Guide · Risk and position sizing · Updated 2026-09-19

Kelly sizes a stake from parameters nobody can estimate, and guessing high is punished far harder than guessing low

Everything below is computed from a stated wager with a stated seed, so it can be redone. Nothing here is a claim about what any method produces. The subject is how one mathematical rule behaves under assumptions that are set out and then taken apart.

The criterion answers a narrower question than the one it is asked

Most of the confusion lives in the setup. Take a wager that can be repeated indefinitely. Before each round you stake a fraction f of your current balance. With probability p the favourable outcome occurs and the amount staked is multiplied by 1 plus b. Otherwise the amount staked is lost. Call b the odds and call pb minus one minus p the edge per unit staked. Throughout this page the odds are 2 to 1 and the implied probability of the favourable outcome is 0.40, which makes the edge 0.20. Those are parameters of a mathematical construction chosen to make the arithmetic land on round figures, not a description of any market.

Now the question. It is not "how much should I stake", which is about preferences and has no mathematical answer. It is much narrower: which fixed fraction, restaked every round out of whatever balance then exists, makes the long run growth rate of that balance as large as possible? That question has a unique answer and the criterion is it. Almost every misuse comes from the gap between the question people think they asked and the one that was answered.

Why the logarithm and not the level

The obvious objective is the expected balance after a round, and the obvious objective is wrong. The expected multiplier for one round is one plus the edge times the fraction staked, which rises in a straight line as the fraction rises, so it is maximised by staking everything. Under that instruction a sequence survives only if the favourable outcome occurs on every single round, and after 200 rounds the probability of that is about ten to the power of -80. The expectation is not wrong. It is carried almost entirely by a set of sequences so rare that no participant will be in it.

The reason is that capital compounds. Rounds multiply rather than add, so the quantity that adds across rounds is the logarithm of the multiplier. Because the logarithm of a product is the sum of the logarithms, and a sum of many similar terms converges to its own mean, the growth rate a single long sequence actually experiences converges to the expected logarithm of the per round multiplier. That is the quantity to maximise, not because logarithms are a preference but because the logarithm is what accumulates. The average across parallel worlds and the average along one path are different quantities for a multiplicative process, and only the second is available to anyone.

The derivation, in four lines

The expected logarithm for one round is p times the logarithm of one plus bf, plus one minus p times the logarithm of one minus f. Differentiate with respect to f. Set the derivative to zero. The terms rearrange to give f equal to p minus one minus p divided by b, which is the same as the edge divided by the odds.

For the worked parameters that is 0.20 divided by 2, or exactly 0.10. The growth rate at that fraction is 0.009712 per round.

Read down the last two columns. The wager at 12 to 1 carries a larger edge than the one at 2 to 1 and calls for a much smaller fraction of capital, because the same edge arrives in rarer, larger pieces and the sequence has to survive the gaps. The fraction tracks the edge divided by the odds, not the edge.

The continuous version is the one that gets applied to trading. If each round is a small change with a mean and a variance rather than a two way outcome, the same differentiation gives the optimal fraction as the mean divided by the variance. That form is where the trouble starts, because it makes the answer directly proportional to an estimated mean.

What the criterion deliberately ignores

Four things, all deliberate, and all of them reasons the raw output should not be staked.

It ignores the path. The objective is the growth rate in the limit, so two sequences with the same growth rate and wildly different declines along the way are equally good to it. Nobody trades in the limit and everybody trades along a path.

It ignores the horizon. The fraction is derived for an indefinitely long sequence. Over any finite number of rounds it is not optimal for most preferences, and the shorter the horizon the further it sits from what a person would choose.

It ignores withdrawals. The derivation assumes the whole balance is restaked every round. Any consumption from the balance changes the problem, in the direction of a smaller fraction.

It ignores parameter uncertainty entirely. The edge and the odds enter as known constants, and there is no term anywhere in the derivation for the possibility that they were estimated from a sample and are wrong. The rest of this page repairs that omission.

The curve is not symmetric, and that is the whole result

Plot the growth rate against the fraction staked and the shape does the arguing. It rises from zero, reaches a single peak at the optimal fraction, then falls back through zero at 2.039 times the optimum and continues downward with no limit. Near the peak the curve is flat, because the growth rate behaves locally like a downward parabola, so the loss grows with the square of the distance from the peak.

