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Why Zero-Coupon Treasury Yields May Differ from the Treasury Curve

Article Quant Q&A · Author: George Wolfe
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Summary

The document explains why a zero-coupon Treasury’s yield to maturity does not generally equal the Treasury yield curve value at the same maturity. The key distinction is how the curve is constructed: it is fitted to yields from selected on-the-run Treasury securities, which typically trade near par, and those securities serve as curve anchor points. A spline then interpolates between them, so the curve’s values need not match yields for other securities or maturities.

The cited methodology describes a quasi-cubic Hermite spline fitted to closing bid yields, using recently auctioned bills, notes, and bonds as inputs. The curve is treated as a par curve. This is a conceptual explanation rather than an empirical comparison of zero-coupon securities. Its practical implication is that a curve value at a given maturity should not automatically be interpreted as the yield of a zero-coupon Treasury with that maturity.

Key ideas

  • The Treasury curve is fitted from yields on selected on-the-run securities.
  • Those securities act as spline anchor points, while values between them are interpolated.
  • The resulting curve is a par curve, so it need not equal a zero-coupon security’s yield to maturity.
  • The described methodology uses closing bid yields for recently auctioned Treasury instruments.

Tags

Fixed incomeUS marketsStatistics
Full text
# Does the YTM for a zero coupon treasury equal the treasury yield curve value for it's maturity?


# Does the YTM for a zero coupon treasury equal the treasury yield curve value for it's maturity?












The title is my question. I think the answer is yes, but I am unsure about it.

## Answer by msitt (score 1, accepted)

https://quant.stackexchange.com/a/33539

No, generally this will not be the case. The yields of the on-the-run treasuries for specific tenors are collected and a cubic spline is fit to these points. Thus, the yields will only match for the on-the-run 1m, 3m, 6m, 12m, 2y, 3y, 5y, 7y, 10y, and 30y treasuries.

For details please see the page on Treasury Yield Curve Methodology.

> The Treasury's yield curve is derived using a quasi-cubic hermite spline function. Our inputs are the Close of Business (COB) bid yields for the on-the-run securities. Because the on-the-run securities typically trade close to par, those securities are designated as the knot points in the quasi-cubic hermite spline algorithm and the resulting yield curve is considered a par curve. ... More specifically, the current inputs are the most recently auctioned 4-, 13-, 26-, and 52-week bills, plus the most recently auctioned 2-, 3-, 5-, 7-, and 10-year notes and the most recently auctioned 30-year bond, plus the composite rate in the 20-year maturity range. The quotes for these securities are obtained at or near the 3:30 PM close each trading day. The inputs for the four bills are their bond equivalent yields.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.