Cardinality constrained index tracking in an end-to-end way: Soft or hard?

Index tracking aims at replicating the benchmark index performance by constructing a tracking portfolio consisting of partial component stocks of the index. This is an NP-hard optimisation question because of the cardinality constraint which only selects partial components from the market index to construct the tracking portfolios. This paper proposes and compares two sparse alternatives of the Softmax function, i.e., Subset-Softmax and Mask-Softmax, to solve this problem in an end-to-end way with Long Short-Term Memory networks via differentiable subset sampling and masking techniques. The first proposed sparse Softmax function is based on reparameterizable subset sampling, which can be regarded as a soft method, while the other is based on hard feature selection and mask learning technique. S&P500 index data is tested with two proposed methods and solving the problem as an exact quadratic programming problem. The empirical results show that the Mask-Softmax based tracking portfolios can present the best tracking performance as well as a lower risk level in most cases. The robustness of tracking performance of the proposed strategies is investigated in detail and the results show that both sparse Softmax based tracking portfolios have good robustness. Then, the tracking portfolio characteristics are analysed by exploring the exposure to systematic risk sources, while the influence of US-China trading friction and Covid-19 on the tracking performance is also analyzed via sub-period analysis. Last, the normalized entropy of tracking portfolios is computed and results show that the proposed methods can have good sector diversification.