DCC: Differentiable Cardinality Constraints for Partial Index Tracking
This work provides a more efficient and interpretable solution for financial investors and analysts dealing with high transaction costs in passive investment strategies, though it is incremental as it builds on existing partial replication methods.
The paper tackles the problem of partial index tracking in portfolio optimization by introducing a differentiable cardinality constraint (DCC) and a floating-point precision-aware method (DCC_fpp) to address non-convex and NP-hard challenges, achieving improved performance over baseline methods across various datasets.
Index tracking is a popular passive investment strategy aimed at optimizing portfolios, but fully replicating an index can lead to high transaction costs. To address this, partial replication have been proposed. However, the cardinality constraint renders the problem non-convex, non-differentiable, and often NP-hard, leading to the use of heuristic or neural network-based methods, which can be non-interpretable or have NP-hard complexity. To overcome these limitations, we propose a Differentiable Cardinality Constraint ($\textbf{DCC}$) for index tracking and introduce a floating-point precision-aware method ($\textbf{DCC}_{fpp}$) to address implementation issues. We theoretically prove our methods calculate cardinality accurately and enforce actual cardinality with polynomial time complexity. We propose the range of the hyperparameter $a$ ensures that $\textbf{DCC}_{fpp}$ has no error in real implementations, based on theoretical proof and experiment. Our method applied to mathematical method outperforms baseline methods across various datasets, demonstrating the effectiveness of the identified hyperparameter $a$.