Rich-Cheap Analysis From Fitted Curve Residuals

Lay a smooth curve through a scatter of Treasury yields plotted against maturity, and almost no bond sits exactly on it. Some yield a few basis points more than the curve says they should, some a few basis points less. Those gaps are not noise to be smoothed away — they are the entire trade. A relative-value desk's job is to find bonds sitting off the curve, buy the ones yielding more than they "should" (cheap) and sell the ones yielding less (rich), duration-matched so the position does not care which way rates move overall.

Fitting the curve

You cannot connect every bond's exact yield with a straight line and call it a curve — that would just be the bonds themselves. Instead you fit a small number of smooth curves through the whole set (commonly a cubic spline or a parametric form like Nelson-Siegel), letting a handful of parameters explain the general shape of level, slope and curvature while treating each bond's individual deviation from that shape as noise to be measured, not fitted. The fitting is usually done on on-the-run and liquid off-the-run issues together, minimizing the sum of squared yield errors, which is exactly what leaves interesting residuals on the less liquid names.

Worked example: spotting a cheap five-year note

A fitted curve puts the fair five-year yield at 4.10%. Three actual five-year-area notes trade at:

Bond	Actual yield	Fitted yield	Residual
Note A (5.0y)	4.16%	4.10%	+6 bp
Note B (5.0y)	4.04%	4.10%	−6 bp
Note C (4.9y)	4.11%	4.09%	+2 bp

Note A is cheap by 6 basis points and Note B is rich by 6 basis points relative to the fitted curve, even though both are labeled "five-year" — the difference usually comes down to liquidity (a recently issued, heavily traded note trades rich because everyone wants it for hedging and repo) or a slightly unusual coupon that skews its cash-flow weighting.

The trade: buy $100 million face of Note A, sell an equal-DV01 amount of Note B. Note A's DV01 is $47,800 per $100 million face and Note B's is $47,500, so the offsetting face amount is $100 \times (47{,}800/47{,}500) \approx 100.6$, i.e. about $100.6 million face of Note B. A parallel move in five-year rates moves both legs by roughly the same dollar amount and cancels out; the position only makes money if the 6-basis-point gap closes — Note A's yield falls relative to Note B's, or equivalently their prices converge.

Why residuals exist at all

If markets priced every bond off a perfectly smooth curve, there would be nothing here to trade. Residuals persist because bonds are not perfectly fungible: a specific bond can be in demand for delivery into a futures contract (making it trade special in repo and rich in cash), a recently auctioned on-the-run note carries a liquidity premium the curve does not know about, and small differences in coupon change how much of a bond's value sits in near versus far cash flows. None of that is "mispricing" in the sense of a free lunch — it is the cost of finding a genuinely matched, high-conviction pair before the position earns anything.

Where it shows up

Curve-fitting residuals are the daily bread of government bond relative-value desks and are also how dealers mark off-the-run bonds they don't actively quote — pricing them off the fitted curve plus a liquidity adjustment. The same technique, applied to swap curves or corporate credit curves, is how traders flag individual bonds or swaps that have drifted away from their peers and need to converge or be arbitraged back in.

Key terms

• Residual — actual yield minus fitted curve yield at that maturity; positive means cheap, negative means rich.
• Curve fit — a spline or parametric model (e.g. Nelson-Siegel) summarizing the whole curve in a handful of parameters.
• On-the-run / off-the-run — the most recently issued benchmark bond versus older issues of similar maturity, which typically trade rich and cheap to each other respectively.
• DV01-matched pair trade — long the cheap bond, short the rich bond, sized so parallel rate moves cancel.