The Square-Root Impact Law

Of all the empirical regularities in finance, few are as robust, or as theoretically puzzling, as the square-root law of market impact. Measure how much a large "metaorder" (a parent order sliced into many child trades) moves the price, and across equities, futures, FX, and even crypto, across decades and across the size of the participant, the impact scales as the square root of the order's size relative to daily volume. It is the empirical fact that Market Impact models must reproduce and that Optimal Execution must price.

The law

For a metaorder of size $Q$ executed in a market with daily volume $V$ and daily volatility $\sigma$, the average impact, the price move from the start of execution to its end, is

$\boxed{\;\frac{\Delta P}{P} \;\approx\; Y\,\sigma\,\left(\frac{Q}{V}\right)^{\delta}, \qquad \delta \approx \tfrac12,\;}$

where $Y$ is a dimensionless constant of order one (empirically often $0.3$–$1$) and the exponent $\delta$ is consistently measured near $0.5$ (typically $0.4$–$0.6$). The dependence on $\sigma$ makes the impact dimensionally a price move; the dependence on the participation ratio $Q/V$ is the heart of it. Note the surprises baked in:

• Concavity. Because $\delta < 1$, impact per share decreases with size, doubling the order does not double the impact, it multiplies it by $\sqrt{2} \approx 1.41$. Small orders are expensive per share; large orders enjoy economies of scale in impact.
• Rate-independence (approximately). To first order the total impact depends on $Q/V$, not strongly on how fast within a day you trade, a striking and much-debated feature.
• Universality. The same functional form and similar $Y$ hold across wildly different assets and eras. This is what makes the law feel like a law rather than a fit.

Why square root? Three arguments

The concave, near-one-half exponent is not obvious, Kyle's model, recall, predicts linear impact. Several complementary explanations converge on the square root.

1. Latent liquidity (locally linear supply). Model the true, mostly-hidden supply/demand as a latent order book whose density of resting volume near the current price grows linearly with distance from the mid, $\rho(p) \propto |p - p_0|$ (a "V-shaped" latent book that is locally empty at the touch). To absorb a metaorder of $Q$ shares you must eat into the book until the cumulative volume equals $Q$: $\int_0^{\Delta P}\rho(p)\,dp \propto (\Delta P)^2 = Q$, hence $\Delta P \propto \sqrt{Q}$. The square root falls out of a linearly vanishing liquidity profile, the market is locally thin and gets deeper as you push, so pushing twice as far absorbs four times the volume.

2. Propagator / no-dynamic-arbitrage. In the transient-impact Market Impact propagator $m_t = \sum_{s<t}G(t-s)\epsilon_s f(v_s)$, order-flow signs are strongly autocorrelated ($C(\ell)\sim \ell^{-\gamma}$) because metaorders are split. For the price to stay a martingale (no predictable trend from predictable flow), the propagator must decay as a matched power law $G(\ell)\sim \ell^{-\beta}$ with $\beta = (1-\gamma)/2$. Summing the decaying impact of a persistent stream of same-signed child trades yields aggregate impact that grows sublinearly, concave, and near square-root for realistic autocorrelation exponents. Concavity is the market's device for reconciling persistent flow with unpredictable prices.

3. Fair-pricing / information equilibrium (Farmer et al.). If informed traders' order sizes are power-law distributed and the market prices each metaorder so that its impact equals the informational content that a rational counterparty would infer, the martingale condition again forces impact to be concave in size, with the square root emerging as the natural exponent. Impact is "fair" in that it just recovers the information the trade reveals.

That three very different starting points, a geometric latent book, a dynamic no-arbitrage condition, and an information-equilibrium argument, all land near $\delta = \tfrac12$ is why the law is taken so seriously.

Worked example

A fund must buy $Q = 500{,}000$ shares of a stock with ADV $V = 5{,}000{,}000$ shares ($Q/V = 10\%$ participation) and daily volatility $\sigma = 150$ bps. Take $Y = 0.5$. Then

$\frac{\Delta P}{P} \approx 0.5 \times 0.015 \times \sqrt{0.10} = 0.5 \times 0.015 \times 0.316 = 0.00237,$

about 24 bps of impact. Now compare execution plans. If instead the fund trades a smaller child participation but the total $Q/V$ is unchanged, the square-root law says the total impact is roughly the same, impact is set by the metaorder's total footprint, not the slicing. But push the total order to $Q/V = 40\%$ and impact rises only to $0.5\times 0.015\times\sqrt{0.4} = 47$ bps, four times the size for twice the cost. This concavity is exactly why large funds can move surprisingly large blocks and why splitting an order across days (lowering each day's $Q/V$) reduces impact more than proportionally.

Failure modes and caveats

• Impact decays, the peak is not the mark. The square-root value is measured at the end of execution; afterward a substantial fraction (often 1/2 to 2/3) reverts as temporary impact relaxes. The permanent impact is smaller and closer to linear. Confusing peak impact with permanent cost overstates permanent price moves.
• Calibration is unstable. $Y$ and even $\delta$ drift with regime, liquidity, and the metaorder-detection methodology; the law is a robust shape, not a precise cost calculator.
• Selection bias. Metaorders are placed because the trader expects a move; disentangling impact from alpha inflates measured impact. Careful studies condition on this.
• Breakdown at extremes. For tiny orders the spread dominates and the law is irrelevant; for enormous orders ($Q/V$ approaching one) liquidity is exhausted and impact steepens beyond square root.
• Linear models still used. The Almgren-Chriss Model assumes linear temporary impact for tractability; this contradicts the square-root law and overstates the cost of large slices. Practitioners often patch in a concave impact term (a "power-law Almgren–Chriss").

In interviews

Know the formula $\Delta P/P \approx Y\sigma\sqrt{Q/V}$ and, crucially, be able to give a mechanism for the square root, the latent-order-book argument is the cleanest: a V-shaped (linearly vanishing) liquidity density means absorbing $Q$ shares pushes the price by $\sqrt{Q}$ since cumulative volume grows as $(\Delta P)^2$. Be ready to state the two headline consequences: impact is concave (economies of scale, quadrupling size only doubles impact) and roughly rate-independent in total. A sharp follow-up contrasts this with Kyle's linear $\lambda$, the reconciliation is that Kyle prices a single auction while the square-root law aggregates a persistent, split metaorder through a transient (propagator) impact, which is covered in Market Impact.