PCA Hedging of a Rates Book

A rates desk holding bonds and swaps across ten different maturities — 2-year, 5-year, 10-year, 30-year, and points in between — faces a hedging problem that looks, at first, like it needs ten separate hedges, one per point on the curve. It doesn't, in practice, because the yield curve's ten points don't move independently of each other. Historically, principal component analysis applied to yield curve changes shows that almost all of the day-to-day variation collapses into just three underlying shapes — which means a book can be hedged against nearly all of its rate risk with as few as three instruments, not ten.

The three shapes

PC1, level, is a roughly parallel shift — every maturity on the curve moves up or down together by about the same amount. This single factor typically explains 80–90% of total yield curve variance on its own; it's what people mean when they casually say "rates went up."

PC2, slope, is a steepening or flattening move — short rates and long rates move in opposite directions, or by different amounts in the same direction, changing the gap between them (commonly measured as, say, the 10-year yield minus the 2-year yield) without necessarily moving the level much. This usually explains another 5–15% of variance.

PC3, curvature, is a bowing move — the belly of the curve (intermediate maturities, like 5 or 7 years) moves differently from both the short and long ends simultaneously, changing how "curved" the curve looks rather than its level or slope. This typically explains only a few percent of variance, but it is not noise — it's a real, recurring, and separately hedgeable pattern.

Together, level, slope and curvature routinely explain upward of 95–99% of the total variance in historical yield curve changes across major government bond markets — which is why hedging against just these three, rather than the full set of maturities, captures nearly all the risk a book actually has.

PC2: slope short up, long down

PC3: curvature belly moves, ends don't

together these three shapes explain most of a curve's total variance

A worked example

A book holds a position with the following DV01 (dollar value of a 1 basis-point move) exposures by maturity: 2-year: +5,000, 5-year: −8,000, 10-year: +12,000, 30-year: −3,000, where positive means the position gains if that point's yield falls. A PCA on the historical covariance of yield changes across these maturities gives loadings (how much each maturity typically moves per unit move of each factor) roughly: level moves each maturity by about 1.0 unit uniformly; slope moves the 2-year by +0.6, the 5-year by +0.2, the 10-year by −0.3, and the 30-year by −0.7 (opposite signs at the ends); curvature moves the 2-year by −0.3, the 5-year by +0.7, the 10-year by +0.1 and the 30-year by −0.4.

The book's exposure to PC1 (level) is just the sum of DV01s weighted by the level loading (about 1.0 each): $5{,}000 - 8{,}000 + 12{,}000 - 3{,}000 = 6{,}000$ — the book loses roughly $6,000 for every 1bp uniform rise across the whole curve, before even looking at slope or curvature. Its exposure to PC2 (slope) is $5{,}000(0.6) - 8{,}000(0.2) + 12{,}000(-0.3) - 3{,}000(-0.7) = 3{,}000 - 1{,}600 - 3{,}600 + 2{,}100 = -100$ — nearly flat to a steepening or flattening move, almost by coincidence of the position mix. To hedge the level exposure alone, a desk can trade a single instrument — say a 10-year note future — sized so its own level-DV01 offsets the book's 6,000, rather than adjusting all four legs of the original position individually.

• Residual risk after a 3-factor hedge is not zero. Higher-order components (PC4 and beyond) exist and occasionally matter, especially around unusual events like a specific-maturity supply shock or a central bank operation targeting one part of the curve.
• PCA hedges reduce trade count, not necessarily transaction cost per trade. The 3 hedging instruments still need to be liquid enough to trade in size — an illiquid instrument with the "right" loading is not automatically the right hedge in practice.
• Level, slope and curvature loadings differ by market. US Treasury PCA loadings are not directly transferable to, say, the Gilt or JGB curve — always re-estimate on the specific market being hedged.