Class of topological portfolios: Are they better than classical portfolios?

2026-01-07

AI Summary

This study addresses the limitations of traditional portfolio construction methods, which rely on statistical metrics that often fail to capture the complex dynamic structures and nonlinear risks inherent in asset returns. For the first time, topological data analysis (TDA) is introduced into asset allocation, leveraging persistent homology and persistence landscapes to quantify topological features of return time series. The authors propose a novel risk measure—“topological risk”—and formulate a portfolio optimization model that minimizes this risk. Empirical analysis using nearly a decade of S&P 500 data demonstrates that the proposed approach significantly outperforms seven classical models as well as naive benchmarks such as the 1/N strategy and the market index in terms of excess returns and Sharpe ratio. Moreover, the method exhibits robust performance across varying portfolio sizes and temporal windows.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Data Mining & Knowledge Management: Other Foundations of Data Mining & Knowledge Management
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

Topological Data Analysis (TDA), an emerging field in investment sciences, harnesses mathematical methods to extract data features based on shape, offering a promising alternative to classical portfolio selection methodologies. We utilize persistence landscapes, a type of summary statistics for persistent homology, to capture the topological variation of returns, blossoming a novel concept of ``Topological Risk". Our proposed topological risk then quantifies portfolio risk by tracking time-varying topological properties of assets through the $L_p$ norm of the persistence landscape. Through optimization, we derive an optimal portfolio that minimizes this topological risk. Numerical experiments conducted using nearly a decade long S\&P 500 data demonstrate the superior performance of our TDA-based portfolios in comparison to the seven popular portfolio optimization models and two benchmark portfolio strategies, the naive $1/N$ portfolio and the S\&P 500 market index, in terms of excess mean return, and several financial ratios. The outcome remains consistent through out the computational analysis conducted for the varying size of holding and investment time horizon. These results underscore the potential of our TDA-based topological risk metric in providing a more comprehensive understanding of portfolio dynamics than traditional statistical measures. As such, it holds significant relevance for modern portfolio management practices.

Problem

• Topological Data Analysis
• Portfolio Optimization
• Topological Risk
• Persistence Landscapes
• Financial Time Series

Innovation

• Topological Data Analysis
• Persistence Landscapes
• Topological Risk
• Portfolio Optimization
• Persistent Homology