Skew-elliptical distributions with applications in risk theory

In this paper we derive important properties of the well-known skew-elliptical (SE) distributions which was introduced in Azzalini and Capitanio (J R Stat Soc Ser B (Stat Methodol) 65:367–389, 2003), and includes the more familiar skew-normal, skew-Student-t and skew-logistic distributions. We then derive the tail value at risk (TVaR) for a portfolio of SE risks. We provide the portfolio risk decomposition with TVaR. Furthermore, we obtain the Esscher premium principle, the weighted-premium principle, and the entropic risk measure with the underlying SE distributions. We also provide an explicit closed-form solution to the optimal portfolio selection with the SE distributions, and provide a numerical simulation of the results.