A stock has 3% daily volatility and 20m shares of average daily volume. Under an impact model of 0.5*sigma*sqrt(Q/V), what is the largest single day order whose impact stays under 25 basis points, and how does that capacity move with the cost budget?

A stock has 3% daily volatility and 20m shares of average daily volume. Under an impact model of 0.5*sigma*sqrt(Q/V), what is the largest single day order whose impact stays under 25 basis points, and how does that capacity move with the cost budget?

Approach: Invert the square root law for Q rather than evaluating it, then read the exponent on the cost budget out of the inverted form.

555,556. The largest order is 555,556 shares. Setting 0.5*sigma*sqrt(Q/V) equal to the budget c gives sqrt(Q/V) = c/(0.5*sigma), which here is 0.0025/0.015, or 1/6, so Q/V = 1/36 and Q = 20,000,000/36, which is 555,556 shares, about 2.8% of the daily volume. Inverting the law leaves Q = V * (c/(0.5*sigma))^2, so capacity is linear in volume and quadratic in the cost budget: allowing 50 basis points instead of 25 buys four times the size, while a stock at twice the volatility carries a quarter of it. The practical reading is that a signal worth 25 basis points cannot be traded at 25 basis points of impact, since the whole edge goes to the market, so the size that matters is the one where impact is a fraction of the forecast and the rest is the profit.

Follow-up: How does that capacity change if the signal decays over five days instead of one?

Key concepts: square root law, participation rate, capacity, cost budget, market impact.