Attribution that reconciles exactly, and still tells the right story.

Executive overview

Attribution effects that explain each period well do not add up to the compounded cumulative active return. Existing linkers smooth the effects or spread the residual by a fixed rule. This paper reframes linking as a controlled optimization: reconciliation is non-negotiable, and the attribution narrative is preserved as much as the math allows.

LD7S and LD8S are convex quadratic programs that start from the raw cumulative effects and solve directly for a linked vector that reconciles exactly. Distortion is measured relative to each effect’s size, and sign, dominant-driver and top-band preservation enter as governed soft penalties.

Findings, in order

1. Linking is residual allocation. The gap q = D − ΣG is distributed across buckets by a convex QP instead of a fixed formula.
2. Distortion is measured relative to size. Weights of 1/(|G|+ε)² stop small effects from being overwhelmed by adjustments that are trivial for large ones.
3. LD7S protects the dominant driver. A soft penalty discourages any outcome in which the largest raw contributor is displaced.
4. LD8S protects the top band. Pairwise penalties extend preservation to the leading group of contributors.
5. A reproducible worked example. Three periods and five effects show raw effects of 90.0000 bps against a compounded target of 90.2865 bps, compared with Cariño, Frongello, Menchero, modified Frongello and reverse Frongello.
6. Validated for production. Monte Carlo, adversarial, out-of-distribution, metamorphic, numerical-conditioning and production-control suites, with a governed fallback from LD8S to LD7S at high bucket counts.

Novelty

Executive overview

Real attribution reports are hierarchies: total, asset class, region or sector, and security. A linked report is defensible only if every parent equals the sum of its children and the top equals cumulative active return. Linking each level independently routinely breaks that, because the adjustments disagree between levels.

HLD parametrizes the linked report by its bottom level, so coherence holds by construction. Top-level reconciliation is a hard constraint, distortion is controlled at every node, and narrative penalties apply level by level. The base formulation is a weighted projection of the raw tree onto the coherent, reconciled set.

Findings, in order

1. Per-level linking does not compose. Independent optimization at each level violated coherence in 99.6% of 500 coherent Monte Carlo scenarios.
2. Coherence becomes structural. Solving for the bottom level through a summing matrix guarantees that parents equal the sum of their children.
3. Base-HLD is a weighted projection. This ties attribution linking to the hierarchical forecast reconciliation literature (Hyndman et al.; Wickramasuriya et al.).
4. Strong comparative results. Under reconciliation- and coherence-gated loss, HLD8S won or tied in 69.4% of scenarios and HLD7S in 65.8%.
5. Multi-system inputs expose leaf-only methods. When levels come from different systems, leaf-only methods fail reconciliation itself.
6. Hardened at production scale. 1,500 mixed-depth and ragged hierarchies, 3,000 production solves and 400 independent objective-oracle comparisons.

Novelty

Executive overview

Should linking before or after combining attribution buckets give the same answer? This paper studies that requirement, aggregation consistency, axiomatically and shows that it is far more restrictive than it looks.

Together with bucket anonymity and continuity, it forces every linker into a common-period-coefficient form. Methods that keep only aggregate effects collapse to signed pro-rata and cannot be defined everywhere. Keeping the return path removes the obstruction, and the Aumann–Shapley value of the compounding function is the principled resolution.

Findings, in order

1. Characterization. Aggregation consistency, anonymity and continuity are equivalent to F_k = c(A,R)ᵀx_k, with one coefficient vector shared by all buckets.
2. Reconciliation becomes one condition. Exact reconciliation reduces to the single scalar condition cᵀA = D.
3. Reduced-form uniqueness. If a method may use only aggregate effects G and total D, the only exactly reconciling choice is signed pro-rata, F_k = (D/S)G_k.
4. Totality obstruction. Admissible paths with S = 0 and D ≠ 0 exist, so no everywhere-defined regular reduced-form method can meet the same requirements.
5. Aumann–Shapley resolves it. The cost-sharing value of the compounding function is total, continuous, exactly reconciling, aggregation-consistent and anonymous.
6. Methods classified. A deterministic battery classifies established and experimental linkers and preserves the counterexamples.

Novelty

Executive overview

A multiperiod report can be coherent in two directions. Merging sectors before or after linking should agree, and collapsing a block of periods and continuing should agree with linking the whole horizon. Exact reconciliation guarantees neither.

This paper proves that, inside the common-coefficient form, temporal associativity is exactly a block-factorization law for the coefficients. It then builds a continuous one-parameter family of linkers that satisfy every structural requirement, with reverse Frongello, Cariño and forward Frongello as three points on the same line.

