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Abstract.

The General Minimum Variance Portfolio (GMVP) has the smallest variance of all the portfolios. The weight of this portfolio depends solely on the inverse of the population covariance matrix, which is an unknown object in practice and must be replaced by an estimator, when the GMV portfolio is constructed. Several estimators of the GMV port- folio weights exist in the literature. The preliminary aim of the thesis is their comparison with respect to the out-of-sample performance and their asymptotic behaviors based on Random Matrix Theory (RMT). We base our analysis on numerical simulations using synthetic and real data. The performance measures we use are the out-of-sample vari- ance and the relative-loss corresponding to their GMVP estimators. Their asymptotic properties are analyzed when the number of assets p and the sample size n are going together to infinity at the same convergence rate p/n, which is called in the literature as a double-limit regime or high-dimensional asymptotics. The different estimators we are interested in are based on the Sample Covariance Matrix (SCM) and Tyler’s robust M-estimator. We use their non-regularized and regularized (shrinkage) forms. There is a conjecture that the variances and relative losses of the non-regularized SCM and Tyler’s estimator as well the different regularized SCM estimators we are considering and the Tyler’s regularized estimator are converging in the double limit regime and their perfor- mances depend on convergence rate. The following four approaches are considered.    1- Frahm and Memmel (2010) [1] They treat the case of the linear shrinkage estimators under assumption of serially in- dependent and identically normally distributed asset returns. They present it for the small sample, and the large sample cases with fixed number of assets p and the sample size n going into infinity (standard asymptotic), as well the case when p and n together are going to infinity, whereas p/n remains finite (high dimensional asymptotics).   2- Bodnar et al. (2018) [2] They improve the estimator of Frahm and Memmel and suggest shrinking the sample estimator for the portfolio weights directly and not the whole sample covariance matrix, which is dominant but not necessarily optimal. A new estimator for the GMV portfo- lio based on Random Matrix Theory is derived, which is optimal and distribution-free. The shrinkage target used is an arbitrary non-random matrix and the asymptotic of the optimal shrinkage intensities are found and estimated consistently. 3-Rubio et al. (2012) [3] They regularize (shrink) the SCM estimator, where the shrinkage target is a nonrandom positive-definite matrix. Thereafter they invert the sum of the SCM estimator and the target matrix and in GMVP implementation the portfolio weights are found. To find the minimum realized variance the shrinkage intensity is optimized. 4-Yang et al. (2015) [4] They are following in principle the same procedure as in Rubio et al. But are using instead for the SCM estimator the Tyler’s M-estimator.