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Black litterman model discussion


  • JOIMwww.joim.com

    Journal Of Investment Management, Vol. 11, No. 1, (2013), pp. 620 JOIM 2013


    The Markowitz (1952, 1959) meanvariance (MV) efficient frontier has been the theo-retical standard for defining portfolio optimality for more than half a century. However,MV optimized portfolios are highly susceptible to estimation error and difficult to managein practice (Jobson and Korkie, 1980, 1981; Michaud, 1989). The Black and Litterman(BL) (1992) proposal to solve MV optimization limitations produces a single maximumSharpe ratio (MSR) optimal portfolio on the unconstrained MV efficient frontier based onan assumed MSR optimal benchmark portfolio and active views. The BL portfolio is oftenuninvestable in applications due to large leveraged or short allocations. BL use an inputtuning process for computing acceptable sign constrained solutions. We compare con-strained BL with MV and Michaud (1998) optimization for a simple dataset. We show thatconstrained BL is identical to Markowitz and that Michaud portfolios are better diversi-fied under identical inputs and optimality criteria. The attractiveness of the BL procedureis due to convenience rather than effective asset management and not recommendablerelative to alternatives.

    1 Introduction

    For more than half a century, the Markowitz(1952, 1959) meanvariance (MV) efficient fron-tier has been the theoretical standard fordefining linear constrained portfolio optimal-ity. Markowitz optimization is a convenientframework for computing MV optimal portfo-lios that are designed to meet practical investmentmandates.

    aNew Frontier Advisors, LLC, Boston, MA 02110, USA.

    MV optimization, however, has a number ofwell-known investment limitations in practice.Optimized portfolios are unstable and ambigu-ous and highly sensitive to estimation errorin risk-return estimates. The procedure tendsto overweight/underweight assets with estimateerrors in high/low means, low/high variances, andsmall/large correlations, often resulting in poorperformance out-of-sample (Jobson and Korkie,1980, 1981). MV optimized portfolios in practiceare often investment unintuitive and inconsistentwith marketing mandates and management priors.

    6 First Quarter 2013

  • Deconstructing BlackLitterman: How to Get the PortfolioYou Already KnewYou Wanted 7

    Ad hoc input revisions and constraints result in anMV optimization process that is largely an exer-cise in finding acceptable rather than optimalportfolios (Michaud, 1989).To address estimation error issues in MV opti-mization, Black and Litterman (BL) (1992) pro-pose a single maximum Sharpe ratio (MSR)portfolio. This portfolio is constructed assuming:an unconstrained MV optimization framework;an assumed MSR optimal benchmark or mar-ket portfolio; an estimation error-free covariancematrix; and active investor views. BL optimalportfolios often have large leveraged and/or shortallocations that may make them uninvestablein applications. Moreover, Jobson and Korkiedocument severe out-of-sample investment lim-itations for the unconstrained MV optimizationframework, of which BL optimization is anexample.1

    BL introduce an input tuning parameter thatenables sign constrained MSR optimal solutions.2We describe the mathematical properties of BL,including -adjustment, and use a simple datasetto illustrate the procedures. We show that theBL sign constrained portfolio is identical toMarkowitz MSR for the same inputs and con-sequently no less estimation error sensitive. BLoptimality is also benchmark centric and subjectto the Roll (1992) critique of optimization onthe wrong efficient frontier. The Michaud (1998)proposal to address estimation error uses MonteCarlo resampling and frontier averaging methodsto generalize the Markowitz efficient frontier.3 Wecompare BL and Michaud MSR optimized port-folios under identical assumptions and show thatthe Michaud portfolios are better diversified andrisk managed.4

    Section 2 describes the mathematical charac-teristics and statistical issues associated withthe BlackLitterman procedure including -adjustment. Section 3 illustrates BL optimization

    with a simple dataset, compares the portfolioswith Markowitz and Michaud alternatives, anddemonstrates the sensitivity to covariance esti-mation error. Section 4 discusses BL relative tobenchmark-centric optimization, unconstrainedMV framework, and investor risk aversion. Sec-tion 5 summarizes and concludes.

