This thesis focuses on portfolio selection problems under the classic mean-variance framework proposed by Markowitz (1952). The "curse of dimensionality" brings new difficulties to Markowitz optimization, motivating the research for this thesis.

Starting from basic settings, in Chapter 2, a new approach is proposed, which combines sparse regression and estimation of the maximum expected return for a given risk level. It is proved that under some sparsity assumptions of the underlying optimal portfolio, our estimated portfolio, the Response-estimated Sparse Regression Portfolio (ReSReP), asymptotically reaches the maximum expected return while satisfying the risk constraint. To the best of our knowledge, this is the first time that these two goals are simultaneously achieved in...[ Read more ]

This thesis focuses on portfolio selection problems under the classic mean-variance framework proposed by Markowitz (1952). The "curse of dimensionality" brings new difficulties to Markowitz optimization, motivating the research for this thesis.

Starting from basic settings, in Chapter 2, a new approach is proposed, which combines sparse regression and estimation of the maximum expected return for a given risk level. It is proved that under some sparsity assumptions of the underlying optimal portfolio, our estimated portfolio, the Response-estimated Sparse Regression Portfolio (ReSReP), asymptotically reaches the maximum expected return while satisfying the risk constraint. To the best of our knowledge, this is the first time that these two goals are simultaneously achieved in high-dimensional setting. The superior properties of ReSReP are demonstrated through simulation and extensive empirical studies.

To extend ReSReP to situations that are more general, a new method is proposed for solving the high-dimensional Markowitz optimization problem when asset returns may exhibit heteroscedasticity. Our method overcomes the difficulties brought by non-normal and non-independent properties of asset returns. The new estimator, ReSReP-H, asymptotically achieves the maximum expected return in the meanwhile satisfies the risk constraint. We also propose a consistent estimator of the maximum expected return with a high convergence rate, provide the convergence rate of a threshold estimator of the mean of returns, and present methods for estimating the mixing densities and changing volatilities. The results of simulation and empirical studies strongly support our theoretical results.