A sizing formula returns a fraction. The market fills whole contracts.

A sizing method is a division. You take a risk budget denominated in currency and divide it by the currency you lose per unit of position if the trade reaches its stop. The quotient is your size. Every method in common use shares that structure. They differ in exactly one place: how the numerator is obtained — and, for one of the four, what is substituted for the denominator.

Fixed dollar risk sets the numerator to a constant. Fixed fractional sets it to a fraction of current equity, so it rescales itself as the account moves. A volatility-normalised method does something structurally different: it does not set a permitted loss at all. It sets a target fluctuation and divides by the instrument's own measured range, which means the stop plays no part in the size. A Kelly fraction computes the numerator from a claimed win rate and payoff ratio rather than choosing it.

Because they take different quantities as given, they disagree, and the disagreement is information rather than error. Then the quotient meets the market. Contracts trade in integers; spot lots trade in fixed steps. The size you computed is not the size you can send, and the risk you carry is the rounded size multiplied by the per-unit loss — not the budget you started from.

The granularity error is bounded by one increment divided by the ideal size. It is not a property of your method, your discipline or your broker; it is arithmetic, and it is largest exactly where accounts are smallest.

What follows states each method's numerator, names the assumption behind it, shows what the number does when that assumption fails, and then derives the rounding error precisely. A live comparator computes all four sizes from inputs you set. No method is presented as the correct one, because they are not answers to the same question.

It computes a single division, and every method named in this article computes the same one. Let R be the risk budget in currency and let U be the currency lost per unit of position if the trade reaches its stop. The ideal size is N* = R / U. The methods differ only in how R is obtained — and, in the volatility case, in what replaces U.

U itself factorises. A stop is a distance in the instrument's quoted units, S, and one of those units is worth some amount of currency for one unit of position, V. So U = S × V. Both factors are properties of the trade and the venue; neither is a preference.

Writing it this way makes an easy confusion visible. The three arguments traders have about sizing — how much to risk, how to handle a volatile instrument, how to convert an edge into a bet — are all arguments about R. None of them is an argument about the division. And the fourth thing, the one almost nobody argues about, is the gap between N* and N, which is where a surprising amount of realised risk goes.

Choosing a sizing method is not choosing a formula. It is choosing which quantity you are willing to hold constant — and therefore which ones you have agreed to let float.

One notational discipline matters for the rest of the article. V is the currency value of one quoted unit for one unit of position. Venues call the quoted unit a point, a tick or a pip, and the position unit a contract, a lot or a share; the arithmetic does not care what they are called, only that V and q are read from the contract specification of the instrument in front of you rather than assumed. This article never asserts a value for either: every worked number below is illustrative and labelled as such.

Fixed dollar risk sets R = D, a constant chosen once. Fixed fractional sets R = f · E, a fraction of current equity, so the budget moves with the account without any decision being taken. Volatility-normalised sizing is the odd one: it sets a target fluctuation T and divides by the instrument's own average range, N* = T / (ATR · V) — the stop does not appear, because the method is not answering a question about the stop. A Kelly fraction sets R = f* · E where f* is computed from a claimed edge rather than chosen.

The fourth numerator is the only one that is solved for rather than chosen, so it is worth doing the work in place. Risk a fraction f of equity on each trade. A win multiplies equity by 1 + b·f; a loss multiplies it by 1 − f. Over many trades the multipliers compound, so the quantity that accumulates is the logarithm of equity, and its expected increment per trade is

Read the last line as edge divided by odds: the numerator W·b − (1 − W) is the expected profit per unit staked, and the denominator is the payoff. The result is a genuine maximum, not a turning point of unknown type — the second derivative, −W·b²/(1 + b·f)² − (1 − W)/(1 − f)², is negative everywhere the expression is defined, so g is strictly concave and f* is its only peak. At W = 44% and b = 2, f* = 0.16 and the peak growth rate is 0.02452 per trade — about 2.48% of equity per trade, compounded. The derivation above is the article's own, and it is what the rest of this clause rests on. It agrees with the published result [1]: Kelly maximised the same quantity — he called it G, the exponential rate of growth of the gambler's capital, and used logarithms to base two, which is a constant factor and cannot move the maximum. He worked the even-money case explicitly and obtained a fraction equal to the win probability minus the loss probability. Setting b = 1 above gives W − (1 − W), which is that same quantity — so the general form used here reduces to his published one exactly.

