Can anyone explain to me the correct method for constructing the par yield curve from the actual treasury curve. This was never quite clear during the CFA program because the Par Yield Curve was always given.
I understand that the Par Yield is the yield that exists if a bond is priced at par, i.e. the coupon and the YTM are equal. However, when deriving the theoretical par yield, do you set the coupon rate equal to the YTM or set the YTM equal to the coupon to determine the par yield. For example, what if you have an unlikely situation where the 10 year on-the-run treasury note is trading at 2.5% and its coupon is 3%. Is the par yield and assumed coupon = 2.5% or 3.0%?
The par curve is the YTM on coupon-paying bonds; the coupon rate is irrelevant. So, if the 10-year OTR T-Note is trading at a YTM of 2.5%, then the 10-year point on the par curve is 2.5%. The fact that the bond has a 3% coupon simply means that it’s trading at a premium.
Right, that is what I was thinking. However everything that I read has said that the “par yield equals the coupon”. However I think it would make more sense to say that the coupon is set equal to the YTM to set the par yield and then to be used for bootstrapping. Does this make sense?
S200, I have a question a friend called and asked me yesterday.
“10-year OTR T-Note is trading at a YTM of 2.5%, then the 10-year point on the par curve is 2.5%”
What happens when the on-the-run note isn’t actually at 10 year maturity anymore, but 9 years left? Do they adjust for that or is it just simply the yield?
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. However, Treasury reserves the option to input additional bid yields if there is no on-the-run security available for a given maturity range that we deem necessary for deriving a good fit for the quasi-cubic hermite spline curve. For example, we are using composites of off-the-run bonds in the 20-year range reflecting market yields available in that time tranche.
Specifically in Fabozzi’s book, “Fixed Income Analysis for the CFA Program”, he states:
“Since all the coupon bonds are selling at par, as explained in the previous section, the yield to maturity for each bond is the coupon rate.”
To me, the more important yield is what the bond at currently trading at in the market and therefore when constructing the “Par Yield Curve” it makes more sense to set the coupon rate equal to the actual YTM? So if you have an OTR treasury with a coupon of 3.5% but the YTM is 2%, (unlikely I know), you would assume the bond is priced at par with a coupon of 2% when you go to bootstrap the spot curve. Does this make sense or is it the other way around? Thanks for your help.
Is the par yield curve any different from the regular old yield curve?
I remember that the official definition of the par yield curve is that it is the coupon rate for bonds selling at par. But since bonds selling at par have a coupon rate equal to the YTM, then you can actually just take regular YTM which is the ordinary yield curve.
Maybe there are some things about reinvestment rate assumptions and shorter maturities that figure in to this. I don’t remember.
Yes: the par yield curve gives the YTM for coupon-paying bonds. If a bond’s coupon equals its YTM, then it sells at par, so you could say that the par yield curve gives the coupon rate for coupon-paying bonds selling at par: _ same thing _.
Finance people tend to use the term “yield curve” rather loosely (i.e., sloppily). (Perhaps this shouldn’t come as a surprise: finance people tend to use a lot of terms sloppily.)
I would imagine that when most finance people say, “regular (or plain) old yield curve”, they mean the par curve.
Let me first say that I had to figure out what a quasi-cubic hermite spline was. Very fascinating though and worth the effort.
To answer your question “clamped or unclamped?” - I believe the answer is neither.
With a natural spline, there’s likely to be too much oscillation, and you can’t use a clamped spline because there’s no function at the first and last tenor points. I think the Treasury is using the label “quasi” because it’s likely approximating the degree and knot vector at the first and last node/tenor point.
I don’t know enough about quasi-cubic hermite splines to say for certain, but if they’re specifying the knot vector at the first and last nodes, it sounds like a clamped spline.
I sort of figured you were being a bit facetious but glad the subject came up.
I’ve been putting together corporate yield curves narrowed to industry & rating groups and have used basic linear interpolation to find whole number tenor points. I’m going to try using this method on my own charts and will hopefully get some brownie points from my manager.
Everyone gets 15 minutes of technobabble. That treasury guy just got theirs.
It’s true that you need some way to smooth a zillion transactions into some kind of curve, and in the (appropriate) absence of a theory saying what the shape “should” be, a spline is a reasonable approach. Since a “normal” yield curve should deliver some kind of term premium, a cubic spline would make sense, since it is the lowest order that can weave around a bit but still require high interest rates for very very long lockups.
As for the quasi- and hermite modifiers. I don’t know enough about splines to know whether they are appropriate, but it sure sounds like it’s designed to scare off questions from anyone who isn’t a mathematics Ph.D…