The smaller sleeve sets the capital boundary

Define concentration through the required trade

Use the same hypothetical allocation as the balanced-rebalance case: 60% A, 30% B and 10% untraded cash. A and B each trade 20% of their sleeve values. Their daily dollar volume assumptions are $50 million and $5 million respectively, over two sessions. No actual instrument, investor or managed account is represented. The question is how much common portfolio capital can satisfy every row's stated constraint.

B is smaller by portfolio weight, yet its order consumes five times as much of its assumed volume as A's order at any common capital level. Its weight-times-traded-fraction is 0.06 versus A's 0.12, while its window volume is only one tenth as large. Portfolio concentration alone therefore does not identify the execution bottleneck; the required trade must be compared with its own available-volume assumption.

Invert participation one row at a time

For a positive traded weight, the participation-only capital boundary is participation cap × window volume ÷ traded weight. A gives 0.10 × $100 million ÷ 0.12 = $83,333,333.33. B gives 0.10 × $10 million ÷ 0.06 = $16,666,666.67. Cash has zero traded weight and adds no trading constraint; division by zero is neither needed nor meaningful.

The 12-basis-point cost budget also needs checking. With fee plus half spread equal to 3 and impact coefficient 20, allowable participation from cost alone is ((12 − 3) ÷ 20) squared = 0.2025. That is 20.25%, above the selected 10% participation cap. The participation rule therefore binds first for both sleeves, and the basket boundary is the smaller of their capital limits.

Check the boundary by calculating forward

At exactly $50 million divided by three of capital, A's order is $2 million and B's order is $1 million. Their participations are 2% and 10%. Costs are 5.82842712 and 9.32455532 basis points per traded dollar. These imply $1,165.685425 and $932.455532 of modeled cost, for $2,098.140957 in total and 1.25888457 basis points of portfolio drag.

This forward substitution matters: an inverted formula can look plausible despite a wrong denominator. Here the binding row reaches its cap, the other remains below it, and both remain below the cost budget. Keep full precision for comparisons and round only the report. A displayed boundary rounded upward to cents may differ infinitesimally from the exact mathematical threshold.

Do not spend liquidity belonging to another sleeve

Pooling window volume would give $110 million; pooling traded weights gives 0.18. Applying the same cap to these totals produces $61,111,111.11. At that capital level, B alone would require about 36.67% of its own window volume, well above 10%. The aggregation falsely assumes unused room in A can absorb an order in B. It cannot unless the actual investment instruction changes.

The useful response is to examine B's trade size, permissible timing and evidence about attainable liquidity. Changing its allocation may change the strategy; extending the window may change the decision's value. Market-impact research also distinguishes order duration from participation. The fixed square-root curve here does not model those channels separately. Record an alternative as a new scenario, preserving the original constraint failure rather than silently relabeling the larger number as capacity.

Keep in the decision record

• The binding sleeve can be smaller by portfolio weight.
• Calculate each row's boundary before taking the minimum.
• Pooling volume across unrelated orders can hide a material breach.

Source notes

1. Zarinelli et al.: Beyond the square root, 2014 paper, revised 2015

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.