Paper: arXiv 2603.01232

Authors: Ruodu Wang, Jingcheng Yu

Abstract

We study submodularity for law-invariant functionals, with particular attention to convex risk measures. Expected losses are modular, and certainty equivalents are submodular exactly when the loss function is convex. Law-invariant coherent risk measures are submodular exactly when they are coherent distortion risk measures, including Expected Shortfall (ES), and several deviation measures are also submodular. Beyond positive homogeneity, submodularity is restrictive for convex risk measures. We give a complete characterization for shortfall risk measures via the Arrow–Pratt measure of risk aversion, show that optimized certainty equivalents are always submodular, and prove that adjusted Expected Shortfall (AES) is submodular only when it reduces to ES. An empirical illustration for daily US equity returns finds no ES submodularity violations, many Value-at-Risk (VaR) violations, and relatively few AES violations.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous and deep theoretical analysis of submodularity across key classes of risk measures with detailed mathematical proofs. While it includes a small empirical illustration using US equity data, the core contribution is purely theoretical rather than empirical or computational.

Research Flowchart

  flowchart TD
    A["Research Goal: Characterize submodularity for law-invariant & convex risk measures"] --> B["Theoretical Framework: Evaluate Certainty Equivalents, Distortion, Shortfall, & AES Measures"]
    A --> C["Empirical Input: Daily US Equity Returns Data"]
    
    B --> D["Mathematical Characterization: Apply Arrow-Pratt measure & convexity constraints"]
    C --> E["Empirical Testing: Check submodularity violations for VaR, ES, & AES"]
    
    D --> F["Theoretical Outcomes: ES & OCE are always submodular; AES reduces to ES; VaR is non-submodular"]
    E --> G["Empirical Outcomes: 0 ES violations, many VaR violations, few AES violations"]

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1 of 32 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.