction. c = 0.5 is half Kelly, c = 2 is double Kelly. Put f = c·f* into the formula and the algebra collapses to something you can hold in your head:

Meanwhile the size of your swings is just c times the full Kelly swings. A straight line. Put the two together and every argument about bet sizing turns into arithmetic:

Two rows deserve a stare.

Half Kelly keeps three quarters of the growth for half the risk. That is the best deal on the table.

1.5× Kelly gives you the same growth as half Kelly while taking three times the risk. Same destination, three times the turbulence, for nothing.

The curve is also lopsided, and that should shape your instincts. Near the peak it is almost flat, so sitting a bit under Kelly barely costs you. Move the same distance the other way and you are walking toward the cliff. You are never sure of your numbers, and the mistakes are not symmetric. Guess low.

Drawdowns tell the same story. Bet full Kelly and the chance your money ever falls to half of what you started with is roughly one in two. Bet half Kelly and that chance drops to about one in eight. Halving the bet does not halve the pain. It cubes it away.

See it happen

Theory is easy to nod along to. Watching a healthy bankroll get wiped out by nothing but sizing is more convincing.

The sandbox below starts three bankrolls at $1,000 and puts them through the same run of coin flips. One bets half Kelly. One bets full Kelly. One bets whatever you set. Nothing else differs, so every gap you see comes from sizing alone.

It opens on the 60/40 coin from the proof, with your slider parked at 45%. That is well past the point where growth turns negative. Drag it to 20% to sit on the peak, or 10% for half Kelly, and watch the table change. Then try dropping the win chance to 55% and see how fast the safe fraction shrinks.

Things worth trying:

• Leave your slider at 45% and read the “middle end” column. Six hundred profitable bets, and the typical bankroll ends at about six cents. On the exact same flips, half Kelly is up thousands of times over.
• Press “New luck” a few times. The three lines reshuffle. Their order does not.
• Set the win chance to 50% with even money. Kelly says bet nothing, because there is nothing to bet on. Any share above zero bleeds.
• Compare 10% and 30%. Both are the same distance from the 20% peak. They are not equally bad. The right hand side always hurts more.

Why we still bet less than Kelly

All of the above assumes you know p, b and a exactly. In a casino you sometimes do. In markets you never do, and that changes the practical answer.

Your edge is a guess, and guesses are wrong. Kelly is very sensitive to the top line. Think your win rate is 60% when it is really 55%, and the true Kelly fraction is 10%, not 20%. You are now betting double Kelly while believing you are at the peak, which puts you exactly on the line where growth is zero. You did not decide to over-bet. You over-estimated, and over-betting came free with it. This is the most common way the formula hurts people.

Edges fade. Kelly assumes every bet is independent and drawn from the same pot forever. Real edges decay, regimes change, and the numbers you fitted last year describe a market that has since noticed.

Real losses are not capped at your stake. A losing trade can cost more than planned through gaps, slippage, or a fill that never arrives. Fat tails make the real σ² bigger than your sample suggests, which makes the true f* smaller than your sample suggests.

Bets do not arrive one at a time. Hold several positions that move together and the risk of the whole book is much larger than the sum of its parts. Size each leg at its own Kelly fraction and you have over-bet the portfolio, sometimes badly.

Drawdowns cost more than arithmetic. Kelly does not mind a 60% fall, because the maths has endless patience and never faces a redemption. Real money has neither. A sizing rule that is optimal but impossible to stick to is not optimal.

So most people who use this run half Kelly or less. Look at the parabola again and the reason is obvious. It costs a quarter of the theoretical growth, halves the swings, cuts the drawdown odds sharply, and buys a wide margin for having got the inputs wrong. It is not a close call.

For our own systems the lesson is less about the formula and more about the shape of the curve. Position size is not a dial you turn up when you feel confident. There is a ceiling, the ceiling is lower than instinct suggests, and going past it does not shrink your return. It reverses it. That is also why we test sizing rules on walk forward data rather than in sample. An over-fitted edge produces an over-confident f, and an over-confident f is how a good strategy takes you to zero.

What to keep in mind

1. Direction and size are two different questions. A good signal answers the first one. Kelly answers the second.
2. Money compounds, so work in logs. The average of your returns is not what you get to spend.
3. Kelly is edge over odds: f* = (pb − qa)/ab. If the edge is not positive, the answer is zero.
4. The growth curve has a peak, and past roughly twice that peak it turns negative. A profitable bet, sized badly, still shrinks your money.
5. Reward grows in a straight line with size. The penalty grows with the square. That gap is the whole argument.
6. Half Kelly keeps 75% of the growth for 50% of the risk. 1.5× Kelly keeps the same 75% for 300% of the risk. Only one of those is a deal.
7. Bad estimates push you right without telling you. Over-estimate the edge and you are over-betting while feeling optimal.
8. When in doubt, bet smaller. The curve is flat on the left of the peak and steep on the right. The mistakes are not symmetric, so your caution should not be either.

Know what your edge is worth. Then bet less than that.

This article is educational. It describes a mathematical result and a simulation of it. It is not advice about how to invest. Please see our disclaimer.