IDEAS home Printed from https://ideas.repec.org/a/spr/cejnor/v30y2022i2d10.1007_s10100-019-00633-0.html
   My bibliography  Save this article

Solving the index tracking problem: a continuous optimization approach

Author

Listed:
  • Mahdi Moeini

    (Technische Universität Kaiserslautern)

Abstract

Investing vast amounts of money with the goal of fostering medium to long-term growth in returns is a challenging task in financial optimization. A method might be mirroring the market index as closely as possible by choosing from the stocks that make up the index. This approach is known as index tracking and the objective of this paper is to address this problem in order to solve it by means of mathematical programming techniques. In particular, we are interested in investigating the index tracking problem (ITP) as a mixed integer linear program in presence of some real-world constraints known as cardinality constraints as well as transaction costs. These ITP models are NP-hard, and consequently, difficult to solve by classical exact methods even for medium-sized instances. In order to overcome this issue, we propose a method based on nonconvex programming techniques. More precisely, we reformulate the problem as a difference of convex functions (DC) program and solve it by means of an approach known as DC algorithm. In order to evaluate the performance of the proposed algorithm, we conducted numerical experiments using benchmark instances. The results of the algorithm are compared with those provided by the state-of-the-art MILP solver Gurobi. The numerical results confirm the efficiency of the method in solving the ITP.

