risk management · 9 min read
The Kelly Criterion: The Math of How Much to Bet
There's an optimal amount to risk per trade that maximizes long-run growth. Bet more and you go broke; bet less and you leave money on the table. Here's the formula — and why most traders should use a fraction of it.
By Quantinger Research
The Question Behind Every Trade
You have an edge — a strategy that wins more than it loses, or wins bigger than it loses. The hardest question in trading isn't whether to take the trade. It's how much to risk on it.
Bet too much, and even a winning strategy can ruin you — a string of losses (which always comes eventually) wipes out your account before your edge can play out. Bet too little, and you leave enormous long-run growth on the table, compounding far slower than your edge allows.
There's a mathematically optimal answer, and it's called the Kelly Criterion. Understanding it — and understanding why you should usually bet less than it suggests — is one of the deepest ideas in risk management.
What Kelly Solves
The Kelly Criterion, developed by John Kelly at Bell Labs in 1956, answers a precise question: given a known edge, what fraction of your capital should you bet to maximize the long-run growth rate of your account?
The key insight is that maximizing growth is not the same as maximizing expected value. A bet that maximizes how much you make on average can also carry a high chance of ruin — and if you go broke, you can't keep playing, so your long-run growth is zero. Kelly finds the bet size that maximizes geometric growth — the actual compounding of your account over many trades — which inherently respects the risk of ruin.
The Formula
For a simple bet, the Kelly fraction is:
f = (bp − q) / b
Where:
- f = the fraction of your capital to bet
- b = the payoff ratio (how much you win per unit risked — your reward-to-risk)
- p = the probability of winning
- q = the probability of losing (1 − p)
In trading terms, with a win rate W and a reward-to-risk ratio R:
Kelly % = W − (1 − W) / R
An example. Suppose your strategy wins 50% of the time (W = 0.5) and your winners are twice the size of your losers (R = 2):
Kelly % = 0.5 − (0.5 / 2) = 0.5 − 0.25 = 0.25
The formula says risk 25% of your capital per trade. That's the mathematically optimal fraction for this edge to maximize long-run growth.
Why 25% Sounds Insane (and Mostly Is)
Risking 25% per trade violates every rule of conservative position sizing — and your instinct that it's reckless is correct. Here's why full Kelly is dangerous in practice, even though it's mathematically "optimal."
Kelly assumes you know your edge precisely. The formula needs your exact win rate and exact payoff ratio. In trading, you don't know these — you estimate them from a backtest, which is itself uncertain and may be overfit. If your true win rate is lower than your estimate, full Kelly massively over-bets, and the consequences are severe.
Full Kelly produces gut-wrenching volatility. Even when your edge estimate is correct, betting full Kelly produces enormous drawdowns — routinely 50% or more. Mathematically optimal for growth, yes; psychologically unbearable for almost everyone, and practically dangerous because a 50% drawdown invites panic-abandonment at the worst moment.
The penalty for over-betting is asymmetric and brutal. Bet slightly under Kelly and you sacrifice a little growth. Bet slightly over Kelly and your growth rate collapses — and far enough over, your expected growth turns negative even with a positive edge. Over-betting is far more dangerous than under-betting. Because your edge estimate is uncertain, you should deliberately err low.
Fractional Kelly: What Practitioners Actually Use
The professional solution is to bet a fraction of the Kelly amount — commonly half-Kelly or quarter-Kelly.
Half-Kelly captures about 75% of the growth rate of full Kelly while cutting the volatility roughly in half. That's an extraordinary trade: you give up a quarter of your growth to halve your drawdowns. For most traders, that's clearly worth it. Quarter-Kelly is even more conservative — less growth, but far smoother and far more robust to errors in your edge estimate.
In our example, full Kelly said 25%. Half-Kelly says 12.5%. Quarter-Kelly says 6.25%. Even quarter-Kelly is aggressive by conventional standards — which tells you something: either the example's edge (50% win, 2:1 reward) is unusually strong, or most traders are wildly over-estimating their edge when they feel comfortable risking even 2%.
Reconciling Kelly With the 1% Rule
If Kelly suggests 6-12% even at its conservative end, why do disciplined traders preach risking only 1-2% per trade?
The reconciliation is in the uncertainty of the edge. The 1% rule is what Kelly collapses to when you're deeply uncertain about your true win rate and payoff — which describes most real traders most of the time. When you can't trust your edge estimate, the correct response is to bet far below the theoretical optimum, because the downside of over-betting is catastrophic and the downside of under-betting is merely slower growth.
The 1% rule is, in effect, deep fractional Kelly — the position size of someone appropriately humble about how well they actually know their edge. As you accumulate real evidence of your edge across hundreds of trades, you might justify edging up toward fuller Kelly. Until then, betting small is the rational response to uncertainty, not timidity.
Kelly Across Multiple Positions
Kelly gets more complex — and more important — when you hold several positions at once. The formula above is for a single bet. With multiple simultaneous trades, especially correlated ones, the combined Kelly fraction across all positions must stay controlled.
Five crypto longs that each look fine individually can represent a massively over-leveraged bet on "crypto goes up" when correlation is accounted for. The portfolio-level Kelly fraction is what matters, and correlation dramatically reduces how much you can safely allocate to any single position. This is the mathematical backbone of "portfolio heat" — the sum of your open risks must respect the combined edge, not each trade in isolation.
The Bottom Line
The Kelly Criterion gives the mathematically optimal bet size for maximizing long-run growth: f = W − (1 − W) / R. It's a profound idea because it respects the risk of ruin — maximizing compounding, not just expected value.
But full Kelly assumes you know your edge precisely (you don't) and produces unbearable volatility (you can't stomach it). The practical answer is fractional Kelly — half or quarter — and for most traders, given how uncertain real edges are, the appropriate fraction collapses all the way down to the familiar 1-2% rule.
The deepest lesson isn't the formula. It's the asymmetry: over-betting is catastrophic, under-betting is merely slow. When in doubt about your edge — and you should usually be in doubt — bet small. The math itself tells you to be humble.
Size positions with confidence: Quantinger's position-size calculator and backtester help you find risk levels that survive the inevitable losing streaks while still compounding your edge.