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I have:

  • 3M,6M,1Y,2Y,3Y....bonds
  • 1W,2W,1M,2M,3M.... bills

To build the yield curve what is better:

  1. build a single curve using bonds+bills
  2. build 2 separate yield curves, 1 to price bonds made out of bonds only, and to price bills, bill only curve
  3. something else?

EDIT just to clarify, I am concerned about whether I should use bill rates in building the bond curve? Will it improve or distort my result?

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  • $\begingroup$ Hi and welcome. As usual: It depends on what type of cash flows (underlying instrument, time to maturity) you want to price off that curve. If the underlying instruments/flows are somewhat alike (e.g. same issuer, same seniority), you may build a curve using both types of instruments. $\endgroup$
    – Kermittfrog
    Dec 9, 2022 at 9:53
  • $\begingroup$ @Kermittfrog yes they are the same gov issued. $\endgroup$
    – darkuss
    Dec 11, 2022 at 10:20

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before you get to curve fitting, you need to decide whether these instruments are on the same curve or not. And that's not a quantitative finance question, but a product question.

For example, in the U.S., they generally are. U.S. treasury sells zero-coupon T-bills with maturities up to 1 year. And there are lots of coupon-paying notes and bonds that U.S. treasury issued years ago and that have less than 1 year left to maturity. Many market participants watch their yields like hawks and, whenever an instruments trades at an anomalous yield different from this single curve, will pounce and make free mouney from arbitrage until the anomaly goes away. Tangentially to your question, beyond the maturity of U.S. T-bills, for technical reasons, there is more demand for on-the-run coupon-paying notes and bonds, so some people split off the on-the-run and off-the-run curves.

But other markets differ, for example, in their taxation, so betting that the yields of their zero-coupon instruments would converge with coupon-paying instruments would not necessarily pay out. I can think of Mexico (cetes and mbonos) and Brazil as good examples.

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As a student, this is how I would have done it.

  1. Using bootstrapping methodology, convert Coupon Bond into zero coupon bond so you have zero coupon bond for all maturities.

  2. you will find that you have some "missing Data". e.g. you don't have the zero coupon bond for certain Coupon date. You can address this issue by choosing an appropriate interpolation methodology [1]

  3. Convert your Zero coupon bond (discount rate) into spot rate then plot your curve.

Basic idea / principle used:

No arbitrage e.g. a bond with coupon C that matures in 3 years (even if it was issued 5 years ago), must have the same price as a 3 years bond (with same coupon) issued today that has the same maturity.[2]

Please note: Zero coupon Bond and Discount factor are the same thing.

Another approach you might want to consider are Curve fitting, using for example the Nelson Siegel Model (or the extended version). I.e. we assume that the discount factor is a function of maturity T (i.e. Z(0,T)).

I hope this help, but I want to emphasis that I don't know what are best practices, I just shared my view on how I would do it if I was still a student.

Best, Jules

[1] Great paper from Patrick S. Hagan and Graeme West. You have a summary on page 10 which compare all methodology. I personally like Raw (linear on log of discount) as it is simple to implement. Draw back being: Not continuous. http://web.math.ku.dk/~rolf/HaganWest.pdf

[2]Fixed Income securities - Pietro Veronesi || p62

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  • $\begingroup$ thank you for the answer, maybe my question wasn't clear enough, I am more concerned about whether I should use bill rates in building the bond curve, so should I keep those 2 curves separated or unite them as a single curve? Will bill rates distort my bond yield curve or improve it? $\endgroup$
    – darkuss
    Dec 8, 2022 at 22:06
  • $\begingroup$ I would include bill rates in building the bond curve (unite them as a single curve). Rationale: Bill rates are more representative for short maturity. Your curve might not be as "smooth", but it will be more accurate. ps: your question was clear, it is just that English is not my first language so sometime I misinterpret the question, sorry! $\endgroup$
    – LvM_
    Dec 8, 2022 at 22:17
  • $\begingroup$ thank you for the answer, I was worried whether mixing those two in e.g. Nelson Siegel model, would result in distorted results, so long-term bonds would distort short-term rates, and my result for bill pricing would be less accurate (less fitted) than using only bills. $\endgroup$
    – darkuss
    Dec 8, 2022 at 22:26

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