Exact and Heuristic Approaches for the Index Tracking Problem with UCITS Constraints

Index tracking aims at determining an optimal portfolio that replicates the performance
of an index or benchmark by investing in a smaller number of constituents or assets. The
tracking portfolio should be cheap to maintain and update, i.e., invest in a smaller number of
constituents than the index, have low turnover and low transaction costs, and should avoid
large positions in few assets, as required by the European Union Directive UCITS (Undertaking for Collective Investments in Transferable Securities) rules. The UCITS rules make
the problem hard to be satisfactorily modeled and solved to optimality: no exact methods
but only heuristics have been proposed so far.
The aim of this paper is twofold. First, we present the first Mixed Integer Quadratic Programming (MIQP) formulation for the constrained index tracking problem with the UCITS rules
compliance. This allows us to obtain exact solutions for small- and medium-size problems
based on real-world datasets. Second, we compare these solutions with the ones provided by
the state-of-art heuristic Differential Evolution and Combinatorial Search for Index Tracking
(DECS-IT), obtaining information about the heuristic performance and its reliability for the
solution of large-size problems that cannot be solved with the exact approach. Empirical
results show that DECS-IT is indeed appropriate to tackle the index tracking problem in
such cases. Furthermore, we propose a method that combines the good characteristics of the
exact and of the heuristic approaches