Law-Invariant Return and Star-Shaped Risk Measures

Paper: arXiv 2310.19552

Authors: Roger J. A. Laeven, Emanuela Rosazza Gianin, Marco Zullino

Abstract

This paper presents novel characterization results for classes of law-invariant star-shaped functionals. We begin by establishing characterizations for positively homogeneous and star-shaped functionals that exhibit second- or convex-order stochastic dominance consistency. Building on these characterizations, we proceed to derive Kusuoka-type representations for these functionals, shedding light on their mathematical structure and intimate connections to Value-at-Risk and Expected Shortfall. Furthermore, we offer representations of general law-invariant star-shaped functionals as robustifications of Value-at-Risk. Notably, our results are versatile, accommodating settings that may, or may not, involve monotonicity and/or cash-additivity. All of these characterizations are developed within a general locally convex topological space of random variables, ensuring the broad applicability of our results in various financial, insurance and probabilistic contexts.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper is a purely theoretical contribution to quantitative risk management, providing advanced functional analysis proofs for law-invariant star-shaped risk measures and Kusuoka-type representations in locally convex spaces. While highly novel and mathematically rigorous, it lacks empirical validation, real-world data application, or backtesting execution.

Research Flowchart

  flowchart TD
    A[Goal: Characterize Law-Invariant<br>Star-Shaped Risk Measures] --> B[Input: Locally Convex Topological Space<br>of Random Variables]
    B --> C[Method: Establish Consistency with 2nd/Convex<br>Stochastic Dominance]
    C --> D[Compute Kusuoka-Type<br>Representations]
    D --> E[Derive General Star-Shaped<br>Robustifications of VaR]
    E --> F[Outcomes: Unified Framework for VaR & ES<br>without Requiring Monotonicity or Cash-Additivity]