What a Million Shares Takes

An institution does not place an order. It starts a schedule. The binding constraint is not money, it is days, because a desk taking much more than a tenth of the daily volume starts moving the price against itself. At a tenth, the time to finish is ten times the order’s size measured in days of volume, and that quantity runs by a factor of nearly seven hundred across perfectly ordinary instruments: a million shares is a seventh of a session in a large exchange-traded fund and sixty-seven sessions in a micro cap. Over sixty-seven sessions the typical move is 31 per cent of the price. Set that against the spread this course spent a whole module on, and the entire spread cost of the same million shares is between twelve and four hundred and forty-six times smaller than the price risk of needing the time. For size, the spread is a rounding error and the clock is the bill.

Prerequisites: Lesson 44, for the 15.87 and the square root of time, which price every wait below, lesson 53, for the half-spread as the whole of what a round trip costs at the quote, and lesson 56, which measured what an off-exchange print does and does not say, and left the order itself unopened.

Turning the one dial there is

The participation rate is the only thing on the desk that can be changed, so it is worth seeing what changing it buys. Below is the same million shares in the small cap from the table above, at $30 with 45 per cent annual volatility and 800,000 shares a day, run at five rates. The daily standard deviation is 30 times 0.45 divided by 15.87, or $0.8507, and every move below is that figure times the square root of the sessions.

Participation rate	Sessions to finish	Typical move over the span	As a share of the price
5 per cent	25.00	$4.253	14.18%
10 per cent	12.50	$3.008	10.03%
20 per cent	6.25	$2.127	7.09%
33 per cent	3.79	$1.656	5.52%
50 per cent	2.50	$1.345	4.48%

Doubling the rate halves the days and divides the exposure by the square root of two, so going from five per cent to fifty — a tenfold change in aggression, and the difference between a patient schedule and a raid — cuts the typical move only from $4.25 to $1.35. That is a factor of 3.16, which is the square root of ten, and it is the whole return on being aggressive. Everything aggression buys beyond that is a cost, in the price you push, and this table does not contain it.

Which is the honest shape of the execution problem, and the reason it has a literature. One side of the trade-off falls as the square root of the days and the other rises with something like the participation rate. Trading faster buys you a certain, immediate, growing cost to avoid an uncertain one that shrinks slowly. There is an optimum in there, it depends on numbers a retail reader does not have, and lesson 59 is where the second half of it gets priced.

The part worth carrying away needs no optimisation. An order that is six days of volume cannot be executed on any schedule that makes the timing risk small, because at every rate a desk will actually run it takes weeks, and weeks of a 60 per cent volatility is a third of the price. No algorithm fixes that. The decision that mattered was made when somebody chose to own six days of the thing.

What this does not settle

That average daily volume is a constant. It is a trailing average of a quantity that doubles on news and halves in August, and it is highest exactly on the days a desk would rather not be trading. A schedule set against last month’s figure finishes early on a busy week and late on a quiet one, and the error is not symmetric: the quiet weeks are the ones that hurt, because they extend the very exposure the schedule exists to manage.

That participation is measured the way the table measures it. Ten per cent of yesterday’s volume and ten per cent of a volume that includes your own trading are different numbers. Solving the second gives shares a day of the rate over one minus the rate, so a genuine tenth finishes in 11.25 sessions where the table says 12.50, and a genuine third finishes in 2.54 where the table says 3.79. The table overstates the days by 11 per cent at a tenth and by 49 per cent at a third, so it overstates the exposure too, and by more the harder you push.

That the square root of time holds over twelve sessions. Lesson 44 measured that exact scaling on this course’s own series and found it overstated the true twenty-bar range by more than half again, because the bars leaned on each other. The same objection applies here and in the same direction, so the moves in both tables are more likely too big than too small. The ordering across the five instruments survives it, because every row is scaled the same way; the absolute figures should be read loosely.

