Statistical analysis on markowitz portfolio mean variance principle

Statistical Analysis on Markowitz Portfolio Mean-Variance Principle LIU HUIXIA (Master of Science, Tsinghua University, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF STATISTICS AND APPLIED PROBABILITY NATIONAL UNIVERSITY OF SINGAPORE 2007 i Acknowledgements I would like to express my deepest gratitude to my supervisor, Professor Bai Zhidong and Associate Professor Wong Wing-Keung Their insights and suggestions helped me improve my research skills Their patience and encouragement carried me on through difficult times And their valuable feedback has been contributing greatly to this dissertation I am grateful to my former advisor Associate Professor Wang Yougan and Associate Professor Bruce Brown whose energetic working style and serious attitude toward research have influenced me greatly Many thanks to Associate Professor Chen Zehua for his teaching, helpful suggestions and kindly help during this period of study Special thanks to Dr Pan Guangming and former graduate Xiao Han for discussions on this topic Thanks to all my friends and former classmates Mr Zhao Yudong, Ms Li Wenyun, Mr Zong Jianping, Ms Hao Ying and Ms Ma Lanfang whom I have spent ii more than three years with and who gave me a lot of help not only in study but also in everyday life Finally, I am forever indebted to my parents and my sisters for their endless love and encouragement during the entire period of my study CONTENTS i Contents Introduction 1.1 Markowitz’s Mean-Variance Principle 1.2 The Markowitz Optimization Enigma 1.3 Existing Approaches In Literature 1.3.1 Bayes-Stein Estimation 1.3.2 Black-Litterman Model 10 1.3.3 Single-Index and Multi-Index Model 12 1.3.4 Shrinkage Estimator of the Covariance Matrix 13 1.3.5 Random Matrix Approach 14 Organization of the Thesis 15 1.4 CONTENTS ii Random Matrix Theory 17 2.1 Basic Concepts 18 2.2 Results Potentially Applicable to Finance 20 Bootstrap Method 3.1 24 24 3.1.1 Nonparametric Bootstrap 25 3.1.2 3.2 Two Basic Bootstrap Methods Parametric Bootstrap 25 The Principle of Bootstrap Method 26 Bootstrap-Corrected Estimation 28 4.1 Plug-in Estimator 29 4.2 Bootstrap Estimator 41 4.3 Simulation Study 44 4.3.1 Over Prediction 45 4.3.2 Bootstrap-Correction Method 48 4.3.3 Illustration 56 CONTENTS iii Asymptotic Normality and hypothesis testing 60 5.1 Introduction of Asymptotic Normality Properties of eigenvectors of large sample covariance matrix 5.2 61 Main Results 65 5.2.1 5.2.2 5.3 Finite Dimensional Convergence and Asymptotic Covariances 71 Tightness 79 Asymptotic Normality of The Bootstrap-Corrected Estimator and hypothesis testing 5.3.1 88 The hypothesis testing based on The Bootstrap-Corrected Estimator 5.3.2 88 Simulation Results 95 Conclusions and Further Research 98 6.1 Conclusions 98 6.2 Further Research 100 SUMMARY iv Summary The Markowitz mean-variance optimization procedure is highly appreciated as a theoretical result in literature Given a set of assets, it enables investors to find the best allocation of wealth incorporating their preferences as well as their expectation of return and risk It is expected to be a powerful tool for investors to allocate their wealth efficiently However, it has been demonstrated to be less applicable in practice The portfolio formed by using the classical Mean-Variance approach always results in extreme portfolio weights that fluctuate substantially over time and perform poorly in the out-of-sample forecasting The reason for this problem is due to the substantial estimation error of the inputs of the optimization procedure The classical meanvariance approach which uses the sample mean and sample covariance matrix as inputs always results in serious departure of its estimated optimal portfolio allocation from its theoretical counterpart In this thesis, applying large dimensional data analysis, we first theoretically SUMMARY v explain that this phenomenon is natural when the number of asset is large In addition, we theoretically prove that the estimated optimal return is always larger than the theoretical value when the number of assets is large To circumvent this problem, we employ large dimensional random matrix theory again to develop a bootstrap method to correct the overprediction and reduce the estimation error Our simulation results show that the bootsrap correction method can significantly improve the accuracy of the estimation Therefore the essence of the portfolio analysis problem could be adequately captured by our proposed estimates This greatly enhances the practical use of Markowitz mean-variance optimization procedure Furthermore, we investigate the asymptotic normality property of our bootstrap corrected estimator This will be useful in performing the hypothesis testing for the theoretical return by using our bootstrap corrected estimator Towards this end, we first generalize the results of the asymptotic properties of the eigenvectors of large sample covariance matrix In addition, we provide the proofs of the asymptotic properties of our bootstrap corrected estimator LIST OF TABLES vi List of Tables ˆ ˆ ˆ Performance of Rp and Rp over the Optimal Return R for different values of p and for different values of p/n 47 Comparison between the Empirical and Corrected Portfolio Returns and Allocations 51 4.3 MSE and Relative Efficiency Comparison 54 4.4 Plug-in Returns and Bootstrap-Corrected Returns 57 5.1 Simulated Power 97 5.2 Simulated Type I Error 97 4.1 4.2 LIST OF FIGURES vii List of Figures 4.1 4.2 4.3 4.4 Empirical and theoretical optimal returns for different numbers of assets 46 Comparison between the Empirical and Corrected Portfolio Allocations and Returns 50 MSE Comparison between the Empirical and Corrected Portfolio Allocations/Returns 55 Comparison between the Plug-in Returns and Bootstrap-Corrected Returns 58 Chapter 6: Conclusions and Further Work 98 Chapter Conclusions and Further Research 6.1 Conclusions The purpose of this thesis is to solve the “Markowitz optimization enigma” by developing a new optimal return estimate to capture the essence of