In the practical business environment, portfolio managers often face business-driven requirements that limit the number of constituents in their tracking portfolio. A natural index tracking model is thus to minimize a tracking error measure while enforcing an upper bound on the number of assets in the portfolio. In this paper we consider such a cardinality-constrained index tracking model. In particular, we propose an efficient nonmonotone projected gradient (NPG) method for solving this problem. At each iteration, this method usually solves several projected gradient subproblems. We show that each subproblem has a closed-form solution, which can be computed in linear time. Under some suitable assumptions, we establish that any accumulation point of the sequence generated by the NPG method is a local minimizer of the cardinality-constrained index tracking problem. We also conduct empirical tests to compare our method with the hybrid evolutionary algorithm [P.R. Torrubiano and S. Alberto. A hybrid optimization approach to index tracking. Ann Oper Res. 166(1) (2009), pp. 57-71] and the hybrid half thresholding algorithm [F. Xu, Z. Xu and H Xue. Sparse index tracking: an L1/2 regularization based model and solution, Submitted, 2012] for index tracking. The computational results demonstrate that our approach generally produces sparse portfolios with smaller out-of-sample tracking error and higher consistency between in-sample and out-of-sample tracking errors. Moreover, our method outperforms the other two approaches in terms of speed.
Bibliographical noteFunding Information:
The authors would like to thank the two anonymous referees for their constructive comments which substantially improved the presentation of the paper. The first author was supported by China NSFC projects (No.11101325) and (No.71371152). The second author was supported in part by NSERC Discovery Grant. The third author was supported by National 973 Program of China (No.2007CB311002) and China NSFC projects (No.70531030). In addition, the first two authors were supported by "the fundamental research funds for the central Universities" (No.2014gjhz04).
- cardinality constraint
- index tracking
- nonmonotone projected gradient method