The square-root impact formula worked on a concentrated position, with the trade-off between impact cost and timing risk and the mistakes that understate it.
The square-root law estimates the cost of a large trade as cost per unit = Y × daily volatility × √(size ÷ daily volume), with Y a constant of order one calibrated to your own fills. Selling an illustrative $60m position worth five days of volume in one session, with 2.0 per cent daily volatility and Y of 0.8, costs about 3.58 per cent, or $2.15m. Spread over 25 days at 20 per cent of volume, impact falls to 0.72 per cent ($0.43m), but the fund carries the price risk for five weeks.
Worked in full in Financial Risk Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A quoted bid-ask spread is good for a quoted size. A fund that has to sell several days of a security's volume moves the price, and the move is the largest liquidity cost it will pay. Liquidity risk reports that stop at the spread understate it by an order of magnitude for concentrated positions. The square-root law is the standard first estimate: an empirical regularity, documented across equity and futures markets, that impact grows with the square root of size relative to volume, not in proportion to it.
| Input | Value |
|---|---|
| Position to sell (X) | $60,000,000 |
| Average daily volume (V) | $12,000,000 |
| Position in days of volume | 5 |
| Daily volatility (σ) | 2.0% |
| Impact constant (Y) | 0.8 |
| Quoted bid-ask spread | 20 bp |
Cost per unit = Y × σ × √(Q ÷ V) = 0.8 × 2.0% × √5 = 0.8 × 2.0% × 2.236 = 3.58% (358 bp)
Cost = 3.58% × 60,000,000 = $2,146,625
Excel: =Y*Sigma*SQRT(Q/ADV)*Q
Crossing half the quoted spread on the same position would cost $60,000. The impact estimate is 36 times larger, and it is the number a redemption actually pays.
Selling Q ÷ N a day: cost per unit = Y × σ × √(Q ÷ (N × V)), so total impact falls with √N
At 20% of daily volume, N = 25 days: 0.8 × 2.0% × √0.20 = 0.72%, or $429,325
Timing risk, one standard deviation = σ × X × √Σ(average holding on day k ÷ X)²
The saving is $1,717,300, a factor of 5.0, the square root of 25. The price is exposure: the position is still mostly held while the market moves. Assuming an even sale each day, so that on day k the fund is exposed on average to X × (1 − (k − 0.5) ÷ N), the one-standard-deviation price risk over the programme is $3,463,409, against $600,000 when everything goes in a day. The sketch assumes each day's impact has decayed by the next morning; in practice part of it persists, so the multi-day saving is somewhat smaller than shown.
| Days | Share of daily volume | Impact per unit | Impact cost ($m) | Timing risk, 1 sd ($m) |
|---|---|---|---|---|
| 1 | 500% | 3.58% | 2.15 | 0.60 |
| 2 | 250% | 2.53% | 1.52 | 0.95 |
| 5 | 100% | 1.60% | 0.96 | 1.54 |
| 10 | 50% | 1.13% | 0.68 | 2.19 |
| 25 | 20% | 0.72% | 0.43 | 3.46 |
Impact cost is an expected loss; timing risk is a dispersion around zero. That is the trade-off formalised by optimal execution models such as Almgren and Chriss: a risk-neutral seller trades slowly, a seller who must meet a redemption with certainty, or who expects the price to keep falling, trades faster and pays for it. The table gives both sides on one line so that the choice can be made explicitly rather than left to the trading desk on the day.
| Position ($) | Days of volume | Impact per unit | Impact cost ($) |
|---|---|---|---|
| 30,000,000 | 2.5 | 2.53% | 758,947 |
| 60,000,000 | 5.0 | 3.58% | 2,146,625 |
| 120,000,000 | 10.0 | 5.06% | 6,071,573 |
| 240,000,000 | 20.0 | 7.16% | 17,173,002 |
Doubling the position raises the cost per unit by 41 per cent (√2 = 1.41) and the total cost 2.83 times (2 to the power 1.5). Concentration is therefore penalised more than proportionally, which is why position limits expressed in days of volume are more useful than limits in per cent of NAV.
Y matters as much as the formula. At Y of 0.5 the one-day cost is $1,341,641; at 1.0 it is $2,683,282. Fit Y on the fund's own large trades, and stress it.
Impact cost = Y × σ × √(Q ÷ V) × Q. On the illustrative $60m position that is $2.15m in a day or $0.43m over 25 days, with five weeks of price risk as the price of patience. Pair it with days to liquidate and the redemption policy, and the cost of getting out becomes a number the board can see. The book prices impact position by position on a fund facing successive redemptions in the free workbook for this case.
Y is an empirical constant usually of order one, and it should be fitted to the fund's own execution data. It moves the answer in proportion: on a $60m sale of five days of volume with 2.0 per cent daily volatility, Y of 0.5 gives $1,341,641 and Y of 1.0 gives $2,683,282.
It is an empirical regularity across equity and futures markets: per-unit impact rises with the square root of the order relative to daily volume. Doubling the order raises per-unit cost by 41 per cent and total cost 2.83 times, so concentrated positions cost more than proportionally to exit.
Impact falls with the square root of the number of days, but price risk rises. Selling $60m over 25 days instead of one cuts expected impact from $2.15m to $0.43m, while one standard deviation of timing risk rises from $0.60m to $3.46m. The right horizon depends on urgency and risk appetite.
This article is one calculation from Financial Risk Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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