PCA of the Yield Curve: Understanding Shift, Twist, and Butterfly

If you track daily rate changes across 20 tenors, you'll notice they're highly correlated — when the 2-year rate moves, the 5-year and 10-year usually move too. This suggests there are fewer independent drivers than there are tenors. Principal Component Analysis (PCA) extracts those drivers, and what it finds is remarkably consistent: three factors — shift, twist, and butterfly — explain over 95% of all daily yield curve dynamics.

The Observation

Consider a dataset of daily changes in the USD OIS curve over 5 years, at tenors from 3 months to 30 years. On any given day, you observe changes at ~20 tenors. That's 20 time series — but they are far from independent.

When the 5Y rate rises by 3bp, the 3Y rate typically rises by 2–4bp and the 10Y by 2–3bp. Short-end rates are more volatile but highly correlated with each other. Long-end rates move less but in the same direction.

This correlation structure implies dimensional reduction — the 20 time series can be explained by a much smaller number of underlying factors. PCA is the statistical tool that finds those factors.

What PCA Does

PCA decomposes the covariance matrix of daily rate changes into eigenvalues and eigenvectors:

Each eigenvector is a curve shape — a vector of loadings, one per tenor, that describes how much each tenor participates in that factor. Each eigenvalue tells you how much total variance that factor explains.

The Three Factors

Across currencies, time periods, and curve types, PCA consistently reveals three dominant factors:

These three shapes are not arbitrarily named — they correspond to the fundamental economic forces acting on the yield curve:

• Shift reflects changes in the overall level of rates — driven by inflation expectations, central bank policy, and global risk appetite.
• Twist reflects changes in the term premium — the compensation investors demand for holding longer-duration bonds. When the twist factor moves, the spread between short and long rates changes.
• Butterfly reflects supply-demand imbalances at specific maturities — often driven by central bank purchases (QE), pension fund demand at the long end, or money market flows at the short end.

Variance Explained

The eigenvalues tell us how much of the total daily curve variance each factor captures:

The dominance of the shift factor explains why parallel-shock scenarios capture the majority of IRRBB risk — but the 5–15% captured by twist and butterfly can translate to hundreds of millions of dollars of ΔEVE for a large bank. Ignoring these factors means ignoring real risk.

The Volatility Term Structure

Before running PCA, it's instructive to look at the annualized volatility of daily changes by tenor:

Short-end rates are more volatile because they are directly influenced by central bank policy, which changes in discrete, large steps (25–75bp per meeting). Long-end rates are anchored by long-term inflation expectations and term premiums, which evolve more gradually.

This declining volatility profile is consistent with the PCA shift eigenvector — the loadings are larger at the short end, reflecting the fact that short rates contribute more to total curve variance.

The Correlation Structure

The correlation matrix of daily rate changes reveals a characteristic pattern:

• Adjacent tenors are highly correlated (3M vs 1Y: ρ ≈ 0.90–0.95; 5Y vs 10Y: ρ ≈ 0.92–0.97).
• Distant tenors are less correlated (3M vs 30Y: ρ ≈ 0.50–0.70).
• The correlation decays with distance along the curve, but never reaches zero — there is always a common factor (the shift) linking all tenors.

This declining correlation structure is precisely what PCA captures. The shift factor explains the high common correlation. The twist factor explains why short-end and long-end correlations are lower — sometimes the curve steepens (short down, long up), reducing the observed correlation between distant tenors.

Practical Applications

PCA is not just an academic exercise — it has direct applications in IRRBB management:

1. Scenario generation. PCA eigenvectors define the "natural" shapes of curve movement. Scaling them by their eigenvalues (standard deviations) produces statistically calibrated scenarios: a 3σ shift, a 2σ twist, a combined shift-and-steepener. These scenarios are more realistic than arbitrary parallel shocks. (See Stress Testing the Yield Curve.)

2. Risk decomposition. Express total ΔEVE as the sum of contributions from each PC factor: ΔEVE_total = ΔEVE_shift + ΔEVE_twist + ΔEVE_butterfly + residual. This tells the risk manager which type of curve movement the bank is most exposed to — and whether the existing hedges address it.

3. Model justification. The fact that three factors capture 95%+ of curve variance provides empirical justification for three-factor term structure models (like G2++) in pricing and risk. If three factors suffice empirically, building a 20-factor model adds complexity without meaningful accuracy.

4. Hedge efficiency. If 90% of the risk comes from the shift factor, a duration-matched swap hedge addresses 90% of the risk. The remaining twist and butterfly exposures may or may not be worth hedging — the cost of additional instruments versus the reduction in residual risk is a quantitative trade-off that PCA makes explicit.

5. Partial-information scenarios. PCA enables the algebraic completion of partially specified scenarios — specify shocks at a few tenors and infer the rest using the eigenvector structure. This bridges expert judgment and statistical consistency.

The Bigger Picture

PCA of the yield curve is one of the most elegant results in financial risk management. The finding that three simple shapes — shift, twist, butterfly — explain essentially all of daily curve dynamics is both empirically robust (it holds across currencies, decades, and market regimes) and economically interpretable (each factor maps to identifiable macroeconomic drivers).

The three-factor structure also highlights the limitation of parallel-only scenarios: they test the dominant factor (shift) but ignore the 5–15% of variance that lives in twist and butterfly. For a bank with $50B in assets and a complex maturity profile, that 5–15% can easily translate to $100M–$500M of ΔEVE — enough to matter for capital planning, hedging, and outlier test compliance.