1. The Dimensionality Challenge
Fixed income markets suffer from the "curse of dimensionality." A portfolio manager isn't just exposed to one price, but to a continuous curve of rates spanning from overnight to 30+ years.
The Problem: High Correlation
Modeling risk using every single tenor (2Y, 3Y, 5Y, etc.) leads to unstable models due to Multicollinearity.
Issue: Corr(2Y, 3Y) ≈ 0.98
⚠ Result: Unstable Betas & Overfitting
The Solution: Factor Reduction
PCA re-maps these correlated rates into orthogonal (independent) factors.
Output: 3 Independent Factors
✓ Result: Clean, Additive Risk Drivers
Litterman and Scheinkman (1991) empirically proved that 98% of yield curve variance can be explained by just three movements. This transforms a chaotic system of 30 variables into a manageable set of 3 "bets."
2. The Mathematical Engine
Decomposing the Covariance Matrix
The input is a matrix X of historical yield changes. PCA relies on the Spectral Theorem to decompose the covariance matrix S. The Eigenvalues (λ) tell us how much "energy" (variance) each factor explains.
Variance Explained (The Scree Plot)
Relative magnitude of eigenvalues typically observed in UST markets.
3. The Big Three Factors
Regardless of the market (US, EU, JP), the first three principal components always take these distinct geometric shapes. They correlate strongly with specific macroeconomic drivers.
Level
PC1 • ~90% Var
Parallel Shift
Drivers: Inflation Expectations & Fed Targets
Slope
PC2 • ~8% Var
Steepening / Flattening
Drivers: Business Cycle (Recession = Inverted)
Curvature
PC3 • ~2% Var
Convexity / Butterfly
Drivers: Volatility & Supply/Demand Segmentation
4. Vector Hedging vs. Duration
Why Duration Fails
Traditional Duration (DV01) assumes parallel shifts. If you are Long 30Y and Short 2Y to be "duration neutral," you are actually massively exposed to Slope Risk.
If the curve steepens, you lose on the Long 30Y (yields up, price down) and lose on the Short 2Y (yields down, price up). A "hedged" book can bleed capital.
The Immunization Equation
[SensH,2 ... ] [w2] = [-SensP,2]
[SensH,3 ... ] [w3] [-SensP,3]
Solving this linear system gives the exact weights (w) to neutralize Level, Slope, and Curvature simultaneously.
| Strategy | Hedge Instrument | Risk Exposure |
|---|---|---|
| DV01 Neutral | Any bond (e.g., 10Y) | Slope Risk & Curvature Risk |
| PCA Level + Slope | Two bonds (e.g., 2Y and 10Y) | Curvature Risk only |
| Full Vector Hedge | Three bonds (e.g., 2Y, 5Y, 30Y) | Idiosyncratic Risk only |
5. Alpha Strategy: The PCA Butterfly
Trading the Residuals
A "Butterfly" trade involves buying the "Body" (e.g., 5Y) and selling the "Wings" (2Y and 10Y). Using PCA weights ensures the trade is immune to broad market moves (Level) and tilts (Slope), isolating pure Relative Value.
The Algorithm:
- Regress bond yield on PC1, PC2, PC3 scores.
- Calculate Model Yield: ymodel = β₁P₁ + β₂P₂ + β₃P₃
- Compute Residual: ε = ymarket - ymodel
- Signal: If ε > 1.75σ, the bond is Cheap. Buy.
6. Regime Change & Risks
PCA is a statistical description of history, not a physical law. The 2022 Inflation Shock broke many PCA models.
Correlation Break
Historically, Stocks and Bonds were negatively correlated ("Flight to Quality"). In 2022, they fell together due to inflation. Models relying on Bonds to hedge Stocks failed catastrophically.
Slope Inversion
Normally, recessions cause steepening (Fed cuts rates). In 2022, recession fears coincided with Fed hiking, causing deep inversion. The classic "Steepener" bet lost money.
7. Implementation Recipe
Step-by-Step Implementation
Fetch daily yield data (e.g., FRED or Bloomberg) for key tenors: 1Y, 2Y, 3Y, 5Y, 7Y, 10Y, 20Y, 30Y.
Compute daily changes (absolute or log changes). Yield_Diff = Yield(t) - Yield(t-1). Do NOT run PCA on raw yield levels (they are non-stationary).
Calculate the covariance matrix of the differenced data. Use a rolling window (e.g., 1-year lookback) to capture changing regimes.
Compute Eigenvectors (Loadings) and Eigenvalues. The first 3 eigenvectors form your factor basis.