In Rockafellar-Uryasev 2001 paper the mean-CVaR optimization can be written as a linear programming optimization problem as:
$$P_{\text{CVaR}} = \arg \min_w \text{VaR}_\alpha+\frac{1}{(1-\beta)S}\sum_{s=1}^{S}y_s $$ subject to: $$y_s \geq f(\mathbf{w},\mathbf{r_s})-\text{VaR}_\alpha$$ $$y_s\geq 0$$ where $$y_s = [f(\mathbf{w},\mathbf{r_s})-\text{VaR}_\alpha]^+$$ I have 2 questions regarding this optimization:
- How is the VaR computed? or while programming the optimization the user has to program the way the VaR is going to be computed. Here (http://past.rinfinance.com/agenda/2009/yollin_slides.pdf) is an R code to do the optimization but I don't see anywhere VaR computation.
- Isn't the first restriction obvious? given the definition of $y_s$ or my understanding of $y_s$ is incorrect?