Improvement Of Estimating Covariance Matrix In Asset Allocation Problem

The Mean-Variance framework that proposed by Doctor Markowitz in 1952 is the cornerstone of modern financial theory.The Mean-Variance model,also known as the Markowitz model,assumes that investors are risk averse.So they'll maximize the risk-adjusted return to maximize their utility.Thus,the portfolio selection problem transfers into the problem of maximizing return in given risk or minimizing risk in given return.Here,Markowitz uses the standard deviation of the portfolio returns as measure of risk.So,the question further becomes how to hold the assets so that there is the smallest standard deviation under the given expected return.The Mean-Variance framework has a beautiful mathematical form and a fascinating foundation of financial theory.But two problems cannot be ignored here.First,is it reasonable to use standard deviation as a measure of risk? Secondly,Markowitz introduces the expected return of assets and the covariance matrix as known parameters,but both of these statistics is unobservable and need to be estimated.And the quality of estimation will directly determine whether the model is reliable or not.This paper is dedicated to the second type of problem.Specifically,our purpose is to improve the estimation of the covariance matrix.The traditional estimator of covariance matrix is sample covariance matrix.It is unbiased,but it is singular when the number of observation lesser than that of assets.And it is also unstable very much.To solve these problems,the previous researchers posed some methods to estimate covariance matrix,such as factor model based covariance matrix proposed by Sharpe(1963)and constant correlation covariance matrix proposed by Elton and Gruber(1973).Though,they are successful in dealing with singular problem and model variant problem,the estimators are biased.Therefore,Olivier Ledoit and Michael Wolf(2002,2004)proposed a method that can take into account both the unbiased and the low Variance property,namely the shrinkage method.Shrinkage estimators will compromise between the two estimate methods,resulting in an estimate that is acceptable on both ways.This paper proposes an improvement of estimating covariance matrix on another perspective.We know that we can divide the financial market into bull,bear and non-directional state.And the covariance matrices of asset return in those three states are systematically different.So,the traditional way,using all kinds of data to estimate covariance matrix,is not reasonable in a mixed state situation.To avoid this,we propose a weighted covariance matrix based on market states.Specifically,we divide data in to three parts according to its state,and then we estimate a covariance matrix for each of state.Finally we get the weighted average of those three estimates by the probabilities of these states.We will prove that the weighted covariance matrix based on market states do have positive effect on asset allocation problem by empirical analysis.The structure of this paper is as follows: The first chapter introduces the research background,research significance and our contribution and flaw.Chapter2 introduces the research results of the predecessors in these fields.Chapter3 is the introduction to theoretical models.One part is about the probabilistic classifier in the supervised learning field.And the other part is about some improved covariance matrix estimators that will be used in this paper.Chapter4 is empirical analysis.Chapter5 is a summary of the full text and some advises to other researcher who would like to explore on the basis of my work.