Paper: arXiv 2401.17265

Authors: Yi Shen, Zachary Van Oosten, Ruodu Wang

Abstract

We introduce the concept of partial law invariance, generalizing the concepts of law invariance and probabilistic sophistication widely used in decision theory, as well as statistical and financial applications. This new concept is motivated by practical considerations of decision making under uncertainty, thus connecting the literature on decision theory and that on financial risk management. We fully characterize partially law-invariant coherent risk measures via a novel representation formula. Strong partial law invariance is defined to bridge the gap between the above characterization and the classic representation formula of Kusuoka. We propose a few classes of new risk measures, including partially law-invariant versions of the Expected Shortfall and the entropic risk measures, and illustrate their applications in risk assessment under different types of uncertainty. We provide a tractable optimization formula for computing a class of partially law-invariant coherent risk measures and give a numerical example.

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 strong theoretical contribution by generalizing law invariance in risk measures and establishing novel representation theorems using robust mathematical formalism. However, as a predominantly theoretical paper, it relies on stylized numerical examples rather than empirical backtests or market data analysis.

Research Flowchart

  flowchart TD
    A[Research Goal: Generalize Law Invariance for Risk Measures under Uncertainty] --> B[Introduce Partial Law Invariance & Strong Partial Law Invariance]
    B --> C[Inputs: Financial Portfolios & Multi-Asset Risk Scenarios]
    C --> D[Methodology: Derive Novel Representation Formulas & Bridge to Kusuoka]
    D --> E[Develop Partially Law-Invariant Versions of Expected Shortfall & Entropic Risk Measures]
    E --> F[Tractable Optimization & Numerical Computation]
    F --> G[Outcomes: Unified Framework Connecting Decision Theory & Financial Risk Management]