Principal Component Analysis (PCA) on the Yield Curve: Level, Slope & Curvature
Decomposing the term structure of interest rates into three orthogonal factors: Level (PC1), Slope (PC2), and Curvature/Butterfly (PC3) for rates modeling.
1. Decomposing the Yield Curve via Linear Algebra
The U.S. Treasury yield curve consists of multiple correlated maturity yields ($3\text{M}, 2\text{Y}, 5\text{Y}, 10\text{Y}, 30\text{Y}$). To eliminate collinearity and model interest rate risk efficiently, quantitative fixed income desks apply Principal Component Analysis (PCA) to the covariance matrix of daily yield changes ($\Delta y$):
2. The Three Fundamental Orthogonal Factors
| Component | Variance Explained | Factor Interpretation | Macro Driver |
|---|---|---|---|
| PC1: Level | 80% – 88% | Parallel Shift (all yields move up/down together) | General inflation expectations and long-term neutral policy rate ($r^*$) revisions. |
| PC2: Slope | 8% – 12% | Steepening vs. Flattening (2Y vs 10Y/30Y spread) | FOMC monetary policy cycle stance (hiking/cutting near-term rates). |
| PC3: Curvature | 2% – 5% | Butterfly (5Y belly relative to 2Y & 30Y wings) | Term premium supply distortions and intermediary duration matching. |
3. Institutional Applications in Relative Value Trading
Fixed income relative-value hedge funds use PCA factor scores to execute duration-neutral butterfly spread trades ($2\text{Y} - 2 \times 5\text{Y} + 30\text{Y}$). When PC3 deviates significantly from its historical mean, traders capture pure mean-reverting curvature alpha without exposing capital to directional rate moves.