How does principal component analysis (PCA) decompose yield curve movements into factors?

Principal Component Analysis (PCA) is a statistical technique that decomposes observed yield curve changes into a small number of uncorrelated factors. For government yield curves worldwide, three factors consistently explain over 95% of all variation.

The Three Factors:

Factor 1: Level (Parallel Shift)

• Explains ~85-90% of yield curve movements
• All maturities move in the same direction by roughly equal amounts
• When the 'level factor' increases by 1 unit, all yields rise
• Driven by: monetary policy, inflation expectations, global risk appetite

Factor 2: Slope (Twist/Steepening)

• Explains ~8-10% of movements
• Short rates move one way while long rates move the other (or stay flat)
• A positive slope factor means the curve steepens (long rates rise relative to short)
• Driven by: business cycle expectations, term premium changes

Factor 3: Curvature (Butterfly)

• Explains ~2-3% of movements
• Medium-term rates move opposite to short and long rates
• Creates a 'butterfly' pattern — wings go up, belly goes down (or vice versa)
• Driven by: supply-demand imbalances at specific maturities, convexity hedging

Example — Greenfield Bond Fund (fictional):

On a given day, yield changes across maturities:

The level factor (+10 bps everywhere) dominates. The slope factor captures the steepening, and the curvature factor captures the belly underperformance.

Portfolio Implications:

• Duration hedges against Factor 1 (level)
• Key rate durations or barbell/bullet positioning hedges against Factors 2 and 3
• Most interest rate risk management focuses on Factor 1, but Factors 2 and 3 can be significant for relative value strategies

Exam Tip: Know that three factors explain 95%+ of yield curve movements, with level dominating. The CFA exam may ask which factor explains the most variance (always level) or how a specific curve movement maps to factors.

Explore yield curve strategies in our CFA Level II practice bank.