# Roll down Treasury curve (Coupon effects)

I'm currently working on roll down calculations for the Treasury curve (3-month roll, 6-month roll, etc..). One of the senior guys (I just started out of college) asked me to adjust for the coupon effects for some of the long dated bonds. I understand that higher coupon bonds have lower yields than other bonds with the same maturity. However, I am not sure the proper way to adjust for the coupon effects.

I've read literature where people use asset swap spreads to observe the coupon effects.

• Specifically what are you computing ? The price change based on assuming a some fixed yield curve and then calculating the price now and in three (six) months ? I don't see why longer dated bonds should be treated differently. They have a higher duration and convexity. But using a standard price-yield calculator will take that into account. – Dom Aug 26 '17 at 12:42
• Are you calculating rolldown using a fitted curve? If so, your rolls are already coupon-adjusted. – Helin Aug 27 '17 at 1:15
• I'm using the live Treasury curve to calculate roll down. – VanillaCall Aug 27 '17 at 3:09
• @VanillaCall high coupon bonds will have higher yields under most circumstance (other than bid for convexity) – rrg May 24 '18 at 15:35

To calculate rolldown that accounts for the coupon effect requires a fitted curve. Assuming such a curve is available, then the following procedure is usually followed:

First, calculate the z-spread of the bond in question relative to the fitted curve: $$P = \sum_{i=1}^n c_i \cdot d(t_i) \cdot e^{-s t_i},$$ where $P$ is the current quoted dirty price (inclusive of accrued interest), $c_i$ is the $i$th upcoming cash flow, $d(t_i)$ is the discount factor corresponding to the $i$th cash flow (obtained from the fitted curve), and $s$ is the z-spread we are solving for. Conceptually, we are looking for how much of a parallel shift we need to apply to the zero coupon curve, so that the shifted curve reprices the bond to its current market price. (I'm using continuous compounding here, but you can use semi-annual compounding if you want; doesn't really matter.)

Assuming that we are calculating 3-month rolldown, we then reprice the bond using discount factors that are three months shorter and using the same z-spread from the previous step: $$P' = \sum_{i=1}^n c_i \cdot d(t_i - 0.25) \cdot e^{-s \cdot (t_i - 0.25)}.$$ (Note that "0.25" is me being lazy. In practice, you should get the correct day count fraction corresponding to the true "3 months.")

We can then calculate a new yield to maturity $y'$ from $P'$ (assuming that maturity has shortened by three months). The difference between the current market yield and $y'$ is rolldown – coupon adjusted.

• I would like to add that this procedure is easier than what it seems with currently available quantitative finance libraries. As instance, if you're using QuantLib you can amend the reference date of the yield term structure and - thanks to relinkable handles - this will reprice you bond "rolling down" the term structure. – Lisa Ann Aug 27 '17 at 16:22
• What is the concept of a fitted curve? Is it the theoretical fair value of the Treasury curve? If a bond is priced fair against the fitted curve then the Z-spread should be zero correct? Any deviations from the fitted curve would be the result of coupon effects? – VanillaCall Aug 31 '17 at 11:55
• Yes, a fitted curve is a theoretical fair value curve. Z-spread IS a coupon-adjusted spread. Any non-zero z-spread signals richness/cheapness relative to the curve, not coupon effect. – Helin Aug 31 '17 at 17:00
• Thanks, one last question and I think i'll finally get it. how is the Z-spread a coupon adjusted spread? – VanillaCall Sep 1 '17 at 0:31
• I think the best proof is as follows: take two bonds maturing on the same day but with different coupons. Price these two bonds with the fitted curve @ zero spread. You'll get two different yields. Whenever a fitted curve is involved to discount cash flows, your result would be immune to coupon effect – because you have priced the bonds with their actual cash flows. – Helin Sep 1 '17 at 2:19