Bivariate Tail Conditional Co-Expectation for elliptical distributions

In this paper, we consider a random vector $X=(X_1,X_2)$ following a multivariate Elliptical distribution and we provide an explicit formula for $E\left(X\right|X\le \tilde{X})$, i.e., the expected value of the bivariate random variable X conditioned to the event $X\le \tilde{X}$, with $\tilde{X}\in R^2$. Such a conditional expectation has an intuitive interpretation in the context of risk measures. Specifically, $E\left(X\right|X\le \tilde{X})$ 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.