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A covariance matrix of asset returns plays an important role in modern portfolio analysis and risk management. Despite the recent interests in improving the estimation of a return covariance matrix, there remain many areas for further investigation. This thesis studies several issues related to obtaining a better estimation of the covariance matrix for the returns of a reasonably large number of stocks for portfolio risk management. \ud The thesis consists of five essays. The first essay, Chapter 3, provides a comprehensive analysis of both old and new covariance estimation methods and the standard comparison criteria. We use empirical data to compare their performances. We also examine the standard comparisons and find they provide limited information regarding the abilities of the covariance estimators in predicting portfolio variances. It therefore suggests that we need more powerful comparison criteria to assess covariance estimators.\ud The second and third essays, Chapter 4 and 5, are concerned with the alternative appraisal methods of return covariance estimators for portfolio risk management purposes. Chapter 4 introduces a portfolio distance measure based on eigen decomposition (eigen-distance) to compare two covariance estimators in terms of the most different portfolio variances they predict. The eigen-distance measures the ratio of the two extreme variance predictions under one covariance estimator for the portfolios that are constructed to have the same variances under the other covariance estimator. We show that the eigen-distance can be used to assess a risk measurement system as a whole, where any kind of the portfolios may need to be considered. Our simulation results show that it is a powerful measure to distinguish two covariance estimators even in small samples.\ud Chapter 5 proposes a0 measure to distinguish two similar estimated covariance matrices from the observed covariance matrix. 0 is constructed based on the essential difference of the two similar covariance matrices: the two extreme portfolios that are predicted to have the most different variances under these two matrices. We show that 0 is very useful in evaluating refinements to covariance estimators, particularly a modest refinement, where the refined covariance matrix is close to the original matrix.\ud The last two essays, Chapter 6 and 7, are concerned with improving the best covariance estimators within the literature. Chapter 6 explores alternative Bayesian shrinkage methods that directly shrink the eigenvalues (and in one case the principal eigenvector) of the sample covariance matrix. We use simulations to compare the performance of these shrinkage estimators with the two best existing estimators, namely, the Ledoit and Wolf (2003a) estimator and the Jagannathan and Ma (2003) estimator using both RMSE and eigen-distance criteria. We find that our shrinkage estimators consistently out-perform the Ledoit and Wolf estimator. They also out-perform the Jagannathan and Ma estimator except in one case where they are not much worse off either.\ud Finally, Chapter 7 extends the analysis of Chapter 6, which is under an unchanging multivariate normal world, to consider implications of both fat-tails and time variation. We use a multivariate normal inverse Gaussian (MNIG) distribution to model the log returns of stock prices. This family of distributions has proven to fit the heavy tails observed in financial time series extremely well. For the time varying situation, we use a tractable mean reverting Ornstein- Uhlenbeck (OU) process to develop a new model to measure an interesting and economically motivated time varying structure where the risks remain unchanged but stocks migrate among different risk categories during their life circles. We find that our shrinkage methods are also useful in both situations and become even more important in the time varying case

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