When the cost budget becomes tighter than the participation cap

Keep the economic question explicit

This case changes the maximum modeled cost per traded dollar from 12 to 7 basis points. It retains the synthetic 60% A, 30% B and 10% untraded cash allocation, a 20% traded fraction in both invested sleeves, two sessions, per-session dollar ADV of $50 million and $5 million, volume multiplier one and a 10% participation cap. The fee remains 1 basis point, half spread 2 and impact coefficient 20.

The budget is a chosen scenario constraint. It is not an observed expected return, a forecast of alpha or a promised execution price. Selecting seven because it creates a desired capital limit would reverse the analysis. A real decision should document why the cost budget is tolerable and how uncertainty, incomplete fills and risks outside this simple equation affect that judgment.

Invert the cost equation before sizing capital

The fixed modeled component is 1 + 2 = 3 basis points. A budget of 7 leaves 4 basis points for the impact term. Solving 20 × square root of participation ≤ 4 gives participation ≤ (4 ÷ 20) squared = 0.04. Use the smaller of this 4% allowance and the separate 10% participation cap for every traded row.

A's capital boundary becomes 0.04 × $100 million ÷ 0.12 = $33,333,333.33. B's is 0.04 × $10 million ÷ 0.06 = $6,666,666.67. B remains the bottleneck, but the reason has changed from participation policy to cost budget. An unchanged binding sleeve can therefore conceal a different binding rule; the report should preserve both the row and the reason.

Reconcile the tighter boundary

At exactly $20 million divided by three of capital, A trades $800,000 and B trades $400,000. Their participation levels are 0.8% and 4%. A's cost is 3 + 20 × square root of 0.008 = 4.78885438 basis points; B's is exactly 7 basis points. Both meet the participation cap, while B sits precisely on the cost budget.

Multiplying those rates by their order notionals gives $383.108351 and $280. The basket cost is $663.108351. Relative to $1.2 million traded, that is 5.52590292 basis points. Relative to starting capital, it is 0.99466253 basis points. The average traded-dollar cost is below seven because A is cheaper; the binding decision still comes from B's individual cost.

Handle the floor and calibration limits

A budget below the 3-basis-point fixed component allows no positive trade under these assumptions. With a positive impact coefficient, a budget equal to that floor also permits no positive participation. An all-cash scenario with no trading has no defined trading-capacity boundary; it does not demonstrate unlimited economic capacity. These edge cases belong in both implementation checks and the interpretation of exported results.

Published market-impact research studies how size and participation relate to price response under particular assumptions. It does not supply this worksheet's coefficient of 20 or validate its application to every market. Changing the coefficient can alter the cost-driven boundary sharply: with the same budget gap, doubling the coefficient quarters the cost-implied participation. Preserve calibration evidence before giving a numerical boundary operational significance, and retain the original hypothetical label when sharing this example.

Keep in the decision record

• A cost budget applies per traded dollar, not per dollar of portfolio capital.
• The lower of the cost-implied participation and the explicit cap controls each row.
• The impact coefficient is an assumption that needs separate calibration evidence.

Source notes

1. Durin, Rosenbaum and Szymanski: The two square root laws of market impact, 2023 paper

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.