Soto, A. B. (2026). What Return Attribution Cannot Tell Us About Decision Risk: Information retained by realized attribution reports. Working paper. https://adambsoto.com/research/decision-risk-information
What Return Attribution Cannot Tell Us About Decision Risk
Information retained by realized attribution reports
Executive overview
Knowing average allocation and selection effects and the complete active-return history can still be insufficient to recover each decision’s signed contribution to realized tracking error. The missing information is how each effect moves with the total.
A classical row-space recovery criterion identifies the linear observations needed to reconstruct those contributions. Exact Brinson examples show opposite risk roles under identical reported means and active returns. A generated sixteen-cell ledger on public industry returns illustrates what monthly compression and stated volatility ceilings retain or lose.
Findings, in order
- Reconciliation does not ensure reconstruction. Means and a complete active-return path can leave decision covariance contributions unidentified.
- Recovery has an exact criterion. For a fixed nonconstant total and at least two unrestricted effects, the weighted centered total must lie in the report’s augmented row space.
- Identical summaries allow opposite roles. The two-period Brinson example keeps means and totals fixed while allocation and selection exchange hedge and risk-adder roles.
- Restrictions can bound ambiguity. Explicit standalone-volatility ceilings and omitted-path norm budgets yield sharp bounds under their stated domains.
- A generated study tests compression. A sixteen-cell policy ledger uses 2,766 public-return observations from 2015–2025. The decisions and fees are generated; the alternatives are report-compatible constructions.
- Cross-moments preserve the fixed task. Means, component cross-moments with the total, and the total’s first two moments reconstruct this realized covariance ledger. Other totals, horizons, weights or risk measures may need more information.
Novelty
Covariance risk allocation and linear-functional estimability are established mathematics. The research contribution examined here is their reporting formulation, explicit ambiguity witnesses, conditional bounds and reproducible compression study. The study does not establish that a manager made a particular decision, that compatible ledgers are feasible trades, or that this information loss is common in institutional systems.
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