Paper: arXiv 2411.08763

Authors: Christian Laudagé, Felix-Benedikt Liebrich, Jörn Sass

Abstract

We revisit the recently introduced concept of return risk measures (RRMs) and extend it by incorporating risk management via multiple so-called eligible assets. The resulting new class of risk measures, termed multi-asset return risk measures (MARRMs), introduces a novel economic model for multiplicative risk sharing. We analyze properties of these risk measures. In particular, we prove that a positively homogeneous MARRM is quasi-convex if and only if it is convex. Furthermore, we provide conditions to avoid inconsistent risk evaluations. Then, we point out the connection between MARRMs and the well-known concept of multi-asset risk measures (MARMs). This is used to obtain various dual representations of MARRMs. Moreover, we conduct a series of case studies, in which we use typical continuous-time financial markets and different notions of acceptability of losses to compare RRMs, MARMs, and MARRMs and draw conclusions about the cost of risk mitigation.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical axiomatic framework extending return risk measures to a multi-asset setting, complete with dual representations and theoretical proofs. While it includes continuous-time case studies and an empirical illustrative section using US automobile insurance and stock index data, it lacks exhaustive statistical backtesting and open-source code implementation.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend Return Risk Measures to Multi-Asset Markets] --> B[Model Formulation: Develop MARRMs for Multiplicative Risk Sharing]
    B --> C[Inputs: Continuous-Time Financial Markets & Acceptability of Loss Criteria]
    C --> D[Theoretical Analysis & Axiomatization]
    D --> E[Computational Case Studies: Compare RRMs, MARMs & MARRMs]
    E --> F[Key Outcomes: Quasi-Convexity Equals Convexity, Dual Representations & Cost Analysis]