Minimizing the tracking error of cardinality constrained portfolios

•Proves NP-hardness of an index tracking problem without bounds on asset weights.•Convex hull formulation with quadratic objective function.•Running time analysis of greedy construction and local improvement methods.•Computational experiments proving superiority over a general purpose B&B-solver. We study the problem of selecting a restricted number of shares included in a stock market index, such that the portfolio resembles the index as closely as possible. To measure the difference between the portfolio and the index, referred to as the tracking error, we use a quadratic function with the covariance matrix of the index returns as coefficient matrix. The problem is proved to be strongly NP-hard, and we give theoretical evidence that continuous relaxations of mixed integer quadratic programming (MIQP) formulations are likely to produce poor lower bounds on the tracking error. For fast computation of near-optimal portfolios, we demonstrate how the best-extension-by-one construction heuristic can be designed to run in time bounded by a fourth order polynomial. We also show that the running time of one iteration of the best-exchange-by one improvement heuristic is of the same order. Computational experiments applied to real-life stock market indices show that in instances where an index of less than 500 assets is to be tracked by a portfolio of 10 assets, a commercially available MIQP solver fails to reduce the integrality gap below 94% in 30 CPU-minutes. In contrast, the construction heuristic under study needs less than 30 CPU-seconds to produce a portfolio of 100 assets tracking an index of nearly 2000 assets.