Solving the high-dimensional Markowitz Optimization Problem: a tale of sparse solutions
We consider the high-dimensional Markowitz optimization problem. A new
approach combining sparse regression and estimation of optimal returns
based on random matrix theory is proposed to solve the problem. We prove
that under some sparsity assumptions on the underlying optimal portfolio,
our novel approach asymptotically yields the theoretical optimal return,
and in the meanwhile satisfies the risk constraint. To the best of our
knowledge, this is the first method that can achieve these two goals
simultaneously in the high-dimensional setting. We further conduct
simulation and empirical studies to compare our method with some benchmark
methods, including the equally weighted portfolio, the bootstrap-corrected
method by Bai et al. (2009) and the covariance-shrinkage method by Ledoit
and Wolf (2004). The results demonstrate substantial advantage of our
method, which attains high level of returns while keeping the risk well
controlled by the given constraint.
Based on joint work with Mengmeng Ao and Yingying Li.