Example: 912797JA6 is a tbill with a maturity of 28 days. It was sold for $99.5905 on 2023-12-21. What is the yield?

Ans: For tbills of not more than half-year maturity, the yield is calculated as

i = ((100 - P)/P) * (y/r)

where y is the days in year, r is the days to maturity.

In [1]: 
def tbill_yield_short_maturity(P, r, y):
    i = ((100 - P) / P) * (y / r)
    i = round(i * 100, 3)
    return i
 

In [2]: 
tbill_yield_short_maturity(99.5905, 28, 366)
Out[2]: 
5.375

so the yield is 5.375%

Ref:

• https://www.treasurydirect.gov/instit/annceresult/press/preanre/2004/ofcalc6decbill.pdf shows examples to compute price, yield and rate for tbills.
  • rate == discount rate?
  • yield == coupon equivalent yield?
  • pg-2 shows how to compute the yield for bills of not more than one half-year to maturity.
• https://www.treasurydirect.gov/auctions/announcements-data-results/announcement-results-press-releases/auction-results/ gives “today's auction results”
• https://www.treasurydirect.gov/instit/annceresult/press/preanre/2023/R_20231221_1.pdf shows that it is 28-day tbill, was sold on 2023-12-21 at a price of $99.590500 for an “Investment Rate” of 5.375%. This “Investment Rate” is same as the yield above.

On https://www.treasurydirect.gov/auctions/announcements-data-results/ showed

It was bought on 12/28/2023, settlement date = 1/2/2024 for a price of 99.585833. The yield on it is

In [1]: 
from tbill_yield import tbill_yield_short_maturity
tbill_yield_short_maturity(99.585833, 28, 366)
Out[1]: 
5.436

The 5.436 matches with the Investment Rate in the table.

So 'Issue Date' in the table is the settlement date.

On https://www.treasurydirect.gov/auctions/announcements-data-results/ showed

It was bought on 01/04/2024, settlement date = 1/9/2024 for a price of 99.588556. The yield on it is

In [2]: 
from tbill_yield import tbill_yield_short_maturity
tbill_yield_short_maturity(99.588556, 28, 366)
Out[2]: 
5.4

The precision in the price matters. If you only have 4 significant digits after the decimal, the result will not match with the 'Investment Rate'

In [3]: 
tbill_yield_short_maturity(99.5885, 28, 366)
Out[3]: 
5.401

Conclusion: Price should have 6 significant digits after the decimal.

https://www.treasurydirect.gov/auctions/upcoming/ → Auction Results shows

https://www.treasurydirect.gov/auctions/announcements-data-results/ → CMBs tab shows

Combining both, we get

Notice how yield (Investment Rate) went down (from 5.401% to 5.397%) even though price went down (from \$99.384000 to \$99.382833) for the first entry? This is because the 'days in year' changes from 366 to 365.

$ ipython

In [1]:
from tbill_yield import tbill_yield_short_maturity
tbill_yield_short_maturity(99.384000, 42, 366)
Out[1]:
5.401

In [2]:
tbill_yield_short_maturity(99.382833, 42, 365)
Out[2]:
5.397

Ref: https://github.com/KamarajuKusumanchi/market_data_processor/blob/master/src/tbills/tbill_yield.py → tbill_yield_short_maturity()