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On-Line Portfolio Selection Using Multiplicative Updates


  • David P. Helmbold
  • Robert E. Schapire
  • Yoram Singer
  • Manfred K. Warmuth


We present an on-line investment algorithm that achieves almost the same wealth as the best constant-rebalanced portfolio determined in hindsight from the actual market outcomes. The algorithm employs a multiplicative update rule derived using a framework introduced by Kivinen and Warmuth. Our algorithm is very simple to implement and requires only constant storage and computing time per stock in each trading period. We tested the performance of our algorithm on real stock data from the New York Stock Exchange accumulated during a 22-year period. On these data, our algorithm clearly outperforms the best single stock as well as Cover's universal portfolio selection algorithm. We also present results for the situation in which the investor has access to additional "side information." Copyright Blackwell Publishers Inc 1998.

Suggested Citation

  • David P. Helmbold & Robert E. Schapire & Yoram Singer & Manfred K. Warmuth, 1998. "On-Line Portfolio Selection Using Multiplicative Updates," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 325-347.
  • Handle: RePEc:bla:mathfi:v:8:y:1998:i:4:p:325-347

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    Cited by:

    1. Zhengyao Jiang & Dixing Xu & Jinjun Liang, 2017. "A Deep Reinforcement Learning Framework for the Financial Portfolio Management Problem," Papers 1706.10059,, revised Jul 2017.
    2. Ting-Kam Leonard Wong, 2015. "Universal portfolios in stochastic portfolio theory," Papers 1510.02808,, revised Dec 2016.
    3. Guy Uziel & Ran El-Yaniv, 2017. "Growth-Optimal Portfolio Selection under CVaR Constraints," Papers 1705.09800,
    4. Panpan Ren & Jiang-Lun Wu, 2017. "Foreign exchange market modelling and an on-line portfolio selection algorithm," Papers 1707.00203,
    5. Yang Wang & Dong Wang & Yaodong Wang & You Zhang, 2018. "RACORN-K: Risk-Aversion Pattern Matching-based Portfolio Selection," Papers 1802.10244,
    6. Vajda, István & Ottucsák, György, 2006. "Empirikus portfólióstratégiák
      [Empirical portfolio strategies]
      ," Közgazdasági Szemle (Economic Review - monthly of the Hungarian Academy of Sciences), Közgazdasági Szemle Alapítvány (Economic Review Foundation), vol. 0(7), pages 624-640.
    7. Ottucsák György & Vajda István, 2007. "An asymptotic analysis of the mean-variance portfolio selection," Statistics & Risk Modeling, De Gruyter, vol. 25(1/2007), pages 1-24, January.
    8. Theodoros Tsagaris & Ajay Jasra & Niall Adams, 2010. "Robust and Adaptive Algorithms for Online Portfolio Selection," Papers 1005.2979,
    9. repec:kap:compec:v:50:y:2017:i:1:d:10.1007_s10614-016-9585-0 is not listed on IDEAS
    10. Freund, Yoav & Schapire, Robert E., 1999. "Adaptive Game Playing Using Multiplicative Weights," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 79-103, October.
    11. Bin Li & Steven C. H. Hoi, 2012. "On-Line Portfolio Selection with Moving Average Reversion," Papers 1206.4626,
    12. Parkes, David C. & Huberman, Bernardo A., 2001. "Multiagent Cooperative Search for Portfolio Selection," Games and Economic Behavior, Elsevier, vol. 35(1-2), pages 124-165, April.
    13. repec:spr:annopr:v:256:y:2017:i:1:d:10.1007_s10479-016-2176-6 is not listed on IDEAS
    14. Bin Li & Steven C. H. Hoi, 2012. "Online Portfolio Selection: A Survey," Papers 1212.2129,, revised May 2013.
    15. A. Borodin & R. El-Yaniv & V. Gogan, 2011. "Can We Learn to Beat the Best Stock," Papers 1107.0036,
    16. Xiaoguang Huo & Feng Fu, 2017. "Risk-Aware Multi-Armed Bandit Problem with Application to Portfolio Selection," Papers 1709.04415,
    17. MacLean, Leonard C. & Zhao, Yonggan & Ziemba, William T., 2016. "Optimal capital growth with convex shortfall penalties," LSE Research Online Documents on Economics 65486, London School of Economics and Political Science, LSE Library.

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