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  • OPERATIONS RESEARCHVol. 60, No. 6, NovemberDecember 2012, pp. 13891403ISSN 0030-364X (print) ISSN 1526-5463 (online) http://dx.doi.org/10.1287/opre.1120.1115

    2012 INFORMS

    Inverse Optimization: A New Perspective on theBlack-Litterman Model

    Dimitris BertsimasSloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139,

    [email protected]

    Vishal GuptaOperations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139,

    [email protected]

    Ioannis Ch. PaschalidisDepartment of Electrical and Computer Engineering, Boston University, Boston, Massachusetts 02215,

    [email protected]

    The Black-Litterman (BL) model is a widely used asset allocation model in the financial industry. In this paper, we providea new perspective. The key insight is to replace the statistical framework in the original approach with ideas from inverseoptimization. This insight allows us to significantly expand the scope and applicability of the BL model. We provide aricher formulation that, unlike the original model, is flexible enough to incorporate investor information on volatility andmarket dynamics. Equally importantly, our approach allows us to move beyond the traditional mean-variance paradigm ofthe original model and construct BL-type estimators for more general notions of risk such as coherent risk measures.Computationally, we introduce and study two new BL-type estimators and their corresponding portfolios: a mean varianceinverse optimization (MV-IO) portfolio and a robust mean variance inverse optimization (RMV-IO) portfolio. These twoapproaches are motivated by ideas from arbitrage pricing theory and volatility uncertainty. Using numerical simulationand historical backtesting, we show that both methods often demonstrate a better risk-reward trade-off than their BLcounterparts and are more robust to incorrect investor views.

    Subject classifications : finance: portfolio optimization; programming: inverse optimization; statistics: estimation.Area of review : Financial Engineering.History : Received May 2011; revisions received November 2011, January 2012; accepted June 2012. Published online in

    Articles in Advance November 20, 2012.

    1. IntroductionThe Black-Litterman (BL) model is a widely used assetallocation model in the financial industry. Introduced inBlack and Litterman (1992), the model uses an equilibriumanalysis to estimate the returns of uncertain investmentsand employs a Bayesian methodology to blend theseequilibrium estimates with an investors private informa-tion, or views, about the investments. Computational expe-rience has shown that the portfolios constructed by thismethod are more stable and better diversified than thoseconstructed from the conventional mean-variance approach.Consequently, the model has found much favor with practi-tioners. The U.S. investment bank Goldman Sachs regularlypublishes recommendations for investor allocations basedon the BL model and has issued reports describing thefirms experience using the model (Bevan and Winkelmann1998). A host of other firms (Zephyr Analytics, BlackRock,Neuberger Berman, etc.) also use the BL model at the coreof many of their investment analytics.

    The model, however, does have its shortcomings. First, itis somewhat limited in the way it allows investors to specifyprivate information. Namely, it only allows investors tospecify views on asset returns, but not on their volatility

    or market dynamics. Secondly, and more restrictively, themodel is predicated upon the mean-variance approach toportfolio allocation. A host of theoretical and empiricalwork suggests variance may not be a suitable proxy forrisk. As a consequence, other risk measures such as valueat risk (VaR) or conditional value at risk (CVaR) have beenexplored. (See, for example, Artzner et al. 1999, Bertsimaset al. 2004, Grootveld and Hallerbach 1999, Harlow 1991,Jorion 1997, Rockafellar and Uryasev 2002.)

    Subsequent research has tried to address these short-comings. We mention only a few examples and refer thereader to Walters (2010) and the references therein for amore complete survey. Giacometti et al. (2007), Martelliniand Ziemann (2007), and Meucci (2005) extend beyondthe mean-variance paradigm, using fat-tailed distributionsto model asset returns, coupled with VaR or CVaR, andviews on the tail behavior of returns. By contrast, Pstor(2000) and Pstor and Stambaugh (2000) build upon theBayesian interpretation of the original model, using a gen-eral pricing model as their prior. Finally, Meucci (2008)and Meucci et al. (2011) formulate a further generalizationwithin the statistical framework, modeling risk factors (inlieu of asset returns) by a general distribution and solving

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  • Bertsimas, Gupta, and Paschalidis: Inverse Optimization: Black-Litterman Model1390 Operations Research 60(6), pp. 13891403, 2012 INFORMS

    a minimum entropy optimization to update this distribu-tion to one that incorporates a very general set of views. Inmost cases, this optimization is solved numerically, and inthe case when the distribution is constrained to belong to aparametric family, the optimization is usually nonconvex.

