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FDR-Controlled Portfolio Optimization for Sparse Financial Index Tracking

Jasin Machkour, Daniel P. Palomar, Michael Muma

TL;DR

The paper addresses sparse index tracking in high-dimensional finance where stock returns exhibit strong dependencies, aiming to control false discoveries via FDR. It extends the dependency-aware T-Rex framework with a nearest-neighbors group design that allows overlapping groups and proves FDR control at the target level $\alpha$. Using a rolling-window training scheme and a constrained quadratic program, it yields sparse portfolios that closely track the S&P $500$ over $20$ years with a small number of stocks. Empirical results show superior wealth tracking and reduced portfolio size compared with baselines like ordinary T-Rex, model-X knockoff+, and ALAIT-E, and the method is implemented in the open-source R package TRexSelector.

Abstract

In high-dimensional data analysis, such as financial index tracking or biomedical applications, it is crucial to select the few relevant variables while maintaining control over the false discovery rate (FDR). In these applications, strong dependencies often exist among the variables (e.g., stock returns), which can undermine the FDR control property of existing methods like the model-X knockoff method or the T-Rex selector. To address this issue, we have expanded the T-Rex framework to accommodate overlapping groups of highly correlated variables. This is achieved by integrating a nearest neighbors penalization mechanism into the framework, which provably controls the FDR at the user-defined target level. A real-world example of sparse index tracking demonstrates the proposed method's ability to accurately track the S&P 500 index over the past 20 years based on a small number of stocks. An open-source implementation is provided within the R package TRexSelector on CRAN.

FDR-Controlled Portfolio Optimization for Sparse Financial Index Tracking

TL;DR

The paper addresses sparse index tracking in high-dimensional finance where stock returns exhibit strong dependencies, aiming to control false discoveries via FDR. It extends the dependency-aware T-Rex framework with a nearest-neighbors group design that allows overlapping groups and proves FDR control at the target level . Using a rolling-window training scheme and a constrained quadratic program, it yields sparse portfolios that closely track the S&P over years with a small number of stocks. Empirical results show superior wealth tracking and reduced portfolio size compared with baselines like ordinary T-Rex, model-X knockoff+, and ALAIT-E, and the method is implemented in the open-source R package TRexSelector.

Abstract

In high-dimensional data analysis, such as financial index tracking or biomedical applications, it is crucial to select the few relevant variables while maintaining control over the false discovery rate (FDR). In these applications, strong dependencies often exist among the variables (e.g., stock returns), which can undermine the FDR control property of existing methods like the model-X knockoff method or the T-Rex selector. To address this issue, we have expanded the T-Rex framework to accommodate overlapping groups of highly correlated variables. This is achieved by integrating a nearest neighbors penalization mechanism into the framework, which provably controls the FDR at the user-defined target level. A real-world example of sparse index tracking demonstrates the proposed method's ability to accurately track the S&P 500 index over the past 20 years based on a small number of stocks. An open-source implementation is provided within the R package TRexSelector on CRAN.
Paper Structure (8 sections, 1 theorem, 18 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 8 sections, 1 theorem, 18 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Dependency-Aware T-Rex Selector
  3. Proposed: FDR-Controlled Index Tracking
  4. Group Design and FDR Control
  5. Stock Returns Data Model
  6. Algorithm: FDR-Controlled Index Tracking
  7. Tracking the S&P 500 Index
  8. Conclusion

Key Result

Theorem 1

Let $\mathop{\mathrm{Gr}}\nolimits(j, \rho_{\mathop{\mathrm{thr}}\nolimits})$ be as defined in eq: nearest neighbors group design principle and $K \rightarrow \infty$. Suppose that $\widehat{V}_{T, L}^{\prime}(v, \rho_{\mathop{\mathrm{thr}}\nolimits}) > 0$. Then, for any quadruple $(v, \rho_{\mathop

Figures (2)

  • Figure 1: Simplified T-Rex selector framework machkour2021terminatingmachkour2022TRexGVS.
  • Figure 2: The proposed T-Rex+DA+NN selector closely follows the S&P $500$ index using the fewest number of stocks in almost all quarters.

Theorems & Definitions (2)

  • Theorem 1: FDR control
  • proof