# Worst-Case Value-At-Risk and Robust Portfolio Optimization: A Conic Programming Approach

@article{Ghaoui2003WorstCaseVA, title={Worst-Case Value-At-Risk and Robust Portfolio Optimization: A Conic Programming Approach}, author={Laurent El Ghaoui and Maksim Oks and François Oustry}, journal={Oper. Res.}, year={2003}, volume={51}, pages={543-556} }

Classical formulations of the portfolio optimization problem, such as mean-variance or Value-at-Risk (VaR) approaches, can result in a portfolio extremely sensitive to errors in the data, such as mean and covariance matrix of the returns. In this paper we propose a way to alleviate this problem in a tractable manner. We assume that the distribution of returns is partially known, in the sense that onlybounds on the mean and covariance matrix are available. We define the worst-case Value-at-Risk…

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## References

SHOWING 1-10 OF 30 REFERENCES

### On the Relation Between Option and Stock Prices: A Convex Optimization Approach

- Economics, Computer ScienceOper. Res.
- 2002

Convex and semidefinite optimization methods, duality, and complexity theory are introduced to shed new light on the relation of option and stock prices based just on the no-arbitrage assumption, and it is shown that it is NP-hard to find best possible bounds in multiple dimensions.

### Robust Portfolio Selection Problems

- EconomicsMath. Oper. Res.
- 2003

This paper introduces "uncertainty structures" for the market parameters and shows that the robust portfolio selection problems corresponding to these uncertainty structures can be reformulated as second-order cone programs and, therefore, the computational effort required to solve them is comparable to that required for solving convex quadratic programs.

### Using Value-at-Risk to Control Risk Taking: How Wrong Can You Be?

- Economics
- 1998

We study a source of bias in value-at-risk estimates that has not previously been recognized. Because value-at-risk estimates are based on past data, a trader will often have a good understanding of…

### Generalized Chebychev Inequalities: Theory and Applications in Decision Analysis

- Computer Science, MathematicsOper. Res.
- 1995

A very general framework for analyzing these kinds of problems where, given certain "moments" of a distribution, the authors can compute bounds on the expected value of an arbitrary "objective" function.

### CORC Technical Report TR-2002-03 Robust portfolio selection problems

- Economics
- 2002

In this paper we show how to formulate and solve robust portfolio selection problems. The objectiveoftheserobustformulationsistosystematicallycombatthesensitivityoftheoptimal portfolio to statistical…

### Optimal Inequalities in Probability Theory: A Convex Optimization Approach

- Mathematics, Computer ScienceSIAM J. Optim.
- 2005

It is shown that it is NP-hard to find tight bounds for k = 2 and $\Omega={\cal R}^n$, and an efficient algorithm for finding tight bounds when S is a union of convex sets, over which convex quadratic functions can be optimized efficiently.

### Robust Portfolio Selection

- Computer Science
- 2000

The problem of statistical robustness of the Markowitz optimizer is discussed and it is shown that the latter is not robust, meaning that a few extreme assets prices or returns can lead to irrelevant 'optimal' portfolios.

### Evaluating Value at Risk Methodologies: Accuracy versus Computational Time

- Business
- 1996

Recent research has shown that different methods of computing Value at Risk (VAR) generate widely varying results, suggesting the choice of VAR method is very important. This article examines six VAR…

### An Interior-Point Method for a Class of Saddle-Point Problems

- Mathematics
- 2003

We present a polynomial-time interior-point algorithm for a class of nonlinear saddle-point problems that involve semidefiniteness constraints on matrix variables. These problems originate from…