Solve a conditional AUM boundary from participation and cost limits

State what is fixed

For one order, define eligible volume V = ADV × volume multiplier × window days, and traded portfolio fraction a = weight × trade fraction. Then order dollars Q = AUM × a and participation p = AUM × a / V. The inversion assumes a and V do not change as AUM changes.

For a participation ceiling p*, the boundary is AUM = p* × V / a, provided a and V are positive. A zero-trade sleeve does not impose a finite participation boundary. Missing or zero volume for a positive order is an invalid scenario, not an invitation to report unlimited size.

Invert the chosen cost rule

Let f equal fees plus half-spread in basis points, k be the impact coefficient, and b be the allowed cost in basis points per traded dollar. Under cost = f + k × √p, and with k positive and b above f, the cost-limited participation is ((b − f) / k) squared.

This is a limit on a modeled order-cost rate. It is not an annual portfolio-drag budget. If b is below f, no positive order meets the budget under the assumptions. If b equals f and k is positive, only zero participation reaches it. If k is zero, the cost rate is fixed and the impact formula supplies no finite size boundary.

A reproducible boundary example

Take hypothetical eligible volume of $20 million and traded portfolio fraction a = 4%. Let the participation ceiling be 10%, fixed charges 3 bps, k = 20 bps and cost budget 8 bps per traded dollar. The participation-only AUM boundary is 0.10 × $20 million / 0.04 = $50 million.

The cost budget permits participation of ((8 − 3) / 20)² = 6.25%. Its AUM boundary is 0.0625 × $20 million / 0.04 = $31.25 million. The lower boundary binds. At that AUM, the order is $1.25 million, modeled cost is $1,000 and portfolio drag is 0.32 bps.

Test the boundary rather than rounding it into authority

Re-enter the resulting AUM into the original formula. The example should reproduce 6.25% participation and 8 bps total cost. A slightly larger AUM should exceed the cost budget, while a slightly smaller one should fall below it. These are useful checks on the implementation and units.

Then vary uncertain inputs. Halving eligible volume halves the boundary to $15.625 million without changing the selected cost-limited participation. Raising k from 20 to 25 bps lowers cost-limited participation to 4% and the base-volume boundary to $20 million. The sensitivity shows why a precise single number can overstate the evidence.

Keep economic validation separate

The Federal Reserve’s current model-risk guidance distinguishes assessing conceptual soundness from comparing outputs with real outcomes. Used here as an analytical reference, that distinction explains why correctly inverting the heuristic does not establish executable capacity. This publication is not claiming compliance with bank supervisory requirements.

For multiple positions, inspect each binding constraint and any shared demand. Preserve omitted instruments and unsupported execution features in the conclusion. Describe the result as a conditional scenario boundary with dated inputs, chosen thresholds and an uncalibrated cost function, rather than a provider’s actual AUM limit.

Keep in the decision record

• Cost budgets here are basis points per traded dollar.
• Take the more restrictive applicable boundary and verify it in the forward formula.
• Sensitivity and arithmetic correctness do not establish real-world capacity.

Source notes

1. Federal Reserve — Supervisory Guidance on Model Risk Management, April 2026
2. Almgren and Chriss — Optimal Execution of Portfolio Transactions, December 2000 (paper mirror)

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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