    In this paper, we provide a new optimization-driven per-spective on the BL model that avoids many of the statis-tical assumptions of previous approaches. The key insightis to characterize the BL estimator as the solution to aparticular convex optimization problem. More specifically,we formulate a problem in inverse optimizationa setupwhere one is given an optimal solution to an optimizationproblem and seeks to characterize the cost function and/orother problem data. Initial research on inverse optimizationfocused on specific combinatorial optimization problems(see Heuberger 2004 for a survey). Later, Ahuja and Orlin(2001) provided a unified approach for linear programmingproblems using duality. Iyengar and Kang (2005) extendedthese ideas to conic optimization with an application tomanaging portfolio rebalancing costs.

    Previous research, starting with Black and Litterman(1992), has drawn some connections between the BL modeland what many authors call reverse optimization. The keyidea, formalized in He and Litterman (1999), is to solvethe first-order optimality condition of a Lagrangean relax-ation of the Markowitz problem to derive the equilibriumreturns. It may not be clear how to generalize this approachto other constrained asset allocation problems. Often, asin Herold (2005), which considers a budget constraint andDa Silva et al. (2009), which considers information-ratioportfolio optimization, the authors are forced to rely on adhoc arguments. Furthermore, reverse optimization onlyprovides the equilibrium estimates. Most authors still use astatistical approach to blend in the views.

    To the best of our knowledge, our work is the first tooffer a rigorous inverse optimization interpretation of theBL model. This has several advantages. First, it allowsus to completely characterize the set of input data inthe Markowitz problem in equilibrium and to incorporatemore general views, e.g., on volatility. Second, it enablesus to move beyond mean-variance and adopt more gen-eral risk metrics by leveraging robust optimization ideas.Further, our approach does not rely on specific distribu-tional assumptions on market returns; we construct theequilibrium estimates and blend them with views usingoptimization techniques. As such, our approach unifies sev-eral threads in the literature (Herold 2005, Krishnan andMains 2005, Martellini and Ziemann 2007). It can also bepotentially combined with the ideas in Meucci (2008) andMeucci et al. (2011) of modelling risk factors instead ofasset returns.

    We summarize the major contributions of our paper asfollows:

    1. By linking the BL model with inverse optimization,we provide a novel formulation that constructs equilibrium-based estimators of the mean returns and the covariancematrix.

    2. Our framework permits greater freedom in the typeof private information and investor views expressible inthe model. Specifically, our formulation allows investors toincorporate views on volatility and market dynamics. Weleverage this freedom to create a new asset allocation pro-cedure we call the mean-variance inverse optimization(MV-IO) approach, which uses BL-type estimators in a set-ting motivated by the arbitrage pricing theory.

    3. By drawing on ideas from robust optimization, wegeneralize the traditional Markowitz portfolio problem toone that can encompass risk metrics such as VaR, CVaR,and generic coherent risk measures.

    4. We then use inverse optimization to construct BL-type estimators in this new model. We illustrate the powerof this method with what we call the robust mean varianceinverse optimization (RMV-IO) approach, which accommo-dates volatility uncertainty.

    5. We provide computational evidence to show that MV-IO and RMV-IO portfolios often provide better risk-rewardprofiles and are more robust to incorrect or extreme viewsthan their BL counterparts.

    The remainder of the paper is organized as follows:Section 2 briefly reviews the BL model. Section 3 derivesthe BL model from principles in inverse optimization andshows how it can be extended to include volatility estima-tion and views on volatility. Section 4 introduces a moregeneral portfolio allocation problem and constructs a BL-type estimator in this setting. Section 5 provides numericalresults illustrating th