Market Impact & the Square-Root Law

You found a beautiful signal. On paper it earns 30 basis points per trade. So you size it up — bigger position, more profit, obviously. Except the moment you start buying in size, the price runs away from you. The fill you wanted at 100.00 arrives at 100.40, and your 30-bp edge is now a loss. Welcome to market impact: the brutal, non-negotiable tax that the market charges you for the privilege of trading at size.

Impact is the single most important concept in execution, because it is the thing that turns a profitable idea into an unprofitable one as you scale. The good news: it follows a remarkably stable, almost universal pattern — the square-root law — and once you understand its shape, you can predict your costs and size your trades like a professional instead of getting steamrolled.

Temporary vs permanent impact

Analogy. Picture pushing a boat through water. As you move, you throw up a bow wave — water piles up in front, churns, then settles back flat once you slow down. That is temporary impact: a disturbance you create by demanding to move right now, which dissipates once you stop. But if the boat is so heavy that it actually sits lower and raises the waterline of the whole harbor, that change stays even after you dock. That is permanent impact: a lasting repricing of the asset itself.

Definitions.

• Temporary impact is the price concession you pay to obtain immediate liquidity. You are consuming the limit order book — eating through resting sell orders to buy — and pushing the price up the book. The book is finite, so each successive share costs more. But the book refills: market makers and other liquidity providers replenish their quotes once you stop, and the price reverts toward where it was. You paid for immediacy, not for a permanent view.
• Permanent impact is the lasting price move caused by the information your trade leaks. When the market sees persistent one-sided buying, it infers that someone might know something — that fair value is genuinely higher — and re-anchors its quotes upward. This component does not revert, because the market has updated its belief about what the asset is worth.

The total cost of trading splits cleanly into these two pieces:

$\text{total impact} \approx \underbrace{\text{temporary}}_{\text{reverts}} + \underbrace{\text{permanent}}_{\text{stays}}$

When to use which lens

If you are trading a liquid, low-information order (e.g., rebalancing an index fund), your cost is almost entirely temporary, and patience is a near-free lunch — wait, and the book refills around you. If you are trading on a sharp, perishable signal, much of your cost becomes permanent (you are the information), and waiting just lets the price drift away before you finish. Diagnosing the mix decides your whole strategy.

The square-root law

Analogy. Filling a swimming pool from a single hose: the first bucket goes in instantly, but to fill the whole pool you wait far longer than “one bucket times pool-size” — yet not catastrophically longer, because the work grows smoothly, not explosively. Impact behaves the same way: bigger orders cost more, but with diminishing marginal pain, not runaway escalation.

Definition / formula. Across markets, asset classes, and decades, the cost of executing an order of size $Q$ is well approximated by:

$\text{impact} \approx Y\,\sigma\,\sqrt{Q/V}$

where $\sigma$ is the asset’s daily volatility, $V$ is the average daily volume (ADV), $Q$ is your order size (same units as $V$, e.g. shares), and $Y$ is a dimensionless constant of order one ($Y \approx 0.5\text{–}1$ empirically). The headline feature is the square root: the function is concave. Doubling $Q$ multiplies impact by only $\sqrt{2} \approx 1.41$, not 2.

Worked example. Take $\sigma = 2\%$ per day, an order equal to $10\%$ of ADV (so $Q/V = 0.10$), and $Y \approx 1$:

$\text{impact} \approx 1 \times 0.02 \times \sqrt{0.10} = 0.02 \times 0.3162 \approx 0.00632 \approx 63\ \text{bps}.$

Now double the order to $20\%$ of ADV. A naive linear thinker expects 126 bps. Reality:

$\text{impact} \approx 1 \times 0.02 \times \sqrt{0.20} = 0.02 \times 0.4472 \approx 0.00894 \approx 89\ \text{bps}.$

Doubling the size raised cost from 63 to 89 bps, a factor of $\sqrt{2} \approx 1.41$ — not 126 bps. The square root just saved you 37 bps relative to the linear guess.

