Three $$l_1$$ Based Nonconvex Methods in Constructing Sparse Mean Reverting Portfolios

Abstract

We study the problem of constructing sparse and fast mean reverting portfolios. The problem is motivated by convergence trading and formulated as a generalized eigenvalue problem with a cardinality constraint (d’Aspremont in Quant Finance 11(3):351–364, 2011). We use a proxy of mean reversion coefficient, the direct Ornstein–Uhlenbeck estimator, which can be applied to both stationary and nonstationary data. In addition, we introduce three different methods to enforce the sparsity of the solutions. One method uses the ratio of $$l_1$$ and $$l_2$$ norms and the other two use $$l_1$$ norm. We analyze various formulations of the resulting non-convex optimization problems and develop efficient algorithms to solve them for portfolio sizes as large as hundreds. By adopting a simple convergence trading strategy, we test the performance of our sparse mean reverting portfolios on both synthetic and historical real market data. In particular, the $$l_1$$ regularization method, in combination with quadratic program formulation as well as difference of convex functions and least angle regression treatment, gives fast and robust performance on large out-of-sample data set.