A stock has 2% daily volatility and 10m shares of average daily volume. Using the square root law with a coefficient of 0.5, estimate the impact of buying 400,000 shares in a day in basis points, and say how total cost scales if the order is quadrupled.

A stock has 2% daily volatility and 10m shares of average daily volume. Using the square root law with a coefficient of 0.5, estimate the impact of buying 400,000 shares in a day in basis points, and say how total cost scales if the order is quadrupled.

Approach: Apply impact = Y * sigma * sqrt(Q/V) to get the price move, then multiply impact by size to see how the dollar cost scales with the order.

20. The participation rate is Q/V = 400,000/10,000,000 = 0.04, so sqrt(Q/V) = 0.2 and the impact is 0.5 * 0.02 * 0.2, or 0.002, which is 20 basis points of the price. Quadrupling the order to 1.6m shares doubles the impact to 40 basis points, and because total cost is impact times size, the dollar cost rises by a factor of eight: under the square root law cost scales as Q^1.5 in the fraction of average daily volume. That convexity in size is why large orders are split across several days instead of pushed through in one session.

Follow-up: The same order is worked over five days rather than one. What happens to total impact cost under the square root law, and what risk does the slower schedule add?

Key concepts: square root law, market impact, participation rate, average daily volume.