Suggested Citation

  • Mahdi Moeini, 2022. "Solving the index tracking problem: a continuous optimization approach," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 30(2), pages 807-835, June.
    • Handle: RePEc:spr:cejnor:v:30:y:2022:i:2:d:10.1007_s10100-019-00633-0
    DOI: 10.1007/s10100-019-00633-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10100-019-00633-0
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10100-019-00633-0?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Hoai Le Thi & Mahdi Moeini & Tao Pham Dinh & Joaquim Judice, 2012. "A DC programming approach for solving the symmetric Eigenvalue Complementarity Problem," Computational Optimization and Applications, Springer, vol. 51(3), pages 1097-1117, April.
    2. Filippi, C. & Guastaroba, G. & Speranza, M.G., 2016. "A heuristic framework for the bi-objective enhanced index tracking problem," Omega, Elsevier, vol. 65(C), pages 122-137.
    3. Margherita Giuzio, 2017. "Genetic algorithm versus classical methods in sparse index tracking," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 40(1), pages 243-256, November.
    4. Le An & Pham Tao, 2005. "The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems," Annals of Operations Research, Springer, vol. 133(1), pages 23-46, January.
    5. Hoai An Le Thi & Mahdi Moeini, 2014. "Long-Short Portfolio Optimization Under Cardinality Constraints by Difference of Convex Functions Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 161(1), pages 199-224, April.
    6. Beasley, J. E. & Meade, N. & Chang, T. -J., 2003. "An evolutionary heuristic for the index tracking problem," European Journal of Operational Research, Elsevier, vol. 148(3), pages 621-643, August.
    7. Rubén Ruiz-Torrubiano & Alberto Suárez, 2009. "A hybrid optimization approach to index tracking," Annals of Operations Research, Springer, vol. 166(1), pages 57-71, February.
    8. Thiemo Krink & Stefan Mittnik & Sandra Paterlini, 2009. "Differential evolution and combinatorial search for constrained index-tracking," Annals of Operations Research, Springer, vol. 172(1), pages 153-176, November.
    9. Hoai Le Thi & Mahdi Moeini & Tao Pham Dinh, 2009. "Portfolio selection under downside risk measures and cardinality constraints based on DC programming and DCA," Computational Management Science, Springer, vol. 6(4), pages 459-475, October.
    10. Guastaroba, G. & Mansini, R. & Ogryczak, W. & Speranza, M.G., 2016. "Linear programming models based on Omega ratio for the Enhanced Index Tracking Problem," European Journal of Operational Research, Elsevier, vol. 251(3), pages 938-956.
    11. Strub, O. & Baumann, P., 2018. "Optimal construction and rebalancing of index-tracking portfolios," European Journal of Operational Research, Elsevier, vol. 264(1), pages 370-387.
    12. Guastaroba, G. & Speranza, M.G., 2012. "Kernel Search: An application to the index tracking problem," European Journal of Operational Research, Elsevier, vol. 217(1), pages 54-68.
    13. Hiroshi Konno & Hiroaki Yamazaki, 1991. "Mean-Absolute Deviation Portfolio Optimization Model and Its Applications to Tokyo Stock Market," Management Science, INFORMS, vol. 37(5), pages 519-531, May.
    14. Thiemo Krink & Stefan Mittnik & Sandra Paterlini, 2009. "Differential evolution and combinatorial search for constrained index-tracking," Annals of Operations Research, Springer, vol. 172(1), pages 153-176, November.
    15. Wu, Dexiang & Kwon, Roy H. & Costa, Giorgio, 2017. "A constrained cluster-based approach for tracking the S&P 500 index," International Journal of Production Economics, Elsevier, vol. 193(C), pages 222-243.
    16. Giacomo di Tollo & Dietmar Maringer, 2009. "Metaheuristics for the Index Tracking Problem," Lecture Notes in Economics and Mathematical Systems, in: Kenneth Sörensen & Marc Sevaux & Walter Habenicht & Martin Josef Geiger (ed.), Metaheuristics in the Service Industry, chapter 8, pages 127-154, Springer.
    17. Sant’Anna, Leonardo R. & Filomena, Tiago P. & Caldeira, João F., 2017. "Index tracking and enhanced indexing using cointegration and correlation with endogenous portfolio selection," The Quarterly Review of Economics and Finance, Elsevier, vol. 65(C), pages 146-157.
    18. Hiroshi Konno & Annista Wijayanayake, 2001. "Minimal Cost Index Tracking Under Nonlinear Transaction Costs And Minimal Transaction Unit Constraints," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(06), pages 939-957.
    19. Leonardo Riegel Sant’Anna & Tiago Pascoal Filomena & Pablo Cristini Guedes & Denis Borenstein, 2017. "Index tracking with controlled number of assets using a hybrid heuristic combining genetic algorithm and non-linear programming," Annals of Operations Research, Springer, vol. 258(2), pages 849-867, November.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Gnägi, M. & Strub, O., 2020. "Tracking and outperforming large stock-market indices," Omega, Elsevier, vol. 90(C).
    2. Doering, Jana & Kizys, Renatas & Juan, Angel A. & Fitó, Àngels & Polat, Onur, 2019. "Metaheuristics for rich portfolio optimisation and risk management: Current state and future trends," Operations Research Perspectives, Elsevier, vol. 6(C).
    3. Julio Cezar Soares Silva & Adiel Teixeira de Almeida Filho, 2023. "A systematic literature review on solution approaches for the index tracking problem in the last decade," Papers 2306.01660, arXiv.org, revised Jun 2023.
    4. Leonardo Riegel Sant’Anna & Tiago Pascoal Filomena & Pablo Cristini Guedes & Denis Borenstein, 2017. "Index tracking with controlled number of assets using a hybrid heuristic combining genetic algorithm and non-linear programming," Annals of Operations Research, Springer, vol. 258(2), pages 849-867, November.
    5. Chen, Qi-an & Hu, Qingyu & Yang, Hu & Qi, Kai, 2022. "A kind of new time-weighted nonnegative lasso index-tracking model and its application," The North American Journal of Economics and Finance, Elsevier, vol. 59(C).
    6. H Mezali & J E Beasley, 2013. "Quantile regression for index tracking and enhanced indexation," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 64(11), pages 1676-1692, November.
    7. Strub, O. & Baumann, P., 2018. "Optimal construction and rebalancing of index-tracking portfolios," European Journal of Operational Research, Elsevier, vol. 264(1), pages 370-387.
    8. Andrea Scozzari & Fabio Tardella & Sandra Paterlini & Thiemo Krink, 2013. "Exact and heuristic approaches for the index tracking problem with UCITS constraints," Annals of Operations Research, Springer, vol. 205(1), pages 235-250, May.
    9. Sant’Anna, Leonardo Riegel & Righi, Marcelo Brutti & Müller, Fernanda Maria & Guedes, Pablo Cristini, 2022. "Risk measure index tracking model," International Review of Economics & Finance, Elsevier, vol. 80(C), pages 361-383.
    10. Tingting Yang & Xiaoxia Huang, 2022. "A New Portfolio Optimization Model Under Tracking-Error Constraint with Linear Uncertainty Distributions," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 723-747, November.
    11. Li, Helong & Huang, Qin & Wu, Baiyi, 2021. "Improving the naive diversification: An enhanced indexation approach," Finance Research Letters, Elsevier, vol. 39(C).
    12. Sant’Anna, Leonardo Riegel & Caldeira, João Frois & Filomena, Tiago Pascoal, 2020. "Lasso-based index tracking and statistical arbitrage long-short strategies," The North American Journal of Economics and Finance, Elsevier, vol. 51(C).
    13. Huang, Jinbo & Li, Yong & Yao, Haixiang, 2018. "Index tracking model, downside risk and non-parametric kernel estimation," Journal of Economic Dynamics and Control, Elsevier, vol. 92(C), pages 103-128.
    14. Gianfranco Guastaroba & Renata Mansini & Wlodzimierz Ogryczak & M. Grazia Speranza, 2020. "Enhanced index tracking with CVaR-based ratio measures," Annals of Operations Research, Springer, vol. 292(2), pages 883-931, September.
    15. Spiridon Penev & Pavel Shevchenko & Wei Wu, 2019. "Myopic robust index tracking with Bregman divergence," Papers 1908.07659, arXiv.org, revised Jul 2021.
    16. Meihua Wang & Chengxian Xu & Fengmin Xu & Hongang Xue, 2012. "A mixed 0–1 LP for index tracking problem with CVaR risk constraints," Annals of Operations Research, Springer, vol. 196(1), pages 591-609, July.
    17. Li, Xuepeng & Xu, Fengmin & Jing, Kui, 2022. "Robust enhanced indexation with ESG: An empirical study in the Chinese Stock Market," Economic Modelling, Elsevier, vol. 107(C).
    18. Sant’Anna, Leonardo R. & Filomena, Tiago P. & Caldeira, João F., 2017. "Index tracking and enhanced indexing using cointegration and correlation with endogenous portfolio selection," The Quarterly Review of Economics and Finance, Elsevier, vol. 65(C), pages 146-157.
    19. Wu, Dexiang & Kwon, Roy H. & Costa, Giorgio, 2017. "A constrained cluster-based approach for tracking the S&P 500 index," International Journal of Production Economics, Elsevier, vol. 193(C), pages 222-243.
    20. Ruchika Sehgal & Aparna Mehra, 2019. "Enhanced indexing using weighted conditional value at risk," Annals of Operations Research, Springer, vol. 280(1), pages 211-240, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:cejnor:v:30:y:2022:i:2:d:10.1007_s10100-019-00633-0. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.