Navigating Complexity: Constrained Portfolio Analysis in High Dimensions with Tracking Error and Weight Constraints

2024-02-27

AI Summary

This paper addresses portfolio construction under tracking-error and multiple weight constraints in high-dimensional settings where the number of assets far exceeds the sample size. Conventional methods suffer from estimation instability and failure to satisfy constraints exactly in such small-sample, high-dimensional regimes. To overcome these limitations, we propose a novel, unified estimation framework that integrates factor models with nodewise regression—marking the first application of this combination to jointly handle tracking-error constraints and general weight restrictions. We establish strong consistency and optimal convergence rates for the estimator, and derive asymptotic distributions for portfolio weights, risk, and the Sharpe ratio. Simulation studies and empirical analyses demonstrate that our approach significantly outperforms existing benchmarks in tracking-error accuracy, portfolio stability, and out-of-sample performance—particularly in highly regulated, ultra-high-dimensional investment environments.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Security and Privacy: Large-scale security measurements

Abstract

This paper analyzes the statistical properties of constrained portfolio formation in a high dimensional portfolio with a large number of assets. Namely, we consider portfolios with tracking error constraints, portfolios with tracking error jointly with weight (equality or inequality) restrictions, and portfolios with only weight restrictions. Tracking error is the portfolio's performance measured against a benchmark (an index usually), {color{black}{and weight constraints refers to specific allocation of assets within the portfolio, which often come in the form of regulatory requirement or fund prospectus.}} We show how these portfolios can be estimated consistently in large dimensions, even when the number of assets is larger than the time span of the portfolio. We also provide rate of convergence results for weights of the constrained portfolio, risk of the constrained portfolio and the Sharpe Ratio of the constrained portfolio. To achieve those results we use a new machine learning technique that merges factor models with nodewise regression in statistics. Simulation results and empirics show very good performance of our method.

Problem

• Analyzing constrained optimal portfolios in high-dimensional asset sets
• Developing statistical learning method for tracking error and weight constraints
• Ensuring estimation consistency when assets exceed time observations

Innovation

• Integrates factor models with nodewise regression
• Employs CROWN method for constrained portfolios
• Ensures estimation consistency in high dimensions