Exact and heuristic approaches for the index tracking problem with UCITS constraints

Abstract

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.

A two-stage approach to the UCITS-constrained index-tracking problem

“…The proposed representation of subsets should be of general interest because it can be used in genetic algorithms for any optimization problem that involves the selection of a subset, such as the feature-selection problem in machine learning. Second, we present a new MIQP formulation of the UCITP that requires fewer constraints than the existing formulation of Scozzari et al [49]. Third, we present a new two-stage approach to the UCITP that is able to devise very good feasible solutions for UCITP instances of arbitrary size in a short CPU time.…”

“…In the literature, two approaches to the UCITP have been proposed: a mixed-integer quadratic programming (MIQP) approach (cf. Scozzari et al [49]) and an approach based on differential evolution and combinatorial search (cf. Krink et al [29]).…”

“…In Subsection 2.1, we present an overview of existing approaches to the index-tracking problem without regulatory constraints. In Subsection 2.2, we present the MIQP formulation of the UCITP introduced by Scozzari et al [49]. In Subsection 2.3, we discuss the existing subset representations used in genetic algorithms.…”

Portfolio optimization and index tracking for the shipping stock and freight markets using evolutionary algorithms

“…The DE algorithm has also been used in other recent studies using hybrid and multi-objective schemes Krink and Paterlini, 2011), as well as in the context of loss aversion (Maringer, 2008) and mutual fund replication (Zhang and Maringer, 2010). Other recently proposed algorithmic procedures include immune systems (Li et al, 2011), hybrid algorithms (Ruiz-Torrubiano and Suárez, 2009;Scozzari et al 2012), robust optimization (Chen and Kwon, 2012) and mixed-integer programming formulations (Canakgoz and Beasley, 2008;Stoyan and Kwon, 2010). An overview of different methods can be found in Woodside-Oriakhi et al (2011).…”

Lasso-based index tracking and statistical arbitrage long-short strategies

“…Different formulations for the IT problem have been proposed in the literature, such as optimization (Konno and Wijayanayake, 2001;Mezali and Beasley, 2013), optimization combined with simulation (Consiglio and Zenios, 2001), heuristic methods (Beasley et al, 2003;Scozzari et al, 2013), cointegration (Alexander, 1999;Alexander and Dimitriu, 2005), and lasso-type regression (Wu et al, 2014;Yang and Wu, 2016). In spite of their methodological distinctions, however, past studies usually carry out their analysis considering a standard feature, which is the use of a cardinality constraint to limit the size of the tracking portfolios and diminish transaction costs.…”