What is the square root law of market impact and how is it used?

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AcadiFi · Accepted Answer

The square root law of market impact posits that the price impact of a trade scales approximately with the square root of trade size relative to average daily volume (ADV): Impact (bps) ≈ Y × σ × √(Q / ADV) where Q is order size, ADV is average daily volume, σ is daily volatility, and Y is a stock-specific coefficient (empirically around 0.5–1.0 for US equities). Example: stock with σ = 2% daily vol and ADV = 5 million shares. A trader named Viktoriya Mielnikova wants to execute 500,000 shares (10% ADV). Impact ≈ 0.7 × 200 bps × √0.10 ≈ 0.7 × 200 × 0.316 ≈ 44 bps. Empirical origins: Loeb, Kyle (1985), Bouchaud, Almgren, and others documented the relationship across asset classes and decades — it's remarkably robust across equities, futures, FX, and even crypto. ```mermaid flowchart LR A[Order Size Q] --> B[Impact ~
sqrt Q/ADV] C[Daily Vol sigma] --> B D[Coef Y] --> B B --> E[bps Cost] ``` **Practical uses**: 1. **Pre-trade cost estimation** — plug in expected size to get a bps impact estimate for TCA and pre-trade analytics. 2. **Capacity analysis** — strategy capacity limits are often derived by capping impact cost at a fraction of expected alpha. Square-root implies 4× size ≈ 2× cost — gentler than linear. 3. **Scheduling** — splitting an order into smaller child orders reduces impact proportionally to √N. 4. **Position sizing for illiquid books** — firms cap daily traded % ADV at ~5–20% to keep impact manageable. **Caveats**: - Breaks down for extreme sizes (>30–50% ADV). - Coefficient Y varies by liquidity regime — crisis days can see 2–3× normal Y. - Doesn't capture path dependence or news-driven regime shifts. For FRM, the law motivates why liquidation-horizon assumptions matter: a portfolio liquidated in one day vs. five days has dramatically different cost profiles under square-root scaling.

What is the square root law of market impact and how is it used?

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