To make the $\sqrt{\cdot}$ shape unmistakable, here is a table at sizes chosen to be perfect squares of ADV (with $\sigma = 2\%$, $Y = 1$, so impact $= 0.02 \times \sqrt{Q/V}$):

Read down the √ column: the perfect squares give the clean sequence 0.1, 0.2, 0.3, 0.4, 0.5, so impact marches up in tidy 20-bp steps even as size leaps 1 → 4 → 9 → 16 → 25. Size grows 25×; impact grows only 5×. (The “naive linear guess” column uses a tiny per-unit cost to show the shape mismatch — at small sizes linear wildly under-charges, which is exactly why a linear model lulls you into oversizing.)

Why concave, intuitively

You don’t need heavy math to feel why the curve bends down rather than up. Three forces conspire:

• Liquidity replenishes as you trade. You are not draining a fixed tank. While you work the order over minutes and hours, market makers post fresh quotes and natural sellers arrive. So the average price you pay rises far more slowly than if you had to clear the entire book in one gulp — the denominator of “how much liquidity met me” keeps growing.
• Informed-trading equilibrium. In the theory (Kyle-style models), a rational informed trader and the liquidity providers reach a balance where price responds to order flow but does not over-react: the market knows some flow is uninformed noise, so it only partially updates per share. That partial, diffusing response is exactly what produces a sub-linear, concave aggregate impact.
• The first shares aren’t actually cheap. Crucially, concave does not mean “small orders are free.” The √ curve is steepest at the origin — the very first slice already moves the price. What flattens is the marginal cost of each additional unit. So small orders are surprisingly expensive per share, and large orders are surprisingly cheap per extra share. That asymmetry is the law’s signature.

Implications: capacity & order splitting

Here is where the law stops being trivia and starts deciding whether your fund makes money. Because impact grows like $\sqrt{Q}$ but you are spreading a fixed alpha across that size, the cost per dollar traded rises as you scale — and at some point it eats your edge entirely. This caps your capacity: the maximum size you can trade before the strategy stops being profitable.

Analogy. Your alpha is a fixed-size cake. The more guests (dollars) you invite, the thinner each slice. With square-root impact, the cake doesn’t just get sliced thinner — the knife itself gets more expensive to swing. Beyond a certain headcount, the cost of serving exceeds the cake.

Worked example — solving for max size. Suppose your signal is worth a gross 30 bps per trade. Your edge survives only while impact stays below 30 bps. Using $\sigma = 2\%$, $Y = 1$, set impact equal to alpha and solve for $Q/V$:

$0.0030 = 1 \times 0.02 \times \sqrt{Q/V} \;\Rightarrow\; \sqrt{Q/V} = \frac{0.0030}{0.02} = 0.15 \;\Rightarrow\; Q/V = 0.15^2 = 0.0225.$

So your break-even size is about 2.25% of ADV. Trade more than that and impact exceeds your 30-bp edge — you are paying the market for the right to lose money. (And that’s the break-even point; you’d actually cap well below it to keep net edge positive.)

Notice capacity scales sub-linearly with alpha too: because impact is √-shaped, doubling your alpha to 60 bps quadruples the tradable size (√ inverts to a square), while halving it to 15 bps cuts capacity to a quarter. Edge is precious precisely because capacity rewards it quadratically.

Putting it together

Market impact is the cost of demanding that the market do something for you now and in size. It comes in two flavors — temporary (reverts, you paid for immediacy) and permanent (stays, you leaked information) — and its total magnitude follows the concave square-root law, impact ≈ Y·σ·√(Q/V). That concavity is the master key: it makes small orders surprisingly costly per share, large orders surprisingly cheap at the margin, and every strategy’s capacity strictly finite.

You now know how much trading costs and why it caps your size. The obvious next question is what to do about it: if one big order is brutally expensive but many small slices ride the cheap, flat part of the √ curve, how exactly should you carve the order up and dribble it into the market over time? That is the job of execution algorithms — VWAP, TWAP, implementation shortfall, and the participation schedules that turn the square-root law from a tax into a plan. On to the next lesson.