Black Litterman

A STEP-BY-STEP GUIDE TO THE BLACK-LITTERMAN MODEL Incorporating user-specified confidence levels
Thomas M. Idzorek

Thomas M. Idzorek, CFA Director of Research Ibbotson Associates 225 North Michigan Avenue Chicago, Illinois 60601-7676

Original Draft: January 1, 2002 This Draft: April 26, 2005

A STEP-BY-STEP GUIDE TO THE BLACK-LITTERMAN MODEL Incorporating user-specified confidencelevels
ABSTRACT

The Black-Litterman model enables investors to combine their unique views regarding the performance of various assets with the market equilibrium in a manner that results in intuitive, diversified portfolios. This paper consolidates insights from the relatively few works on the model and provides step-by-step instructions that enable the reader to implement this complex model. A newmethod for controlling the tilts and the final portfolio weights caused by views is introduced. The new method asserts that the magnitude of the tilts should be controlled by the user-specified confidence level based on an intuitive 0% to 100% confidence level. This is an intuitive technique for specifying one of most abstract mathematical parameters of the Black-Litterman model.

ASTEP-BY-STEP GUIDE TO THE BLACK-LITTERMAN MODEL Incorporating user-specified confidence levels
Having attempted to decipher many of the articles about the Black-Litterman model, none of the relatively few articles provide enough step-by-step instructions for the average practitioner to derive the new vector of expected returns.1 This article touches on the intuition of the Black-Litterman model,consolidates insights contained in the various works on the Black-Litterman model, and focuses on the details of actually combining market equilibrium expected returns with “investor views” to generate a new vector of expected returns. Finally, I make a new contribution to the model by presenting a method for controlling the magnitude of the tilts caused by the views that is based on an intuitive 0% to 100%confidence level, which should broaden the usability of the model beyond quantitative managers.

Introduction The Black-Litterman asset allocation model, created by Fischer Black and Robert Litterman, is a sophisticated portfolio construction method that overcomes the problem of unintuitive, highly-concentrated portfolios, input-sensitivity, and estimation error maximization. These threerelated and well-documented problems with mean-variance optimization are the most likely reasons that more practitioners do not use the Markowitz paradigm, in which return is maximized for a given level of risk. The Black-Litterman model uses a Bayesian approach to combine the subjective views of an investor regarding the expected returns of one or more assets with the market equilibrium vector ofexpected returns (the prior distribution) to form a new, mixed estimate of expected returns. The

A STEP-BY-STEP GUIDE TO THE BLACK-LITTERMAN MODEL

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resulting new vector of returns (the posterior distribution), leads to intuitive portfolios with sensible portfolio weights. Unfortunately, the building of the required inputs is complex and has not been thoroughly explained in the literature.The Black-Litterman asset allocation model was introduced in Black and Litterman (1990), expanded in Black and Litterman (1991, 1992), and discussed in greater detail in Bevan and Winkelmann (1998), He and Litterman (1999), and Litterman (2003).2 The Black Litterman model combines the CAPM (see Sharpe (1964)), reverse optimization (see Sharpe (1974)), mixed estimation (see Theil (1971, 1978)), theuniversal hedge ratio / Black’s global CAPM (see Black (1989a, 1989b) and Litterman (2003)), and mean-variance optimization (see Markowitz (1952)). Section 1 illustrates the sensitivity of mean-variance optimization and how reverse optimization mitigates this problem. Section 2 presents the Black-Litterman model and the process of building the required inputs. Section 3 develops an implied...