How is PCA used to decompose yield curve risk into principal components?

Principal Component Analysis applied to the yield curve is one of the most elegant applications of linear algebra in risk management. Let's connect the math to the practical intuition.

The Problem:

A yield curve has many points (3-month, 6-month, 1-year, 2-year, 5-year, 10-year, 30-year). If you want to measure your portfolio's interest rate risk, tracking every single point creates a high-dimensional problem. PCA reduces this to a few meaningful factors.

How PCA Works on the Yield Curve:

1. Collect historical daily yield changes across all maturities
2. Compute the covariance matrix of these changes
3. Find eigenvalues and eigenvectors
4. The eigenvectors (principal components) are the independent factors driving yield curve movements

The Three Key Components:

Example — Atlas Fixed Income Group:

Atlas runs PCA on 2 years of daily US Treasury yield changes and finds:

PC1 eigenvector: [0.35, 0.36, 0.37, 0.38, 0.39, 0.38, 0.37]

(All positive, roughly equal — this IS a parallel shift)

PC2 eigenvector: [0.55, 0.40, 0.20, 0.00, -0.25, -0.42, -0.52]

(Positive for short maturities, negative for long — this IS a slope change)

PC3 eigenvector: [-0.35, -0.15, 0.30, 0.55, 0.30, -0.15, -0.35]

(Negative at ends, positive in middle — this IS curvature)

Risk Management Application:

Instead of tracking DV01 at 7+ maturity points, Atlas can express its risk as:

• PC1 exposure: $1.2M per 1 standard deviation level shift
• PC2 exposure: $0.3M per 1 std dev slope change
• PC3 exposure: -$0.05M per 1 std dev curvature change

This tells them they're heavily exposed to parallel shifts but have a small natural hedge against curvature moves.

Key Exam Insights:

• PC1 alone explains ~85-90% of yield curve variance — this is why duration (a parallel-shift measure) is such a powerful risk metric
• The remaining 10-15% is non-parallel risk, which key rate durations or PCA-based measures capture
• PCA components are orthogonal (uncorrelated) by construction, so risks don't double-count

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