Portfolio Management Using Black-Litterman

Table of Contents

Table of Figures

Table of Abbreviations

Table of Symbols

1 Introduction

2 Basic Concepts – Foundation for Black-Litterman
2.1 Criticism of Classical Portfolio Optimization
2.2 Market Equilibrium Implied by CAPM
2.3 Bayes’ Theorem

3 The Black-Litterman Model
3.1 Assumptions of the Model
3.2 Putting the Approach into Practice
3.2.1 Intuition
3.2.2 Equilibrium Market Implied Returns
3.2.3 Investors’ Views
3.2.4 Revised Implied Returns
3.2.5 Revised Portfolio Weights
3.3 The Equations Behind the Model
3.3.1 Calculating Implied Returns
3.3.2 Defining the Black-Litterman Optimization Problem
3.3.3 Implementing Views with Uncertainty
3.3.4 Computing Revised Implied Returns
3.3.5 Obtaining Revised Portfolio Weights
3.4 Illustration of the Model

4 Critical Review of the Black-Litterman Model
4.1 Advantages and Benefits
4.2 Weaknesses and Limitations
4.3 Extensions and Enhancements
4.4 A Behavioral Finance Viewpoint
4.5 A Practical Viewpoint

5 Conclusion

Appendix

References

Table of Figures

Fig. 1 Illustration of Black-Litterman Return Distribution

Fig. 2 Black-Litterman as Building Block of the Asset Allocation Process

Fig. 3 Black-Litterman as Building Block of the Asset Allocation Process

Table of Abbreviations

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Table of Symbols

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1 Introduction

Portfolio management usually comprises asset allocation decisions with the goal of creating diversified portfolios. Managers can consult quantitative mod­els to support their decision-making process.

Fischer Black and Robert Litterman (1992) developed the Black-Litterman (BL) optimization model. It is based on the idea of efficient markets, the capi­tal asset pricing model of Sharpe (1964) and Lintner (1965), as well as the es­tablished mean-variance optimization (MVO) developed by Markowitz (1952), and conditional probability theory dating back to Bayes (1763).

Starting point of the BL model is the assumption that equilibrium markets and market cap. weights provide the investor with Implied Returns. The BL model uses a mixed estimation technique to incorporate investors’ Views into return forecasts. It is possible to implement relative and absolute opinions regarding expected returns of assets with different levels of confidence. These Views en­able an adjustment of equilibrium Implied Returns, which forms a new expec­tation of BL Revised Implied Returns. As a result of optimization with BL in­put data, the investor gets new optimal portfolio weights.

The motivation of Black and Litterman (1992) to develop a new portfolio op­timization tool was a lack of acceptance of the Markowitz algorithm within professional asset managers.^[1] There aim was to shape a model which can over­come the weaknesses of MVO and which combines a quantitative and qualita­tive approach.^[2] Consequently, the BL model tackles the weakest point of MVO, its sensitivity to the return forecasts and allows taking active Views.

This paper is structured in the following sections: First, it shows the basic prin­ciples on which the BL model is founded. Then, it illustrates the model by means of its assumptions, the general approach, and the math involved. Finally, it evaluates the model in a critical review, provides an overview of applicable extensions, and addresses the issues of practicability and behavioral finance.

2 Basic Concepts – Foundation for Black-Litterman

2.1 Criticism of Classical Portfolio Optimization

Harry Markowitz’s (1952) MVO is seen as the cornerstone for many models in modern portfolio theory.^[3] The aim of this paper is not to reproduce the well established MVO technique but to show its main weaknesses which motivated Black and Litterman (1992) to create their model.

The purely mathematical MVO algorithm is very sensitive to the return fore­casts and requires expected returns for all available assets.^[4] The high input sen­sitivity can produce highly concentrated, poorly diversified portfolios that dis­accord with the common idea of diversification. Moreover, small changes in returns can result in large changes in the optimal portfolio weights which entail unstable portfolios.^[5] A main critique is that the portfolio outcome of MVO of­ten includes large, unjustified short positions.^[6] MVO operates as an error maxi­mization routine since more weight is assigned to assets with higher expected returns which are highly qualified for estimation risk.^[7] The input sensitivity also makes it difficult for investors to incorporate their own views or confi­dence levels on anticipated developments into the optimization process.

Another problematic aspect is that MVO makes no consideration of market cap., but small capitalized assets can imply some limitations.^[8]

Overall the fund manager does not get intuitive results when applying the stan­dard MVO approach.

2.2 Market Equilibrium Implied by CAPM

Another acknowledged concept in financial theory is the capital asset pricing model (CAPM), simultaneously developed by Sharpe (1964) and Lintner (1965) among others. The standard CAPM starts from the idea of efficient capital markets where supply equals demand, and market-clearing security prices appear.^[9] Alongside its many implications, the most relevant in the con­text of the BL model is the occurring market equilibrium. The market portfolio is on the efficient frontier, and has the maximum Sharpe Ratio.^[10] In equilibrium investors should hold the market portfolio as their risky asset, and adjust their risk exposure by borrowing or lending at the risk free rate (Tobin Separation).^[11]

The market cap. of the different assets determine their weights in the market portfolio at equilibrium. This appealing result of equilibrium weights is used as “a neutral reference point” for investors in the BL model.^[12]

[...]

^[1] cf. He and Litterman (1999), p. 2,3

^[2] cf. Satchell and Scowcroft (2000), p. 147

^[3] cf. Markowitz (1987) for a general introduction into portfolio theory.

^[4] cf. Drobetz (2003), pp. 206-213

^[5] cf. Drobetz (2001), p. 60

^[6] cf. Black and Litterman (1992), p. 28

^[7] cf. Michaud (1989), p. 33, 34, Harvey et al. (2004) address est. error and higher moments.

^[8] cf. Markert (2006), p. 20

^[9] cf. Market efficiency was defined by Fama (1970)

^[10] The Sharpe Ratio is a measure of the mean excess return per unit of risk, Sharpe (1981)

^[11] cf. Tobin (1958)

^[12] cf. Black and Litterman (1992), p. 29