Capital allocation and tail central moments for the multivariate normal mean-variance mixture distribution

Capital allocation is a procedure used to assess the risk contributions of individual risk components to the total risk of a portfolio. While the conditional tail expectation (CTE)-based capital allocation is arguably the most popular capital allocation method, its inability to reflect important tail behaviour of losses necessitates a more accurate approach. In this paper, we introduce a new capital allocation method based on the tail central moments (TCM), generalising the tail covariance allocation informed by the tail variance. We develop analytical expressions of the TCM as well as the TCM-based capital allocation for the class of normal mean-variance mixture distributions, which is widely used to model asymmetric and heavy-tailed data in finance and insurance. As demonstrated by a numerical analysis, the TCM-based capital allocation captures several significant patterns in the tail region of equity losses that remain undetected by the CTE, enhancing the understanding of the tail risk contributions of risk components.