Application of Robust Statistics to Asset Allocation Models

107 Pages·2006·4.91 MB·English

Application of Robust Statistics to Asset Allocation Models
by
Xinfeng Zhou
M.S., Chemistry
University of Washington, 2001 MASSACHUSETTS INSTTfUTE
OF TECHNOLOGY
Submitted to Sloan School of Management
JUL 2 4 2006
in partial fulfillment of the requirements for the degree of
Master of Science in Operations Research
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2006 ARCHIVES
@ 2006 Massachusetts Institute of Technology. All rights reserved
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a' c// Sloan School of Management
Operations Research Center
May 2006
Certifiedb y .......... ... ........................-...-.. ....................
f-d Roy E. Welsch
Professor of Management Science, Statistics, and Engineering Systems
Thesis Supervisor
Certifiedb y ................... ................. .......... . .... ..................
George Verghese
Professor of Electrical Engineering
Thesis Supervisor
Accepted by ..............................................................
James B. Orlin
Edward Pennell Brooks Professor of Operations Research
Codirector, Operations Research Center

Application of Robust Statistics to Asset Allocation Models
By
Xinfeng Zhou
Submitted to Sloan School of Management
On May 17, 2006 in partial fulfillment of the requirements
for the degree of Master of Science in Operations Research
ABSTRACT
Many strategies for asset allocation involve the computation of expected returns and the
covariance or correlation matrix of financial instruments returns. How much of each instrument
to own is determined by an attempt to minimize risk (the variance of linear combinations of
investments in these financial assets) subject to various constraints such as a given level of return,
concentration limits, etc. The expected returns and the covariance matrix contain many
parameters to estimate and two main problems arise. First, the data will very likely have outliers
that will seriously affect the covariance matrix. Second, with so many parameters to estimate, a
large number of observations are required and the nature of markets may change substantially
over such a long period. In this thesis we use robust covariance procedures, such as FAST-MCD,
quadrant-correlation-based covariance and 2D-Huber-based covariance, to address the first
problem and regularization (Bayesian) methods that fully utilize the market weights of all assets
for the second. High breakdown affine equivariant robust methods are effective, but tend to be
costly when cross-validation is required to determine regularization parameters. We, therefore,
also consider non-affine invariant robust covariance estimation. When back-tested on market
data, these methods appear to be effective in improving portfolio performance. In conclusion,
robust asset allocation methods have great potential to improve risk-adjusted portfolio returns
and therefore deserve further exploration in investment management research.
Thesis Supervisor: Roy E. Welsch
Title: Professor of Management Science, Statistics, and Engineering Systems
Thesis Supervisor: George Verghese
Title: Professor of Electrical Engineering

Acknowledgements
I would like to express my greatest gratitude to my advisors, Professor Roy Welsch and
Professor George Verghese for their constant guidance, advice and encouragement. Working
with Professor Welsch was a really rewarding experience. He truly shaped my understanding of
statistics. Professor Verghese has been an inspiring mentor over the years who continuously
encouraged and supported my efforts in knowledge-building and research. I am also deeply
indebted to my PhD advisor, Professor Peter Dedon from the department of Biological
Engineering Division, for his understanding and support over the last five years. I also want to
thank Sameer Jain for his expertise, support and inputs into my work. Last but not at least, I
would like to express many thanks to Dr. Jan-Hein Cremers at State Street Corporation for
providing me opportunities to work on asset management projects and for inspiring my interests
in topics covered in this thesis.
This thesis is dedicated to the memory of my sister, Xinli Zhou.

