A Tractable Parametric Approach to Value-at-Risk Optimization

Operations Research, 2014

This paper introduces a framework for robustifying convex, law invariant risk measures. The robustified risk measures are defined as the worst case portfolio risk over neighborhoods of a reference probability measure, which represent the investors beliefs about the distribution of future asset losses. It is shown that under mild conditions, the infinite dimensional optimization problem of finding the worst-case risk can be solved analytically and closed-form expressions for the robust risk measures are obtained. Using these results, robust versions of several risk measures including the standard deviation, the Conditional Value-at-Risk, and the general class of distortion functionals are derived. The resulting robust risk measures are convex and can be easily incorporated into portfolio optimization problems and a numerical study shows that in most cases they perform significantly better out-of-sample than their non-robust variants in terms of risk, expected losses, and turnover.

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.

Mathematical and Statistical Methods for Actuarial Sciences and Finance, 2010

The portfolio optimisation problem is modelled as a mean-risk bicriteria optimisation problem where the expected return is maximised and some (scalar) risk measure is minimised. In the original Markowitz model the risk is measured by the variance while several polyhedral risk measures have been introduced leading to Linear Programming (LP) computable portfolio optimisation models in the case of discrete random variables represented by their realisations under specified scenarios. Recently, the second order quantile risk measures have been introduced and become popular in finance and banking. The simplest such measure, now commonly called the Conditional Value at Risk (CVaR) or Tail VaR, represents the mean shortfall at a specified confidence level. The corresponding portfolio optimisation models can be solved with general purpose LP solvers. However, in the case of more advanced simulation models employed for scenario generation one may get several thousands of scenarios. This may lead to the LP model with a huge number of variables and constraints, thus decreasing the computational efficiency of the model. We show that the computational efficiency can be then dramatically improved with an alternative model taking advantages of the LP duality. Moreover, similar reformulation can be applied to more complex quantile risk measures like Gini's mean difference as well as to the mean absolute deviation.

Annals of Operations Research, 2021

Robust optimization (RO) models have attracted a lot of interest in the area of portfolio selection. RO extends the framework of traditional portfolio optimization models, incorporating uncertainty through a formal and analytical approach into the modeling process. Although several RO models have been proposed in the literature, comprehensive empirical assessments of their performance are rather lacking. The objective of this study is to fill in this gap in the literature. To this end, we consider different types of RO models based on popular risk measures and conduct an extensive comparative analysis of their performance using data from the US market during the period 2005–2020. For the analysis, two different robust versions of the mean–variance model are considered, together with robust models for conditional value-at-risk and the Omega ratio. The robust versions are compared against the nominal ones through various portfolio performance metrics, focusing on out-of-sample results.

2009

Value-at-Risk, despite being adopted as the standard risk measure in finance, suffers severe objections from a practical point of view, due to a lack of convexity, and since it does not reward diversification (which is an essential feature in portfolio optimization). Furthermore, it is also known as having poor behavior in risk estimation (which has been justified to impose the use of parametric models, but which induces then model errors). The aim of this paper is to chose in favor or against the use of VaR but to add some more information to this discussion, especially from the estimation point of view. Here we propose a simple method not only to estimate the optimal allocation based on a Value-at-Risk minimization constraint, but also to derive-empirical-confidence intervals based on the fact that the underlying distribution is unknown, and can be estimated based on past observations.

In the portfolio optimization, the goal is to distribute the fixed capital on a set of investment opportunities to maximize return while managing risk. Risk and return are quantities that are used as input parameters for the optimal allocation of the capital in the suggested models. But these quantities are not known at the time of the formulation and solving the problem. Thus they should be estimated to solve the problem which might lead to large error. One of the widely used approaches to deal with such a situation, is robust optimization. In this paper we study the robust models of the Mean-Conditional Value at Risk (Mean-CVaR) portfolio selection problem under the estimation risk in mean return for both interval and ellipsoidal uncertainty sets. The corresponding robust models are a linear programming problem and a second order conic programming problem (SOCP) respectively. At the end an example is given to demonstrate the impact of uncertainty.

Annals of Operations Research, 2007

Many risk measures have been recently introduced which (for discrete random variables) result in Linear Programming (LP) models. While some LP computable risk measures may be viewed as approximations to the variance (e..g., the mean absolute deviation or the Gini's mean absolute difference), shortfall or quantile risk measures are recently gaining more popularity in various financial applications. In this paper we study LP solvable portfolio optimization models based on extensions of the Conditional Value at Risk (CVaR) measure. The models use multiple CVaR measures thus allowing for more detailed risk aversion modeling. We study both the theoretical properties of the models and their performance on real-life data.

2010

Recently, a new approach for optimization of Conditional Value-at-Risk (CVaR) was suggested and tested with several applications. For continuous distributions, CVaR is defined as the expected loss exceeding Value-at Risk (VaR). However, generally, CVaR is the weighted average of VaR and losses exceeding VaR. Central to the approach is an optimization technique for calculating VaR and optimizing CVaR simultaneously. This paper extends this approach to the optimization problems with CVaR constraints. In particular, the approach can be used for maximizing expected returns under CVaR constraints. Multiple CVaR constraints with various confidence levels can be used to shape the profit/loss distribution. A case study for the portfolio of S&P 100 stocks is performed to demonstrate how the new optimization techniques can be implemented.

Financial Engineering, E-commerce and Supply Chain, 2002

This paper suggests two new heuristic algorithms for optimization of Value-at-Risk (VaR). By definition, VaR is an estimate of the maximum portfolio loss during a standardized period with some confidence level. The optimization algorithms are based on the minimization of the closely related risk measure Condi-£ This paper is based on the research project conducted by Nicklas Larsen during his visit to the University of Florida in Spring 2000. This research project has not been conducted in the framework of any legal agreement between the