Statistical Inference For Markowitz Efficient Portfolios

This dissertation, "Statistical Inference for Markowitz Efficient Portfolios" by Yuanyuan, Zhu, 朱淵遠, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of the thesis entitled ST A TISTICAL INFERENCE FOR MARKOWITZ EFFICIENT POR TFOLIOS Submitted by ZHU, YUANYUAN for the degree of Do ctor of Philosophy at The University of Hong Kong in September 2015 Markowitz mean-v ariance mo del has been the foundation of modern portfolio theory . The Markowitz model attempts to maximize the portfolio expected return for a given level of portfolio risk, or equiv alently to minimize portfolio risk for a given level of expected return. Assuming multivariate normality of the asset returns, the optimal portfolio weights can be treated as a function of the unknown mean vector and covariance matrix. However it has b een criti- cized by many researchers the ineective and unstable performance of the op- timal portfolio under the model. This thesis intends to improve the Markowitz mean-variance model through two new methods. The rst method is to make use of generalized pivotal quantity (GPQ). The GPQ approach is widely used in constructing hypothesis tests and condence interv als. In this thesis, the GPQ approach is used to make statistical inference on the optimal portfolio weights. Dierent approaches are proposed for con- structing point estimator and simultaneous condence interv als for the optimal portfolio weights. Simulation studies has been conducted to compare the GPQ estimators with existing estimators based on Markowitz model, bootstrap andshrinkage methods. The results show that the GPQ based approach results in a smallest mean squared error for the point estimate of the portfolio weights in most cases and satisfactory coverage rate for the simultaneous condence interv als. F urthermore, an application on portfolio re-balancing problem is considered. Results show that the condence intervals help investors decide whether or not to update the p ortfolio weights so as to achieve a higher prot. This thesis not only focuses on the portfolio optimal weights, but also proposes a new estimator for the Sharpe ratio. Sharpe ratio serves as an important measure of the portfolio performance measure. Some researches have been done on the estimation of the distribution of Sharpe ratio when the number of assets is not too large but the sample size is big. This thesis makes use of GPQ to estimate the Sharpe ratio for high-dimensional data or small-sample-size data. The second method attempts to improve the estimation of the unknown cov ariance matrix. Note that the plug-in estimator for the optimal portfolio weights is biased and p erforms po orly due to the estimation error, especially in the cases of high dimensions. Instead of the sample covariance matrix, we consider the scaled sample cov ariance matrix to construct the new estimator for weights. The explicit formulae for both the mean and v ariance of the new estimator are derived. T wo approaches are prop osed to determine the optimal scale parameter of the covariance matrix estimator. Simulation studies show that the new estimators outperform the existing ones, especially when the number of assets is large. In addition, we illustrate the new estimators with an example from the US stock market. DOI: 10.5353/th_b5689290 Subjects: Portfolio management - Statistical methods

The composition of portfolios is one of the most fundamental and important methods in financial engineering, used to control the risk of investments. This book provides a comprehensive overview of statistical inference for portfolios and their various applications. A variety of asset processes are introduced, including non-Gaussian stationary processes, nonlinear processes, non-stationary processes, and the book provides a framework for statistical inference using local asymptotic normality (LAN). The approach is generalized for portfolio estimation, so that many important problems can be covered. This book can primarily be used as a reference by researchers from statistics, mathematics, finance, econometrics, and genomics. It can also be used as a textbook by senior undergraduate and graduate students in these fields.

Until now, few systematic studies of optimal statistical inference for stochastic processes had existed in the financial engineering literature, even though this idea is fundamental to the field. Balancing statistical theory with data analysis, Optimal Statistical Inference in Financial Engineering examines how stochastic models can effectively des

