Are Shortfall Systemic Risk Measures One Dimensional?

Paper: arXiv 2306.10752

Authors: Alessandro Doldi, Marco Frittelli, Emanuela Rosazza Gianin

Abstract

Shortfall systemic (multivariate) risk measures $ρ$ defined through an $N$-dimensional multivariate utility function $U$ and random allocations can be represented as classical (one dimensional) shortfall risk measures associated to an explicitly determined $1$-dimensional function constructed from $U$. This finding allows for simplifying the study of several properties of $ρ$, such as dual representations, law invariance and stability.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a strong theoretical contribution by proving that multivariate shortfall systemic risk measures can be reduced to a single-dimensional representation. While mathematically sophisticated and novel in its simplification, it is purely theoretical and lacks empirical backtesting or real-world financial data validation.

Research Flowchart

  flowchart TD
    A["Research Goal: Are Shortfall Systemic Risk Measures One Dimensional?"] --> B["Inputs: N-dimensional Multivariate Utility Function (U) & Random Allocations"]
    B --> C["Methodology: Mathematical Equivalence & Allocation Vector Analysis"]
    C --> D["Process: Construct 1-Dimensional Utility Function from Multivariate U"]
    D --> E["Outcome: Systemic Risk Measure (ρ) Reduced to 1D Shortfall Risk Measure"]
    E --> F["Key Findings: Simplified Dual Representations, Law Invariance & Stability Analysis"]