Square-root law of market impact

What it means

A larger stock order can move the market price as it is executed, but impact does not necessarily rise in direct proportion to order size. The square-root law is a commonly used empirical approximation: holding other inputs fixed, an order four times as large has roughly twice the estimated impact, not four times. Actual outcomes can differ substantially.

The rule concerns market impact—price movement associated with executing the order—not every cost of trading. It does not, by itself, account for the bid-ask spread, commissions or subsequent price changes unrelated to the order.

How it is calculated

One common form is I = Y × σ × √(Q / V).

• I is estimated impact as a fraction of the stock price; multiply by 10,000 to express it in basis points (bps).
• Y is a dimensionless calibration coefficient. There is no universal value; its choice depends on the model, data and definition of impact.
• σ is daily price volatility expressed as a decimal, such as 0.02 for 2%.
• Q is the number of shares in the order.
• V is average daily trading volume in shares per day. Q / V is the order’s size relative to one typical day’s volume.

The formula estimates an impact magnitude under specified assumptions, not an exact fill price. A model must also specify what its estimate measures—for example, average execution impact or the price displacement around completion—because those are different quantities. Using an order’s dollar value for Q and share volume for V would make the ratio invalid; use matching share units, or matching dollar-volume units.

Worked example

Suppose, hypothetically, a stock trades at US$50 per share, has average daily volume of 1,000,000 shares and daily volatility of 2%. For illustration only, set Y = 1.

1. An order for 10,000 shares is 10,000 / 1,000,000 = 0.01, or 1% of average daily volume.
2. Its estimated impact fraction is 1 × 0.02 × √0.01 = 0.002: 0.20%, or 20 bps. At a US$50 reference price, 0.20% is US$0.10 per share as a price-equivalent impact magnitude—not a promised fill or a complete cost estimate.
3. Increase the order to 40,000 shares, or 4% of average daily volume. The estimate becomes 1 × 0.02 × √0.04 = 0.004: 0.40%, or 40 bps.

The order is four times larger, while the estimated impact per share is twice as large. The result depends on the illustrative coefficient and assumptions, not just the arithmetic.

Why it matters before you trade

Order size relative to available trading volume can affect execution costs. A pre-trade check can put a proposed quantity beside average daily volume and volatility, then consider whether the resulting impact estimate is material alongside the spread and other costs. It can also compare quantities under the same assumptions without implying that the eventual market will behave exactly as modeled.

Average daily volume is an imperfect stand-in for liquidity at the moment of execution. Intraday volume, order-book depth, news, trading halts and execution speed can all matter. Splitting an order across time may change its participation rate and impact, but it also changes exposure to price movements while the order remains unfinished.

How PreTrAIde uses it

PreTrAIde runs pre-trade analysis on a user-built stock basket before an order is placed. Its quantitative layer estimates market impact using an Almgren-Chriss-style model and measures participation in daily volume; it also estimates spread and volatility. Its trading cost calculator covers spread plus market impact.

The product facts do not say that PreTrAIde implements this square-root formula specifically. The connection is the pre-trade question it illustrates: how an order’s size and a stock’s trading conditions may affect estimated execution costs. PreTrAIde provides decision support; it does not place orders.

Common mistakes

• Treating the estimate as a guarantee. The coefficient and market conditions can change; realized impact may be higher or lower.
• Confusing impact with total trading cost. Spread, commissions and other sources of slippage need separate consideration.
• Mixing units or time horizons. Share orders need share-volume inputs, and daily volatility must match the model’s daily-volume convention.
• Assuming the rule holds at every size. Tiny orders, unusually large orders and stressed markets may not follow the approximation.