Estimated covariance matrix for portfolio selection: Application of the minimum variance portfolio in ISE

Investors in the financial markets would usually like to incorporate into their portfolios more than one financial tool that is traded within the capital market of interest so as to reduce their risks. By doing so, they not only alleviate their risks, but they also increase their potential returns. It is for this reason that portfolio diversification is quite an important concept for investors. Diversification can be performed by the use of both risky and riskless investment tools, and the types of securities to appear in the portfolios depend on the investors. However, the common objective of all investors in the market is to be subject to minimal risk while achieving their target returns. Therefore, focusing on minimum-risk portfolio options, this study is concerned with minimum-variance portfolio options in the ISE.Portfolio optimization basically requires two inputs: an expected return vector estimate and a covariance matrix estimate. The estimation of the expected return vector is performed via models such as CAPM and APT, whereas that of the covariance matrix is commonly done using sample covariance matrix estimators. Because this study focuses only on the estimation of the covariance matrix to be used in the portfolio optimization, the study does not deal with expected return vector estimation, and therefore considers only the minimum-variance portfolios in the time period of concern.The objective of this study is to determine how the minimum-variance portfolios output by optimization are affected by the use of various covariance matrix estimators. More specifically, two types of covariance matrix estimators are used: sample covariance matrix estimators and shrinkage estimators. The estimators were compared on basis of the results associated with the minimum-variance portfolios in the ISE. The main purpose of shrinkage estimation, the alternative way of covariance matrix estimation, is to lessen or completely eliminate the estimation errors caused by the sample covariance matrix estimators which are commonly used and easy to compute.This study, which is concerned with the formation of minimum-variance portfolios by means of various estimators, consists of five sections. In the first section, performance evaluation criteria are analyzed in the framework of portfolio management and approaches in general, portfolio theories, portfolio management strategies and portfolio management processes. In the second section, the studies in the Turkish and other literature are reviewed in two main groups: those dealing with portfolio selection models and those dealing with covariance matrix estimation. The third section dwells upon the concept of risk that constitutes the objective function in this study and analyzes the covariance matrix estimators used in the literature for the estimation of risk. The fourth section of the study provides an application of minimum-variance portfolio formation using two estimators on the ISE data pertaining to years 1986 through 2009: the sample covariance estimator and the shrinkage estimator due to Ledoit and Wolf (2004). The portfolios obtained using these estimators are compared based on the number of stocks, risk and return. These portfolios are also compared with the equal-weight portfolio, the market portfolio and the return rates of the government securities. In the last section, assessments and suggestions are made based on the results. Furthermore, evaluations are performed regarding criteria such as number of stocks, risk, return, and especially the effect of each additional constraint on the relationship between the resulting minimum-variance portfolio and the market portfolio.Based on the results associated with the minimum-variance portfolios formed in the ISE, the study reveals options that provide the investor with minimum risk in the ISE under certain constraints through the use of two different estimators. The results suggest that the use of shrinkage estimators which are used as alternatives to the commonly-used sample covariance estimators is more preferable in achieving portfolios with lower risk levels. The findings also make it possible to draw general conclusions about the ISE in terms of the portfolio options obtained by the use of different estimators and the performance of the managers of these portfolios.