Expected Shortfall

Assuming sufficient differentiability properties, results of Tasche (2000) show that

$$\frac{\partial \text{ES}_{\alpha}}{\partial u_i} (u_1,\ldots,u_n) = - \mathbf{E}[X_i \vert X \leq - \text{VaR}_{\alpha}(X)] .$$ (22)

We refer to Tasche (2000) but also Lemma A.1 and A.2 in Appendix A for further information on the differentiation of ES. For $\rho(u) =$   ES$_{\alpha}(X(u))$, equation (21) can be written as

$$\frac{\partial \text{ES}_{\alpha}'}{\partial u_i} (u_1,\ldots,u_{n-1}) = - \mathbf{E}[X_i \vert X \leq - _{\alpha}(X)]$$ (23)

$$+ \frac{V_i}{V_n}\mathbf{E}[X_n \vert X \leq - _{\alpha}(X)] .$$

In fact, this simple expression of expectations is very suitable for numerical computations by Monte-Carlo methods.