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

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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.

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Notes

  1. If the optimization software permits, it is better to replace the constraints \(-M y_i \le \pi _i\le M y_i\) with logical constraints \(((\pi _i==0)||(y_i == 1))\), because such BigM formulations (7) including large values of \(M\) sometimes behave in unstable ways.

  2. Normalized means that \(f\) satisfies \(q(\emptyset )=0\).

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Authors and Affiliations

  1. Department of Administration Engineering, Keio University, 3-14-1 Hiyoshi, Kouhoku, Yokohama, Kanagawa, 223-8522, Japan

    Akiko Takeda

  2. School of Electronics and Computer Science, University of Southampton, Southampton, SO17 1BJ, UK

    Mahesan Niranjan

  3. Department of Industrial and Systems Engineering, Chuo University, 2-13-27 Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan

    Jun-ya Gotoh

  4. The Institute of Scientific and Industrial Research, Osaka University, 8-1 Mihogaoka, Ibaraki-shi, Osaka, 567-0047, Japan

    Yoshinobu Kawahara

Authors
  1. Akiko Takeda
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  2. Mahesan Niranjan
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  3. Jun-ya Gotoh
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  4. Yoshinobu Kawahara
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Correspondence to Akiko Takeda.

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Takeda, A., Niranjan, M., Gotoh, Jy. et al. Simultaneous pursuit of out-of-sample performance and sparsity in index tracking portfolios. Comput Manag Sci 10, 21–49 (2013). https://doi.org/10.1007/s10287-012-0158-y

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  • DOI: https://doi.org/10.1007/s10287-012-0158-y

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