Robust VaR and CVaR optimization under joint ambiguity in distributions, means, and covariances

Highlights

• Develop robust optimization models for risk management under general data ambiguity.

• An algorithm and a heuristic calculate ellipsoids in robust optimization.

• Overcome the well-known conservatism of robust optimization models.

• Robust models alleviate sensitivity of risk optimization to higher moments.

• Testing on credit default swaps during eurozone crisis shows that robustness can be achieved without sacrificing portfolio performance.

Abstract

We develop robust models for optimization of the VaR (value at risk) and CVaR (conditional value at risk) risk measures with a minimum expected return constraint under joint ambiguity in distribution, mean returns, and covariance matrix. We formulate models for ellipsoidal, polytopic, and interval ambiguity sets of the means and covariances. The models unify and/or extend several existing models. We also show how to overcome the well-known conservativeness of robust optimization models by proposing an algorithm and a heuristic for constructing joint ellipsoidal ambiguity sets from point estimates given by multiple securities analysts. 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. Finally, applying the model to the active management of portfolios of sovereign credit default swaps (CDS) from Eurozone core and periphery, and Central, Eastern and South-Eastern Europe countries, we illustrate that investment strategies using robust optimization models perform well out-of-sample, even during the eurozone crisis. We consider both buy-and-hold and active management strategies.

Introduction

Mean-variance portfolio optimization from the seminal thesis of Harry Markowitz provided the basis for a descriptive theory of portfolio choice: how investors make decisions. This led to further research in financial economics, with the development of a theory on price formation for financial assets (by William Sharpe) and on corporate finance taxation, bankruptcy and dividend policies (by Merton Miller). These descriptive contributions of the behavior of financial agents were recognized by a joint Nobel Prize in 1990. The prescriptive part of the theory – how investors should make decisions – was also acclaimed by practitioners, and mean-variance models proliferated. Here, however, problems surfaced: mean-variance portfolio optimization is sensitive to perturbations of input data (Best, Grauer, 1991, Chopra, Ziemba, 1993). Since the estimation of market parameters is error prone, the models are severely handicapped. In theory they produce well diversified portfolios but in practice they generate portfolios biased towards estimation errors.

With advances in financial engineering, variance was replaced by more sophisticated risk measures. Value-at-risk (VaR) became an industry standard and written into the Basel II accords to calculate capital adequacy, or calculate insurance premia, or set margin requirements. However, value-at-risk is criticized for being non-convex and it is also computationally intractable to optimize. In a seminal paper Artzner, Delbaen, Eber, and Heath (1999) provided an axiomatic characterization of coherent risk measures, and conditional value-at-risk (CVaR) emerged as one such measure. CVaR rose to prominence with the work of Rockafellar and Uryasev (2000) who showed that it can be minimized as a linear program. CVaR optimization emerged as a credible successor to mean-variance models: it is coherent, computationally tractable, and found numerous applications (Zenios, 2007, ch. 5). Basel III shifted from VaR to an expected shortfall measure of risk, to capture tail risk, especially during periods of financial market stress.

With the increased attention placed on CVaR an argument against its use and in favor of VaR also surfaced. VaR estimated from a set of sampled scenarios is a robust statistic, i.e., it is insensitive to small deviations of the underlying distribution from the observed distribution, whereas CVaR is not. Kou, Peng, and Heyde (2013) argue that risk measures should be robust but coherent risk measures are not and CVaR lacks a key property.

This paper contributes to an extensive body of literature that seeks to eliminate the sensitivity of CVaR by incorporating data ambiguity in the optimization model. We develop robust models for VaR and CVaR optimization under general ellipsoidal ambiguity sets and for joint ambiguity in means and covariances. These models touch on previous works by others, reviewed below, and extend and/or unify some of the previous contributions. We also develop an algorithm and a heuristic to construct an ellipsoid ambiguity set from point estimates given by multiple securities analysts, and to control the ambiguity set to avoid too conservative solutions. This contribution is, to the best of our knowledge, novel in the literature. We use numerical experiments to highlighting that the models can be robust under first, second, and higher-moment ambiguity, and also that robustness does not necessarily come by sacrificing portfolio performance.

