RORC

The partial derivatives (21) of the ES-RORC under the portfolio constraint (16) are obtained using standard rules of differentiation:

$$\frac{\partial \varphi_{\alpha}'}{\partial u_i} = - \mathbf{E}[X]\cdot \frac{\mathbf{E}[X_i \vert X \leq - \text{Va... ...}(X)]}{\text{ES}_{\alpha}(X)^2} + \frac{\mathbf{E}[X_i]}{\text{ES}_{\alpha}(X)}$$ (24)

$$- \frac{V_i}{V_n} \left( - \mathbf{E}[X]\cdot \frac{\mathbf{E}[X_... ...xt{ES}_{\alpha}(X)^2} + \frac{\mathbf{E}[X_n]}{\text{ES}_{\alpha}(X)} \right) .$$

As in the case of (23), we see from the definition (10) of ES that (24) is a relatively simple expression of expectations.