Why do the first three PCA components of yield curves represent level, slope, and curvature?

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AcadiFi · Accepted Answer

Across developed-market yield data (US, UK, Germany, Japan), PCA of yield changes consistently produces three dominant eigenvectors with recognizable shapes. This empirical regularity has economic meaning.

Factor 1 — Level (≈70-80% of variance):
Loadings approximately equal and positive across all maturities (~0.30-0.40 each for a 10-tenor curve). A shock to this factor shifts the whole curve up or down in parallel. Drivers: Fed policy, inflation expectations, global rates.

Factor 2 — Slope (≈12-18% of variance):
Loadings run from negative at short maturities to positive at long maturities (e.g., −0.45 at 2Y, 0 at 7Y, +0.40 at 30Y). Shock steepens or flattens. Drivers: growth expectations, term premium, QE/QT.

Factor 3 — Curvature (≈3-5% of variance):
Loadings form a hump: negative at short and long ends, positive in the middle (or vice versa). Shock produces a butterfly move. Drivers: liquidity preferences at specific maturities, pension demand at long end.

Economic interpretation: policy operates via short rates (affects slope), inflation expectations shift levels, supply/demand dislocations at specific tenors create curvature.

Practical use:

1. Portfolio risk: compute DV01 exposure along each factor. A bond portfolio might be neutral to level but heavily exposed to slope.
2. Trade construction: a steepener trades slope, a butterfly trades curvature. PCA gives you factor-neutral hedge ratios.
3. Term-premium decomposition: separate expected-rates component (level+slope) from term premium (curvature residual).

The stability of three factors across markets justifies parsimonious term structure models like Nelson-Siegel, which parameterizes yields with exactly three shape parameters.

Why do the first three PCA components of yield curves represent level, slope, and curvature?

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