MEAN-VARIANCE FUNCTIONAL ESTIMATION FOR OPTIMAL PORTFOLIOS

Abstract

Motivated by Markowitz’s portfolio optimization problem, this thesis aims at esti-mating functionals Σ −1 μ, μΣ −1 μ involving both the mean vector μ and covariance matrix Σ. These functionals are closely related to the optimal portfolio allocation and Sharpe ratio. The estimation problem is studied under the high-dimensional setting, and two different underlying structure are considered. In the first structure, sparsity of Σ −1 μ is assumed. Minimax estimators are obtained, and the optimal rate for estimating the functional μΣ −1 μ undergoes a phase transi- tion between regular parametric rate and some form of high-dimensional estimation rate. It is further shown that the optimal rate is attained by a carefully designed plug-in estimator based on de-biasing, while a family of naive plug-in estimators are proved to fall short. The second structure is the approximate factor model. In this setting, we only assume finite fourth-moment. A robust procedure is proposed for estimating these function- als, and adaptive tuning is employed for implementation. These structures are well justified by empirical evidence, and they are suitable for practical implementation in different situation. Extensive numerical studies are pre- sented which lend further support to the results.