A Progressive Resampling Algorithm for Finding Very Sparse Investment Portfolios

The mean-variance framework by Markowitz is a classical approach to portfolio selection. Earlier work has shown that the basic Markowitz portfolios obtained by solving a quadratic program tend to have poor out-of-sample performance. These issues have been addressed by devising sparse variants of Markowitz portfolios in which the number of active positions is reduced either by applying a no-short-selling constraint or L1-regularisation. In this work we consider a combinatorial approach for finding sparse portfolios, which we call naive k-portfolios, that allocate available capital uniformly on a fixed number of k assets, and only take long positions. We present a novel randomised algorithm, progressive resampling, that efficiently finds such portfolios, and compare this with a number of well-known portfolio selection strategies using public stock price data. We find that naive k-portfolios can be a viable alternative to L1-regularisation when constructing sparse portfolios.