That a typical move is a cost. It is not. A standard deviation is symmetric, and half the time the drift over those twelve sessions runs in the buyer’s favour and the order finishes at a better average price than it started. What this lesson has measured is the size of the uncertainty, which is the thing a desk is judged on and the thing that gets a portfolio manager a phone call, not an expected loss. The expected loss is the part that only moves one way, and that part is impact, which lesson 59 prices and this page does not.

That ten per cent is the number. It is a convention with a wide range around it: a desk may run three per cent in a name it is worried about and thirty in one it is not, and the closing auction is a single print a schedule can lean on rather than a rate it has to hold to. The identity in the claim survives any of those choices — the days are the size in days of volume divided by the rate — but the specific sessions in the tables move with the rate you assume, and the second table exists to show by how much.

And the concession that costs this lesson most: it has priced the clock and not the alternative. Every figure here treats the position as given and asks what getting into it costs, which is the execution desk’s question and not the interesting one. The interesting one was settled months earlier by whoever decided to own six days of a micro cap’s volume, and no schedule and no algorithm reverses that decision. So the honest conclusion is that most of what is called an execution problem is a sizing problem in an execution costume, and the arithmetic that would have prevented it — lesson 9’s, applied to liquidity instead of to risk — appears nowhere on this page. It is a real omission and it is not accidental: it is the part that would have made the tables unnecessary.

Problems

1. Put your own position on the days-of-volume scale. Take your largest holding, divide the share count by the instrument’s average daily volume, and multiply by ten. That is roughly how many sessions a desk running a tenth of volume would need to unwind it. For almost every retail account the answer is a fraction of a minute, and knowing that is the point: it tells you which of the last four lessons describe a problem you have. Five minutes, and do it for the least liquid thing you own rather than the largest.
2. Price the clock on a name you actually watch. Take its annual volatility by lesson 44’s four steps, divide by 15.87 for a day, then compute the typical move over one, five and twenty sessions. Compare each with the half-spread. The ratio crosses from spread-dominated to time-dominated somewhere in the first hour, and finding where it crosses for your instrument tells you which costs are worth your attention at your holding period. Twenty minutes.
3. Watch a schedule finish. Pick a mid-cap name and record its volume in each five-minute bucket for one session, then compute what a tenth of each bucket would be. That is the shape of a participation order: a few hundred shares a minute at midday and several thousand into the close. Then look at the actual prints in the quiet middle hours and see how many exceed it. In most names, almost none do. An afternoon, and it converts lesson 56’s argument from a claim into something you have seen.

Sources. André Perold, “The Implementation Shortfall: Paper versus Reality” (The Journal of Portfolio Management, 1988), for the measurement this whole lesson is a component of: the gap between the price when the decision was made and the price actually achieved. Robert Almgren and Neil Chriss, “Optimal Execution of Portfolio Transactions” (Journal of Risk, 2001), for the trade-off between timing risk and impact and for why the answer is a schedule rather than a trade. Albert Kyle, “Continuous Auctions and Insider Trading” (Econometrica, 1985), for why an informed trader spreads the order out at all, which is the model everything above quietly assumes. Michael Barclay and Jerold Warner, “Stealth Trading and Volatility” (Journal of Financial Economics, 1993), for the finding that most of the price movement associated with informed trading arrives in medium-sized trades rather than large ones, which is what a schedule looks like from the outside.

An order of size is a schedule, and the schedule is set by one number: the order divided by the daily volume, times ten. A million shares is a seventh of a session in a large fund and sixty-seven sessions in a micro cap, and over those sixty-seven sessions the typical move is 31 per cent of the price. Against that, the entire spread cost of the same order is between twelve and four hundred and forty-six times smaller. Turning the participation dial from a twentieth of volume to a half cuts the exposure only by the square root of ten, which is the whole prize for aggression and the reason there is an argument about it. None of this is a problem a retail account has, and the use of knowing it is to stop reading a tape of two-hundred-share prints as evidence of anything except a desk running its schedule. Lesson 58 goes to the instruments that schedule is written in: the order types a professional desk actually uses, and what each of them gives away.

Educational only. Trading involves substantial risk of loss. Not financial advice. Past performance does not guarantee future results.