Two readings matter. First, the concession for restraint is small: 0.50 times the optimum retains 75.7 per cent of the growth rate and 0.75 times retains 94.0 per cent. Second, the penalty for excess is not merely larger, it is of a different kind: 2.00 times the optimum retains 7.2 per cent, 2.25 times gives -44.4 per cent, and 2.50 times gives -107.3 per cent. A negative share is a negative growth rate, which is not a smaller version of a positive one. It is a sequence that declines towards zero with probability one while the underlying wager stays favourable throughout. Nothing about the edge changed. Only the stake did.

That is the structural asymmetry, and it needs no reference to sampling at all. Understaking is bounded: the worst it can do is stake nothing, forgo the whole growth rate and stop at zero. Overstaking is unbounded: there is no worst case, only a curve that keeps descending. Equal errors in opposite directions are not comparable mistakes.

Misestimate the edge and the growth rate changes sign

Now repair the omission. Suppose the parameters were estimated rather than known, the stake was chosen from that estimate, and the truth is something else. The stake is fixed, because it was already placed. Only the truth moves.

The row to read is the first. When the true edge is 40 per cent of what was assumed, the full fraction produces -0.001795, which is negative, while the half fraction produces +0.001484 and the quarter fraction +0.001361, both positive. Against the 0.001580 that was attainable under that truth, the half fraction retains 93.9 per cent and the quarter fraction 86.1 per cent. A stake chosen from an assumption wrong by a large margin, at a fraction of the criterion, lands close to what a perfectly informed participant would have chosen.

Now the mirror image. When the true edge is 160 per cent of what was assumed, an error of comparable size the other way, the full fraction retains 86.5 per cent of the attainable growth rate. Below the optimum, and firmly positive. Optimism changes the sign. Pessimism changes the size.

That table is the cleanest statement of the whole subject. A participant staking a fraction k of the criterion stays in positive growth for as long as the true edge is more than roughly k divided by two of the edge assumed. The full fraction tolerates the truth being 49.4 per cent of the assumption and no less; the quarter fraction tolerates 12.5 per cent. The exact solutions and the approximation agree to within a fraction of a percentage point, which is a check that the reasoning and the arithmetic describe the same object.

The same result, simulated along the path

The growth rate is asymptotic, so it is worth seeing what it does to sequences of realistic length. The simulation runs 20,000 independent sequences of 200 rounds under three truths, seed 20260919, with the stake fixed at the full, half or quarter of the fraction implied by the assumed edge of 0.20. Draws are shared across the three stakes within a scenario, so the comparison is paired.

Three things fall out of it. The simulated medians match the growth rate multiplied by the number of rounds to three decimal places across all nine cells, which is the instrument agreeing with the closed form and the reason to trust the rest of the table. In the scenario where the assumption was exactly right, the full fraction is best on the median and worst on the path: its median largest decline is 78.7 per cent of the running peak, against 49.8 per cent at half and 28.0 per cent at a quarter. And where the edge was overstated, 59.2 per cent of sequences at the full fraction finish below where they started, against 35.7 per cent at half.

The decline figures are the part practitioners feel. Even with the parameters exactly correct, the fraction that maximises the growth rate puts the median sequence through a decline of around 79 per cent from its own peak within 200 rounds. That is not a malfunction. It is the criterion working as specified, on an objective that treats the path as irrelevant. That a largest decline observed in a sample is itself a badly behaved sample statistic is worked through in the article on maximum drawdown.

The inputs are exactly the quantities that cannot be estimated

Everything so far has treated the true edge as a scenario. In practice it is an estimate from a sample, and the whole difficulty is that the sample is short.

Take the continuous form, where the optimal fraction is the mean divided by the variance. The variance is estimated well from modest samples, because second moments converge quickly. The mean is not. The standard error of an estimated effect size, meaning the mean divided by the standard deviation, is one divided by the square root of the number of observations, and it does not depend on how large the effect is. So the relative error in the estimated mean, which is exactly the relative error in the fraction the criterion prints, is one divided by the effect size times the square root of the sample. That makes the next quantity computable: the probability that a stake derived from a sample lands past the point where the growth rate turns negative.

At 250 observations, a correctly executed estimation procedure applied to a genuine effect of 0.05 produces a full stake on the wrong side of zero 21.5 per cent of the time. Not because the method was bad or the estimate careless, but because 250 observations do not pin down a mean that small. At the same sample the half fraction lands there 0.89 per cent of the time and the quarter fraction effectively never.

That connects the argument directly to the sample size question. The computation of how many trades are needed to distinguish an edge from luck establishes that an effect of 0.05 per observation needs roughly 2,474 observations merely to conclude it is not zero. Sizing is a stricter requirement than detection, and the gap can be computed exactly.

