# CVar optimization algorithms

An alternative measure of losses to Var, with more attractive properties, is Conditional Value-at-risk or CVar which is also called Mean Excess Loss, Mean Shortfall, or Tail Var. CVar is a more consistent measure of risk since it is sub-additive and convex. Moreover, it can be optimized using linear programming and non smooth optimization algorithms, which allow handling portfolios with very large numbers of instruments and scenarios. Numerical experiments show that the minimization CVar also leads to near optimal solutions in VaR terms because CVar is always greater than or equal to Var. CVar can be used in return-risk analyses. For instance, we can calculate a portfolio with a specified return [...].

[Conditional Value-at-risk: Optimization Algorithms and Applications - Stanislav Uryasev - 2002]

I am currently learning about the "vanilla" CVar optimization (no expected returns estimates, long only portfolio, minimize CVar of the entire portfolio, no returns constrain).

Considering the research article "CVaR Robust Mean-CVaR Portfolio Optimization" by Maziar Salahi, Farshid Mehrdoust, and Farzaneh Piri:

1. The optimization problem can be expressed through linear programming. Please note that $$\alpha$$ is the given confidence level (in relation to CVar), $$T$$ is the number of generated random scenario/returns (i.e. via Monte Carlo simulation), $$z$$ is an array of artificial variables (please refer to the paper).

$$w^* = {{\underset{w}{\mathrm{arg\ min}}} = \gamma + \frac{1}{(1-\alpha )\cdot T}\sum z}\\ s.t.,\ z\geq f(w, y) - \gamma,\ i=1,...,T \ and \ z\geq 0$$

1. $$f(w,y)$$ is the loss function which is equal to $$-y^Tw$$. Please note that $$y$$ is the realization of the generated random events (the vector of the $$T$$ scenarios/returns of $$N$$ assets).

I read other papers too but still it is not very clear to me what are $$\gamma$$ and $$z$$. For what I understand they are not given parameters and they are not constants. Is $$\gamma$$ the probability density function of the distribution of the entire portfolio returns $$f(w,y)$$? In any case it would be great if someone could briefly explain to me what they are.

EDIT: clarification

$$w^* = {{\underset{w,\ z, \ \gamma}{\mathrm{arg\ min}}} = \gamma + \frac{1}{(1-\alpha )\cdot T}\sum z}\\ s.t.,\ z\geq f(w, y) - \gamma,\ i=1,...,T \ and \ z\geq 0$$

• Although the notation you use is different, this question resembles another question about CVar Optimizaton here quant.stackexchange.com/questions/38095/… Aug 3, 2020 at 15:34
• What you call $\gamma,z$ are "decision variables" (outputs) of the optimization. When the optimizer terminates and returns the optimal $\gamma$ it can be interpreted as the VaR of the optimal portfolio. Aug 3, 2020 at 15:44
• Oh I see. So it is a three variable optimization problem (indeed in the paper is $min_{x,\ z,\ \gamma}$). Moreover, is the constraint about the sum of the weights be equal to 1 implicit in that formulation of the problem? Aug 3, 2020 at 21:40
• Perhaps this is useful as well quant.stackexchange.com/questions/51724/… Aug 4, 2020 at 9:17