Table of Contents
, Introduction of Asset Allocation Models ................................................... 9
1.1 Overview ................................................... 9
1.2 Foundation of Mean-Variance Portfolio Optimization. .................................................. 11
1.3 The Mean-Variance Efficient Frontier ................................................... 15
1.4 Advantages and Disadvantages of Mean-Variance Optimization ...................................... 20
2. Shrinkage Models and GARCH Models ................................................... 22
2.1 Introduction ................................................... 22
2.2 Sample Mean and Covariance Matrix. .................................................. 23
2.3 Factor Models ................................................... 25
2.3.1 Single-factor Model ................................................... 25
2.3.2. Multi-factor (Barra) model ................................................... 26
2.3.3. Principal component analysis (PCA) ................................................... 28
2.4. Shrinkage methods. .................................................. 29
2.5. Exponentially weighted moving average (EWMA) model ............................................... 33
2.6. GARCH Model ................................................... 35
2.6.1. Constant conditional correlation GARCH ................................................... 39
2.6.2. Dynamic conditional correlation GARCH ................................................... 40
2.6.3. BEKK GARCH ................................................... 42
2.6.4. Scalar GARCH. .................................................. 42
2.7. Application to Historical Data ................................................... 44
2.8. Conclusion ................................................... 53
3. Robust Estimation of the Mean and Covariance Matrix. .................................................. 55
3.1. Introduction ................................................... 55
3.2. Traditional Robust Estimators ................................................... 56
3.2.1. LAD portfolio estimator ................................................... 56
3.2.2. Huber portfolio estimator ................................................... 58
3.2.3. Rank-correlation portfolio estimator. .................................................. 59
3.3. M-estimator ................................................... 61
3.4. Fast-MCD ................................................... 62
3.5. Pair-wise Robust Estimation. .................................................. 65
3.5.1. Quadrant correlation method ................................................... 67
3.5.2. Two-dimensional Winsorization method ............................ ..................68
3.5.3. Two-dimensional Huber method ................................................... 71
3.6. Application to Historical Data ................................................... 73
3.7. C onclusion ......................................................................................................................... 80
4. Market-weight Bayesian Approaches ................................................... 81
4.1. Introduction........................................................................................................................ 81
4.2. Black-Litterman model ...................................................................................................... 83
4.3. Regularization models ................................................... 87
4.4. Application of Regularization models ................................................... 89
4.5. Co nclusion ......................................................................................................................... 95
5. Summary and Future Research Directions ................................................... 96
References ............................................................................................................................... 105
List of Figures
Figure 1.1. Efficient frontier and Capital Market Line
Figure 2.1. Optimal shrinkage intensity estimated using either Mahalanobis distance or
asymptotic Frobenius norm estimation
Figure 3.1. Adjusted Winsorization (for initial covariance) with c, = 1.96
List of Tables
Table 2.1. Application of shrinkage and GARCH methods
Table 2.2. Performance of V, CAPM, Principal, Mahalanobis, Frobenius, CCC-GARCH, DCC-
GARCH models and Market index for up = 15%
Table 2.3. Performance of V, CAPM, Principal, Mahalanobis, Frobenius, CCC-GARCH, DCC-
GARCH models and Market index for p = 20%
Table 2.4. GMVP performance of V, CAPM, Principal, Mahalanobis, Frobenius, CCC-GARCH,
DCC-GARCH models and Market index
Table 3.1. Application of robust estimation to expected returns and covariance matrix
Table 3.2. Performance of V, LAD, Huber, Rnakcov, FAST-MCD, QCIQR, 2D-Winsorization,
2D-Huber models and Market index for p = 15%
Table 3.3. Performance of V, LAD, Huber, Rnakcov, FAST-MCD, QCIQR, 2D-Winsorization,
2D-Huber models and Market index for p = 20%
Table 3.4. GMVP performance of V, FAST-MCD, QCIQR, 2D-Winsorization, 2D-Huber
models and Market index
Table 4.1. Performance of V, V1, V2, LADI, LAD2, Hi, H2 models and Market index for
,p = 15%
Table 4.2. Performance of V, V1, V2, LAD1, LAD2, Hi, H2 models and Market index for
Up= 20%
Chapter 1
Introduction of Asset Allocation Models
1.1 Overview
Asset allocation is the process that investors use to determine the asset classes in which to invest
in and the weight for each asset class. Past studies have shown that asset allocation explains 75 -
90°,' of the return variation and is the single most important factor determining the variability of
portfolio performance. The objective of an asset allocation model is to find the right asset mix
that provides the appropriate combination of expected return and risk that allows investors to
achieve their financial goals. Harry Markowitz's mean-variance portfolio theory is by far the
most well-known and well-studied asset allocation model for both academic researchers and
practitioners alike (1, 2). The crux of mean-variance portfolio theory assumes that investors
prefer (1) higher expected returns for a given level of standard deviation/variance and (2) lower
standard deviations/variances for a given level of expected return. Portfolios that provide the
maximum expected return for a given standard deviation and the minimum standard deviation
for a given expected return are termed efficient portfolios and those that don't are termed
inefficient portfolios.
Although intuitively and theoretically appealing, the application of mean-variance portfolio
optimization has been hindered by the difficulty in accurately estimating model inputs, the
expected returns and the covariance matrix of the assets. The goal of this thesis is to address this
critical problem from different perspectives, with an emphasis on robust statistics and Bayesian
approaches. In doing so, the thesis is organized as five Chapters:
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In Chapter 1, we first present the economic and mathematical background of the mean-variance
portfolio optimization model and discuss the importance as well as the difficulty in estimating
the model inputs, the expected return and the covariance matrix.
In Chapter 2, we investigate and apply some of the existing factor models and Bayesian
shrinkage models as well as Generalized Autoregressive Conditional Heteroskedastic (GARCH)
models to estimate the expected returns and the covariance matrix. Using the performance results
of a US industrial selection portfolio, we show that "optimal" portfolios selected by shrinkage
models have limited success when the number of assets N is of the same order as the number of
return observations T. The GARCH models, which automatically include all historical
information by exponential weighting, yield much better results.
In Chapter 3, we investigate and expand some of the robust statistical approaches to estimate the
expected returns and the covariance matrix. Beside traditional robust methods, such as the least
absolute deviation (LAD) method and the Spearman rank correlation, we focus more on recent
developments such as the quadrant-correlation-based covariance and the 2D-Huber-based
covariance to reduce or eliminate the effect of outliers. These new models prove to be more
valuable than shrinkage models and GARCH models in dramatically improving risk-adjusted
portfolio performance and reducing asset turnovers.
In Chapter 4, we investigate some of the more complex Bayesian approaches with emphasis on
regularization models. We show that regularization models yield portfolios with significant
increases in risk-adjusted portfolio performance, especially when transaction costs are taken into
consideration. Our results also indicate that L regularization methods may yield better results
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than L regularization methods. Overall L regularization methods also outperform the robust
2 1
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