On a multivariate extension for Copula-based Conditional Value at Risk

TL;DR

This paper extends Copula-based Conditional Value at Risk (CCVaR) to multivariate cases using Archimedean copulas, providing a near-closed form expression and analyzing its coherence, with empirical validation on real data.

Contribution

It introduces a multivariate extension of CCVaR with an explicit expression under Archimedean copulas, filling a gap in existing research.

Findings

Derived an almost closed-form expression for multivariate CCVaR.

Identified conditions for CCVaR to satisfy coherence.

Numerical experiments show CCVaR's performance compared to classical VaR and CVaR.

Abstract

Copula-based Conditional Value at Risk (CCVaR) is defined as an alternative version of the classical Conditional Value at Risk (CVaR) for multivariate random vectors intended to be real-valued. We aim to generalize CCVaR to several dimensions (d>=2) when the dependence structure is given by an Archimedean copula. While previous research focused on the bivariate case, leaving the multivariate version unexplored, an almost closed-form expression for CCVaR under an Archimedean copula is derived. The conditions under which this risk measure satisfies coherence are then examined. Finally, numerical experiments based on real data are conducted to estimate CCVaR, and the results are compared with classical measures of Value at Risk (VaR) and Conditional Value at Risk (CVaR).