Paper: arXiv 2404.08475

Authors: Hirbod Assa, Peng Liu

Abstract

This paper introduces and studies factor risk measures. While risk measures only rely on the distribution of a loss random variable, in many cases risk needs to be measured relative to some major factors. In this paper, we introduce a double-argument mapping as a risk measure to assess the risk relative to a vector of factors, called factor risk measure. The factor risk measure only depends on the joint distribution of the risk and the factors. A set of natural axioms are discussed, and particularly distortion, quantile, linear and coherent factor risk measures are introduced and characterized. Moreover, we introduce a large set of concrete factor risk measures and many of them are new to the literature, which are interpreted in the context of regulatory capital requirement. Finally, the distortion factor risk measures are applied in the risk-sharing problem and some numerical examples are presented to show the difference between the Value-at-Risk and the quantile factor risk measures.

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 provides a rigorous mathematical framework extending standard risk measures to joint distributions with factor vectors, introducing novel axiomatic characterizations and theoretical concepts like coherent factor risk measures. However, it relies predominantly on theoretical derivations and simple numerical toy examples rather than extensive empirical backtesting with real market data.

Research Flowchart

  flowchart TD
    A[Goal: Measure portfolio loss risk relative to key macro/systemic factors] --> B[Data/Inputs: Joint distribution of loss random variable and factor vector]
    B --> C[Methodology: Axiomatize Factor Risk Measures via double-argument mapping]
    C --> D[Develop Concrete Classes: Quantile, Distortion, Linear & Coherent FRMs]
    D --> E[Applications & Computation: Solve Risk-Sharing Problems & Regulatory Capital Models]
    E --> F[Outcomes: New systemic risk measures & empirical proof of VaR vs Quantile FRM differences]

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0 of 45 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.