The Intuition Behind Black-Litterman Model Portfolios

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Investment Management Division

The Intuition Behind Black-Litterman Model Portfolios
s In this article and as our title suggests, we demonstrate a method for understanding the intuition behind the Black-Litterman asset allocation model. s To do this, we use examples to show the difference between the traditional meanvariance optimization process and the Black-Litterman process. We show that the mean-variance optimization process, while academically sound, can produce results that are extreme and not particularly intuitive. In contrast, we show that the optimal portfolios generated by the Black-Litterman process have a simple, intuitive property: − The unconstrained optimal portfolio is the market equilibrium portfolio plus a weighted sum of portfolios representing an investor’s views. − The weight on a portfolio representing a view is positive when the view is more bullish than the one implied by the equilibrium and other views. − The weight increases as the investor becomes more bullish on the

view as well as when the investor becomes more confident about the view.

December 1999

Goldman Sachs Investment Management

Investment Management Research
Goldman Sachs Quantitative Resources Group
Guangliang He Robert Litterman (212) 357-3210 (212) 902-1677

Copyright 1999 Goldman, Sachs & Co. All rights reserved. The information in this publication is for your private information and is not intended as an offer or solicitation to buy or sell any securities. The information in this publication is for your private information and should not be construed as financial advice and it is not intended as an offer or solicitation to buy or sell any securities. The sole purpose of this publication is to inform the reader about the intuition behind the Black-Litterman model portfolios and is not intended as a solicitation for any Goldman Sachs product or service.

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Investment Management Research

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The Intuition Behind Black-Litterman Model Portfolios

Goldman Sachs Investment Management

Executive Summary

Since publication in 1990, the Black-Litterman asset allocation model has gained wide application in many financial institutions. As developed in the original paper, the Black-Litterman model provides the flexibility to combine the market equilibrium with additional market views of the investor. In the Black-Litterman model, the user inputs any number of views or statements about the expected returns of arbitrary portfolios, and the model combines the views with equilibrium, producing both the set of expected returns of assets as well as the optimal portfolio weights. In contrast to the Black-Litterman model, in the traditional mean-variance approach the user inputs a complete set of expected returns1, and the portfolio optimizer generates the optimal portfolio weights. However, users of the standard portfolio optimizers often find that their specification of expected returns produces output portfolio weights which may not make sense (due to the complex mapping between expected returns and portfolio weights and the absence of a natural starting point for the expected return assumptions). In this article, we use examples to illustrate the difference between the traditional mean-variance optimization process and the Black-Litterman process. In so doing, we demonstrate how the Black-Litterman approach2 provides both a reference point for expected return assumptions as well as a systematic approach to deviating from this point to express one’s market views.

The Traditional MeanVariance Approach

The Markowitz formulation of the portfolio optimization problem is a brilliant quantification of the two basic objectives of investing: maximizing expected return and minimizing risk. Having formed the foundation of portfolio theory for the nearly half a century since its publication, this framework has stood the test of...
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