Kelly Criterion

What the Kelly criterion is

Suppose you can make the same bet over and over, and each time you choose what fraction of the account to put at risk. Bet too little and the account grows slowly. Bet too much and the losses cut the account so hard that the wins cannot repair it. Somewhere in between is the size that makes the account grow fastest in the long run. John Kelly published the formula for that size in 1956, and it is called the Kelly criterion or Kelly fraction.

It is a statement about growth rate, not about comfort. It does not ask how deep a drop you can stand, whether the odds are right, or whether the account has other uses. It answers one narrow question, and it answers it assuming you already know the odds.

The formula

Here p is the probability a bet wins, 1 - p the probability it loses, and b the average win divided by the average loss. The result f* is the share of the account to risk on each bet. If f* comes out at zero or below, the formula says the edge is zero or negative and the best bet size is nothing. The second line is the average growth of the logarithm of the account per bet, which is the number the Kelly fraction maximises. The calculation works only for repeated, independent bets of the same type, which real trading rarely is. For your own numbers, use the Kelly criterion calculator. This page is about what the result means and how it breaks.

Why full Kelly is violent

Take a made-up bet: it wins 55 percent of the time, and a win is the same size as a loss, so p = 0.55 and b = 1. Then f* = 0.55 - 0.45 / 1 = 0.10. Full Kelly says risk 10 percent of the account each time.

Now look at what 10 percent means as a path. After a loss the account is 0.90 of what it was. A run of seven losses leaves 0.9 to the power 7, which is 0.4783, or 47.8%. With a 45 percent chance of losing each bet, a run of seven is not rare over a few hundred bets. Full Kelly accepts deep drops as the price of the fastest growth, and drops of more than half are part of the plan, not a failure of it. Most people, and most prop-firm rules, cannot live with that. See drawdown for what a large drop does to the climb back.

The growth curve in the picture at the top also shows an asymmetry. Going from 10 percent to 20 percent bets takes growth from +0.50 percent per bet to about zero. Going from 10 percent down to 5 percent costs a quarter of the growth. Too small is cheap, too big is expensive.

Fractional Kelly

Half Kelly means betting half of f*, and quarter Kelly a quarter. People use these because the growth curve is flat near its peak. In the made-up example half Kelly gives about three quarters of the full growth rate, with much shallower drops. The table below is for p = 0.55 and b = 1, and then repeats the same bet sizes if the true win probability is lower than believed. It shows arithmetic, and it is not a recommendation of any fraction.

This is a made-up example. Growth per bet is the average log growth, so "after 100 bets" is the typical multiple of the starting account, e^(100 x growth). It is the median outcome, not a promise: the actual path is random and some paths end well below or well above it. Calculated by us on 29 Oct 2026, exact for the model.

Read the table in two halves. In the true p 0.55 columns, full Kelly gives the highest growth, 0.501% per bet, and 1.5 x Kelly is already worse than half Kelly. At 2 x Kelly growth is -0.014% per bet, about zero, so doubling the "best" size takes all the edge away. In the true p 0.52 columns, the same bet sizes were chosen for p = 0.55, but the real edge is smaller. Full Kelly for 0.55 is now a losing size: -0.101% per bet. Half Kelly still grows, at 0.075% per bet. Fractions are a cushion against being wrong about p.

How an uncertain win rate breaks it

The formula wants the true p and b. What you have is an estimate from a finite list of past trades. If a backtest shows 55 wins in 100 trades, the honest range for the true rate is wide: a few percentage points either way is ordinary noise. The true value of f* for p = 0.52 is only 0.04, not 0.10, because f* = 0.52 - 0.48 = 0.04. Betting 10 percent is then 2.5 times the real Kelly fraction, which is the zone where growth is negative. The error does not cancel: overestimating the edge costs more than underestimating it by the same amount.

It gets worse for two reasons. First, backtest numbers are usually flattered by overfitting and by ignored costs, so the estimate leans high. Second, the win rate and payoff of a real bot change with the market, so there is no single p to know. Any Kelly number calculated from one backtest is an upper bound on what the maths allows, and often a poor guess at that.

How it hits bots

• Martingale and recovery bots raise size after a loss, which is the opposite of Kelly. Their size grows exactly when the account is smallest.
• Grid and DCA bots have no single bet. They hold a changing pile of positions, so p and b do not describe them, and the formula does not apply without a rough translation.
• Fixed-fraction EAs are the closest fit. Their risk setting per trade is a Kelly-style choice, and the table shows how steeply the result depends on it.
• Costs cut b and p at once. A spread that eats a fifth of a win moves f* a long way.

How to check a Kelly number

1. Take p and b from trades the bot has not been tuned on, such as forward test results.
2. Recalculate with p a few points lower and b a little lower. If f* falls to near zero, the edge is not measured well enough to size from.
3. Compare your actual risk per trade with f*. If it is above f*, the arithmetic says growth is already falling, and near 2 x f* it is about zero.
4. Look at the risk of ruin at the size you use, which asks a different question from Kelly: how likely is the worst case, not how fast is the best case.

What it does not tell you

Kelly assumes bets are independent, sizes are free to adjust, losses are bounded by the bet, and nothing else needs the money. None of these is true for a leveraged account with a margin level. It also says nothing about whether the strategy has an edge at all. A positive f* from a bad estimate is a wrong answer given with confidence.