The 2008 global crisis revived the work of Chicago economist Knight (1921) that considers financial and economic data as ambiguous instead of uncertain. Under uncertainty a probability model is known but the random variables are observed with some measurement error, whereas under ambiguity the probability model is unknown. Hence, data mis-specification is not only due to measurement error, that can be reduced with improved estimation techniques, but is an integral part of financial decision making. Data ambiguity deserves attention as an issue to be modeled, not as a problem to be eliminated. It is from this perspective that we develop this study.

Early suggestions in dealing with the sensitivity of portfolio optimization models to data estimation errors use Bayesian or James–Stein estimators, resampling, or restricting portfolio choices with ad hoc constraints. We do not review this literature as it is outside the scope of our work. We build, instead, on recent research that brings developments in robust optimization to bear on portfolio selection under data ambiguity.

Robust optimization models require constraints to be satisfied even with ambiguous data, and the objective value to be insensitive to the ambiguity. Concepts of robustness in optimization have been developed independently in the fields of operations research and engineering design. Mulvey, Vanderbei, and Zenios (1995) proposed the robust optimization of large scale systems when data take values from a discrete scenario set, using a regularization of the objective function to control its sensitivity, and penalty functions to control constraint violation. This approach spurred numerous applications in facility location, power capacity planning, disaster response, agribusiness, supply chain management, production and process planning, network design, and so on. Robust convex optimization was developed by Ben-Tal and Nemirovski (1998) for optimization problems with data ambiguity described by an ellipsoid. They showed that important convex optimization problems admit a tractable robust counterpart. The foundational paper spurred extensive theoretical and applied research (Ben-Tal, Ghaoui, Nemirovski, 2009, Bertsimas, Brown, Caramanis, 2011). In a way, robust portfolio optimization brings ideas from Taguchi robust engineering design to the design of portfolios. Authors usually adopt the robust convex optimization framework over an appropriate ambiguity set, and it is in this domain that our paper makes a contribution. Fabozzi, Huang, and Zhou (2010) review robust portfolio optimization using mean, VaR, and CVaR risk measures.

The robust counterpart to mean-variance optimization was developed by Goldfarb and Iyengar (2003). Using a linear factor model for asset returns they introduce “uncertainty structures” – the confidence regions associated with parameter estimation – and formulate robust portfolio selection models corresponding to these uncertainty structures as second order cone programs (SOCP). They also develop robust counterparts for VaR and CVaR optimization under the normality assumption of mean-variance models. Schottle and Werner (2009); Tütüncü and Koenig (2004); Ye, Parpas, and Rustem (2012) develop further robust mean-standard deviation and mean-variance models, removing some of the assumptions of the Goldfarb–Iyengar paper.

Our paper develops a robust counterpart of CVaR optimization (RCVaR) and finds it identical to robust VaR optimization (RVaR). Hence, we give a detailed review of previous works on RVaR and RCVaR optimization, so we can place the innovations of our own contribution.

Current literature addresses the following sources of ambiguity of model parameters: (i) ambiguity in mean return estimates, (ii) ambiguity in covariance matrix estimate, and (iii) ambiguity in the distribution of the data. Ambiguity can be independent for each parameter or joint for multiple parameters. If ambiguity is independent for each parameter we have simple sets of parameter values, e.g., a (sub)vector of parameters lies in some interval. For joint ambiguity the parameters belong to sets such as ellipsoids or convex polytopes. Models based on discrete scenarios may have ambiguity in the scenario values or the scenario probabilities or both. For models with continuous distributions, ambiguity is in the moments. Another important distinction in understanding the various contributions is captured in the terminology used. VaR and CVaR minimization models lack a minimum return constraint, and ambiguity is restricted to the objective function. VaR and CVaR optimization trade off the risk measure against an expected return target. The difference between risk-minimization and risk-optimization is not innocuous: risk-optimization models with a target expected return are difficult to analyze if the means are ambiguous, as ambiguity appears in the constraints, and not only in the obective.

