Conditions for Combining Causal Risk Mandates into a Pooled Portfolio

Summary

This theoretical study asks when several causal risk mandates, each built from overlapping information, can be implemented by one self-financing portfolio that is optimal given pooled information. In an incomplete continuous Brownian market, it projects signal-responsive exposures onto traded risk directions and represents them in a predictable Hilbert space. The analysis separates implementation error into non-traded structural response, incompatibility among local mandates, and distortion of the shared traded portfolio. A fusion operator is introduced to correct duplicated common risk directions.

The central result gives conditions for exact decentralization: pooled information must preserve reference martingales, the pooled optimum must lie in the additive span of local desks, local mandates must be compatible with a common portfolio, and that portfolio must match the additive pooled optimum. The paper also studies log-growth regret, quadratic mandate penalties, comparative statics in a two-desk model, joint interventions, and extensions. Its conclusions are theoretical, partial-equilibrium statements for continuous markets; the excerpt provides no empirical validation or direct evidence about discrete-market trading performance.

Key ideas

• The study defines when local causal risk mandates can be implemented by one pooled self-financing portfolio.
• Implementation error separates into non-attainability, incompatibility across mandates, and distortion of the common traded portfolio.
• Exact decentralization requires specific compatibility and span conditions, as well as preservation of reference martingales.
• The analysis derives a quadratic-penalty allocation and comparative statics in a two-desk model.
• The results are theoretical and limited to continuous-market partial equilibrium.

Tags

# Causal Price-of-Risk Mandates under Overlapping Information


# Causal Price-of-Risk Mandates under Overlapping Information









We study whether causal risk mandates constructed from overlapping information blocks can be implemented by one self-financing portfolio that is optimal under pooled information. In an incomplete continuous Brownian market, signal-responsive exposures are projected onto traded Brownian directions and embedded as closed subspaces of a predictable Hilbert space. Each causal mandate is the attainable projection of an identified, baseline-centered intervention surface and is therefore fixed upstream of the allocator. A first exact decomposition separates non-traded structural response, incompatibility across local mandates, and distortion of the common traded book. A fusion operator corrects duplication of common risk directions. The main theorem shows that exact causal decentralization holds if and only if the pooled enlargement preserves reference martingales, the pooled optimum has no component outside the additive desk span, the local causal shadows are compatible with one common book, and that book equals the additive pooled optimum. Under immersion, log-growth regret separates into pooled-interaction loss and common-book distortion, while raw causal implementation error also contains non-attainability and incompatibility. No common benchmark can reduce incompatibility. We derive the unique allocation under a quadratic causal-mandate penalty and exact comparative statics in a two-desk model with shared, desk-specific, and pooled-interaction risk. Additional results cover joint interventions, representation stability, admissible coalitions, and a covariance-weighted continuous-semimartingale formulation. The conclusions are partial-equilibrium statements for continuous markets.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.