Bootstrapping Treasury Spot Curves with Duplicate Maturities
Summary
The document discusses building a short Treasury spot-rate curve when several securities mature on the same date. For bootstrapping, the answer recommends choosing one bond per maturity, with a newer issue offered as a practical selection rule. A fitted spline can instead use multiple bonds at a maturity because it estimates a curve that best matches the prices across the set.
The reported negative short-end spot rates may reflect the supplied prices, but the answer questions their accuracy and distinguishes clean from dirty prices. It illustrates this concern by comparing the quoted price of the earliest-maturing bond with a lower clean-price estimate for the stated valuation date. The responses also give a coupon-bond discount-factor recursion for bootstrapping. The discussion is limited: it does not work through a complete curve calculation, and it cautions that very short-dated Treasuries may be too illiquid to provide useful curve information.
Key ideas
- For bootstrapping, use one bond per maturity; a newer issue is suggested when choosing between duplicates.
- A spline can fit prices from multiple bonds with the same maturity.
- Negative short-end rates can result from the inputs, but quoted prices and yield conventions should be checked.
- Clean and dirty prices differ, which can affect yield calculations.
- Very short-dated Treasuries may be too illiquid for reliable curve construction.
Tags
Full text
# Build spot rate curve with multiple treasuries for each maturity
# Build spot rate curve with multiple treasuries for each maturity
I have the following treasuries:
- T 0 1/4 01/31/15 at 100.1236
- T 2 1/4 01/31/15 at 101.1257
- T 0 1/4 02/15/15 at 100.1251
- T 4 02/15/15 at 101.9994
- T 11 1/4 02/15/15 at 105.6269
- T 0 1/4 02/28/15 at 100.1237
- T 2 3/8 02/28/15 at 101.1878
- T 0 3/8 03/15/15 at 100.1866
- T 0 1/4 03/31/15 at 100.1182
- T 2 1/2 03/31/15 at 101.2421
- T 0 3/8 04/15/15 at 100.1784
- T 0 1/8 04/30/15 at 100.0554
- T 2 1/2 04/30/15 at 101.2375
- T 0 1/4 05/15/15 at 100.1103
- T 4 1/8 05/15/15 at 102.0451
- T 2 1/8 05/31/15 at 101.0417
- T 0 1/4 05/31/15 at 100.1095
- T 0 3/8 06/15/15 at 100.1644
- T 0 3/8 06/30/15 at 100.1617
- T 1 7/8 06/30/15 at 100.9101
And I want to calculate the 6 month spot rate curve from today date. When I do this I get negative returns for the spot rate. I followed the BEY convention and used this question as reference. I got negative spot rates for the first part of the curve. Is this correct? Another point to consider is that I have multiple securities for the same expiration date (i.e. 1 and 2, 6 and 7) so when I build the spot rate I get two of them for one maturity. Which method should I use to ponder this.
## Answer by Helin (score 1)
https://quant.stackexchange.com/a/16322
If you're bootstrapping and if there are bonds maturing on the same date, you should use only one. A good rule is to discard the older issue and keep the more recently issued securities.
If you're building a spline, then it really doesn't matter since you're building a best fit curve that best approximates the prices of all bonds.
Assuming the quotes you provided are correct, then some of the yields are indeed negative. But your prices might just be off. For example, the first bond 0.25s of Jan15, has a clean price of 100.00390625 today (2015-01-23), significantly lower than the 100.1236 you have. (It's possible you're listing dirty prices, in which case your yield calculation might be off...)
Either way, shorted dated Treasuries are almost never used for building curves, since they're not liquid enough to provide useful information.
## Answer by james42 (score -1)
https://quant.stackexchange.com/a/17967
I'm new in using bootstrapping, but the relationship used to recover the discount function $v(t,t_m)$ from the price of the bond $P(t,T:c)$ and the coupon $c$ is
$v(t,t_m)=\frac{P(t,T:c)-c\sum_{i=1}^{m-1}v(t,t_i)}{1+c}$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.