The thesis addresses the probabilistic-statistical analysis of sample estimators of the characteristics of efficient portfolios under a multivariate elliptical distribution and a weakly stationary Gaussian process for the return vector, and the construction of shrinkage estimators of the weights of the global minimum variance (GMV) and the maximum Sharpe ratio portfolios when the number of assets is comparable to the sample size.
Relevance. The classical Markowitz mean-variance theory and its extensions determine the optimal portfolio weights through the unknown parameters of the return distribution, replaced in practice by their sample estimators. The resulting sample estimators of the portfolio characteristics are generally biased, especially in the high-dimensional setting where the number of assets k is comparable to the sample size n. Hence the study of their statistical properties and the construction of bias-corrected and shrinkage estimators is a relevant problem of applied financial mathematics.
The object of the study is the sample estimators of the characteristics of efficient portfolios; the subject is the asymptotic distributions, bias, and bias-corrected and shrinkage estimators of the Sharpe ratio, Value-at-Risk and portfolio weights. The methods used include probability theory and mathematical statistics, the theory of elliptical distributions, the delta method, random matrix theory, shrinkage estimation and Monte Carlo simulation.
In the second chapter, for the GMV portfolio, the asymptotic distribution of the sample estimators of the Sharpe ratio and the Value-at-Risk is derived, and their asymptotic variances are shown to depend on the kurtosis parameter λ of the elliptical distribution; these results are new both for elliptical and, as a special case, normal distributions. Bias-corrected estimators are also constructed. For the minimum Value-at-Risk portfolio, the asymptotic distribution of the sample estimator of the Sharpe ratio is obtained under an elliptical distribution with independent realizations and under a weakly stationary Gaussian process with autocorrelation; confidence intervals and a significance test are constructed and applied to real Dow Jones index data.
In the third chapter, in the high-dimensional setting, a shrinkage estimator of the GMV portfolio weights is constructed. Since the Sharpe ratio cannot serve as a criterion for the optimal shrinkage intensity, the criterion of maximizing the expected return-to-variance ratio is used instead. The convergence of the optimal intensity to deterministic limiting values is proved in both regimes c<1 and c>1 (for c>1 the Moore-Penrose pseudoinverse is used), and consistent estimators are constructed via random matrix theory.
In the fourth chapter, the analogous problem is solved for the maximum Sharpe ratio portfolio: under the criterion of minimizing the out-of-sample variance, a closed-form optimal shrinkage intensity and its consistent estimators are obtained for the regimes c<1 and c>1.
Scientific novelty. For multivariate elliptical distributions, the asymptotic distributions of the sample estimators of the Sharpe ratio and the Value-at-Risk of the GMV portfolio are derived; the asymptotic distribution of the sample Sharpe ratio of the minimum Value-at-Risk portfolio is obtained under a weakly stationary Gaussian process with autocorrelation, and a test of its significance; shrinkage estimators of the weights are proposed for the GMV portfolio (maximizing the expected return-to-variance ratio) and the maximum Sharpe ratio portfolio (minimizing the out-of-sample variance), and it is proved that the Sharpe ratio cannot be used to determine the shrinkage intensity.
Practical significance. The derived closed-form expressions and confidence intervals for the Sharpe ratio and the Value-at-Risk make it possible to quantify the accuracy of the sample estimators, while the proposed shrinkage estimators of the weights are applicable to capital allocation problems in which the number of assets is comparable to the sample size. A software package comparing the sample and shrinkage estimators by Monte Carlo simulation was developed; the experiments confirmed the substantial superiority of the shrinkage estimator over the sample one for all concentration ratios c=k/n except those close to zero.
Keywords: mathematical modelling, statistical analysis, sample estimator, portfolio, risk, parameter estimation, consistency, asymptotic distribution, variance, asymptotic normality, Value-at-Risk, statistical simulation, mathematical model.
