Black-Litterman and Mean-Variance Efficient Portfolios: An Empirical Comparison (PDF)

Inhaltsangabe:Abstract:
In practice, mean-variance optimization results in non-intuitive and extreme portfolio allocations, which are highly sensitive to variations in the inputs. The Black-Litterman approach overcomes or at least mitigates the major part of these problems. Furthermore, it enables us to incorporate investment views and to assign confidence levels to the views. These features make the Black-Litterman model a strong quantitative tool that provides an ideal framework for tactical asset allocation. This work considers the weak and strong aspects of both models and demonstrates how their optimization procedures are put into practice. In different setups, the performance of the two approaches is compared empirically. The Black-Litterman efficient portfolios achieve a significantly better return-to-risk performance than the mean-variance optimal strategy.
Inhaltsverzeichnis:Table of Contents:
List of Tables v
List of Figuresvi
List of Abbreviationsvii
1Introduction1
2Mean-Variance Portfolio Optimization3
2.1The Model3
2.2Restrictive Assumptions of the Model12
2.2.1Mean-Variance Preferences12
2.2.2Other Assumptions and Drawbacks of the Model22
2.3Problems with Mean-Variance Optimization in Practice23
2.3.1Scope of the Inputs23
2.3.2Estimating the Inputs24
2.3.3Extreme Allocations and Corner Solutions30
2.3.4Instability of Mean-Variance Efficient Portfolios33
3The Black-Litterman Approach36
3.1Overview and Intuition behind the Approach36
3.2Reverse Optimization and Equilibrium Implied Returns37
3.3Long-Term Equilibrium in the Black-Litterman Model40
3.4Setting Up Views and Specifying Confidence Levels40
3.5Combining Equilibrium Returns with Views44
3.5.1Views under Certainty44
3.5.2Views under Uncertainty45
3.6Solving the Model48
3.6.1Calibrating the Model – the Factor tau.48
3.6.2Black-Litterman Efficient Portfolios in Practice49
4Review and Comparison of the Established Results54
5Empirical Return and Risk Analysis of both Approaches55
5.1Framework for the Empirical Comparison55
5.2Performance of Mean-Variance and Black-Litterman Efficient Portfolio Strategies58
5.2.1Backtesting Comparison I58
5.2.2Backtesting Comparison II60
6Conclusion62
Appendix ADerivation of the Solution to the Mean-Variance Optimization Problem64
Appendix BInput Estimates for Portfolios of the Dow Jones Industrial Average Constituents66
Appendix CInput Estimates for Portfolios of the EuroStoxx Sector Indices70
Appendix DQuantiles of the Standard Normal Distribution73
Appendix EReturn Distributions of Optimized Portfolios75
Bibliography78
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Chapter 1, Introduction:
By publishing Portfolio Selection, Harry Markowitz (1952) laid the foundations of modern portfolio theory. In his article, Markowitz described the fundamental relationship between risk and expected return of securities and worked out the basic principles of quantitative portfolio construction. His approach, which is nowadays also known as mean-variance optimization, bases on the idea that an investor has two conflicting objectives in wanting the expected return of a portfolio to be as high as possible and portfolio risk to be as low as possible. Therefore, an investor seeks to maximize expected return for a given level of risk, or to minimize portfolio risk for a given expected return. The latter denotes the potential reward of a portfolio, whereas the risk of a portfolio is measured by the standard deviation of returns.
When Portfolio Selection was first published in the Journal of Finance in 1952, it did not draw a lot of attention initially. But over the years the financial community recognized the potential of Markowitz’s model and numerous extensions and approaches, inspired by his work, followed. Based on the concept of Markowitz’s approach, Sharpe (1964) and Lintner (1965) developed the Capital Asset Pricing Model (CAPM), which describes the pricing of securities in a state of market equilibrium. In 1990 Markowitz was awarded the Nobel Prize for his work, together with Merton Miller and William Sharpe.
However, as elegant as Markowitz’s approach might be in theory, as many problems seem to arise when putting it to work. The mean-variance model relies on assumptions, that require to restrict either the return distributions of all assets or the preferences of the investor. Furthermore, it is necessary to estimate an enormous amount of input parameters, which brings along the problems of estimation errors and error maximization of the optimizer. In practice, mean-variance optimization usually results in rather extreme and non-intuitive portfolios. In addition, the optimal weights are highly-sensitive to variations in the inputs. These characteristics of Markowitz’s approach make it difficult for investment professionals to utilize it in an asset management mandate.
To overcome these problems, Fischer Black and Robert Litterman (1992) developed an approach which bases on the idea of combining market equilibrium with investment views. Unlike in the mean-variance model, there is no need to estimate the expected returns for all assets involved. The Black-Litterman approach works with equilibrium returns that are implied by relative market capitalizations or the weightings of a benchmark portfolio. The model enables the manager to incorporate an arbitrary number of investment views. Uncertainty in these views can be taken into account by specifying confidence levels. These features of the Black-Litterman approach make it possible to efficiently integrate the specific knowledge of researchers into the allocation process. The optimization procedure results in stable and well-diversified portfolios, that match economic intuition and reflect the manager’s investment views. However, the Black-Litterman approach is not able to overcome all of the problems of mean-variance optimization, so that still some difficulties remain unsolved.
In this work I aim at demonstrating the different optimization procedures of the Black-Litterman approach and the mean-variance model, and how both approaches are put into practice. In an empirical comparison, it will be examined how the efficient portfolios of both approaches would have performed in the past.
The thesis is organized as follows. In the ensuing section 2, I give an overview of the mean-variance model. Since the mean-variance optimizer is embedded in the Black-Litterman approach, the shortcomings of this method are discussed in great detail and illustrated on practical examples. In section 3, the Black-Litterman approach is introduced. The procedure is explained by means of a step-by-step guide focussing on items relevant to practical use. A critical review in section 4 considers the weak and strong points of both approaches, as well as their common features and differences. Section 5 contains an empirical analysis of Black-Litterman and mean-variance efficient portfolios. The two approaches are examined and compared with regard to their return and risk performance. A summary of the established results and concluding remarks in section 6 close the thesis.
The following calculations and portfolio optimizations have been carried out using MatLab and Visual Basic Applications for Microsoft Excel.