Rockafellar-Uryasev Mean-CVaR Optimization with Auxiliary Variables
Summary
The document explains how the Rockafellar-Uryasev formulation turns CVaR optimization into a linear program. A reader asks how VaR is calculated within the optimization and whether the constraint defining each scenario's excess loss is redundant because that excess is the positive part of loss above VaR.
The response clarifies that VaR is a scalar decision variable, not a statistic calculated in a separate step; the optimization produces it along with the portfolio solution. The scenario excess variables are also decision variables. Their nonnegativity and lower-bound constraints make them equal to the positive-part excess at an optimum, without imposing that positive-part equality directly. The response explains that directly imposing the equality would destroy convexity and would forfeit the linear-program structure. The exchange focuses on the formulation and the role of its variables; it does not discuss choices of scenarios, portfolio constraints, or empirical performance.
Key ideas
- In the Rockafellar-Uryasev formulation, VaR is optimized as a scalar decision variable.
- Scenario excess losses are represented by auxiliary decision variables.
- Constraints cause those variables to match positive-part excesses at the optimum.
- Imposing the positive-part equality directly would undermine the convex linear-program formulation.
- The formulation does not itself specify scenario construction or portfolio-specific constraints.
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# Rockafellar-Uryasev mean-CVaR optimiztion
# Rockafellar-Uryasev mean-CVaR optimiztion
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?
## Answer by Drew (score 6, accepted)
https://quant.stackexchange.com/a/39753
- $VaR_\alpha$ is a scalar choice variable in the minimization problem. In the Rockafeller-Uryasev paper, it is simply called $\alpha\in R$. (C.f., the program described in Theorem 2 of that paper, or the programming problem described after equation (17); alternatively, look at the structure of the choice vector $x$ on page 16 of the Yollin slides.) $VaR_\alpha$ is thus a by-product of solving the problem.
- The fact that $y_{s}=[f(w,rs)−VaR_\alpha]^{+}$ is also a by-product of solving the problem; it is not imposed. Rather, the $y$ vector is an auxiliary choice variable. (Your $y$ plays the role of $d$ in the Yollin slides, which are embedded in the choice vector $x$ on page 16.) The constraints impose the equality implicitly. But if that equality were imposed directly in the program, the constraint set would fail to be convex (the problem would not be a linear programming one), and the solution would be greatly complicated.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.