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 can 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
identifies 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 flexible 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.