Copula-based Conditional Value at Risk ( \(\textrm{CCVaR}\) ) is a real-valued tail risk measure for multivariate random vectors defined through conditioning on a copula level set. This paper considers the extension of \(\textrm{CCVaR}\) to dimensions \(d\ge 2\) under Archimedean dependence. While existing work has primarily treated the bivariate case, the multivariate setting involves a d-dimensional conditioning region and does not admit the same low-dimensional representations. For d-dimensional Archimedean copulas, we derive an explicit representation that reduces the defining conditional expectation to a one-dimensional form involving the multivariate Kendall distribution function and a dimension-dependent correction term, enabling tractable computation and calibration in moderate dimensions. We also examine coherence-type properties. Finally, numerical experiments using real data illustrate the behavior of \(\textrm{CCVaR}\) and compare it with classical \(\textrm{VaR}\) and \(\textrm{CVaR}\) under several copula families and marginal specifications.