What it implies is that the curve is not symmetric about its peak. Because ln(1 − f) falls away without bound as f approaches 1 while ln(1 + b·f) only creeps upward, the cost of being above f* is far steeper than the cost of being below it:

Halving the fraction keeps three-quarters of the growth rate. Doubling it keeps seven per cent, and a little past twice f* the growth rate reaches zero — a positive edge, compounded at too large a fraction, stops compounding. The asymmetry is a property of the logarithm, not a preference, and it is why the comparator carries a fraction-of-full-Kelly control at all.

Two honest qualifications. The zero crossing sits at 2.04 × f* here, not at exactly twice: the clean "growth vanishes at double Kelly" result holds in the continuous approximation, and in this discrete two-outcome case it lands between about 1.99 and 2.16 × f* depending on W and b. And maximising expected log growth is an objective, not a law. It maximises the long-run growth rate and says nothing whatever about the chance of being wiped out along the way — a separate quantity, worked through at risk of ruin. A trader on a fixed horizon, or under a rule set with a floor, is solving that second problem, and f* is not its answer.

Applying any of this to a trade needs one more assumption, and it is easy to walk past: the loss must be exactly one stop, never more. The derivation is for a bet whose downside is known, because ln(1 − f) assumes exactly f is lost. A stop that can be jumped is not that bet. Nor is a position that cannot be sized continuously — which is the subject of clause 03, and which applies to f* exactly as it applies to the other three.

The comparator below computes all four from inputs you set. Nothing in it is pre-computed. Change the stop and three numbers move while one does not; change the volatility and one number moves while three do not. That asymmetry is the whole argument of this article, and it is easier to see than to describe.

Under the values the comparator opens with, the four ideal sizes are 6.67, 8.33, 3.75 and 133.33 units for one trade. The widest and the narrowest differ by a factor of thirty-five. Nothing in that spread is a mistake: each number is the exact answer to its own question, and they were asked four different questions.

The four methods are not competing estimates of one quantity. They are exact answers to four different questions, and a thirty-fold spread between them is the normal result of asking four different things.

Because N* is a real number and a market accepts only multiples of a fixed increment q. Futures trade in whole contracts; spot lots trade in fixed steps. The size you send is N = q · floor(N*/q) if you round down, or q · round(N*/q) if you round to the nearest — and the risk you then carry is N × U, which is not the budget the method computed.

Define the rounding error as N/N* − 1. Under round-down, N/N* lies in the half-open interval from (N* − q)/N* up to 1, so the shortfall is strictly less than q/N*. That one expression is the entire granularity problem. The error does not depend on your method, your broker or your discipline: it is one increment divided by the size you computed.

Neither extreme is a rounding mistake. Both are the exact consequence of a market that does not sell fractions. And the two behave differently: round-down can only ever take risk away, while round-to-nearest can add it, and it adds the most precisely where the ideal size is smallest.

There is also an average, and it has a closed form. Sweep equity continuously through one step and the fractional part of N* is uniform. The expected fraction of intended risk carried under round-down is then n · ln(1 + 1/n) exactly, where n is the whole number of increments in the step. It evaluates to 0.693 at one increment, 0.811 at two, 0.912 at five, 0.953 at ten and 0.990 at fifty, converging to 1 − 1/(2n). Rounding down is therefore not neutral. It is a systematic under-risk whose magnitude is set by nothing except how many increments you trade.

There is exactly one lever that shrinks the granularity error, and it is not risk tolerance. The bound is q/N*: it falls when the increment falls or when the ideal size rises. A contract at one-tenth the unit value produces ten times the N* for the same budget and one-tenth of the error. A smaller contract is not a smaller bet — it is a finer ruler.

Everything else people reach for works through the same channel or not at all. More capital raises N*. A wider stop raises U, which lowers N* and makes the error worse, not better — which is the opposite of the intuition that a wider stop is the conservative choice for a small account. Lowering the risk fraction lowers N* too, so the trader most careful about risk is also the trader whose realised risk is least faithful to the number they chose.

