Measuring Impact With Real Orders: Almgren, Thum, Hauptmann and Li

The optimal execution literature has a dirty secret. Every model, from Almgren and Chriss onward, tells you how to trade given a market impact function. Not one of them tells you what that function actually is. Plug in the wrong impact model and your beautifully optimal schedule is optimal for a market that does not exist.

So somebody had to go and measure it. Robert Almgren, one of the authors of the model that most needed the numbers, teamed up with Chee Thum, Emmanuel Hauptmann and Hong Li at Citigroup to do exactly that, using something the academic world almost never gets: a large database of real institutional orders where they knew which side the client was on.

The problem: you cannot measure impact from public data

Here is why this is hard, and why it had not been done properly before.

Public market data tells you that a trade happened, at what price, for how many shares. It does not tell you whose order it was part of, or which direction they were going. You see a print. You do not see that it was slice 47 of a 300,000-share buy order that a pension fund started working two hours ago.

That matters enormously, because market impact is a property of the parent order, not of the individual print. The thing a portfolio manager cares about is: I decided to buy 300,000 shares, I worked the order over four hours, how much did the price move against me because of my own trading?

To answer that you need to know:

• The full size of the parent order, not just the individual slices.
• The direction, so you know whether a price rise was for you or against you.
• The duration, because trading the same size over one hour is a completely different animal from trading it over a day.

Public tape data gives you none of this. Broker data gives you all of it. The authors had roughly 700,000 real orders executed by a major equity desk over an eighteen-month period, with the direction known.

The key idea via analogy: the wake behind a boat

Picture a boat crossing a lake, and think about the disturbance it leaves.

There are two distinct effects, and the paper's central move is to insist on separating them.

The bow wave, the water piled up right in front of the boat as it pushes forward. This exists only while the boat is moving and only because of how fast it is going. Stop the boat and the bow wave collapses immediately. This is temporary impact: the extra price you pay for demanding liquidity right now, which depends on your rate of trading. Push harder, pay more.

The wake, the disturbance that spreads out behind the boat and lingers on the lake long after the boat has gone. This depends on how much water the boat displaced in total, not on how fast it was going. This is permanent impact: the lasting shift in the price because the market has concluded that somebody with conviction was buying, and has revised its view accordingly. It depends on your total size, not your speed.

The empirical model the authors fit reflects exactly this structure.

• Permanent impact depends on total order size relative to the stock's normal daily volume, and it does not care how quickly you traded.
• Temporary impact depends on your participation rate, the fraction of the market's volume you represent while you are active. Trade a big fraction, pay a lot.
• Everything scales with volatility. A volatile stock has a wider, more fragile order book and punishes you harder for the same participation.

The headline empirical result concerns the shape of the temporary piece. Rather than the linear relationship the theory models assumed, or the exact square root the physicists championed, the authors' fitted exponent came out at three-fifths, sitting between the two. It is decidedly concave: trading twice as aggressively costs meaningfully less than twice as much.

The practical value of this is enormous. The whole model is expressed in terms of a small handful of things you can look up before you trade: order size, daily volume, volatility, shares outstanding, and how long you plan to take. That is a pre-trade cost estimator you can actually run.

Why it mattered

• It plugged the hole in the optimal execution literature. Almgren and Chriss had given the industry a framework with unknown parameters. This paper supplied the parameters, from real data, and did so in a form that fed straight back into the framework.
• It became a standard broker cost model. Pre-trade transaction cost analysis, the little estimate your broker gives you saying "this order should cost you around 18 basis points," is built on models of essentially this shape.
• It confirmed concavity with clean, direction-labelled institutional data. The econophysics crowd had been arguing for concave impact using public trade data with inferred signs. This paper reached a similar qualitative conclusion using data where the signs were known, which is a much stronger footing.
• It disciplined the temporary-versus-permanent split empirically. Lots of models postulate the split. This one measured both pieces and showed they depend on genuinely different variables, size for one and rate for the other.

The honest limitations

• The data is one broker's flow, and that flow is not random. These are the orders that arrived at one desk. If that desk's clients are systematically different from the market as a whole, or if clients send their easy orders here and their hard orders elsewhere, the estimates are biased in ways that are impossible to check from inside the sample.
• Selection bias runs deep. Orders that get worked to completion are not a random sample of orders. Traders cancel when things go badly. Measuring impact on the surviving orders systematically flatters the numbers.
• Separating your impact from the market's noise is genuinely hard. During the four hours you were buying, the stock moved for a hundred reasons. Attributing part of that move to you requires assumptions, and reasonable people make different ones and get different exponents.
• The three-fifths exponent is not a law of nature. It is what fell out of this dataset, in this period, with this specification. Other researchers on other data get other numbers. The robust finding is concavity, not the specific value.
• The data is from the early 2000s. That market had wider spreads, less electronic fragmentation, and no meaningful high-frequency market making. The structural conclusions have held up better than the numerical ones.

The one-line takeaway

Almgren, Thum, Hauptmann and Li took roughly 700,000 real institutional orders with known direction and measured market impact instead of theorising about it, confirming that impact splits cleanly into a permanent piece driven by total size and a temporary piece driven by how aggressively you trade, and that the temporary piece is distinctly concave.