Read the square-root impact term as an uncalibrated heuristic

Inspect the exact formula

Let Q be order dollars, V be assumed eligible volume, p = Q / V, and k be the user’s impact coefficient in basis points. The modeled total cost rate is fee bps + half-spread bps + k × √p. Participation is entered as a decimal in the square root: 4% means 0.04, not 4.

At k = 20 bps and p = 4%, impact is 20 × 0.2 = 4 bps. Using the number 4 inside the square root would incorrectly produce 40 bps, a tenfold error. The coefficient sets the scale of the scenario; the publication does not infer it from market observations or claim a universally appropriate default.

Doubling size does more than double dollars of impact

Consider hypothetical order sizes of $1 million and $2 million against $25 million of eligible volume. Participation rises from 4% to 8%. With k = 20 bps, the impact rates are 4 and 5.657 bps. Impact dollars rise from $400 to approximately $1,131.37, a factor of 2.828.

This follows directly from the formula: with fixed volume, impact dollars are proportional to Q × √Q, or Q to the power 1.5. Fees and the fixed half-spread allowance scale proportionally with Q in this model. Showing the components separately explains why total dollar cost need not scale linearly with AUM.

Do not attach another model’s authority

Almgren and Chriss analyze execution costs and risk using explicit assumptions about price dynamics and impact. Their original paper supplies a research foundation for thinking about execution tradeoffs, but it does not validate this site’s chosen coefficient or its simple participation formula.

Other impact formulations can produce materially different results. A linear participation term, a saturating response or an instrument-specific empirical model would generate different size sensitivities. The useful question is whether a formulation has evidence for the intended instruments, order sizes, time windows and market conditions. Familiar mathematical shape is not sufficient evidence.

Test sensitivity before interpreting precision

For the $1 million order at 4% participation, coefficients of 10, 20 and 40 bps imply impact rates of 2, 4 and 8 bps. With 3 bps of fixed charges, total rates become 5, 7 and 11 bps, or $500, $700 and $1,100. This range comes from parameter choices, not a statistical confidence interval.

Volume uncertainty interacts with the coefficient. Halving eligible volume doubles participation and multiplies the impact term by √2. A scenario that simultaneously raises k and lowers volume may be a useful stress, but disclose both changes. Otherwise readers cannot distinguish the assumed liquidity reduction from the assumed response to that reduction.

Recognize when the formula is out of scope

At very large participation, the workbench continues to return arithmetic even though the assumed execution may be implausible. It has no order book, price limit, halt process, auction clearing rule or guarantee of completion. A computed result above 100% of assumed volume is a warning about incompatible demand, not evidence that the market can accommodate the order.

At zero order size, cost dollars are zero. At positive order size and zero eligible volume, participation is undefined and the scenario should fail validation rather than produce a reassuring number. These boundary checks establish software correctness; they do not empirically validate the model’s economic response.

Keep in the decision record

• Use decimal participation inside the square root.
• Treat k as a disclosed scenario input unless separately calibrated.
• Formula tests establish arithmetic behavior, not live execution accuracy.

Source notes

1. Almgren and Chriss — Optimal Execution of Portfolio Transactions, December 2000 (paper mirror)
2. BIS — Guidance for Supervisors on Market-Based Indicators of Liquidity, January 2014

Sources inform the definitions and limitations. Scenario inputs and arithmetic are publication illustrations, not extracted market observations. Access checks are dated in the source inventory.

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