# Best method for interpolating yield curve? [Multiple questions]

I'm building a spot curve for US Treasuries. My original selection of cash treasury include all the on-the-run bills, notes, bonds from 6 months to 30 years, as well as some selected off-the-run instruments to fill in-between the on-the-runs.

Because US treasury stopped issuing 30 year cash for around 5 years in the early 2000s, there is a gap in the yield curve roughly between 15 year and 20 year maturities. I did the best I can by providing 14 year and 20 year off-the-run in the original selection. My 14 selection is (in years): 0.5, 1, 1.5, 2, 3, 5, 7, 10, 12, 14, 20, 23, 27, 30.

My question is, if I interpolate this curve, the method of interpolation will have a non-trivial effect on the shape of the curve due to the gap. Therefore when I bootstrap my spot curve based off of coupon yield, the interpolation technique on the long end of the coupon curve builds into the long end spot rate.

So far I've tried Linear Interpolation and Piecewise Cubic Hermite Interpolating Polynomial. I think the Fed Reserve publish their daily yield curve off of the second kind.

Lastly, can someone critique whatever I've said if they have any better idea? My spot curve using PCHIP method on coupon yield curve is drawn below. The spot curve was also interpolated into 1-month interval using PCHIP. I just feel odd looking at this because of the sudden curvature change around 12 year to 22 year sector. If someone thinks this is a good approximation for how the yield curve should look like fundamentally, I would love to hear your explanation. I sincerely apologize for so many questions.

Thanks.

• You haven't said what you are using this for. What is the cost of being wrong? What do you lose from inaccuracy? Jan 28, 2017 at 17:01
• I'm using it to price treasury securities, and then analyzing the residual between market price and the discounted price derived from the spot curve. Jan 29, 2017 at 3:04
• The final application is to determine the cheapness/richness of particular issues in a cross-sectional view. Jan 29, 2017 at 3:14

Typically, the yield curve used for performing relative value analysis should be built from off-the-run bonds.

Different vendors select different bonds, but starting with all outstanding Treasury issues, you'd usually remove the following:

1. Treasury bills: Because of market segmentation concerns, bills are usually excluded, while short-term coupon bonds are preferred.
2. On-the-run bonds: Most vendors remove all on-the-run issues (i.e., the most recently issued 2-, 3-, 5-, 7-, 10-, and 30-year issues) – these issues may command liquidity & financing advantages, so they trade richer than other issues. Some also remove first-off-the-runs and even double old issues for the same reason.
3. Very seasoned issues: For example, original 30-year bonds with only 5 years to maturity would trade differently from recently issued 5-year bonds. Needless to say, the recently issued 5-year notes are likely to provide more meaningful information, so the old 30-year bonds should be removed.

The goal of the process is to retain issues that are likely to reflect the fair value of the Treasury market. Regardless of the filtering procedure you adopt, you're likely to be left with 150-200 notes/bonds that span the entire maturity spectrum.

A spline is then fit through all of these issues. You have the option of modeling the discount curve (by far the most common approach), zero coupon curve, or forward curve. Once you have a functional form for the yield curve, you choose parameters for your model so that it can fit all your input bonds reasonably well -- the fit won't be exact of course; otherwise, you'd have no relative value opportunity to speak of...

A good piece on bond curve construction is published by the Fed: The U.S. Treasury Yield Curve: 1961 to the Present. Almost all the major banks also publish their curve methodologies.