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

TL;DR

This paper introduces a novel topological data analysis approach to portfolio optimization, using persistence landscapes to quantify topological risk, and demonstrates its superior performance over traditional methods on S&P 500 data.

Contribution

It develops a new topological risk measure based on persistence landscapes and shows its effectiveness in portfolio optimization compared to classical models.

Findings

TDA-based portfolios outperform traditional models in excess return.

Topological risk correlates with portfolio performance.

Method remains robust across different investment horizons.

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.