Optimal investment strategies for enhanced indexation problems have attracted considerable attentions over the last decades in the field of fund management. In this paper, a featured difference from the existing literature is that our main concern of the investigation is the development of a sparse-enhanced indexation model to describe the process of assets selection by introducing a sparse ${\ell }_{1/2}$ regularization instead of binary variables, which is expected to avoid the over-fitting and promote a better out-of-sample performance for the resulting tracking portfolio to some extent. An Alternating Quadratic Penalty (AQP) method is proposed to solve the corresponding nonconvex optimisation problem, into which the Block Coordinate Descent (BCD) algorithm is integrated to solve a sequence of penalty subproblems. Under some suitable assumptions, we establish that any accumulation point of the sequence generated by the AQP method is a KKT point of the proposed model. Computational results on five typical data-sets are reported to verify the efficiency of the proposed AQP method, including the superiority of the sparse ${\ell }_{1/2}$ model with the AQP method over one cardinality constrained quadratic programming model with one of its solution methods in terms of computational costs, out-of-sample performances, and the consistency between in-sample and out-of-sample performances of the resulting tracking portfolios.

Enhanced-index-funds have attracted considerable attention from investors over the last decade, which aims at outperforming a benchmark index while maintaining a similar risk level. In this article, we investigate an enhanced indexation methodology using Conditional Value-at-Risk (CVaR). In particular, we adopt CVaR of excess returns as risk measurement subject to cardinality constraint for controlling the tracking portfolio scale precisely and tunable short-selling constraints for adjusting the margin of each risky asset adaptively within the budget of short-selling. As the resulted model is a mixed 0–1 binary program, we propose an improved hybrid heuristic method, where a customized relax-round-polish is embedded to improve the quality of the iterative population. Computational results on five standard data sets from OR-library show that our proposed method is generally superior to the naive portfolio strategy and the CVaR-LASSO method in terms of the out-of-sample excess return, Sharpe ratio and maximum drawdown of the portfolio.