portfolio selection By utilizing the large dimensional data analysis, we first explain, and thereafter, theoretically prove that the plug-in return obtained by plugging the sample mean and the sample covariance into the formulae of the optimal return is always larger than its theoretical value when the number of assets is large We called this phenomena“over-prediction” in this thesis We show that this problem is not due to the “measurement error” but due to poor estimation of the allocation by plugging the sample mean and sample covariance into the theoretical allocation formulae To circumvent this problem, we develop new estimators, the bootstrap-corrected Chapter 6: Conclusions and Further Work 99 return and the bootstrap-corrected allocation for their theoretical counterparts by employing both the large dimensional random matrix theory and the parametric bootstrap method In addition, in order to perform hypothesis testing for the theoretical optimal return using our proposed estimator, we further derive some asymptotic properties of our proposed estimators in the last part of this thesis Our simulation results confirm that the essence of the portfolio analysis problem could be adequately captured by our proposed bootstrap-corrected method which improves the accuracy of the estimation dramatically As our approach is easy to operate and implement in practice, the whole efficient frontier of our estimates can be constructed analytically Thus, our proposed estimator makes the Markowitz MV optimization procedure to be absolutely implementable and practically useful We also note that our model includes situations in which one of the assets is a risk-free asset so that investors can lend and borrow at the same rate In this situation, the separation theorem held and thus our proposed return estimate is the optimal combination of the riskless asset and the optimal risky portfolio We further note that the other assets listed in our model could be common stocks, preferred shares, bonds and other types of assets so that the optimal return estimate proposed in this thesis actually represents the optimal return for the best combination of risk-free rate, bonds, stocks and other assets As the estimate developed in this thesis greatly enhances the Markowitz mean-variance optimization procedure to become practically useful, we will encourage financial institutions to adopt our approach in their quantitative investment processes and employ quantitatively- Chapter 6: Conclusions and Further Work 100 oriented specialists to take key positions in their investment team In addition, we relax the condition of the assets return distributions which usually restrict the implementation of Markowitz optimization procedure to the existence of the second moment for some cases and fourth moment for some other cases Many studies, (for example, see Fama (1963, 1965), Blattberg and Gonedes (1974), Clark (1973), Fielitz and Rozelle (1983)) conclude that the normality assumption in the distribution of a security or portfolio return is always violated Fama (1963, 1965) suggest a family of stable Paretian distributions between normal and Cauchy distributions for stock returns Blattberg and Gonedes (1974) suggest student-t as an alternative distribution Clark (1973) suggests a mixture of normal distributions while Fielitz and Rozelle (1983) suggest that a mixture of non-normal stable distributions would be a better representation of the distribution of the returns The contribution of this thesis is that we not need to assume any distribution but only the existence of some moments which are more easily satisfied by the asset returns For example, all distributions mentioned above could be dealt with by our proposed approach What is more, they are not necessarily identically-distributed 6.2 Further Research There is still much work to be done on the current work Chapter 6: Conclusions and Further Work 101 In this thesis, we study the situation in which short selling is allowed One could extend our approach to estimate the optimal portfolio selection with non-negativity constraints on the weights since short selling sometimes is impossible or is too expensive to carry out Moreover, in this study, we got the analytic solution of optimal return by maximizing return under a given risk level In the literature, however, it is interesting to solve the problem of minimizing risk under a given return level As this is a quadratic optimization problem and this formulation could be nicer from a numerical point of view, especially if further restrictions are added Therefore, one need to extend our approach to this framework On the other hand, the theorems derived and the approach developed in this study are based on the assumption that the returns are independent In practice, however, this is not the case For example, many studies suggest that the returns are autocorrelated rather than independent Hence, one could extend our work by releasing the independent assumption to make the application of the MV theory to be more realistic Finally, except for the short-selling restriction, the optimization problem could also be formulated with other restrictions, like trading costs, liquidity constraints, turnover constraints and budget constraints, see, for example, Detemple and Rindisbacher (2005), Muthuraman and Kumar (2006), and Lakner and Nygren (2006) The attainable efficient frontier could also be Chapter 6: Conclusions and Further Work 102 defined subject to some of these imposed constraints which could then be incorporated to make MV optimization a more flexible tool Appendix 103 Bibliography [1] Alexander, Gordon J., and Resnick, B.G., (1985) More on estimation risk and simple rules for optimal portfolio selection Journal of Finance, 40(1), 125-133 [2] Arnold, L., (1967) On the asymptotic distribution of the eigenvalues of random matrices 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Comparison between the Plug-in Returns and Bootstrap-Corrected Returns 58 Chapter 1: Introduction Chapter Introduction 1.1 Markowitz? ??s Mean- Variance Principle The pioneer