On heavy-tailed risks under Gaussian copula: the effects of marginal transformation

Paper: arXiv 2304.05004

Authors: Bikramjit Das, Vicky Fasen-Hartmann

Abstract

In this paper, we compute multivariate tail risk probabilities where the marginal risks are heavy-tailed and the dependence structure is a Gaussian copula. The marginal heavy-tailed risks are modeled using regular variation which leads to a few interesting consequences. First, as the threshold increases, we note that the rate of decay of probabilities of tail sets vary depending on the type of tail sets considered and the Gaussian correlation matrix. Second, we discover that although any multivariate model with a Gaussian copula admits the so called asymptotic tail independence property, the joint tail behavior under heavier tailed marginal variables is structurally distinct from that under Gaussian marginal variables. The results obtained are illustrated using examples and simulations.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a strong theoretical and mathematical exploration of heavy-tailed risks combined with a Gaussian copula, offering insightful proofs on asymptotic joint tail behavior. However, it relies heavily on theoretical derivations and Monte Carlo simulations rather than empirical validation using real-world market datasets.

Research Flowchart

  flowchart TD
    A[Research Goal: Compute Multivariate Tail Risks<br>under Heavy-Tailed Marginals & Gaussian Copula] --> B[Inputs: Heavy-Tailed Marginals RV &<br>Gaussian Copula Correlation Matrix]
    B --> C[Methodology: Apply Regular Variation &<br>Marginal Transformations]
    C --> D[Computational Simulations &<br>Tail Set Probability Decay Analysis]
    D --> E[Outcome 1: Tail Set Probability Decay Rate<br>Varies by Threshold & Correlation]
    D --> F[Outcome 2: Heavy-Tailed Joint Tail Behavior<br>Structurally Differs from Gaussian Marginals]