In this thesis, we consider a nonlinear shrinkage estimation of a covariance matrix, which was introduced by Ledoit and Wolf (2014). It is a new estimation method of a covariance matrix, which works also when the number of observations is smaller than the dimension of the observed vector. In contrast to many papers that address the problem of estimating the expected future returns of the assets in a portfolio, this thesis strongly focuses on covariance matrix estimation, as it is essential for selecting optimal weights of a portfolio. We show that the method of linear shrinkage estimation of the covariance matrix that was introduced in Ledoit and Wolf (2004b) is always invertible and contains less variance than the sample covariance matrix. We emphasize the motivation of the nonlinear shrinkage estimation of the covariance matrix to estimate O(N) parameters in contrast to the linear shrinkage method, who estimate only O(1) parameters. To show the optimality of the nonlinear shrinkage method, we show asymptotically that a specific loss function in the context of portfolio selection can be minimized by estimating the covariance matrix with this nonlinear shrinkage method. It is evidenced that the nonlinear shrinkage covariance matrix can be computed by a formula that keeps the sample eigenvectors in the spectral decomposition of the sample covariance matrix and that nonlinearly shrinks the sample eigenvalues towards their mean. Finally we show empirically that the nonlinear shrinkage estimator of the covariance matrix is optimal for portfolio selection if the dimension of the observed vector is large. Therefore we do a backtesting on historical stock data. We compute the performance measures standard deviation, expected return and Sharpe- ratio for different portfolios and compare the nonlinear shrinkage method with the linear shrinkage method, which was introduced by Ledoit and Wolf (2004b), and the sample covariance matrix.      «

In this thesis, we consider a nonlinear shrinkage estimation of a covariance matrix, which was introduced by Ledoit and Wolf (2014). It is a new estimation method of a covariance matrix, which works also when the number of observations is smaller than the dimension of the observed vector. In contrast to many papers that address the problem of estimating the expected future returns of the assets in a portfolio, this thesis strongly focuses on covariance matrix estimation, as it is essential for s...     »