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Abstract

Pioneered by Markowitz [3], the economic theory of investor’s optimal portfolio choice is now well understood. And the theory has been one of the most important in finance, both for academic researchers and practitioners. The Markowitz portfolio weights given by mean-variance analysis depend on the mean and covariance of asset returns. When estimat-ing the optimal portfolio weights given by mean-variance analysis, people need to estimate the means and covariance matrix of asset returns first, which will definitely leads to estima-tion error. Normally, people use plug-in maximum likelihood (ML) method to estimate the parameters such as means and covariance matrix when making decisions of optimal portfo-lio, resulting in the maximum likelihood estimations (MLE) of Markowitz optimal portfolio weights and other statistics. Although MLE has appealing asymptotic properties, it is prob-lematic in finite sample. This makes MLE of optimal portfolio weights less trustable in practice. As pointed out by Jobson and Korkie [2], Markowtiz optimal portfolio has se-rious small sample problem. One of the reasons leading to this problem will be the poor estimation of covariance of asset returns in small sample. Thus, improving the estimation of covariance of asset returns might improve small sample property of Markowitz optimal portfolio.

The main aim of this paper is to improve the estimation of Markowitz portfolio weights under mean-variance analysis in finite sample. The method we use here is called shrinkage method, which has been discussed in Ren and Shimotsu [1]. We investigate the performance of shrinkage method in a Monte-Carlo simulation. Also we apply this method to Fama-French’s real portfolio data, i.e. 25 size- and book-to-market-sorted portfolios, 6 size- and book-to-market-sorted portfolios, 10 industry portfolios and 5 industry portfolios, which can be accessed from Kenneth French’s Data Library. From both simulation and empirical aspects, we find the shrinkage estimation of mean-variance optimal portfolio performs better than MLE.

Key Words: Markowitz portfolio, Mean-Variance analysis, covariance estimation, shrink-age method

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Ⅶ

Contents

Abstract (Chinese)

... Ⅰ

Abstract (English)

... Ⅲ

Chapter 1 Introduction

... 1

Chapter 2 Literature Review

... 5

2.1 Portfolio Theory before 1950’s ... 5

2.2 Markowitz Portfolio Theory ... 5

2.3 Model Improvment ... 8

2.4 Estimation Improvment ... 8

Chapter 3 Theoretical Foudation

... 11

3.1 Markowitz Mean-Variance Analysis... 11

3.2 Shrinkage Method ... 21

Chapter 4 Simulation Analysis

... 25

4.1 Feasibility Analysis... 25

4.2 Simulation: Using Shrinkage Method ... 25

Chapter 5 Empirical Analysis

... 35

5.1 Data Selection ... 35 5.2 Model Valuation ... 35 5.3 Empirical Study ... 37 5.4 Empirical Result... 38

Chapter 6 Conclusion

... 43

References

... 45

Appendices

... 48

Acknowledgements

... 51

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