Tail variance for Generalized Skew-Elliptical distributions

Notable changes in financial markets have required the development of a standard structure for risk measurement, and obtaining an appropriate risk measurement from historical data is the challenge that is addressed in this article. In recent years, insurance and investment experts are interested in focusing on the use of the tail conditional expectation (TCE) because it has usable and desirable features in different situations. It is well-known that the tail conditional expectation as a risk measurement provides information about the mean of the tail of the loss distribution, while the tail variance (TV) measures the deviation of the loss from this mean along the tail of the distribution. In this paper, we present a theorem that extends the tail variance formula from the elliptical distributions to a rich class of Generalized Skew-Elliptical (GSE) distributions. We develop this theory for the four main classes of skew-elliptical distributions, including the Skew-Normal, Skew-Student-t, Skew-Logistic and Skew-Laplace distributions and obtain the proposed TV measure for them.