Computational Management Science

, Volume 10, Issue 1, pp 21–49 | Cite as

Simultaneous pursuit of out-of-sample performance and sparsity in index tracking portfolios

  • Akiko Takeda
  • Mahesan Niranjan
  • Jun-ya Gotoh
  • Yoshinobu Kawahara
Original Paper

Abstract

Index tracking is a passive investment strategy in which a fund (e.g., an ETF: exchange traded fund) manager purchases a set of assets to mimic a market index. The tracking error, i.e., the difference between the performances of the index and the portfolio, may be minimized by buying all the assets contained in the index. However, this strategy results in a considerable transaction cost and, accordingly, decreases the return of the constructed portfolio. On the other hand, a portfolio with a small cardinality may result in poor out-of-sample performance. Of interest is, thus, constructing a portfolio with good out-of-sample performance, while keeping the number of assets invested in small (i.e., sparse). In this paper, we develop a tracking portfolio model that addresses the above conflicting requirements by using a combination of L0- and L2-norms. The L2-norm regularizes the overdetermined system to impose smoothness (and hence has better out-of-sample performance), and it shrinks the solution to an equally-weighted dense portfolio. On the other hand, the L0-norm imposes a cardinality constraint that achieves sparsity (and hence a lower transaction cost). We propose a heuristic method for estimating portfolio weights, which combines a greedy search with an analytical formula embedded in it. We demonstrate that the resulting sparse portfolio has good tracking and generalization performance on historic data of weekly and monthly returns on the Nikkei 225 index and its constituent companies.

Keywords

Portfolio optimization Index tracking Norm constraint  Regularization Sparse portfolio Greedy algorithm 

