What the formula says
For a simple bet with two outcomes, Kelly gives the fraction of current equity to put at risk in order to maximise the growth rate of that equity over many repetitions. Write the win probability as p, the loss probability as q, which is one minus p, and the net payoff as b, meaning a win returns b times the amount risked. The fraction is p minus q divided by b. Two features of that expression deserve attention before any arithmetic. It scales with current equity, so the amount risked falls automatically after losses and rises after gains, which is what makes ruin impossible inside the model. And it maximises the growth rate of the logarithm of equity, not the expected profit. Maximising expected profit alone would tell you to risk everything on any favourable bet, which is correct arithmetic and a certain route to zero. Kelly is the answer to the question of compounding, which is the question traders actually face.
A worked calculation
Take the familiar shape from earlier work: a win probability of 0.4 with a payoff of 2 to 1. The loss probability is 0.6, so q divided by b is 0.6 divided by 2, which is 0.3. Kelly is 0.4 minus 0.3, which is 0.1, meaning ten per cent of current equity risked on each trade. Most traders will stop reading at that figure, and they should pause rather than stop, because the formula is not being reckless. It is answering a question with no bankruptcy in it, no gaps, and perfect knowledge of p and b. Check a second case to see the formula behaving sensibly. With a win probability of 0.5 and a payoff of 1 to 1, q divided by b is 0.5, and 0.5 minus 0.5 is zero. No edge, no bet, which is the correct answer. If the subtraction gives a negative number the formula is telling you the bet is unfavourable, and the only useful response is to decline it.
The growth curve and the cost of overbetting
The fraction matters less than the curve it sits on. For a two outcome bet the expected growth per trade is p multiplied by the natural log of one plus b times f, plus q multiplied by the natural log of one minus f. Run that for the example with p of 0.4 and b of 2. At f of 0.05 the growth rate is 0.00735. At f of 0.10, the Kelly fraction, it peaks at 0.00971. At f of 0.15 it has already fallen to 0.00743. At f of 0.20 it is 0.00070, barely above nothing. At f of 0.25 it is minus 0.01042, so the system now shrinks equity despite having a genuine edge. The shape is the lesson. The curve is nearly flat around the peak and then falls off a cliff. Being too small costs you a little growth. Being too large costs you the growth and then the account, which is why the asymmetry argues for erring low every time.
Why half Kelly is the usual answer
The flatness near the peak has a practical consequence. In the worked example, growth at half the Kelly fraction is 0.00735 against 0.00971 at the peak, so halving the position size keeps about 76 per cent of the growth rate. You give up roughly a quarter of the compounding and you halve the size of every swing, which most people will accept without hesitation. The continuous version of the same problem gives a tidier version of the result, where half Kelly retains exactly three quarters of the growth rate, and the two answers agreeing closely is reassuring rather than coincidental. There is a sharper way to see the asymmetry. In the example, growth at f of 0.15 is 0.00743, almost identical to growth at f of 0.05. One of those positions is three times the size of the other for the same expected compounding, and only one of them leaves you comfortable. When two sizes offer the same growth, the smaller one is strictly better.
How a small error in the win rate wrecks it
Kelly assumes you know p and b. You never do, you estimate them from a finite record, and the formula is brutally sensitive to the estimate. Hold the payoff at 2 to 1 and the fraction becomes one and a half times p, minus one half. At a win probability of 0.40 the fraction is 0.10. At 0.45 it is 0.175. At 0.50 it is 0.25. Now go the other way, which is the direction that matters. If the true win probability is 0.35 rather than the 0.40 you believed, the correct fraction is 0.35 minus 0.325, which is 0.025. A five point overestimate of the win rate means the correct size was one quarter of what you calculated, and you would have been risking four times too much while believing you were following a rigorous rule. Given that a win rate measured from a short record carries an error far wider than five points, this sensitivity is the real argument for a fraction rather than any dislike of the maths.
Where trading breaks the assumptions
The model behind Kelly is a repeated bet with a fixed payoff, a known probability, independent outcomes, and a loss that never exceeds the amount staked. Trading violates every clause. Payoffs vary trade by trade because exits are not identical. Probabilities are unknown and drift as conditions change. Outcomes cluster, so a bad patch is not a series of independent draws. Two open positions in the same instrument are one bet wearing two tickets, and Kelly applied to each separately doubles the real exposure. Most importantly, the loss is not bounded by the stake. A gap through a stop or a fast move can cost more than planned, and the model has no concept of that, so it treats a 1R loss as a hard floor when it is an expectation. Any of these alone would argue for using a fraction. Together they mean the printed figure should be read as a ceiling you stay well below, not a target.
What traders use instead, and why
The common practice is fixed fractional sizing at a small percentage of equity, chosen so the worst plausible losing run leaves both the account and the operator intact. That is Kelly in spirit, since it scales with current equity, with the fraction set by tolerance rather than by a formula fed uncertain inputs. Layer the practical limits on top. Cap total open risk across correlated positions rather than per trade. Reduce size while an edge is unproven and after a drawdown, which is the opposite of what the model suggests and the right call when p is in doubt. Respect any external constraint, since a daily loss limit under funded account rules can make ruin arrive long before the model expects it. Get the mechanics right so the intended risk is the actual risk, which means knowing how lot size translates into risk. Keep the whole arrangement written into your risk rules and track the realised sizes on the live chart rather than trusting the plan.
FAQ
Does Kelly really say I should risk ten per cent a trade?
In the worked example with those exact inputs, yes, and the inputs are the problem rather than the formula. The model knows the win rate perfectly, bounds the loss at the stake, and treats trades as independent. Remove any one of those and the correct fraction falls sharply, which is why a fraction of Kelly is standard.
What fraction of Kelly is sensible?
There is no formula for it, since it trades growth against the swings you are willing to endure and the confidence you have in your inputs. Half is the most quoted and a quarter is common when the edge is newly measured. The decision belongs to tolerance and sample size, not to arithmetic.
What happens if I bet more than Kelly?
Growth falls, and past a point it turns negative despite a genuine edge. In the worked example growth is nearly zero at twice the Kelly fraction and clearly negative above that. The curve is asymmetric, so overbetting is punished far harder than underbetting, and it is punished through compounding rather than through any single trade.
Can I apply Kelly to several positions at once?
Not by treating each one separately, because that multiplies exposure to whatever the positions have in common. Correlated trades behave as a single larger bet, and in a single instrument the correlation is close to total. Size the combined exposure rather than each ticket, and treat a cap on total open risk as the binding rule.
Does Kelly work with varying payoffs?
There are extensions for continuous outcomes and varying payoffs, and they need even more information about the distribution of results than the simple version needs. The practical response is to use the simple formula as a rough ceiling, with conservative estimates for both inputs, then stay comfortably underneath it.
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