Tail conditional moments for elliptical and log-elliptical distributions

In this paper we provide the tail conditional moments for the class of elliptical distributions, which was introduced in Kelker (1970) and was widely discussed in Gupta et al. (2013) and for the class of log-elliptical distributions. These families of distributions include some important members such as the normal, Student-t, logistic, Laplace, and log-normal distributions. We give analytic formulae for the nth higher order unconditional moments of elliptical distributions, which has not been provided before. We also propose novel risk measures, the tail conditional skewness and the tail conditional kurtosis, for examining the skewness and the kurtosis of the tail of loss distributions, respectively.