# Build spot rate curve with multiple treasuries for each maturity

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.

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