References

  1. Boer PM, Hafner CM (2005) Ridge regression revisited. Statistica Neerlandica 59(4):498–505CrossRefGoogle Scholar
  2. Brodie J, Daubechies I, De Molc C, Giannone D, Loris I (2009) Sparse and stable markowitz portfolios. PNAS 106:12267–12272CrossRefGoogle Scholar
  3. Chang T, Meade N, Beasley J, Sharaiha Y (2000) Heuristics for cardinality constrained portfolio optimisation. Comput Oper Res 27:1271–1302CrossRefGoogle Scholar
  4. Das A, Kempe D (2008) Algorithms for subset selection in linear regression. In: Proceedings of STOC 2008, pp 45–54Google Scholar
  5. Das A, Kempe D (2011) Submodular meets spectral: Greedy algorithms for subset selection, sparse approximation and dictionary selection. In: Proceedings of international conference on machine learningGoogle Scholar
  6. Davis G, Mallat S, Zhang Z (1994) Adaptive time-frequency decompositions with matching pursuits. Optic Eng 33(7):2183–2191CrossRefGoogle Scholar
  7. DeMiguel V, Garlappi L, Nogales F, Uppal R (2009a) A generalized approach to portfolio optimization: improving performance by constraining portfolio norms. Manag Sci 55:798–812CrossRefGoogle Scholar
  8. DeMiguel V, Garlappi L, Uppal R (2009b) Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy? Rev Financ Stud 22(5):1915–1953CrossRefGoogle Scholar
  9. Fastrich B, Winker P (2012) Robust portfolio optimization with a hybrid heuristic algorithm. Comput Manag Sci 9(1):63–88CrossRefGoogle Scholar
  10. Gilli M, Këllezi E (2002) The threshold accepting heuristic for index tracking. Financial Engineering, E-Commerce, and Supply Chain. Kluwer, Dordrecht, pp 1–18Google Scholar
  11. Gotoh J, Takeda A (2011) On the role of norm constraints in portfolio selection. Comput Manag Sci 8(4):323–353CrossRefGoogle Scholar
  12. Hoerl E, Kennard RW (1970) Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12:55–67CrossRefGoogle Scholar
  13. Hastie T, Tibshirani R, Friedman J (2008) The elements of statistical learning—data mining, inference, and prediction, 2nd edn. Springer, New YorkGoogle Scholar
  14. Konno H, Wijayanayake A (2001) Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Math Progr 89(2):233–250CrossRefGoogle Scholar
  15. Konno H, Wijayanayake A (2002) Portfolio optimization under d.c. transaction costs and minimal transaction unit constraints. J Glob Optim 22(1–4):137–154CrossRefGoogle Scholar
  16. Lobo M, Fazel M, Boyd S (2007) Portfolio optimization with linear and fixed transaction costs. Ann Oper Res 152(1):341–365CrossRefGoogle Scholar
  17. Markowitz H (1952) Portfolio selection. J Financ 7(1):77–91Google Scholar
  18. Minoux M (1978) Accelerated greedy algorithms for maximizing submodular set functions. In: Optimization, techniques, pp 234–243Google Scholar
  19. Nemhauser GL, Wolsey LA, Fisher ML (1978) An analysis of approximations for maximizing submodular set functions. Math Program 14:265–294CrossRefGoogle Scholar
  20. Pati Y, Rezaiifar R, Krishnaprasad P (1993) Orthogonal matching pursuit: recursive function approximation with application to wavelet decomposition. In: Asilomar conference on signals, systems and computersGoogle Scholar
  21. Prigent J-L (2007) Portfolio optimization and performance analysis. Chapman& Hall/CRC, Boca RatonCrossRefGoogle Scholar
  22. Roll R (1992) A mean/variance analysis of tracking error. J Portf Manag 18(4):13–22CrossRefGoogle Scholar
  23. Rudolf M, Wolter HJ, Zimmermann H (1999) A linear model for tracking error minimization. J Bank Financ 23(1):85–103CrossRefGoogle Scholar
  24. Ruiz-Torrubiano R, Suárez A (2009) A hybrid optimization approach to index tracking. Ann Oper Res 166(1):57–71CrossRefGoogle Scholar
  25. Schölkopf B, Smola A, Williamson R, Bartlett P (2000) New support vector algorithms. Neural Comput 12(5):1207–1245CrossRefGoogle Scholar
  26. Takeda A, Gotoh J, Sugiyama M (2010) Support vector regression as conditional value-at-risk minimization with application to financial time-series analysis. In: Proceedings of 2010 IEEE international workshop on machine learning for signal processingGoogle Scholar
  27. Tibshirani R (1996) Optimal reinsertion: regression shrinkage and selection via the lasso. J R Stat Soc Ser B 58(1):267–288Google Scholar
  28. Woodside-Oriakhi M, Lucas C, Beasley J (2011) Heuristic algorithms for the cardinality constrained efficient frontier. Eur J Oper Res 213:538–550CrossRefGoogle Scholar
  29. Xu F, Xu Z, Xue H (2011) Sparse index tracking: an l1/2 regularization based model and solution. http://gr.xjtu.edu.cn:8080/LiferayFCKeditor/UserFiles/File/NewAOR.pdf
  30. Yen Y, Yen T (2011) Solving norm constrained portfolio optimizations via coordinate-wise descent algorithms. Technical report, London School of Economics and Political Science, UKGoogle Scholar
  31. Zhang T (2008) Adaptive forward-backward greedy algorithm for sparse learning with linear models. In: Advances in Neural Information Processing Systems, vol 22, pp 1921–1928Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Akiko Takeda
    • 1
  • Mahesan Niranjan
    • 2
  • Jun-ya Gotoh
    • 3
  • Yoshinobu Kawahara
    • 4
  1. 1.Department of Administration EngineeringKeio UniversityYokohamaJapan
  2. 2.School of Electronics and Computer ScienceUniversity of SouthamptonSouthamptonUK
  3. 3.Department of Industrial and Systems EngineeringChuo UniversityTokyoJapan
  4. 4.The Institute of Scientific and Industrial ResearchOsaka UniversityIbaraki-shiJapan

Personalised recommendations