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

ArXiv ID: 2601.03974 “View on arXiv”

Authors: Anubha Goel, Amita Sharma, Juho Kanniainen

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.

Keywords: topological data analysis, persistence landscapes, persistent homology, topological risk, portfolio optimization, Equities

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper introduces advanced Topological Data Analysis (TDA) concepts like persistent homology and persistence landscapes, requiring significant mathematical abstraction, while the empirical section details nearly a decade of S&P 500 backtesting against multiple baselines with standard financial metrics.

  flowchart TD
    A["Research Goal<br>Are topological portfolios superior<br>to classical portfolios?"] --> B["Data Input<br>S&P 500 Historical Data"]
    B --> C["Methodology: Topological Data Analysis<br>Calculate Persistence Landscapes<br>Quantify Topological Risk via Lp norm"]
    C --> D["Computation: Optimization<br>Minimize Topological Risk<br>Construct Optimal Portfolio"]
    D --> E["Outcome: Validation<br>Compare against 7 classical models<br>and 2 benchmarks"]
    E --> F["Findings<br>TDA portfolios yield higher excess returns<br>and superior financial ratios<br>across varying horizons"]