Principal Component Copulas for Capital Modelling

We introduce a class of copulas that we call Principal Component Copulas. This class intends to combine the strong points of copula-based techniques with principal component-based models, which results in flexibility when modelling tail dependence along the most important directions in multivariate data. The proposed techniques have conceptual similarities and technical differences with the increasingly popular class of factor copulas. Such copulas can generate complex dependence structures and also perform well in high dimensions. We show that Principal Component Copulas give rise to practical and technical advantages compared to other techniques. We perform a simulation study and apply the copula to multivariate return data. The copula class offers the possibility to avoid the curse of dimensionality when estimating very large copula models and it performs particularly well on aggregate measures of tail risk, which is of importance for capital modeling.