Position Sizing
One winner at +2R and two losers at −1R come to exactly nothing in R, and they still cost the account money: 0.030 per cent of it at 1 per cent risk a trade, and nearly nine times that at 3. The edge is identical in both. What the fraction decides is not your return but your depth, and it turns an ordinary bad year from a 15 per cent fall into a 39.5 per cent one that needs a 65 per cent gain to undo.
Prerequisites: Lesson 17, which turned your edge into a number, and lesson 18, which did the same for the bad run you have to sit through to collect it.
Those two lessons are the ends of a bridge and this one is the span. Lesson 17 said what a trade is worth on average; lesson 18 said what the road to that average looks like from inside. Sizing is the single decision that sits between them. It changes nothing about whether the method works. It decides whether you are still there when it does.
The formula, and the two numbers it keeps apart
position size = (account × the fraction you risk) ÷ risk per unit to your stop
The numerator is money: what one R is worth in this account today. The denominator is the distance from your entry to your stop, per share or per contract. The quotient is a quantity to buy.
Most of the value in that line is that it holds apart two numbers people routinely merge: what a position is worth, and what it can cost you. The first is the figure on the screen and it is not a risk. The second is the only one you chose, and it is the same $250 whether the position is worth six thousand dollars or twenty-five thousand.
Note also what the stop distance is doing there. It is an input, not a lever. Move the stop further away and the formula hands you fewer shares, which is not the formula being awkward — it is the correct answer, because a trade that needs more room to be wrong is a trade you take less of. Moving the stop closer in order to buy more shares is not a sizing decision at all. Lesson 21 is where the stop’s position gets settled, and it is settled by the trade rather than by the size you wanted.
Why a fraction and not an amount
“Risk $250 a trade” and “risk 1 per cent of a $25,000 account” are the same instruction exactly once. After a 20 per cent fall the fixed dollar figure is 1.25 per cent of what is left, and after another it is more again. The bet grows in real terms precisely when the account can least carry it, which is also when you are least inclined to notice.
A fixed fraction does the opposite without being asked. It shrinks the loss as the account shrinks, so a run of losses cannot arithmetically reach zero — each one takes its slice of a smaller number. And it grows the bet as the account grows, which is the entire source of compounding. Both behaviours fall out of the same property, and neither is a policy you have to remember while frightened. That is what earns fixed fractional sizing its slight awkwardness in practice: it is a rule that corrects itself.
The sequence that is zero in R and not zero in the account
Here is the part of sizing that surprises people who are comfortable with the rest of it. Take three trades from lesson 18’s system: one winner at +2R and two losers at −1R. In R that is exactly nothing. In the account it is not nothing.
At 1 per cent a trade those three multiply the account by 1.02 × 0.99 × 0.99 = 0.9997, a loss of 0.030 per cent. At 2 per cent the same three cost 0.118 per cent. At 3 per cent, 0.265 per cent. At 10 per cent, 2.8 per cent. The sequence has not changed and no trade in it was a mistake; only the fraction moved, and the cost of standing still grew nearly ninefold, because it goes with the square of the fraction. Double what you risk and you very nearly quadruple what standing still costs you.
The cause is that percentages are not symmetric: a loss of x% needs more than x% to undo, and the gap widens as x does. So sizing larger does not simply scale your result up. It scales your result and then subtracts a tax, and the tax accelerates.
This is also the reason lesson 18’s account columns are deeper than they would be if you compounded its drawdowns as runs of consecutive losses. A 16R drawdown is not sixteen losses. It is a longer stretch with winners scattered through it, and every trade in that stretch pays the tax on the way down.
The one cap that binds across trades
Everything so far sizes one position at a time, and lesson 18’s distribution assumed the trades arrive that way — independently, one after another. It also said, in its own bounds, that real records are not like that.
Positions taken for the same reason are not several trades. They are one trade wearing several tickets, and they stop out together. Three positions at 2 per cent each, all hit in the same hour, takes close to 6 per cent off the account in an afternoon: not a catastrophe, but three of the losses you budgeted arriving at once, which means the distribution you sized against has stopped describing you.
So there is a second number to choose, and it is the total risk you allow to be open at one time among positions that would fail together. No universal figure for it exists, and the way to find yours is to decide what a simultaneous exit ought to be allowed to cost. If it should cost no more than a single bad trade at your chosen size, that is two positions, not five. Counting correlated positions as one is the whole of the rule here; two longs in the same sector on the same thesis are one position for this purpose, whatever the tickers say. How to measure that correlation rather than eyeball it, and how to run the cap across a whole book, is lesson 72 — this much of it belongs here because without it the table below quietly means something other than what it says.
Putting a number on the fraction
An account of $25,000 risking 1 per cent a trade. One R is $250. Entry at $50.00 and a stop at $48.00 puts $2.00 of risk on each share, so the position is $250 ÷ $2.00 = 125 shares. That holding is worth $6,250, a quarter of the account, and it can cost $250, which is one per cent of it.
Now move the stop to $49.50 and change nothing else. The risk per share is $0.50, the position is 500 shares, and it is worth $25,000 — the whole account — while still risking exactly $250. Same one per cent, four times the position.
That second line is where the formula stops protecting you, and it is worth being precise about why. The 1 per cent is honest about one thing only: what happens if the stop is hit as a stop. It says nothing about the price gapping through it, which lesson 15 showed a stop cannot prevent; and it says nothing about having the entire account in one instrument, which is a risk the number does not measure at all. The tighter stop is also hit more often, which is lesson 21’s subject. None of that is an argument for the wider stop. It is an argument for knowing which of your risks the 1 per cent covers.