Which raises the question this article deliberately stops short of. Everything above establishes what risk you are actually carrying once the increment has had its say — a quantity that can sit at half or double the one you chose. What a rule set then does with that quantity, and how the size you end up trading determines how many losing trades a rule leaves room for, is worked through in why evaluations fail. The two pieces meet exactly here: this one tells you the number, that one tells you what the number costs you.

Figure 1's wine rule already marks the equity below which the ideal size is under one increment. That threshold has a closed form. The ideal size reaches exactly one contract at E = U/f, so the smallest account that can hold the intended fraction in a single contract is the per-contract loss divided by the risk fraction — at the illustrative U = $60 and f = 1%, that is $6,000. Below it, the bound this article derives above — the shortfall is strictly less than q/N* — stops being informative, because q/N* itself now exceeds one. The arithmetic still holds. It has simply stopped describing an error and started describing a choice between two outcomes, neither of which is the plan.

At $3,000 — half the floor — the budget is $30 against a $60 contract, so N* = 0.5 exactly. Round down returns zero, which is not a smaller position; it is no trade. Round up returns the one contract that exists, at $60, which on $3,000 is 2.00% of equity. Below the floor — where N* is at most one increment — round-up's ratio to intent collapses to a simple exact form, N/N* = q/N*: the very quantity this article already bounds the round-down shortfall by, now describing round-up's overshoot instead. At N* = 0.5 that is exactly 2. Halving the account from the floor halves N* in turn, so the multiple a single contract forces is exactly the floor divided by the account — smaller accounts forced into larger multiples of intent, without limit as the account shrinks toward zero. There is no size between these two outcomes that respects the fraction, because there is no size between zero contracts and one.

This makes the rounding rule a trade-selection rule, not only a sizing one. Round down keeps every trade whose ideal size clears one increment and deletes every trade that does not — and because a wider stop divides the same budget into a smaller N*, the trades it deletes are systematically the wide-stop trades, never a random sample of them.

A book sized by rounding down and read only for its surviving trades will look narrower-stopped than the strategy that produced it, and nobody decided that: it is a consequence of where each trade's stop happened to put N* relative to one. Above the floor the same mechanism is continuous rather than binary — the shortfall already varies with stop distance, trade to trade, exactly as derived above — but at the floor it goes all the way to exclusion.

Each method is exact given its inputs, so every failure is an input failure. Fixed dollar risk assumes your capacity to absorb a loss does not move with the account. Fixed fractional assumes the stop is the loss. Volatility-normalised sizing assumes measured volatility predicts the next period's. A Kelly fraction assumes you know your edge. Those four sentences are the whole of what separates the methods, and each of them fails in a specific, computable way.

Volatility-normalised sizing assumes the measurement is current. Average true range is a smoothed moving average of true range — the largest of the period's high-to-low span and the two gaps to the previous close — using Wilder's smoothing, which is a moving average with weight 1/n on each new observation, at a recommended n of 14 [2]. That construction has a consequence you can calculate: after a step change in true range, the fraction of the gap the average has closed by bar k is 1 − (1 − 1/n)^k. At n = 14 that is 40.5% after 7 bars, 64.6% after 14 and 87.4% after 28.

So for something like three weeks of daily bars after a regime change, the size is being set from a volatility that is no longer there: too large when volatility rises, too small when it falls. The method is not wrong. Its input is late, and it is late by an amount you can state in advance.

Fixed fractional assumes the stop is the loss. If a fraction p of losing trades realise at m times the stop — a gap through it, slippage, an order that was not filled where it was resting — then the expected loss per losing trade is (1 − p + p·m) · U, and the fraction you are actually running is f · (1 − p + p·m). At p = 5% and m = 2.5, chosen for illustration, that is 1.075 f: a 7.5% understatement that no care taken over the division will ever find, because the error is in U and not in R.

Fixed dollar risk assumes capacity is invariant. D does not move, so after equity falls by a fraction d the fraction at risk is D / (E(1 − d)). At d = 0.20 that is a 25% increase in the fraction risked with no decision taken, and symmetrically it does not scale up when the account grows. Fixed dollar holds currency constant and lets the fraction float; fixed fractional does the exact reverse. Neither is holding "risk" constant, because "risk" is two different quantities here.

