The effect of regularization in portfolio selection problems

Portfolio selection problems have been thoroughly studied under the risk-and-return paradigm introduced by Markowitz. However, the usefulness of this approach has been hindered by some practical considerations that have resulted in poorly diversified portfolios, or, solutions that are extremely sensitive to parameter estimation errors. In this work, we use sampling methods to cope with this issue and compare the merits of two approaches: a sample average approximation approach and a performance-based regularization (PBR) method that appeared recently in the literature. We extend PBR by incorporating three different risk metrics—integrated chance-constraints, quantile deviation, and absolute semi-deviation—and deriving the corresponding regularization formulas. Additionally, a numerical comparison using index-based portfolios is presented using historic data that includes the subprime crisis.