CVaR Optimization: The Threshold and Scenario Excess Losses
Summary
The document presents a long-only portfolio problem that minimizes Conditional Value-at-Risk (CVaR) over simulated return scenarios, without expected-return estimates or a return constraint. It asks how to interpret the optimization variables gamma and z, and whether gamma represents the portfolio-return probability density. In the scenario-based linear formulation, gamma is an optimized threshold for losses, while each nonnegative z represents how far a scenario loss exceeds that threshold. Minimizing the objective selects the threshold and excess-loss values that yield the CVaR estimate at the chosen confidence level.
The discussion points to CVaR's use as a tail-risk measure and notes that the formulation can be handled with linear programming. It supplies no numerical example, solution, or comparison of portfolio outcomes. The setup also leaves implementation details such as portfolio weight constraints and how the scenarios are generated unspecified, so it introduces the variables rather than giving a complete optimization recipe.
Key ideas
- Gamma is an optimized loss threshold in the scenario-based CVaR formulation.
- Each z variable records a scenario loss's nonnegative excess above gamma.
- The objective averages the scaled excess losses and adds the threshold to estimate CVaR.
- The example minimizes portfolio CVaR without expected returns or a return constraint.
- Scenario generation and portfolio weight constraints are not specified.
Tags
Full text
# CVar optimization algorithms
# 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:
- 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).
$$ w^* = {{\underset{w}{\mathrm{arg\ min}}} = \gamma + \frac{1}{(1-\alpha )\cdot T}\sum z}\\ s.t.,\ z\geq f(w, y) - \gamma,\ i=1,...,T \ and \ z\geq 0$$
- $f(w,y)$ 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
$$ w^* = {{\underset{w,\ z, \ \gamma}{\mathrm{arg\ min}}} = \gamma + \frac{1}{(1-\alpha )\cdot T}\sum z}\\ s.t.,\ z\geq f(w, y) - \gamma,\ i=1,...,T \ and \ z\geq 0$$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.