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.
This project synthesizes concepts from advanced mathematics, macroeconomics, and quantitative finance.
- 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.
- 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.
The analysis follows a strict Top-Down quantitative workflow:
- 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.
- 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.
- 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.
- 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. |
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.
- 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)
- Clone the repository:
git clone [https://github.com/yourusername/geometry-of-risk.git](https://github.com/yourusername/geometry-of-risk.git)
- Install dependencies:
pip install pandas numpy yfinance scikit-learn networkx matplotlib seaborn
- Run the Jupyter Notebook:
jupyter notebook geometry_of_risk.ipynb
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.