Then the harder half: which fraction. Take lesson 18’s system — a 45 per cent win rate at b = 2, an expectancy of +0.35R a trade — and price each fraction against the bad runs that system produces. The figures come from the same simulation as lesson 18: two hundred thousand runs of two hundred trades, seed 20260901, and for each row the median account fall among the runs whose worst drawdown was that many R.
| Risk per trade | Typical bad run, 9R | One year in twenty, 16R | One in a hundred, 21R | Gain needed to undo the one-in-twenty |
|---|---|---|---|---|
| 0.5% | 4.4% | 7.7% | 10.0% | 8.4% |
| 1% | 8.7% | 15.0% | 19.2% | 17.7% |
| 2% | 16.8% | 28.1% | 35.3% | 39.1% |
| 3% | 24.4% | 39.5% | 48.5% | 65.4% |
Read the last two columns together, because that is where the tax from earlier arrives all at once. Going from 1 per cent to 3 per cent multiplies the fall by about two and a half. It multiplies the gain you need to climb back by nearly four.
Which reframes the choice, and this is the sentence to keep. Every row here makes money: the edge is identical in all four, and none of them is wrong. So the question is not what return you want. It is what fall you can keep trading through without changing anything — because the row you cannot sit still in is the row where your sizing stops mattering and lesson 24 takes over.
What this does not settle
Where the stop goes. Every figure above takes the stop distance as given, and it is the input that decides both how large your position is and how often you are stopped out at all. That is lesson 21, and it is the other half of this arithmetic rather than a refinement of it.
Whether you go broke. A fixed fraction cannot reach zero by arithmetic, which is not at all the same as being safe: a broker’s minimum size puts a floor under how small you can go, and a gap through the stop makes the loss larger than the one you sized for. The probability that a given fraction ends a given account is computable, and it is lesson 22.
That the growth-optimal fraction is the one to use. It is computable too, and it is startling. For a system that wins 45 per cent of the time at b = 2, the fraction that maximises long-run growth is (p·b − (1−p)) ÷ b, or 17.5 per cent of the account on every trade, about six times the largest row in the table. It is also unusable, and not because of the tail. At 17.5 per cent it is the median bad run that takes 87.4 per cent of the account and needs a gain of 692 per cent to undo — the ordinary one, the one that happens to half of everybody. That fraction is optimal for an edge you know exactly, over unbounded time, with no other claim on the money and no limit to what you can sit through. What you have instead is an estimate of p and b from a sample too small to trust (lesson 19), and overestimating either pushes the optimum past the point where growth turns negative. Every fraction in the table is a small fraction of that number, deliberately.
That sizing by setup quality is available to you. Risking more on your better setups is correct if your better setups genuinely earn more, and that is a claim about your record rather than about your confidence at the time you took them. Lesson 23 is the ten fields that let you test it. Until it has been tested, one fraction for everything is the right size and not a beginner’s compromise.
Nor is the fraction you can sit through something you can know in advance. Problem 2 below asks you to pick the row you could live through without changing anything, and you will answer it by imagining a 39.5 per cent fall, which is not the same act as being inside one with no way of telling whether it has stopped. Everybody who has ever sized too large answered that question exactly the way you are about to. The only honest correction is to take the row below the one you believe you could hold, and to write the choice down with the date on it, so that the next drawdown is tested against a decision rather than against a memory of one.
And that this table is yours. Like lesson 18’s, every cell in it is computed at a 45 per cent win rate and b = 2, so all four rows move when your own two numbers do. Problem 2 builds your version of it; borrowing this one gives you the right shape at the wrong depths.
You cannot size your way into an edge, and you can very easily size your way out of one. This is the only number in the module you choose rather than measure, which is exactly why it is worth choosing on purpose.
Problems
- Size the same idea three ways. Take one trade you would actually take, and with your own account and a 1 per cent risk compute the position for a stop at one, two and four times the distance you would normally use. The amount at risk is identical in all three and the position value is not. Note which of the three you would have called “too small” by eye, because that reaction is the one this formula exists to overrule.
- Find the fraction you can sit in. Run lesson 18’s simulation with your own p and b, take the worst drawdown at the ninety-fifth percentile, and work out what it costs at 0.5 per cent, 1 per cent, 2 per cent and 3 per cent: the fall in the account, and the gain needed to undo it. Then pick the row you could live through without changing anything you do, and compare it with what you are risking now. If those two disagree, the one to move is the risk.
- Measure the tax you are already paying. Take your last fifty trades as a list of R multiples. Add them up for your result in R. Now compound them at the fraction you actually use, multiplying the account by (1 + f × R) for each trade in order. The gap between that final factor minus one and your R total multiplied by f is what your fraction cost you over those fifty trades. Halve f and run it again, and you will have priced the difference between the two rows you are choosing between.
Sources. Leo Breiman, “Optimal Gambling Systems for Favorable Games” (Fourth Berkeley Symposium on Mathematical Statistics and Probability, 1961), which proves that maximising the expected logarithm of wealth maximises the long-run growth rate, and so defines the fraction that the fractions in this lesson are fractions of. Edward Thorp, “The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market” (1997), the practical treatment by someone who used it with real money, and the clearest statement of why practitioners bet a fraction of the optimum and what the drawdowns look like when they do not. Ralph Vince, The Mathematics of Money Management (1992), which works the other side of the same curve: past the optimal fraction a system with a genuine edge loses money, and the loss accelerates the further past it you go.
You can now turn an edge and a stop into a number of shares, and choose the fraction from the drawdown rather than from the hope. The next lesson takes the input this one assumed, and decides where the stop actually belongs.
What an Edge Feels Like
The drawdown table every fraction here is priced against.
Read Lesson →Educational only. Trading involves substantial risk of loss. Not financial advice. Past performance does not guarantee future results.
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