Hierarchies of Archimedean copulas

Abstract

We present a flexible class of hierarchical copulas capable of modelling multidimensional joint distributions of asset returns with a richer rank correlation structure than existing models. We derive estimators and simulation techniques. The methods are applied to an illustrative portfolio consisting of a subset of DAX stocks.

Keywords:

• Copulas
• Portfolio management
• Risk management
• Insurance mathematics

Acknowledgement

We would like to thank Alfred Müller for stimulating discussions and suggestions.

Notes

†We note in passing that, of course, it is possible to construct hierarchies of arbitrary (possibly non-Archimedean) copulas if C _l,j (C _l,j ) is regarded as the copula of the copula values C ^l,j . We suggest calling such constructions ‘copula cascades’. The properties of copula cascades and their statistical inference are different from those set out in this paper for hierarchical Archimedean copulas, and, to our knowledge, have not yet been investigated.

‡The dth-order derivatives of the inverse generator functions are explicitly given by Savu and Trede (2008) for the Gumbel, Clayton, Frank, and Ali-Mikhael-Haq copulas.

†The R code is available from the authors.

†Maximum likelihood estimation can also be performed in two steps (Joe 2005). In the first step, the parameters of the marginal distributions are estimated separately for each dimension. In the second step the U _it are computed using the estimated first-step parameters. We restrict the description to the one-step approach as it is more efficient.

†The method of moments estimates (using Kendall's τ) are similar: on the upper level and , , and .