Kelly assumes the edge is known. From f* = W − (1 − W)/b, the sensitivity is df*/dW = 1 + 1/b. At b = 2 that is 1.5, so a four-point error in the win rate moves the fraction by six points of equity — from 10% to 22% — which under the comparator's opening values is the difference between 83 contracts and 183. The formula is exact and the input is an estimate, and the estimate's error is transmitted at one-and-a-half times. Worth adding: the payoff ratio fed into it is a gross figure unless it was measured net, and execution friction lowers it — the subject of article 04. And the error is transmitted into the steep side of the growth curve derived in clause 02: an over-estimate of W pushes the fraction past f*, where halving costs a quarter of the growth rate and doubling costs almost all of it.

Three of the four methods produce a number that does not contain the instrument's volatility at all. Under a stop set on structure, their size is identical in a calm market and a violent one — which is a defensible choice and an indefensible accident, and the figure below is how you tell which one you made.

The stop distance becomes exogenous: the chart sets it and you do not. Under fixed fractional that has a precise consequence, because N* = f·E / (S·V) makes size inversely proportional to the stop. Two trades taken at an identical 1% risk can differ five-fold in contract count — and therefore five-fold in notional exposure, in sensitivity to slippage, and in what a gap straight through the stop costs.

Both trades were sized to risk $500. One of them loses eight times as much as the other when the stop is not honoured, and the sizing method contains nothing that would have told you so, because the quantity it holds constant is the stop-honoured loss.

Now set the stop from volatility instead: S = k · ATR. Substitute that into the fixed-fractional formula and you get N* = f·E / (k · ATR · V) — which is exactly the volatility-normalised formula with target fluctuation T = f·E/k. Not approximately; identically, to machine precision, for every value of ATR.

Fixed fractional sizing with a volatility-derived stop is volatility-normalised sizing. They are one method wearing two names. Fixed fractional sizing with a structural stop is a genuinely different method — and that is the only case in which the choice between them is a real choice.

Which reframes the whole comparison. The question is not "which method" but "what does each one hold still":

Read the middle column and the right one together and the article's central claim falls out. No method holds risk constant, because there is no single quantity called risk. Each holds one specific risk still and lets the others move — and the one it lets move is, reliably, the one that produces the surprise.

Because the methods differ in which trades receive the most capital — and, once rounding enters, in whether the order of the trades matters at all. The second half of that sentence is the one worth sitting with: order dependence in this setting is not produced by compounding. It is produced by the floor function.

Start from the result rather than re-deriving it. Take a fixed multiset of R-multiples, r1 through rn: the outcome of each trade measured in units of the risk taken on it. Continuous fixed-fractional compounding is order-independent — terminal equity is E0 · (1 + f·r1) · … · (1 + f·rn), multiplication commutes, and article 05 works through what that invariance implies for a capital path. Fixed dollar risk is order-independent for the simpler reason that E0 + N·U·(r1 + … + rn) is a sum. What follows is what happens when the sizes cannot be continuous.

Rounding breaks it. With an integer constraint the fraction actually risked on trade i is not f but q · floor(f·Ei / (q·U)) · U / Ei, a non-linear, non-multiplicative function of the equity standing at that moment. The product no longer factorises into terms that can be reordered, and the same trades in a different order finish somewhere else.

The continuous run is identical to the cent in both orders, as the algebra says it must be. The integer run differs by $120 — exactly two contracts' worth of stop — because the wins-first path crosses a contract step earlier and spends more of the sequence one contract larger. Nothing about the strategy changed. Nothing about the method changed. The only thing that differs is where the floor function landed.

Rounding is what makes trade order matter. The commutativity that leaves a continuous capital path invariant does not survive a floor function — so the path dependence appears on exactly the accounts too small to carry many increments, and is absent from the ones that can.

The other two methods break the symmetry for their own reasons and are worth separating. Fixed dollar risk stays order-independent even with rounding, because N never changes: the floor function is applied once, not per trade. Volatility-normalised sizing is order-dependent under any conditions, because the size depends on ATR at the time, so the same R-multiple contributes different amounts of currency depending on the regime it arrived in. Two traders with the same edge and the same trades can therefore end in three different places for three unrelated structural reasons — and only one of those reasons is about risk appetite.

