Bivariate Tail Conditional Co–Expectation for Elliptical distributions

In this paper, we consider a random vector following a multivariate Elliptical distribution and we provide an explicit formula for , i.e., the expected value of the bivariate random variable X conditioned to the event , with . Such a conditional expectation has an intuitive interpretation in the context of risk measures. Specifically, can be interpreted as the Tail Conditional Co–Expectation of X (TCoES). Our main result analytically proves that for a large number of Elliptical distributions, the TCoES can be written as a function of the probability density function of the Skew Elliptical distributions introduced in the literature by the pioneering work of Azzalini (1985) . Some numerical experiments based on empirical data show the usefulness of the obtained results for real-world applications.