Sparse Markowitz Portfolios

Modern portfolio theory originated from the work of Markowitz, who insisted on the fact that returns should be balanced with risk and established the theoretical basis for portfolio optimization according to this principle. Sparse Markowitz portfolios impose an additional requirement of sparsity to the objectives of risk and expected return in traditional Markowitz portfolios. This chapter overviews the Markowitz portfolio formulation and describes its fragility in high-dimensional settings. It argues that sparsity of the portfolio can alleviate many of the shortcomings, and presents an optimization formulation based on convex relaxations. The chapter then introduces sparse portfolio rebalancing and optimal forecast combination. The sparse portfolio methodology has been validated by an empirical exercise. The aim of the empirical exercise was rather to assess the validity of the investment strategy, as it would be carried out by different investors using the same methodology in different years.