How Much the Trade Holds

The number lesson 59 printed twice

An edge can die four ways. Your own size can outgrow the mispricing; enough other people can find it; somebody faster can reach it first; or the market condition it needs can simply be absent, which is not a death at all and is the subject of lesson 36. The first two are the same piece of arithmetic done twice, and this page does it. The other two are judgement calls dressed up as diagnoses, and this page leaves them alone.

Lesson 59’s law is one line: the fraction the price moves against you is the instrument’s daily standard deviation multiplied by the square root of your order divided by the day’s volume. It took the coefficient in front as one, and so does this page. What lesson 59 priced was a single order. An edge is a round trip, so it pays that move twice, once going in and once coming out, and two crossings is the convention every figure below rests on.

Now watch those two conventions cancel. Capacity is the size at which the round trip costs the whole edge, so it is where twice the impact equals twenty basis points, which is where one crossing costs ten. Ten basis points is a tenth of a per cent. Lesson 59’s first threshold column, computed for an entirely different question, is this page’s capacity column with no new calculation in it at all.

The table below is that column, plus the two things it implies. The second is four ninths of it, and the third follows from the second; both are derived in the worked example. The instruments, their volumes and their volatilities are lesson 59’s, unchanged.

Instrument	Dollars traded a day, millions	The whole trade holds	Best single position	Dollars a trade there
ETF	36,400	358,300,000	159,244,444	106,162.96
Mega cap	4,000	16,128,000	7,168,000	4,778.67
Mid cap	240	493,700	219,422	146.28
Small cap	24	29,900	13,289	8.86
Micro cap	1.5	1,050	467	0.31

Read the bottom row, and then stop reading it. A twenty basis point edge in the micro cap is worth 31 cents a trade at the size that pays best, and lesson 12 priced that instrument’s round-trip spread at 467 basis points, which is 23 times the entire edge. The micro cap is in the table because the arithmetic runs there, not because anybody should trade it. Capacity is not even the binding constraint that far down; the quote is.

The row that matters is the small cap. Twenty-four million dollars a day is an ordinary stock in an ordinary account, and the whole trade holds 29,900 dollars. At the size that pays best, 13,289, it pays 8 dollars 86 a trade, which at the forty trades a month lesson 65 fixed is 354 dollars. That is the ceiling. Not your ceiling: the ceiling, for the sum of everyone in it.

Which is the second thing the number does, and it is the one nobody computes. The mispricing does not care whose order eats it. If four people are long the same setup on the same bar, the price moves against the sum of their four orders, and the square root applies to that sum. So the same figure that caps your position caps the trade, and dividing it by a position size gives the number of people it supports.

Instrument	The whole trade holds	People at 1,000 each	At 10,000 each	At 100,000 each
ETF	358,300,000	358,300	35,830	3,583
Mega cap	16,128,000	16,128	1,613	161
Mid cap	493,700	494	49	4.9
Small cap	29,900	30	3.0	0.3
Micro cap	1,050	1.05	0.10	0.011

Three people at ten thousand dollars each finish the small cap. One person at a thousand dollars takes 95 per cent of the micro cap. You do not need to know who else is in the trade to know that, and no amount of watching your own results would ever have told you, because the crowd does not appear in your record until it has already taken the money.

Almost nobody on the other side of your trades has divided their position size into a day’s volume this week. It is two numbers off a quote page and one square root, and it takes about as long as reading this paragraph.

The sheet this module has been building since lesson 62 gains a seventh column, and it is the capacity of the rule in the instrument it actually trades, in dollars, computed before the first position rather than after the disappointing year. Lesson 70 is where the whole sheet gets spent at once.

The size that pays best is not the biggest one that works

Take the small cap and walk the money, not the cost. What a position earns is its size multiplied by what survives of the edge, and what survives is twenty basis points less the round trip you caused. Size pushes the first term up and the second term down, so the money has a peak somewhere in between and the peak is not at the edge of the cliff.

The impact has a form that makes this easy to check. Written as a fraction of capacity it is exactly the edge times the square root of that fraction, with the volume and the volatility both gone, so the whole curve can be walked in one column. On the small cap’s 29,900:

2,990 dollars, a tenth of it: the round trip costs 6.32 basis points, you keep 13.68, and the position earns 4 dollars 09.

