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The Geometry of Risk: A Spectral Decomposition of Global Markets

Mapping Systemic Fragility via Eigen-Centrality and Rolling Absorption Ratios

Python Finance Math Status

📉 Executive Summary

Modern Portfolio Theory (MPT) relies on the assumption of structural diversification. However, in "supply-side" macro regimes, asset correlations often converge, rendering traditional diversification ineffective.

This research project applies Linear Algebra (Eigen-Decomposition) and Network Theory (PageRank) to decode the hidden market structure of the 2024-2025 macro regime. By moving beyond static correlation matrices to dynamic Spectral Decomposition, this study identifies a fundamental Regime Shift: the transition from an idiosyncratic "Stock Picker's Market" to a unified "Petro-Economy" driven by energy input costs and interest rate sensitivity.

🧠 Theoretical Framework & Skills Applied

This project synthesizes concepts from advanced mathematics, macroeconomics, and quantitative finance.

1. Mathematical & Computational

  • Principal Component Analysis (PCA): utilized Singular Value Decomposition (SVD) to isolate latent market drivers.
    • Application: Calculated "Global Beta" by extracting the First Principal Component (PC1) from a multi-asset covariance matrix.
  • Graph Theory & Centrality: modeled the S&P 500 as a weighted undirected graph.
    • Application: Applied PageRank (Eigenvector Centrality) to identify transmission nodes ("Systemic Hubs") within the correlation network.
    • Application: Constructed Minimum Spanning Trees (MST) to visualize the "backbone" of market structure, filtering out noise.
  • Rolling Window Statistics: Implemented custom rolling PCA algorithms to generate time-series signals for systemic risk.

2. Financial & Economic

  • Systemic Risk Modeling: Implemented the Absorption Ratio (Kritzman et al.) to quantify market fragility.
  • Regime Identification: Diagnosed the shift from "Demand-Pull" (Growth) to "Supply-Push" (Cost) inflation regimes.
  • Asset Pricing Theory: Analyzed "Duration Risk" in equities, modeling Big Tech as long-duration bond proxies sensitive to the 10-Year Treasury Yield.
  • Cross-Asset Correlation: Mapped the transmission mechanism between Commodities (Brent Crude), Sovereign Bonds (US 10Y), and Global Equities.

🛠 Methodology & Research Pipeline

The analysis follows a strict Top-Down quantitative workflow:

Phase 1: The Macro Layer (Global Beta)

  • Input: Log-returns of major indices (SPX, STOXX50, N225, SHCOMP), Commodities (Oil, Gold), and Rates (TNX).
  • Technique: Spectral Decomposition of the Global Correlation Matrix.
  • Finding: Brent Crude Oil emerged as the dominant factor loading (0.88) on PC1.
  • Signal: The Rolling Absorption Ratio (60-Day) identified a "V-Shaped" recovery in systemic risk, signaling a re-coupling of global assets around energy prices in Q4 2025.

Phase 2: The Meso Layer (Sector Rotation)

  • Input: Top 100 constituents of the S&P 500.
  • Technique: 3D PCA Scatter Plots.
  • Finding: A structural merger of Utilities and Technology sectors (96% correlation).
  • Insight: Utilities have transitioned from "Defensive Hedges" to "AI Power Proxies," effectively increasing the aggressive beta of standard diversified portfolios.

Phase 3: The Micro Layer (Systemic Nodes)

  • Input: Adjacency matrix of S&P 500 constituents (Threshold > 0.7).
  • Technique: Google PageRank Algorithm ($d=0.85$).
  • Finding: While media focus remains on "Mag 7" Tech stocks, the mathematical "Center of Gravity" is Visa (V) and Mastercard (MA).
  • Implication: The US market structure is chemically dependent on Consumer Credit Velocity. Shocks to the payment rail system propagate faster and wider than idiosyncratic tech volatility.

Phase 4: The Global "Petro-Sensitivity" Test

  • Input: Cumulative PageRank scoring of Energy Clusters across Europe, Japan, and China.
  • Finding: Global markets exhibit "Convergent Fragility." Despite geopolitical differences, all major regions show >40% structural sensitivity to the Energy complex.
Region Energy Cluster Influence Dominant Player Risk Profile
China (Shanghai) 45.4% PetroChina High. Index behaves as an Energy Proxy despite State Policy buffers.
Europe (Stoxx) 41.9% Eni SpA High. Levered play on global input costs.
Japan (Nikkei) 39.9% ENEOS Holdings High. Driven by Refiner margins and import costs.

🚦 Scenario Analysis & Validation

The project validates the "Fragility" thesis by reconciling mathematical signals with recent price action:

  • The Signal: The Absorption Ratio spiked in late Q4 2025, while Rate Cut probabilities collapsed (<40%).
  • The Price Action: Big Tech suffered a ~30% drawdown while the broader index remained resilient.
  • The Conclusion: The market is caught in a "Duration Trap." With the "Fed Put" removed, capital crowded into "Bond Proxies" (Cash-Rich Tech), creating extreme concentration. The subsequent drawdown validates the model's signal that Liquidity Stress had returned, forcing a mechanical de-leveraging of long-duration assets.

💻 Tech Stack

  • Data Acquisition: yfinance (Yahoo Finance API)
  • Data Processing: pandas, numpy (Log-return normalization, Rolling Windows)
  • Linear Algebra: scikit-learn (PCA), numpy.linalg (SVD, Eigenvalues)
  • Network Analysis: networkx (Graph construction, PageRank, Minimum Spanning Trees)
  • Visualization: matplotlib, seaborn (Heatmaps, 3D Plots, Time-Series Analysis)

🚀 How to Run

  1. Clone the repository: 
    git clone [https://github.com/yourusername/geometry-of-risk.git](https://github.com/yourusername/geometry-of-risk.git)
  2. Install dependencies: 
    pip install pandas numpy yfinance scikit-learn networkx matplotlib seaborn
  3. Run the Jupyter Notebook: 
    jupyter notebook geometry_of_risk.ipynb

📜 Disclaimer

This project is for educational and research purposes only. It demonstrates the application of quantitative methods to financial data and does not constitute investment advice.