Market Impact Models and the Square Root Law

Impact scales with the square root of participation rate, so a 20 percent order costs about double a 5 percent one, not four times as much.

Market impact models describe how much the price of a stock moves in response to an order trading against it, and the central empirical finding across decades of research is that impact grows with the square root of participation rate rather than in a straight line with order size. That single relationship explains why doubling an order rarely doubles its cost.

Impact splits into two pieces that behave differently once trading stops. Temporary impact is the extra cost paid to consume liquidity while the order is actively trading, and it decays once the order finishes as the book refills. Permanent impact is the portion of the price move that persists after trading ends, reflecting information the market extracted from the fact that a large order traded at all.

Almgren and Chriss formalized this split in their work on optimal execution, framing the trader's problem as a trade-off between trading fast, which raises temporary impact, and trading slow, which raises exposure to price risk and opportunity cost over a longer horizon.

The square root law holds that expected market impact is proportional to the square root of the ratio between order size and the volume that trades over the execution horizon, often written informally as impact scaling with the square root of the participation rate rather than with the participation rate itself. Empirical work across equity markets, including research associated with Jean-Philippe Bouchaud and coauthors, has found this relationship holding remarkably consistently across different stocks, time periods and asset classes.

Suppose trading a stock at a 5 percent participation rate is observed to cost 20 basis points of impact. Under the square root law, moving to a 20 percent participation rate, four times the size relative to volume, does not multiply the cost by four. The ratio of impact is the square root of the ratio of participation rates, the square root of 20 divided by 5, which is the square root of 4, which equals 2.