Sparse Portfolios for High-Dimensional Financial Index Tracking

Index tracking is a popular passive portfolio management strategy that aims at constructing a portfolio that replicates or tracks the performance of a financial index. The tracking error can be minimized by purchasing all the assets of the index in appropriate amounts. However, to avoid small and illiquid positions and large transaction costs, it is desired that the tracking portfolio consists of a small number of assets, i.e., a sparse portfolio. The optimal asset selection and capital allocation can be formulated as a combinatorial problem. A commonly used approach is to use mixed-integer programming (MIP) to solve small sized problems. Nevertheless, MIP solvers can fail for high-dimensional problems while the running time can be prohibiting for practical use. In this paper we propose efficient and fast index tracking algorithms that automatically perform asset selection and capital allocation under a set of general convex constraints. A special consideration is given to the case of the non-convex holding constraints and to the downside risk tracking measure. Further, we derive specialized algorithms with closed-form updates for particular sets of constraints. Numerical simulations show that the proposed algorithms match or outperform existing methods in terms of performance, while their running time is lower by many orders of magnitude.