This paper proposes two nested methods—fully nested and partially nested—for the general d-dimensional extension of the Archimedean copula. It has extensively been used to model dependence among the return series, however, its extension to general d-dimension with a proper dependence structure is limited. The paper analyzes the data of six major stock indices collected on a daily basis for the period, March 8, 2002 to March 7, 2007, and finds that partially nested structure is better than a symmetric and fully nested structure for general empirical application. Kendall's tau has been primarily used for determining a nested structure owing to its one-to-one relationship with the Archimedean copula parameter. The paper quantifies the impact of the dependence structure on portfolio risk measured by Value at Risk (VaR) and conducts a backtest for assessing the soundness of the VaR models. It confirms that the dependence structure of the return series is a critical determinant of the portfolio VaR and partially nested structure gives a conservative VaR estimate in comparison to the symmetric case.

A copula is a function that joins or couples the multivariate distribution and marginal distribution function. This means that we can separate a marginal and joint distribution using a copula which, in practice, makes it very useful to model a multivariate distribution. There are two broad classes of copula—elliptical copula and Archimedean copula. The elliptical copula is derived from the multivariate distribution function. The well-known copulas in this class include the Gaussian and t copula. The elliptical copulas have become popular in modeling the dependence of financial return series. However, they are not very effective in modeling important characteristics of financial return data due to the following reasons. First, financial returns typically show a fat tail, excess kurtosis and extreme co-movement in the tail region, which make Gaussian copula unsuitable to model financial returns (Mashal and Zeevi, 2002; and Dobric and Schmid, 2005). Second, even though the t copula has extensively been used on account of its relatively good performance, ease in implementation and so on, the t copula shows symmetric co-movement in the tail area making it unfit for modeling dependence of the financial returns.

The Archimedean copula is derived through a generator and is very useful for dependence modeling because it can model extreme co-movement in the tail regions. Some of the well-known Archimedean copulas are Gumbel, Frank and Clayton. The bivariate Archimedean copula is well-established and can be effectively applied to financial modeling. However, the d-dimensional generalization of the Archimedean copula is limited. In this paper, we highlight the multivariate extension of the Archimedean copula that is extremely effective in financial modeling. The first d-dimensional generalization of the Archimedean copula is known as the symmetric (Joe, 1997) or exchangeable case. A symmetric structure requires only one generator regardless of the dimensions of a joint distribution. Thus, the number of estimated parameters do not increase with increase in dimension. Consequently, the same pair-wise dependence is assumed for all variables in the d-dimensional space. Hence, it is not feasible to reflect dependence among variables, in practice.

Financial Risk Management Journal, Archimedean Copula, Financial Return Modeling, Multivariate Distributions, Gaussian Copula, Financial Assets, Bayesian Information Criterion, Akaike Information Criterion, Sampling Algorithm, Theoretical Structures, Portfolio Risk.