Principal Component Analysis for Fixed Income Risk
Fixed Income Trading and Risk Management by Alexander Düring (ISBN 9781119756354)
Previous: Bond Trading | Next: Bond Index Mechanics
PCA shows up throughout Düring’s book. Chapter 33 is the dedicated tutorial: what it is, when to use levels vs changes, and why it beats ordinary regression for hedge ratios.
From one dimension to many
In one dimension, mean and standard deviation summarize a normal variable. In two dimensions, “standard deviation” depends on direction. Mark every distance from the mean along all ruler angles and you get an ellipse. The longest axis is the first principal component.
Alice wants to send Bob one number that best describes a bivariate sample. Picking x alone wastes information about y. The best linear combo αx + βy minimizes uncertainty. That is PCA as dimension reduction and as estimation.
Why linear is enough (locally)
Portfolio risk is usually “small moves around today’s rates.” Taylor expansion: ΔP ≈ Δθ · ∇P. Linear PCA on changes is the first-order risk story. Non-linearities (convexity, options) sit in higher terms.
How PCA works
Find the linear combination of variables explaining most variance. That eigenvector is factor 1. Factor 2 explains what is left after removing factor 1, and so on. All factors come from the covariance matrix eigen decomposition at once. Largest eigenvalue → first factor.
Levels vs changes: stationary series give similar eigenvectors either way. Non-stationary levels break PCA assumptions; use first differences for yield curves.
Covariance vs correlation: curve hedging wants raw vol differences across maturities (use covariance). Comparing equity and bond index levels with different scales → correlation matrix to strip historical scale artifacts.
PCA as better regression
Ordinary regression y = βx + α assumes x is exact and y is noisy. Flip x and y and you get a different β; β’ ≠ 1/β in practice.
Financial variables often share a latent factor z driving both, plus noise on each leg. When signal dominates, the first PCA eigenvector gives a symmetric hedge ratio. Inverse relationship holds when you swap variables. Closed form exists for 2×2 case via eigenvalues of the covariance matrix.
This is why Chapter 36 sends you back here when regression hedges look unstable.
Measuring curve complexity
Eigenvalues measure variance along each factor direction. Their relative sizes tell you how many dimensions matter.
Herfindahl index on eigenvalues: H = Σeᵢ² / (Σeᵢ)². High H → one factor dominates (simple dynamics, maybe one hedge instrument enough). Low H → complex moves, need more hedges.
Rolling 90-day PCA on US and German splines: 2020 COVID volatility dropped Germany’s H (less single-factor dominance). Same machinery on multiple Japanese issuer curves at 5Y and 10Y flags “interesting” spread periods needing richer hedge structures.
Trade-off: long windows slow to detect regime shifts; short windows make hedge ratios noisy.
Sign conventions and factor labels
Eigenvectors are defined up to sign flip. Düring adjusts signs so PCA factors align with familiar strategies (factor 1 = parallel, factor 2 = slope, factor 3 = curvature). When you import PCA from a stats package into a trading spreadsheet, check signs before wiring hedge ratios. A flipped factor 2 loads your steepener hedge backwards.
Relation to Chapter 31 and 36
Chapter 31 used PCA on spline yields to justify why outright moves dominate daily variance and why butterfly residuals are smaller. Chapter 36 turns the same factors into butterfly hedge systems. Chapter 33 is the shared engine.
If you only run PCA once a year, you get pretty charts. If you run rolling PCA (90-day windows in the complexity section), you get a risk dashboard: when H drops, dust off multi-factor hedges and widen scenario grids.
Chapter 33 is math with a purpose. It is not academic PCA for its own sake. It is the tool behind curve factor trades, hedge ratios, and the complexity gauge Düring uses to say “this market just got harder to neutralize.”