A multivariate tail covariance measure for elliptical distributions

This paper introduces a multivariate tail covariance (MTCov) measure, which is a matrix-valued risk measure designed to explore the tail dispersion of multivariate loss distributions. The MTCov is the second multivariate tail conditional moment around the MTCE, the multivariate tail conditional expectation (MTCE) risk measure. Although MTCE was recently introduced in Landsman et al. (2016a), in this paper we essentially generalize it, allowing for quantile levels to obtain the different values corresponded to each risk. The MTCov measure, which is also defined for the set of different quantile levels, allows us to investigate more deeply the tail of multivariate distributions, since it focuses on the variance–covariance dependence structure of a system of dependent risks. As a natural extension, we also introduced the multivariate tail correlation matrix (MTCorr). The properties of this risk measure are explored and its explicit closed-form expression is derived for the elliptical family of distributions. As a special case, we consider the normal, Student-t and Laplace distributions, prevalent in actuarial science and finance. The results are illustrated numerically with data of some stock returns.