7,475 dollars, a quarter: the round trip costs 10.00, you keep 10.00, and it earns 7 dollars 48.

13,289 dollars: the round trip costs 13.33, you keep 6.67, and it earns 8 dollars 86.

17,940 dollars: the round trip costs 15.49, you keep 4.51, and it earns 8 dollars 09.

23,920 dollars: the round trip costs 17.89, you keep 2.11, and it earns 5 dollars 05.

Look at the fourth line against the third. Seventeen thousand nine hundred and forty dollars is 35 per cent more money on the table and it comes back with less. And the fifth line is worse still: 80 per cent more money on the table comes back with 57 per cent of the return. There is a point past which putting more on takes money off, and it arrives long before the edge runs out.

Where it arrives has a closed form, and it is worth doing because it comes out as a pure number. The money is the size times the edge, less twice the volatility times the size to the power of three halves over the square root of the volume. Differentiate, set to zero, and the square root of your participation is the edge divided by three times the volatility. Capacity had the same quantity with a two where this has a three. Square both and divide, and the best position is four ninths of capacity — in every instrument, at every edge, at every volatility, always.

Put four ninths back into the impact and the second identity falls out. The round trip at the best size costs two thirds of the edge, and you keep one third, and neither of those depends on anything either. Two thirds of your edge is the price of collecting the other third.

Which settles the crowd as well. If everyone in the trade sizes at four ninths of capacity, the trade holds nine quarters of them: 2.25 people, in the exchange-traded fund and in the micro cap alike, because the ratio of two pure numbers cannot know what instrument it is in. The third person to size correctly takes it below breakeven for all three.

So compute four ninths of your own ceiling tonight, and size there rather than at the number that felt bold.

What this does not settle

That the coefficient in front of the law is one. It is not a physical constant, it is a fitted number, and the published estimates for equities run from roughly a half to about one and a half. Capacity moves as its inverse square, so the small cap’s 29,900 becomes 119,467 dollars at a coefficient of a half and 13,274 at one and a half, a factor of nine across the range of a quantity nobody can pin down for your particular instrument. Every dollar figure on this page carries that factor. The two ratios, four ninths and two thirds, do not, which is why the page leans on them and not on the dollars.

That the round trip pays the impact exactly twice. Impact has a part that decays and a part that stays, and a position exited slowly sells into the level its own buying created and recovers some of what it pushed, while one exited in a hurry pays more than twice. Lesson 59 charged one crossing because it was pricing one order; this page charges two because it is pricing a round trip, and the truth for any particular strategy is somewhere between one and rather more than two depending on how long it holds.

That impact is the only charge. It is not, and lesson 12 already showed which one is larger at retail size: the round-trip spread was 0.2 basis points in the fund and 467 in the micro cap. Adding the spread moves every optimum on this page down and every dollar figure down with it, and the last two rows of the first table do not survive it at all. What the page has priced is the cost of being large. Lesson 12 priced the cost of showing up, and at these sizes the second one is bigger.

That the crowd arrives together. The pool bound applies to size that hits the market at the same time, because that is what makes one order out of many. Spread the identical total across a year and the trade supports far more people, none of whom ever meet. So the head counts are the number of participants acting on the same bar, which is the right bound for a published signal everybody receives at once and the wrong one for a slow discretionary read that five people reach in five different weeks.

That 2.25 is a population. It is the number of participants each sized as though they were alone, and a crowd does not do that: once others are in, each one’s best response is smaller than four ninths, so a real crowd fragments into more participants holding less. The dollars are the finding and they do not move — the total is capped at the same number however it is divided. The head count is an illustration of the dollars, not a census.

And the concession that costs most: this page has priced your order and said nothing whatever about whether the twenty basis points was ever there. Lesson 64’s bar and lesson 67’s test both apply before any of this arithmetic starts, and an edge that does not exist has a capacity of zero. The formula will not notice. Hand it a number you got from a search of 253 configurations and it will hand you back a confident ceiling in dollars, which is the most expensive kind of wrong answer, because it looks like a measurement. Lesson 69 asks what the person between the signal and the order costs, and finds that in this same small cap five minutes of hesitation moves the price a typical 32.10 basis points, which is 2.41 times the impact this page spent the whole lesson minimising.