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.
Bibliographical noteFunding Information:
We would like to thank the anonymous referees for their very useful comments. This research was supported by the Israel Science Foundation (Grant No. 1686/17 ).
© 2018 Elsevier B.V.
- Elliptical distributions
- Multivariate risk measures
- Multivariate tail conditional expectation
- Multivariate tail covariance
- Tail confidence ellipsoid
- Tail variance
ASJC Scopus subject areas
- Statistics and Probability
- Economics and Econometrics
- Statistics, Probability and Uncertainty