Robust Discrete Optimization and Downside Risk Measures

European Journal of Operational Research, 2015

In this paper we consider characterizations of the robust uncertainty sets associated with coherent and distortion risk measures. In this context we show that if we are willing to enforce the coherent or distortion axioms only on random variables that are affine or linear functions of the vector of random parameters, we may consider some new variants of the uncertainty sets determined by the classical characterizations. We also show that in the finite probability case these variants are simple transformations of the classical sets. Finally we present results of computational experiments that suggest that the risk measures associated with these new uncertainty sets can help mitigate estimation errors of the Conditional Value-at-Risk.

Operations Research, 2010

In this paper, we propose a framework for robust optimization that relaxes the standard notion of robustness by allowing the decision maker to vary the protection level in a smooth way across the uncertainty set. We apply our approach to the problem of maximizing the expected value of a payoff function when the underlying distribution is ambiguous and therefore robustness is relevant. Our primary objective is to develop this framework and relate it to the standard notion of robustness, which deals with only a single guarantee across one uncertainty set. First, we show that our approach connects closely to the theory of convex risk measures. We show that the complexity of this approach is equivalent to that of solving a small number of standard robust problems. We then investigate the conservatism benefits and downside probability guarantees implied by this approach and compare to the standard robust approach. Finally, we illustrate the methodology on an asset allocation example consisting of historical market data over a 25-year investment horizon and find in every case we explore that relaxing standard robustness with soft robustness yields a seemingly favorable risk-return trade-off: each case results in a higher out-of-sample expected return for a relatively minor degradation of out-of-sample downside performance.

SIAM Review, 2011

In this paper we survey the primary research, both theoretical and applied, in the area of robust optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multistage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.

SSRN Electronic Journal, 2000

We show that robust optimization of the VaR and CVaR risk measures with a minimum return constraint under distribution ambiguity reduce to the same second order cone program. We use this result to formulate models for robust risk optimization under joint ambiguity in distribution, mean returns and covariance matrices, under ellipsoidal ambiguity sets. We also obtain models for robust VaR and CVaR optimization for polytopic and interval ambiguity sets of the means and covariance. The models unify and/or extend several existing models. We also propose an algorithm and a heuristic for constructing an ellipsoidal ambiguity set from point estimates given by multiple securities analysts, and show how to overcome the well-known conservatism of robust optimization models. Using CDS spread return data from eurozone crisis countries we illustrate that investment strategies using robust optimization models perform well even out-of-sample. Finally, using a controlled experiment we show how the well-known sensitivity of CVaR to mis-specifications of the first four moments of the distribution is alleviated with the robust models.

2006

Value-at-Risk (VaR) is the most widely accepted risk measure in the financial and insurance industries, yet efficient optimization of VaR remains a very difficult problem. We propose a computationally tractable parametric approximation method for optimizing the VaR of a portfolio based on robust optimization techniques. The method results in the optimization of a modified VaR measure, Robust Parametric VaR (RPVaR), that is coherent, and is thus also desirable from a financial theory perspective. We show that RPVaR approximates the Conditional VaR of the portfolio as well. Numerical experiments with simulated and real market data indicate that our parametric approach results in lower realized portfolio VaR, better efficient frontier, and lower maximum realized portfolio loss than alternative approaches for quantilebased portfolio risk minimization.

Operations Research, 2011

In this paper we develop tight bounds on the expected values of several risk measures that are of interest to us. This work is motivated by the robust optimization models arising from portfolio selection problems. Indeed, the whole paper is centered around robust portfolio models and solutions. The basic setting is to find a portfolio that maximizes (respectively, minimizes) the expected utility (respectively, disutility) values in the midst of infinitely many possible ambiguous distributions of the investment returns fitting the given mean and variance estimations. First, we show that the single-stage portfolio selection problem within this framework, whenever the disutility function is in the form of lower partial moments (LPM), or conditional value-at-risk (CVaR), or value-at-risk (VaR), can be solved analytically. The results lead to the solutions for single-stage robust portfolio selection models. Furthermore, the results also lead to a multistage adjustable robust optimization...

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 only bounds on the mean and covariance matrix are available. We define the worst-case Value-at-Risk as the largest VaR attainable, given the partial information on the returns' distribution. We consider the problem of computing and optimizing the worst-case VaR, and we show that these problems can be cast as semidefinite programs. We extend our approach to various other partial information on the distribution, including uncertainty in factor models, support constraints, and relative entropy information.

Robust models have a major role in portfolio optimization for resolving the sensitivity issue of the classical mean-variance model. In this paper, we survey developments of worst-case optimization while focusing on approaches for constructing robust portfolios. In addition to the robust formulations for the Markowitz model, we review work on deriving robust counterparts for valueat-risk and conditional value-at-risk problems as well as methods for combining uncertainty in factor models. Recent findings on properties of robust portfolios are introduced and we conclude by presenting our thoughts on future research directions.

Operations Research, 2004

A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. A concern with such an approach is that it might be too conservative. In this paper, we propose an approach that attempts to make this trade-off more attractive; that is, we investigate ways to decrease what we call the price of robustness. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations. An attractive aspect of our method is that the new robust formulation is also a linear optimization problem. Thus we naturally extend our methods to discrete optimization problems in a tractable way. We report numerical results for a portfolio optimi...

Management Science, 2008

Value-at-Risk (VaR) is one of the most widely accepted risk measures in the financial and insurance industries, yet efficient optimization of VaR remains a very difficult problem. We propose a computationally tractable approximation method for minimizing the VaR of a portfolio based on robust optimization techniques. The method results in the optimization of a modified VaR measure, Asymmetry-Robust VaR (AR-VaR), that takes into consideration asymmetries in the distributions of returns and is coherent, which makes it desirable from a financial theory perspective. We show that ARVaR approximates the Conditional VaR of the portfolio as well. Numerical experiments with simulated and real market data indicate that the proposed approach results in lower realized portfolio VaR, better efficient frontier, and lower maximum realized portfolio loss than alternative approaches for quantile-based portfolio risk minimization.