Findings, in order

1. Cross-sectional coherence. A self-contained proof that aggregation consistency on a rich signed domain implies common period coefficients.
2. Associativity is factorization. Temporal associativity holds exactly when each full-history coefficient equals the block coefficient times the within-block coefficient.
3. A continuous family. An infinite one-parameter family of total, exactly reconciling, arbitrarily factorizing linkers on the positive gross-return group.
4. History sits on one line. Reverse Frongello, Cariño and forward Frongello appear at parameters −1, 0 and 1, so reconciliation and factorization alone do not pick a historical formula.
5. Symmetry selects Cariño. Within the family, both reversal symmetry and full order invariance single out Cariño.
6. Order symmetry is not associativity. Published order-symmetric constructions by Reztsov are non-examples of block factorization.

Novelty

Executive overview

Attribution systems routinely compress a closed period into a small summary, its portfolio and benchmark returns and linked effects, and discard the raw path. That is only safe if the discarded information could never change a later calculation.

This paper proves that temporal associativity is exactly a sufficiency condition on that stored state, then classifies five widely used methods. Some are safe with a fixed state, some need a small repair, and Aumann–Shapley requires state that grows with the horizon.

Findings, in order

1. Associativity is sufficiency. Leading-block associativity is equivalent to future sufficiency of the standard stored state.
2. Two strengths of safety. Arbitrary associativity needs a stronger two-sided condition, which matters for interior restatements as well as appends.
3. Five methods classified. Cariño and both Frongello directions are safe; Modified Frongello separates the two notions with an exact rational witness; Menchero and Aumann–Shapley fail ordinary sufficiency.
4. Some failures are cheap to repair. Menchero and Modified Frongello admit small exact states that do not grow with the horizon.
5. Aumann–Shapley needs growing state. An exact polynomial state of dimension KT + 1 composes recursively and is generically minimal, so no fixed-size exact state exists.

Novelty

Executive overview

Practitioners often ask for a linker with every desirable property: exact reconciliation, period symmetry, boundary continuity, block transitivity and principled handling of compounding. This paper tests whether such a "golden" linker can exist.

For the usual design, one return-dependent coefficient per period applied to every effect, four of those requirements are incompatible once continuity covers simultaneous total-loss periods. The result is sharp but bounded: the Davies–Laker notional-fund construction escapes it by carrying extra state.

Findings, in order

1. A common-coefficient impossibility. Reconciliation, permutation symmetry, block transitivity and boundary continuity cannot all hold, shown with a three-period proof.
2. A genuine exception. The Davies–Laker construction carries category-level product state and escapes the common-coefficient premise.
3. A correction. The claim that normalized logarithmic linking is non-transitive is corrected.
4. A polynomial Shapley alternative. Its properties and limitations are proved.
5. Negative results reported. An exploratory superiority screen for minimum-burden projections failed and is reported as such.

Novelty

Executive overview

Several collateral pools may pay overlapping creditors, each pool by its own priority tiers and each creditor up to one aggregate cap. If recoveries are computed by processing the pools one at a time, the answer can depend on which pool goes first, which makes the processing order a hidden modelling input.

This paper gives an exact pairwise test for when the order cannot matter, for any caps and pool values. A creditor eligible in only one of two pools must rank strictly above, in that pool, every shared creditor. For each shared creditor, the two pools must agree on the ranking, including ties, of the shared creditors it weakly outranks in both. The test is sufficient for any common splitting rule satisfying the paper’s axioms, and necessary and sufficient under proportional splitting.

Findings, in order

1. Exact criterion. Under proportional splitting, every creditor’s total recovery is the same for all processing orders, caps and pool values exactly when every pair of pools passes the test. Sufficiency holds for any common splitting rule satisfying the paper’s axioms.
2. Small counterexamples. Every failure contains one of three obstructions on two or three creditors. A polynomial screening test finds one and returns it as a certificate.
3. Strict priorities. With strict priorities and the same eligible creditors in both pools, the test is avoidance of the permutation pattern 132, so compatible second-pool orders are counted by the Catalan numbers.
4. Admissions and reserves. For one open slot and one soft-reserve slot, both accepting all students and using a common master ranking within groups, admitted sets are independent of precedence for every applicant set exactly when no reserve-ineligible student lies between two reserve-eligible students in that ranking.
5. Funding, pool-at-a-time rules and averaging. Equal totals do not imply equal pool-level funding; no pool-at-a-time rule that respects local priority and wastes nothing is order-invariant on every structure; and averaging over processing orders can break priority within a pool.

Novelty