The ratio is 2.59 and it is the same at every effect size, because both requirements are proportional to one over the square of the effect. Detecting an effect of 0.05 takes about 2,474 observations; sizing against it with the full fraction takes about 6,400. If the first figure is already out of reach across a working lifetime, and the companion article shows that it is, the second is not a target at all.

Two further problems make the estimate worse rather than better. The observations are not independent, so a count of trades overstates the evidence in them, and the multi position form of the criterion needs a covariance matrix, which is less stable than a mean; the instability of estimated correlation is its own subject. And the edge being sized against is usually the survivor of a search across many variants, which biases the estimate upward, precisely the direction the criterion punishes most.

A fraction is not timidity

Staking some fraction of what the criterion prints is not a psychological compromise. It is the answer to a better posed problem. The criterion asks what fraction is optimal if the edge is exactly this. The honest question is what fraction is optimal if the edge lies somewhere in a distribution whose centre is this and whose spread is large. Because the penalty is bounded on one side and unbounded on the other, averaging the growth rate over that distribution moves the answer downward, and the more uncertain the input the further down it moves. A fraction is what falls out of the arithmetic once the inputs are treated as estimates, which is what they are.

There is a second reason, visible in the simulation table. The concession on growth rate is small and the concession on the path is large: half the fraction gives up 24.3 per cent of the growth rate when the assumption is right, and takes the median largest decline from 78.7 per cent to 49.8 per cent. A participant who stops during a decline has an effective growth rate of zero from that point regardless of what the formula said, so the decline is not a comfort question. It is part of the objective and the criterion leaves it out.

What follows is what serious users do: treat the printed fraction as a ceiling rather than a target, apply a fraction well under one, recompute the inputs and watch how far the answer moves, and cap the result with a separate constraint that does not depend on the estimate at all. Position sizing from first principles sets out that second constraint, and the risk of ruin arithmetic covers the boundary the sizing has to respect.

Three assumptions Indian market structure breaks outright

Generic treatments stop at the formula. The assumptions underneath it are where it meets the market, and three do not survive contact.

Continuous divisibility. The derivation assumes any fraction can be staked. Exchange traded derivatives trade in contracts of a size fixed by the exchange, so achievable stakes are integer multiples of one contract. The optimal fraction is then unavailable, and the question becomes which reachable multiple is best.

The last two rows are the ones that matter for a smaller balance. Once one contract is 2.50 times the optimal fraction, the smallest position available has a growth rate of -0.010423, which is negative, and the best reachable stake is therefore zero. The rule's own answer, computed from its own objective, is to take no position at all. That is not a statement about anyone's capital but about the arithmetic: below a certain ratio of balance to contract size, the smallest available position is already past the point where the growth rate turns negative, and no amount of conviction about the edge changes it. Contract sizes are set by the exchange and are revised from time to time, so the current specification for any segment has to be checked directly rather than assumed.

Bounded loss. The derivation contains the term one minus f, the fraction of the balance remaining after an unfavourable outcome, and it presumes the loss cannot exceed the stake. On a margined position it can, because the exposure is a multiple of the amount committed. When the loss can exceed the stake, the logarithm's argument can reach zero or go below it, and the objective is not merely mis-parameterised, it is undefined. Applying the formula to a leveraged position requires defining f as the fraction of the balance genuinely at risk, meaning a stop that is actually honoured, rather than the fraction committed as margin.

Independence and repetition. Rounds are assumed independent and identically distributed, and the whole balance is assumed to be restaked each round. Neither holds. Positions held at the same time across related instruments are one bet with a large stake, not several with small ones, so the effective fraction staked is far larger than the sum of the individual sizes suggests. Conditions also change, so the parameters are not merely uncertain, they are non stationary, which is a stronger objection than sampling error and is not repaired by collecting more history.

What the criterion is actually for

It is not a calculator. Nobody can supply the inputs it needs, and this page has computed how far a realistic sample falls short of supplying them. Treating its output as a position size is using it for the one thing it cannot do.

What it does deliver is a boundary, derived rather than asserted. There exists a stake above which the growth rate is negative regardless of how good the underlying method is; that ceiling sits at roughly twice a quantity nobody can measure; and the cost of staying well below it is small and bounded while the cost of crossing it is unbounded and arrives as a change of sign. Those three statements are arithmetic, they hold for every wager of this shape, and none of them requires knowing the edge.

Used that way the criterion stops being a formula to apply and becomes a constraint to respect. It says the failure mode of position sizing is one sided, that the error worth guarding against is optimism about an input rather than caution about it, and that the correct response to not knowing a parameter is to stake less on the assumption that you do. That conclusion survives the fact that the inputs are unknowable, which is more than can be said for the formula itself.