Explicit Expressions for Multidimensional Value-at-Risk under Archimedean Copulas
Dotamana Yéo, Saralees Nadarajah, Amadou Sawadogo
TL;DR
The paper addresses multivariate VaR under dependence by leveraging Archimedean copulas and Kendall's distribution to obtain explicit lower-tail VaR expressions in arbitrary dimensions. It provides closed-form marginal VaR formulas for Clayton, Frank, Gumbel-Hougaard, Joe, and Ali-Mikhail-Haq copulas, linking the dependence parameters to joint risk in a transparent way. Through Monte Carlo validation, the authors demonstrate finite-sample accuracy and show that copulas with the same Kendall tau can yield different VaR levels due to tail dependence. The results offer computationally efficient tools for risk management and systemic risk assessment, with potential extensions to multivariate Expected Shortfall and time-varying dependence for regulatory 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.