Cardinality versus q-Norm Constraints for Index Tracking

Index tracking aims at replicating a given benchmark with a smaller number of its constituents. Different quantitative models cam be set up to determine the optimal index replicating portfolio.In this paper, we propose an alternative based on imposing a constraint on the q-norm, 0 < q < 1, of the replicating portfolios' asset weights: the q-norm constraint regularises the problem and identifes a sparse model. Both approaches are challenging from an optimisation viewpoint due to either the presence of the cardinality constraint or a non-convex constraint on the q-norm .The problem can become even more complex when non-convex distance measures or other real-world constraints are considered.We employ a hybrid heuristic as a fexible tool to tackle both optimisation problems. The empirical analysis on real-world financial data allows to compare the two index tracking approaches.Moreover, we propose a strategy to determine the optimal number of constituents and the corresponding optimal portfolio asset weights.