A fundamental result in quantitative finance is Markowitz's single-period mean-variance portfolio optimization theory that provides optimal asset weights which minimize the variance of the return at a target level of the mean return of the portfolio. This theory assumes known means, variances and covariances of the returns of all assets in the portfolio. Since these are actually unknown and have to be estimated from historical data and since one usually has a large number of assets, resulting in more parameters to be estimated than the sample size, practical implementation of Markowitz's portfolio optimization theory has been a long-standing problem. Different covariance estimators have been proposed in the literature to address this problem. In particular, multi-factor models, shrinkage estimators, thresholding and regularization have been developed as alternatives to the naive sample estimator that has been shown to perform poorly. This thesis first reviews these approaches and proposes a new high-dimensional covariance estimator that can estimate both the covariance matrix and its inverse consistently.; The proposed new estimator is based on the modified Cholesky decomposition of the covariance matrix, and assumes sparsity in this parametrization. It uses a boosting algorithm with a modified Hannan-Quinn-type stopping criterion. For portfolio optimization applications, a factor model can be constructed and the covariance estimator can then be applied to the residuals, and an empirical study shows that this approach outperforms those that use the naive sample covariance matrix or shrinkage estimators. The main theoretical contributions of this thesis are consistency results for the boosting algorithm and stopping criterion and for the new high-dimensional covariance matrix estimator.
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