    2 Black and Litterman optimization

    2.1 BlackLitterman frameworkBL optimization requires three investmentassumptions: (1) unconstrained MV optimiza-tion; (2) capital market portfolio M in equilib-rium on the Markowitz MV efficient frontier;and (3) covariance matrix without estimationerror. Under these conditions M is the MSRportfolio on the MV efficient frontier. Uncon-strained MV optimization and perfectly esti-mated covariance matrix allow computation ofthe implied or inverse returns = M con-sistent with MV Sharpe ratio optimality (Sharpe,1974; Fisher, 1975). The result is a set of esti-mated returns and covariance matrix forwhich the market portfolio M is the MSR effi-cient portfolio on the unconstrained MV efficientfrontier.5

    In elemental form, the BL proposal is a rationalefor the identification of a benchmark portfolio toanchor the optimization and overlay investmentviews. Benchmark anchoring of MV optimizedportfolios has a long tradition in investment prac-tice and is subject to Roll (1992) critiques.6The procedure trivially replicates its input in theabsence of any additional investor views. Investorviews are processed using an adaptation of theTheil and Goldberger (1961) mixed estimationformula relative to the implied return estimates.7 The revised returns with views BL areused to compute the BL MV optimal portfolioB. Deviations from the index weights indicate

    First Quarter 2013 Journal Of Investment Management

  • 8 Richard O. Michaud et al.

    optimal overweights and underweights for eachasset relative to the benchmark portfolio.

    2.2 BlackLitterman mathematical structure

    It is useful to briefly review the mathematicalstructure of the BL optimization framework.8We are given data for N assets with theoreticalmean and known variance . We assume Ma vector of equilibrium market or index port-folios weights. We construct an estimate of theimplied or inverse expected returns =M. represents the returns associated withmarket portfolio M in equilibrium for knowncovariance matrix .

    Views are specified as P N(v,), as where Pis a KN matrix whose rows are portfolios withviews, v is the vector of expected returns for theseportfolios, and is the covariance matrix for theviews. In the terminology of Bayesian statistics,we assign the views as the prior distribution. BLintroduce a tuning parameter to adjust the impactof the views. They express the distribution of theequilibrium mean as N(, ). The parameter may be viewed as a proxy for 1/T , the reciprocalof the number of time periods in the data, or asa measure of the relative importance of the viewsto the equilibrium but is often used simply to findinvestable (long-only) BL optimal portfolios. Theresulting posterior distribution then has a normaldistribution with mean equal to the BL estimateswhich can be expressed as:

    BL = + P (

    + PP

    )1(v P)

    = + V. (1)Equation (1) is useful, since it decomposes theestimate into the original data-based estimate and the contribution from the views V . In fact,if the mean estimate is the vector of equilibriumimplied returns , then the maximum informa-tion ratio unconstrained portfolio optimization

    results in portfolio weights PBL, which are pro-portional to:

    1BL = M + P (

    + PP


    (v P) (2)The second term on the right-hand side of Equa-tion (2) is a multiplication of the matrix P ,whose k columns are the portfolios with views,by the k by 1 vector (

    + PP )1(v P).

    Thus, in mathematical terms, the contribution ofinvestor views to the BL portfolio is confinedto the subspace spanned by the view portfolios.More intuitively, the BL portfolio pushes itselftoward or away from each view as necessary, but islimited to directions specified by the view portfo-lios themselves. Active bets induced by includingviews result in allocations in a direction which issolely a linear combination of those views.

    2.3 Investable BlackLitterman portfoliosBL unconstrained MV optimized portfolios oftenpossess large leveraged and/or short positions.In practice, investors often require that opti-mal portfolios are investable, i.e., that they aresign constrained and/or linear inequality con-strained within some specified range.9 By def-inition the equilibrium or market portfolio issign constrained. BL introduce the input tun-ing parameter 0 1 for finding portfoliosbetween the BL portfolio B and index portfolio Mthat are nonnegative (long-only) or satisfy somesuitable inequalities. The parameter provides amechanism for finding investable BL portfolios.The parameter operates as a scalar that divides thevariances associated with the uncertainty of theviews. Smaller values of cause greater inflationof the views uncertainty and limit their influenceon the results. As is reduced, or the constantmultiplier of the standard deviations of the viewsincreased, the BL portfolio approaches the bench-mark portfolio. The value of may be chosen to

    Journal Of Investment Management First Quarter 2013

  • Deconstructing BlackLitterman: How to Get the PortfolioYou Already KnewYou Wanted 9

    compute an investable BL portfolio when it usesjust enough of the certainty in investors viewsto meet investability constraint boundaries. Thenet effect of the -adjustment is to redu