Read it as a diagnostic instrument rather than as a menu. The four sizes are not candidates to pick between; two of their ratios are statements about things you cannot otherwise read off a screen — your stop, and your edge estimate — and those statements are the output the exercise actually produces.

The first ratio is about your stop. Strip out the two constants you chose and what remains is S / ATR. That is all the comparison contains: where your stop sits relative to the instrument's own range. It is not a verdict on either method — it is a measurement you can only read by computing both.

The second ratio is about your edge estimate. f* / f is literally how many times larger the fraction implied by your claimed win rate and payoff ratio is than the fraction you chose. If it comes out at sixteen, one of two things is true and it is worth knowing which: either the chosen fraction is a deliberate haircut on a number you believe, or the edge estimate is one you would not bet on. The ratio does not distinguish them. It does insist that you do.

The four numbers are not a menu. Two of their ratios are sentences — one about your stop, one about your edge estimate — and reading those two sentences is the point of computing four sizes rather than one.

None of this reduces to a rule, and the arithmetic does not contain one. What the arithmetic does contain is the pair of quantities that no sizing method reports and every sizing method depends on: q/N*, the fidelity of the size you can send to the size you meant, and S/ATR, the relationship between your stop and the instrument. Both are computable before a trade is placed. Neither appears on a platform's order ticket. The simulator carries the same arithmetic into a full rule set; the glossary holds the definitions used here.

Almost everything on this page is derived: the identity, the bound q/N*, the closed form n · ln(1 + 1/n), the equivalence between fixed fractional under a volatility stop and volatility-normalised sizing, the order-independence of continuous compounding and its failure under rounding, and the sensitivity 1 + 1/b. Each is shown with its working and can be checked with a calculator. Two external results are used and are cited below. Every currency figure, point value, increment and win rate on this page is illustrative and is labelled where it appears.

• [1] · the growth-optimal fraction
  Kelly, J. L. (1956). A New Interpretation of Information Rate. Bell System Technical Journal 35(4), 917–926. doi:10.1002/j.1538-7305.1956.tb03809.x. Read directly (author's copy hosted at princeton.edu, retrieved 28 July 2026). Supports, from p. 920: the definition of G as the exponential rate of growth of capital; the explicit rejection of maximising expected capital, on the ground that a gambler doing so "would be broke with probability one if he continued indefinitely"; the wealth relation V(N) = (1+l)^W (1−l)^L V(0) for a bet of fraction l; the resulting G = q·log(1+l) + p·log(1−l); and its maximisation, which for the even-money case gives a fraction equal to q − p. Kelly uses logarithms to base two; this article uses natural logarithms, a constant factor that leaves the maximising fraction unchanged. The arbitrary-odds form used here is the standard generalisation — Kelly's abstract states the result "is generalized to include the case of arbitrary odds" without writing this expression — and it is derived in clause 02 rather than imported. At b = 1 it returns q − p, matching his published case.
• [2] · average true range
  Wilder, J. W. Jr. (1978). New Concepts in Technical Trading Systems. Trend Research. ISBN 978-0-89459-027-6. The definition used here — true range as the largest of the period's high-to-low span and the two absolute gaps to the previous close, smoothed with weight 1/n and a recommended n of 14 — was taken from en.wikipedia.org/wiki/Average_true_range, retrieved 27 July 2026, which cites the book as the origin. The book itself remains unread: it was sought again on 28 July 2026 and the only full-text copy found is lending-restricted, so the definition above rests on the secondary source and is labelled accordingly. The consequence computed from that definition, 1 − (1 − 1/n)^k, is derived here and is not attributed to either source.
• Not sourced, and deliberately so
  No contract specification, point value, minimum increment, margin requirement or venue rule is asserted anywhere on this page. Those values change and are venue-specific; the comparator takes them as inputs precisely so that the article never has to claim one. No trading firm is named, and no pass, failure or survival statistic appears, because we hold none that would survive being cited.
• Reproducibility
  No random numbers are used on this page. Every figure and every table is computed deterministically from the comparator's current values, so any two readers with the same inputs see the same numbers, and every arithmetic claim in the prose can be checked against the live output above.