# 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 '17 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 '17 at 3:04
• The final application is to determine the cheapness/richness of particular issues in a cross-sectional view. Jan 29 '17 at 3:14

## 2 Answers

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.
4. Ad-hoc filters: Some traders remove cheapest-to-delivers; others remove issues trading rich in the repo market.

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.

Some additional comments that might be helpful:

1. Most textbooks ask you to interpolate between bond yields, and then bootstrap from that interpolated curve. That's great for pedagogical purposes, but no one does that in real life. Unless all the bonds are par bonds (coupon rate = yield, price = 100), you cannot just connect their yields.
2. Regarding the dip you mentioned around the 15-year sector, there are two things of note: 1) are you plotting par yields? if you are merely plotting quoted bond yields, then remember that the 15-year bond is a high coupon issue, which would have lower yield (than a lower coupon bond, since the yield curve is upward sloping); 2) even the par yield curve have a dip there, because the 15-year part of the curve has indeed been trading very very rich.
• Thanks, can you elaborate what do you mean by discount curve? I assumed discount curve is just another terminology for spot curve/zero coupon curve, etc. Jan 29 '17 at 9:34
• Also, if not bootstrapping spot curve from interpolated par curve, what other curve can I use? The gap in 15-20 years make it impossible to bootstrap without interpolation. Jan 29 '17 at 9:41
• @Helin for treasuries removing on-the-run might be feasible, but for other currencies these will provide the staple structure. I think my opinion would be to keep as many bonds as possible, with each weight as close to unitary as I thought sensible and then analyse the evolution or time series of the residual to the curve of selected bonds. Any opinion?
– Attack68
Apr 2 '18 at 20:18
• @Attack68 Yep, I wouldn't generalize this. Lots of art involved and the best technique depends on the country & evolves over time. For example, in the US, there's not a ton of value to exclude on-the-runs in recent years... For countries like Germany and UK, we really have no choice but to include all issues. For countries like Japan, it's actually customary to OVERweigh on-the-run issues... Apr 3 '18 at 22:58

A few observations: the coupon yield curve is never going to be smooth, because a high coupon Treasury and a low coupon Treasury with the same maturity do not yield the same. That's because in an upward sloping yield curve, then one with the lower coupon has effectively a longer duration and therefore a higher yield. Secondly, recently issued Treasuries often trade "special" in the repo market, meaning you can borrow cheaply to invest in them, so they will be relatively expensive and therefore have a lower yield.

To do this properly, you could graph the zero coupon yields instead of the par yields. They will be somewhat smoother. Alternatively, you could look at the swap curve, which is typically a lot smoother than the Treasury curve.