Ghaoui, Oks, and Oustry (2003) address RVaR minimization with partially known distributions of returns, whereby means and covariance lie within a known uncertainty set, such as an interval, a polytope (polytopic uncertainty), or a convex subset (convex moment uncertainty). Given this information on return distributions they cast RVaR minimization for interval uncertainty as semidefinite program (SDP), and for polytopic uncertainty as SOCP. They also give a general, but potentially intractable, model for convex moment uncertainty. Their model lacks the target expected return constraint.

The first RCVaR optimization model is by Quaranta and Zaffaroni (2008) for interval uncertainty of the means. Zhu and Fukushima (2009) consider RCVaR optimization for box and ellipsoidal uncertainty in distribution, as well as distribution mixtures of convex combination of predetermined distributions and unknown mixture weights. By “distribution” the authors mean the probabilities of the discretized data. Their model can potentially be used to approximate joint ambiguity of means, covariances, and higher moments through the choice of weights of the distributions, using results of Marron and Wand (1992).¹ Distinctly from this work, our models are exact. Furthermore, if there is no information on the distribution of the discretized random variable but there is information on the first two moments of the distribution, then we can use our models but not Zhu–Fukushima.

Delage and Ye (2010) show (as a special case of their work) that RCVaR minimization for ambiguity in the probabilities, mean, and second moment, can be solved in polynomial time. The authors provide bounds and generate confidence regions on the mean and covariance matrix in case of moment uncertainty but stop short from developing RCVaR models and develop, instead, a robust model for expected utility maximization under moment uncertainty.

Chen, He, and Zhang (2011) point out that robust solutions come with a computational price: robust optimization models can be infinite dimensional and, without proper choice of uncertainty sets, the model may be intractable. They obtain bounds on worst case value of lower partial moments and use them to develop RVaR and RCVaR minimization for distribution ambiguity with closed form solution under a normalization constraint. Pac and Pınar (2014) extend further RVaR and RCVaR optimization for distribution and mean returns ambiguity, but fixed covariance matrix. This is one of the papers extended in our work, by allowing covariance matrix ambiguity.

A contribution that filled several gaps is Gotoh, Shinozaki, and Takeda (2013). Scenario based VaR and CVaR minimization models use discrete data observations (i.e., scenarios) and their probabilities to determine the empirical distribution. There are three possible ways to introduce ambiguity and formulate RVaR and RCVaR counterparts using scenarios. The first approach (Zhu & Fukushima, 2009), keeps the scenarios fixed and considers ambiguous probabilities from a box or an ellipsoid. Gotoh et al. (2013) consider a second approach with uncertainty in scenarios but fixed probabilities, and a third approach, where both scenarios and probabilities are ambiguous.

Our work considers ambiguity in the distribution as well as mean returns and the covariance matrix, and joint ambiguity in combinations of the above. These are, to the best of our knowledge, the most general ambiguity sets considered in the literature for RVaR and RCVaR models. Joint uncertainty in means and covariance matrix was also considered by Schottle and Werner (2009) and Ye et al. (2012) but for mean-standard deviation and mean-variance optimization, respectively. We use an ellipsoidal ambiguity set which is general and obtain tractable optimization models as SOCP. We use the term ambiguity sets in the Knightean sense, instead of uncertainty sets, in discussing robust models. Robust optimization literature typically refers to uncertainty sets although usually ambiguity is meant.

The paper is organized as follows. Section 2 defines VaR, CVaR, RVaR, and RCVaR models and discusses the instability of VaR and CVaR optimal portfolios. Section 3 is the main one. It formulates RVaR and RCVaR for ambiguous distributions, and ellipsoidal ambiguity in means and covariance, discusses the construction of ambiguity sets (Section 3.2), extends or unifies existing results (Section 3.3). It also develops models for polytopic and interval ambiguity sets (Section 3.4). In developing our models we also identify some implicit limiting assumptions made in previous works and explain how we overcome the limitations. Section 4 reports on two distinct numerical tests. First, using simulations we test the robustness of optimal portfolios under mis-specification of mean, variance, skewness, and kurtosis. Second, using sovereign CDS spread returns during the eurozone crisis we investigate the robustness of alternative investment strategies. Conclusions are in Section 5. Proofs are gathered in the appendix.