Beyond VaR and CVaR: Topological Risk Measures in Financial Markets

Paper: arXiv 2310.14604

Authors: Amit Kumar Jha

Abstract

This paper introduces a novel approach to financial risk assessment by incorporating topological data analysis (TDA), specifically cohomology groups, into the evaluation of equities portfolios. The study aims to go beyond traditional risk measures like Value at Risk (VaR) and Conditional Value at Risk (CVaR), offering a more nuanced understanding of market complexities. Using last one year daily real-world closing price return data for three equities Apple, Microsoft and Google , we developed a new topological riskmeasure, termed Topological VaR Distance (TVaRD). Preliminary results indicate a significant change in the density of the point cloud representing the financial time series during stress conditions, suggesting that TVaRD may offer additional insights into portfolio risk and has the potential to complement existing risk management tools.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces an innovative theoretical framework applying topological data analysis and cohomology to risk management. However, the empirical evaluation is severely limited, relying on a very small dataset of only three tech stocks over a single year.

Research Flowchart

  flowchart TD
    A[Goal: Enhance Risk Assessment Beyond VaR/CVaR] --> B[Input Data: 1-Year Daily Returns for AAPL, MSFT, GOOG]
    B --> C[Construct Point Cloud from Financial Time Series]
    C --> D[Apply TDA & Cohomology Groups]
    D --> E[Develop Topological VaR Distance Metric]
    E --> F[Key Finding: Significant Point Cloud Density Shift During Stress]
    F --> G[Outcome: TVaRD Complements Existing Risk Management Tools]