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I have the following treasuries:

  1. T 0 1/4 01/31/15 at 100.1236
  2. T 2 1/4 01/31/15 at 101.1257
  3. T 0 1/4 02/15/15 at 100.1251
  4. T 4 02/15/15 at 101.9994
  5. T 11 1/4 02/15/15 at 105.6269
  6. T 0 1/4 02/28/15 at 100.1237
  7. T 2 3/8 02/28/15 at 101.1878
  8. T 0 3/8 03/15/15 at 100.1866
  9. T 0 1/4 03/31/15 at 100.1182
  10. T 2 1/2 03/31/15 at 101.2421
  11. T 0 3/8 04/15/15 at 100.1784
  12. T 0 1/8 04/30/15 at 100.0554
  13. T 2 1/2 04/30/15 at 101.2375
  14. T 0 1/4 05/15/15 at 100.1103
  15. T 4 1/8 05/15/15 at 102.0451
  16. T 2 1/8 05/31/15 at 101.0417
  17. T 0 1/4 05/31/15 at 100.1095
  18. T 0 3/8 06/15/15 at 100.1644
  19. T 0 3/8 06/30/15 at 100.1617
  20. 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.

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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.

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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

$v(t,t_m)=\frac{P(t,T:c)-c\sum_{i=1}^{m-1}v(t,t_i)}{1+c}$

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