How the Treasury curves are fitted

01What is fitted

Five curves, from two functional forms. Four of them are Nelson-Siegel-Svensson: six parameters, four level terms and two decay terms, fitted to observed prices rather than to quoted yields.1 The fifth reduces the same day to three factors with a fixed decay.

• Zero curve, zero-coupon discount rates at ten tenors from one to thirty years, with the par yield and instantaneous forward implied by the same parameters. This is the one to discount a cashflow with.
• Par curve, the coupon a bond issued today would need to price at 100. Comparable with what Treasury and the Federal Reserve publish, and the wrong choice for anything else, because it carries a coupon effect.
• LSC curve, level, slope and curvature as three daily series. The one to regress on: the Svensson coefficients change sign every third day while describing a curve that moved five basis points, so a regression on those measures noise.
• Money market curve, one week to one year, fitted from bills alone on a bond-equivalent basis.
• Real curve, seven tenors from two to thirty years, fitted from inflation-linked securities instead of nominal ones, so its rates are yields above inflation rather than in cash terms. Same functional form; a different universe and a different meaning.

A fifth curve, on USD swaps, is fitted and held internally. It is not published, and not because it is unfinished: the underlying dissemination data is free to obtain and not ours to redistribute.2

02The inputs, and the one that surprises people

Prices come from Treasury's own end-of-day file, taken on the bid side. Yields are then computed from real cashflow schedules, actual coupon dates, actual accrual, rather than from an approximation.

There is no single Treasury day count. There are several, and which applies depends on the segment and on which quantity was asked for.

A day whose prices Treasury has not yet posted is skipped rather than stored. The file is served before end-of-day pricing with every price at zero, and a curve fitted to that would be nonsense that looked like data.

03Which securities enter the fit

The nominal fit uses coupon-bearing notes and bonds. Inflation-linked securities are excluded because they price off a real curve; floating-rate notes because their coupon resets; callable issues because their cashflows are not fixed. Bills are fitted separately, into the money market curve, because money-market effects dominate the very front end.

Excluding a security from the nominal fit is not the same as having nothing to say about it. Inflation-linked securities get a fit of their own. That is the real curve, fitted to the same functional form against their real cashflows with the index ratio applied, which is why its rates are yields above inflation and not comparable with the nominal ones tenor for tenor. A floating-rate note has a discount margin and a spread duration instead. Those are different measures, not missing ones.

That does not mean every excluded type can have a curve of its own. Inflation-linked securities can support one, because 53 of them trade out to thirty years. Floating-rate notes cannot: Treasury issues them only as two-year paper, so the entire market is eight securities inside a two-year span, and has been for a decade.5 Eight points across 1.8 years is a cross-section rather than a term structure, and they all float off the same index, so fitting a curve through them would be fitting the wrong axis. A floater engineers maturity risk away with its reset; what is left is spread, which belongs to each security as a time series rather than to a curve. Whatever term structure there is belongs to the 13-week bill, and that is the money market curve.

04How closely it fits

Every curve page shows its own fit error rather than burying it, because the number moves by a factor of six across the history and a reader quoting a single day should know which kind of day it was.

The worst nominal days in the whole history are in December 2008, at 19 to 20 bp. On those days the market was genuinely dislocated rather than the fit being wrong, which makes them the most interesting days in the dataset and the ones most likely to be quoted out of context. Sixteen days in eighteen years failed to converge at all on the par fit.3

The LSC curve fits least tightly of the nominal forms, at 9.61 bp against the Svensson forms' 3.8, and that is the trade rather than a defect: three factors cannot bend as many ways as six. What they buy is a coefficient you can model.

The real curve is the loosest fit on the page, and structurally so. It is fitted from 24 to 46 inflation-linked securities on a given day against roughly 350 nominal coupon securities, so the same functional form has an order of magnitude less to hold it down. Its mean is 6.21 bp against the par fit's 3.83, its worst day is 48.28 bp on December 2, 2008, and 103 of 4,507 days did not converge against the par fit's sixteen. None of that is hidden behind an average: the fit error and the convergence flag for the day being shown appear on the curve itself.

05Against the Federal Reserve

The Federal Reserve fits its own curve to the same market and publishes the parameters. It is the best available external check, and the easiest one to misuse.

Read that as: typical agreement is a basis point or two on any basis, and the long end has a bad tail. The forwards are the exception, and not a small one.

The real curve has its own external check, against the Federal Reserve's TIPS curve rather than its nominal one, a separate publication, compared zero against zero. Over the same 4,498 overlapping days it agrees to 2.81 bp RMSE with a mean error of 0.00. That is tighter than the real curve's own fit error, which is worth noticing rather than boasting about: the two fits disagree less with each other than either disagrees with the prices it was fitted to, because both are smoothing the same sparse universe in much the same way.

The two-year is the worst of them at 5.27 bp, for the same reason the whole curve fits loosely: there are rarely many linkers with two years left to run, so the front of a real curve is the least constrained part of the least constrained curve. It is also why nothing is published inside two years at all.

06What we will not claim

• The forwards are not usable beyond twenty years, at all. A 61 bp RMSE at thirty years against the Fed's own forwards is not a tail, it is a systematic failure. Forwards are the derivative of the curve, so whatever is loose in the fit shows up there first and largest. The refit improved this from 92 bp and it is still the weakest number on the page, which is why it is stated rather than omitted.
• A single long-end measurement should not be trusted. Three runs of the same code on different day samples gave 26.6, 56.97 and 63.47 bp for the same quantity. The error is episodic and a handful of days dominate it, so any comparison of two long-end numbers has to hold the day sample fixed.
• Nobody outside this firm has reviewed any of it. No independent review, no external audit, and no oversight function separate from the people who do the calculation. The whole thing was built so that an outsider could reproduce it from public data; that nobody has yet is the largest item on our own list.

07Tried, measured, rejected

Several obvious-looking improvements are recorded internally as measured failures, and they are worth stating so that nobody spends a week rediscovering them.

• A Kalman filter on the fitted curve. Rejected on measurement rather than taste: across 148 consecutive days the thirty-year forward moves 1.08× the Federal Reserve's own, so there is no excess to smooth in the body of the distribution. The excess is tail-only, and a constant-noise smoother would degrade the 92% of days that already track.
• Excluding twenty-year bonds, as some published methodologies do. Tested and rejected, see section 3.
• Fixing the decay parameters. Tested, not adopted.
• An arbitrage-free Nelson-Siegel formulation. Accepted in structure, rejected as specified: its level term is Ho-Lee, which is the whole long-end problem rather than a fix for it.