Tail conditional moment for generalized skew-elliptical distributions

Substantial changes in the financial markets and insurance companies have needed the development of the structure of the risk benchmark, which is the challenge addressed in this paper. We propose a theorem that expands the tail conditional moment (TCM) measure from elliptical distributions to wider classes of skew-elliptical distributions. This family of distributions is suitable for modeling asymmetric phenomena. We obtain the analytical formula for the $ {n^{\textrm{th}}} $ nth TCM for skew-elliptical distributions to help well to figure out the risk behavior along the tail of loss distributions. We derive four significant results and generalize the tail conditional skewness (TCS) and the tail conditional kurtosis (TCK) measures for generalized skew-elliptical distributions, which are used to determine the skewness and the kurtosis in the tail of loss distributions. The proposed TCM measure has been applied to well-known families of generalized skew-elliptical distributions. We also provide a practical example of a portfolio problem by calculating the proposed TCM measure for the weighted sum of generalized skew-elliptical distributions.