What Hurry Costs

Market impact is not a penalty for being large. It is the price of being in a hurry, and the standard estimate makes that precise: the move you cause is your instrument’s daily volatility times the square root of your order as a fraction of the day’s volume. Run it backwards and it says how much you can trade before the impact reaches a tenth of a per cent. On the five instruments lesson 57 costed, that threshold is 358 million dollars in the exchange-traded fund and 1,050 dollars in the micro cap — a span of 341,000 to one against a dollar-volume span of 24,300 to one. The extra factor of 14 is exactly the square of the ratio of their volatilities. Then set the impact against the clock lesson 57 priced, and the optimum has no volatility in it at all: the best number of sessions is the square root of the order’s size in days of volume, divided by how much certain cost you will pay to shed a dollar of risk.

Prerequisites: Lesson 11, which showed that a market order larger than the depth in front of it walks the book and pays for every level it eats, lesson 57, for the schedule, the participation rate and the timing exposure that grows with the square root of the sessions, and lesson 12, for the habit of dividing any cost by what the instrument moves in a day before deciding whether it is large.

The two terms are equal at the bottom

Take the small cap and watch the sum. A million shares at 30 dollars, 800,000 shares a day, 45 per cent a year, which is 2.84 per cent and 85 cents a day. The order is 1.25 days of volume. Worked over half a session the impact is 1.34 a share and the exposure 0.60, for 1.95. Over one session, 0.95 and 0.85, for 1.80. Over two, 0.67 and 1.20, for 1.88. Over five, 0.43 and 1.90, for 2.33. Over the 12.5 sessions the ten per cent convention prescribes, 0.27 and 3.01, for 3.28.

The bottom sits at 1.118 sessions, which is the square root of 1.25, and the total there is 1.798 a share. Notice what the two terms are doing at that point: impact is 0.899 and exposure is 0.899. They are equal, and not by construction of these particular numbers.

That equality is the useful part, because it survives when the constants do not. Whatever your volatility, whatever your volume, whatever your willingness to pay for certainty, the schedule that minimises the sum is the one where the impact you pay equals the risk-weighted exposure you carry. You do not need to know either constant to use it. Estimate both terms for the schedule you are actually running, compare them, and the larger one tells you which way to move: more impact than exposure means you are going too fast, more exposure than impact means too slow.

Apply it to the convention and it comes out consistent, which is the honest way to end. At 12.5 sessions the impact is 0.269 and the raw exposure is 3.007. Those are nowhere near equal, so 12.5 sessions is not the bottom at a dollar for a dollar — it costs 1.478 a share more, 82 per cent above the minimum. Weight the exposure by 0.089, the willingness the convention implies, and it becomes 0.269. Equal again. The convention is optimal, for a desk that values a dollar of timing risk at nine cents. Whether your desk does is a question about your desk, not about the market.

What this does not settle

That the coefficient is one. The law as written here puts no constant in front of the volatility term, and published calibrations put one there, generally somewhere between a half and one and a half depending on the market, the period and whose data was used. Every dollar figure in the two tables scales linearly with it, so a coefficient of 0.6 makes the fund’s threshold nearly 1 billion and the micro cap’s nearly 2,900. What does not scale is the ratio between rows, or the identity that the extra factor of 14.06 is the square of the volatility ratio, or the shape of the trade-off. Read the tables for their spread, not for their levels.

That the willingness number is a measurement. It is not. It is the one place in the whole calculation where somebody’s preference enters, and this page has been deliberate about keeping it visible rather than folding it into a constant. The published treatments call it risk aversion and then choose it, and the choice does the same work as the choice of 12.5 sessions it is supposed to justify. Nothing here tells you what yours should be.

That impact is a cost you pay and get back. Part of it is temporary and reverts once you stop, and part of it is permanent, because the market reads sustained one-way flow as somebody knowing something. The estimate here does not separate them, and the separation matters to anybody who is going to hold the position, since the permanent part is a worse entry price and the temporary part is a mark you will see reverse. Everything on this page is the total.