In spite of theoretical benefits, Markowitz mean-variance (MV) optimized portfolios often fail to meet practical investment goals of marketability, usability, and performance, prompting many investors to seek simpler alternatives. Financial experts Richard and Robert Michaud demonstrate that the limitations of MV optimization are not the result of conceptual flaws in Markowitz theory but unrealistic representation of investment information. What is missing is a realistic treatment of estimation error in the optimization and rebalancing process. The text provides a non-technical review of classical Markowitz optimization and traditional objections. The authors demonstrate that in practice the single most important limitation of MV optimization is oversensitivity to estimation error. Portfolio optimization requires a modern statistical perspective. Efficient Asset Management, Second Edition uses Monte Carlo resampling to address information uncertainty and define Resampled Efficiency (RE) technology. RE optimized portfolios represent a new definition of portfolio optimality that is more investment intuitive, robust, and provably investment effective. RE rebalancing provides the first rigorous portfolio trading, monitoring, and asset importance rules, avoiding widespread ad hoc methods in current practice. The Second Edition resolves several open issues and misunderstandings that have emerged since the original edition. The new edition includes new proofs of effectiveness, substantial revisions of statistical estimation, extensive discussion of long-short optimization, and new tools for dealing with estimation error in applications and enhancing computational efficiency. RE optimization is shown to be a Bayesian-based generalization and enhancement of Markowitz's solution. RE technology corrects many current practices that may adversely impact the investment value of trillions of dollars under current asset management. RE optimization technology may also be useful in other financial optimizations and more generally in multivariate estimation contexts of information uncertainty with Bayesian linear constraints. Michaud and Michaud's new book includes numerous additional proposals to enhance investment value including Stein and Bayesian methods for improved input estimation, the use of portfolio priors, and an economic perspective for asset-liability optimization. Applications include investment policy, asset allocation, and equity portfolio optimization. A simple global asset allocation problem illustrates portfolio optimization techniques. A final chapter includes practical advice for avoiding simple portfolio design errors. With its important implications for investment practice, Efficient Asset Management 's highly intuitive yet rigorous approach to defining optimal portfolios will appeal to investment management executives, consultants, brokers, and anyone seeking to stay abreast of current investment technology. Through practical examples and illustrations, Michaud and Michaud update the practice of optimization for modern investment management.

Portfolio construction is fundamental to the investment management process. In the 1950s, Harry Markowitz demonstrated the benefits of efficient diversification by formulating a mathematical program for generating the "efficient frontier" to summarize optimal trade-offs between expected return and risk. The Markowitz framework continues to be used as a basis for both practical portfolio construction and emerging research in financial economics. Such concepts as the Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT), for example, provide the foundation for setting benchmarks, for predicting returns and risk, and for performance measurement. This volume showcases original essays by some of today’s most prominent academics and practitioners in the field on the contemporary application of Markowitz techniques. Covering a wide spectrum of topics, including portfolio selection, data mining tests, and multi-factor risk models, the book presents a comprehensive approach to portfolio construction tools, models, frameworks, and analyses, with both practical and theoretical implications.

​This monograph provides the fundamentals of statistical inference for financial engineering and covers some selected methods suitable for analyzing financial time series data. In order to describe the actual financial data, various stochastic processes, e.g. non-Gaussian linear processes, non-linear processes, long-memory processes, locally stationary processes etc. are introduced and their optimal estimation is considered as well. This book also includes several statistical approaches, e.g., discriminant analysis, the empirical likelihood method, control variate method, quantile regression, realized volatility etc., which have been recently developed and are considered to be powerful tools for analyzing the financial data, establishing a new bridge between time series and financial engineering. This book is well suited as a professional reference book on finance, statistics and statistical financial engineering. Readers are expected to have an undergraduate-level knowledge of statistics.

In 1952, Harry Markowitz published "Portfolio Selection," a paper which revolutionized modern investment theory and practice. The paper proposed that, in selecting investments, the investor should consider both expected return and variability of return on the portfolio as a whole. Portfolios that minimized variance for a given expected return were demonstrated to be the most efficient. Markowitz formulated the full solution of the general mean-variance efficient set problem in 1956 and presented it in the appendix to his 1959 book, Portfolio Selection. Though certain special cases of the general model have become widely known, both in academia and among managers of large institutional portfolios, the characteristics of the general solution were not presented in finance books for students at any level. And although the results of the general solution are used in a few advanced portfolio optimization programs, the solution to the general problem should not be seen merely as a computing procedure. It is a body of propositions and formulas concerning the shapes and properties of mean-variance efficient sets with implications for financial theory and practice beyond those of widely known cases. The purpose of the present book, originally published in 1987, is to present a comprehensive and accessible account of the general mean-variance portfolio analysis, and to illustrate its usefulness in the practice of portfolio management and the theory of capital markets. The portfolio selection program in Part IV of the 1987 edition has been updated and contains exercises and solutions.

This book provides an introduction to the use of statistical concepts and methods to model and analyze financial data. The ten chapters of the book fall naturally into three sections. Chapters 1 to 3 cover some basic concepts of finance, focusing on the properties of returns on an asset. Chapters 4 through 6 cover aspects of portfolio theory and the methods of estimation needed to implement that theory. The remainder of the book, Chapters 7 through 10, discusses several models for financial data, along with the implications of those models for portfolio theory and for understanding the properties of return data. The audience for the book is students majoring in Statistics and Economics as well as in quantitative fields such as Mathematics and Engineering. Readers are assumed to have some background in statistical methods along with courses in multivariate calculus and linear algebra.