Portfolio Analysis in High Dimensions with TE and Weight Constraints

Abstract

This paper explores the statistical properties of forming constrained optimal portfolios within a high-dimensional set of assets. We examine portfolios with tracking error constraints, those with simultaneous tracking error and weight restrictions, and portfolios constrained solely by weight. Tracking error measures portfolio performance against a benchmark (typically an index), while weight constraints determine asset allocation based on regulatory requirements or fund prospectuses. Our approach employs a novel statistical learning technique that integrates factor models with nodewise regression, named the Constrained Residual Nodewise Optimal Weight Regression (CROWN) method. We demonstrate its estimation consistency in large dimensions, even when assets outnumber the portfolio's time span. Convergence rate results for constrained portfolio weights, risk, and Sharpe Ratio are provided, and simulation and empirical evidence highlight the method's outstanding performance.