That the volatility to use is the historical one. The two tables use trailing annual figures divided by the square root of 252, which is the number you can get. The number the law actually wants is the volatility over the window you will be trading in, and those come apart precisely when it matters, which is around earnings, index changes and the days somebody else is working the same name. Lesson 44’s treatment of volatility as a quantity applies to this input as much as to any other, and lesson 43 names the days on which the trailing figure is exactly the wrong one.

That five instruments make a distribution. They do not. They were chosen in lesson 57 to span a range rather than to represent one, and the 341,000 to one figure is a statement about the endpoints of a chosen range, not about the market. The identity underneath it — that the threshold ratio exceeds the volume ratio by the square of the volatility ratio — is exact and holds for any two instruments you like. The 341,000 is an illustration of it.

And the concession that costs this lesson most: it has priced the trade and not the decision. Every number here assumes the position is worth taking and asks only what taking it costs. But the size at which impact eats an edge is the size at which the position should be smaller, and that is a portfolio question this page does not touch. A trader who reads the small cap row as an execution problem will spend an afternoon on algorithms. A trader who reads it as a sizing constraint will spend ten minutes and get further. The arithmetic above cannot tell you which reading is yours, and the second one is right more often than the first.

Problems

1. Find the size at which you become the large trader. For the three instruments you trade most, write down the average daily volume in dollars and the daily volatility as a percentage. Divide 0.1 by the volatility, square it, and multiply by the dollar volume. That is the position at which your estimated impact reaches a tenth of a per cent. Compare it with the position you actually take. Ten minutes, and for at least one of the three the answer will be uncomfortable.
2. Measure the daily volatility rather than assuming it. Take sixty daily closes, compute the bar-to-bar percentage changes, and take their standard deviation. On this course’s series that comes to 1.50 per cent a bar, which annualises to 23.7 per cent. Do it for the same three instruments and check the answers against whatever figure your platform displays, because platforms differ on the window and on whether they annualise. Half an hour, and every number in the first problem depends on getting this one right.
3. Test the equality on your own schedule. Take an order you have actually worked in pieces. Compute the impact term for the number of sessions you used, then the exposure term, then their ratio. If impact is the larger you were going too fast; if exposure is the larger you were going too slow; if they are close you were near the bottom for whatever willingness you were implicitly using. An afternoon, and unlike the tables above it uses your instrument, your size and your habits.

Sources. Albert Kyle, “Continuous Auctions and Insider Trading” (Econometrica, 1985), for permanent impact as the market pricing the information it infers from order flow, which is the term this page folds into a total and does not separate. Robert Almgren and Neil Chriss, “Optimal Execution of Portfolio Transactions” (Journal of Risk, 2000), for the trade-off between impact and timing risk and for the risk-aversion parameter this page calls a willingness and refuses to choose for you. Robert Almgren, Chee Thum, Emmanuel Hauptmann and Hong Li, “Direct Estimation of Equity Market Impact” (Risk, 2005), for the calibration on real order data, which is where the coefficient the first concession complains about comes from. Jean-Philippe Bouchaud, Doyne Farmer and Fabrizio Lillo, “How Markets Slowly Digest Changes in Supply and Demand” (Handbook of Financial Markets, 2009), for the square-root exponent across instruments and decades, and for the competing explanations of why it should be a square root at all.

Impact is the price of speed. The estimate is the daily volatility times the square root of your order as a share of the day’s volume, and inverting it says how large you can be before it bites: 358 million dollars in an index fund, 29,900 in a small cap, 1,050 in a micro cap, with the span between the ends exceeding the volume span by exactly the square of the volatility ratio. On this course’s own series, whose bars carry a 1.50 per cent standard deviation, the threshold arrives at 0.45 per cent of a day. Set that against lesson 57’s clock and the optimum drops both the volatility and the price, leaving the square root of the order’s days of volume over a willingness number that is a preference rather than a measurement — and at that optimum the impact paid and the exposure carried are exactly equal, which is a test you can run without knowing either constant. Lesson 60 turns to a set of participants whose flow is not a choice at all: dealers who are short options and have to buy strength and sell weakness, all day, whatever they think.

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