Sparse Index Tracking With K-Sparsity or ϵ-Deviation Constraint via ℓ₀-Norm Minimization

Area of Science:

• Quantitative Finance
• Computational Finance
• Financial Engineering

Background:

• Sparse index tracking is a passive investment strategy that constructs a portfolio with a few assets to mirror a benchmark financial index.
• Existing algorithms often lack explicit control over the number of assets or the resulting tracking error.
• Parameter learning in adaptive systems, using a sliding window approach, is key to updating portfolios.

Purpose of the Study:

• To address limitations in existing sparse index tracking algorithms by enabling direct control over portfolio parameters.
• To formulate sparse index tracking as constrained optimization problems.
• To propose novel algorithms for enhanced control in passive investment strategies.

Main Methods:

• Formulation of sparse index tracking as two constrained optimization problems.
• Development of Nonnegative Orthogonal Matching Pursuit with Projected Gradient Descent (NNOMP-PGD) for minimizing tracking error with a fixed asset count.
• Development of Alternating Direction Method of Multipliers for l0-norm (ADMM-l0) for minimizing asset count within a specified tracking error threshold.

Main Results:

• NNOMP-PGD allows direct and explicit control over the number of selected assets.
• ADMM-l0 allows direct and explicit control over the tracking error.
• Both algorithms demonstrate convergence properties.
• Numerical experiments show the proposed algorithms outperform existing methods.

Conclusions:

• The proposed NNOMP-PGD and ADMM-l0 algorithms provide investors with explicit control over key portfolio metrics (asset count or tracking error).
• These algorithms enhance the capabilities of sparse index tracking for passive investment.
• The developed methods offer superior performance compared to current approaches.