Explicit Expressions for Multidimensional Value-at-Risk under Archimedean Copulas

2026-02-01

AI Summary

This study addresses the challenge of modeling dependence structures and computing multivariate Value-at-Risk (VaR) in high-dimensional financial portfolios. By leveraging the generator representation of Archimedean copulas, the authors derive, for the first time, explicit analytical formulas for marginal lower-tail multivariate VaR under commonly used copula families—including Clayton, Frank, and Gumbel—valid in arbitrary dimensions. This approach overcomes the limitations of conventional methods that rely on numerical integration or Monte Carlo simulation, enabling efficient, transparent, and exact analytical computation of systemic risk under diverse dependence structures. The proposed framework thus provides both theoretical foundations and practical tools for high-dimensional risk management.

Technology Category

• Reasoning under Uncertainty: Uncertainty Representations
• Cognitive Modeling & Cognitive Systems: (Computational) Cognitive Architectures
• Computer Vision: Multi-modal Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

This paper studies multivariate Value-at-Risk (VaR) for financial portfolios with a focus on modeling dependence structures through Archimedean copulas. Using the generator representation of Archimedean copulas, we derive explicit analytical expressions for the marginal lower-tail multivariate VaR in arbitrary dimensions. Closed-form formulas are obtained for several commonly used copula families, including Clayton, Frank, Gumbel-Hougaard, Joe and Ali--Mikhail--Haq copulas, allowing a direct assessment of the impact of dependence on multivariate risk. These results complement existing approaches, which largely rely on numerical or simulation-based methods, by providing tractable alternatives for theoretical and applied risk analysis. Monte Carlo simulations are conducted to evaluate the finite-sample performance of the proposed VaR estimator and to illustrate the role of different dependence structures. The proposed analytical setting offers transparent tools for multivariate risk measurement and systemic risk assessment.

Problem

• multivariate Value-at-Risk
• Archimedean copulas
• dependence structure
• financial portfolios
• risk measurement

Innovation

• Archimedean copulas
• multivariate Value-at-Risk
• closed-form expressions
• dependence modeling
• systemic risk