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A mixed 0–1 LP for index tracking problem with CVaR risk constraints

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Abstract

Index tracking problems are concerned in this paper. A CVaR risk constraint is introduced into general index tracking model to control the downside risk of tracking portfolios that consist of a subset of component stocks in given index. Resulting problem is a mixed 0–1 and non-differentiable linear programming problem, and can be converted into a mixed 0–1 linear program so that some existing optimization software such as CPLEX can be used to solve the problem. It is shown that adding the CVaR constraint will have no impact on the optimal tracking portfolio when the index has good (return increasing) performance, but can limit the downside risk of the optimal tracking portfolio when index has bad (return decreasing) performance. Numerical tests on Hang Seng index tracking and FTSE 100 index tracking show that the proposed index tracking model is effective in controlling the downside risk of the optimal tracking portfolio.

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Acknowledgements

This research is supported by National Natural Science Foundations, P.R. China (Grant No. 10971162, 11101325 and 71171158).

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Authors and Affiliations

  1. Department of Mathematics of Faculty of Science, Xi’an Jiaotong University, Xi’an, 710049, P.R. China

    Meihua Wang

  2. SKLMSE Lab. and Department of Mathematics, Xi’an Jiaotong University, Xi’an, 710049, P.R. China

    Chengxian Xu

  3. Department of Mathematics, Xi’an Jiaotong University, Xi’an, 710049, P.R. China

    Fengmin Xu

  4. School of Economics and Finance, Xi’an Jiaotong University, Xi’an, 710061, P.R. China

    Hongang Xue

Authors
  1. Meihua Wang
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  2. Chengxian Xu
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  3. Fengmin Xu
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  4. Hongang Xue
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Corresponding author

Correspondence to Fengmin Xu.

Appendix: A hybrid genetic method for index tracking problem

Appendix: A hybrid genetic method for index tracking problem

Ruiz-Torrubiano and Suárez (2009) propose a hybrid optimization approach for index tracking problem, where the tracking error is measured by mean square deviation between the returns of the tracking portfolio and of the index. The resulting problem is a mixed 0–1 quadratic programming problem. The approach has been tested on different practical data sets. It can effectively solve index tracking problems. A hybrid genetic method based on the approach is applied to solve the mixed 0–1 linear programming problem generated from the index tracking problem with CVaR constraint. In the following, we summarize the hybrid genetic method:

  1. 1.

    Initialization.

    Generate initial population of P individuals (candidate solutions), where each individual is a portfolio containing K randomly selected stocks. When the stocks in the tracking portfolio is fixed, the problems (7) and (17) with cardinality constraint deleted are linear programs which can be easily solved to obtain an optimal portfolio with K specified component stocks by optimization software such as CPLEX and Matlab. Then the fitness of the optimal portfolio is calculated. The choice of the fitness function is crucial in the design of a genetic algorithm. In the proposed hybrid genetic method, the fitness is set as the negative value of the tracking error (i.e., −TE in problem (7) or (17)).

  2. 2.

    Evaluation.

    Evaluate the fitness of individuals in the population. If the population satisfy stop criterion, stop. Else, goto Step 3;

  3. 3.

    Selection.

    Select individuals for parents from the population;

  4. 4.

    Recombination.

    Recombine parents to produce new generation (children);

  5. 5.

    Mutation.

    Mutate the children with a given mutation probability and obtain a new child, then goto step 2.

The crossover operator (RAR operator) given in Moral-Escudero et al. (2006) is applied in the recombination step. This strategy choice results in a genetic algorithm with high evolutionary pressure which has shown good performance in index tracking and portfolio optimization problems.

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Wang, M., Xu, C., Xu, F. et al. A mixed 0–1 LP for index tracking problem with CVaR risk constraints. Ann Oper Res 196, 591–609 (2012). https://doi.org/10.1007/s10479-011-1042-9

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  • DOI: https://doi.org/10.1007/s10479-011-1042-9

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