Paper: arXiv 2309.10152

Abstract

Sparse index tracking is a prominent passive portfolio management strategy that constructs a sparse portfolio to track a financial index. A sparse portfolio is preferable to a full portfolio in terms of reducing transaction costs and avoiding illiquid assets. To achieve portfolio sparsity, conventional studies have utilized $\ell_p$-norm regularizations as a continuous surrogate of the $\ell_0$-norm regularization. Although these formulations can construct sparse portfolios, their practical application is challenging due to the intricate and time-consuming process of tuning parameters to define the precise upper limit of assets in the portfolio. In this paper, we propose a new problem formulation of sparse index tracking using an $\ell_0$-norm constraint that enables easy control of the upper bound on the number of assets in the portfolio. Moreover, our approach offers a choice between constraints on portfolio and turnover sparsity, further reducing transaction costs by limiting asset updates at each rebalancing interval. Furthermore, we develop an efficient algorithm for solving this problem based on a primal-dual splitting method. Finally, we illustrate the effectiveness of the proposed method through experiments on the S&P500 and Russell3000 index datasets.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces advanced optimization techniques like ℓ0-norm constraints and primal-dual splitting, demonstrating significant mathematical sophistication. It supports these methods with empirical experiments on real-world datasets (S&P500, Russell3000), though it lacks backtest-ready details like code or transaction cost models.

Research Flowchart

  flowchart TD
  A["Research Goal: Sparse Index Tracking<br/>with L0-Constraint"] --> B{"Data Inputs & Setup"};
  B --> C["Proposed Model Formulation"];
  C --> D["Algorithm Development<br/>Primal-Dual Splitting Method"];
  D --> E["Computational Execution & Tuning"];
  E --> F["Experimental Validation<br/>S&P500 & Russell3000 Datasets"];
  F --> G["Key Findings & Outcomes"];