A whole yield curve in three numbers
US Treasury yields come in a dozen maturities from one month to thirty years, and they all move every day. Yet they do not move independently: a shock to the policy rate ripples smoothly along the curve. That makes the daily yield changes an ideal case for PCA. The principal components of the covariance matrix of yield changes have a stable, readable shape that traders have named for decades.
The first component has loadings of the same sign at every maturity: when it is positive the whole curve shifts up, which traders call the level (the loadings are larger in the middle of the curve than at the very short end, because short rates follow the policy rate more tightly). The second changes sign once, short end one way and long end the other, so it tilts the curve: the slope. The third changes sign twice, the two ends moving against the middle, so it bends the curve: the curvature. Because the curve is smooth, three scores rebuild almost every daily move, and a position can be hedged against these three risks instead of against a dozen separate yields. Run PCA on changes in yields, not on levels: levels are persistent and nearly non-stationary, and their first component would just trace the slow drift of the average rate.
- Level: all loadings share one sign; the whole curve moves up or down together.
- Slope: the loadings change sign once; the short and long ends move in opposite directions.
- Curvature: the loadings change sign twice; the ends move against the middle.
Lab · what's graded
- Name the components15 pts
- Explained variance of three components15 pts
- Rebuild a yield move15 pts
- How many components10 pts
- Levels or changes15 pts
- What a one-factor hedge leaves15 pts
- Why so few factors15 pts