How Treasury Bills Convert Discount Rates into Prices and Yields
Summary
The document explains how a U.S. Treasury bill’s quoted discount rate relates to its price and coupon-equivalent yield. Because a bill has no coupon, the standard coupon-bond valuation formula is not the relevant calculation. The accepted answer says the cited market display uses a converted yield and first applies the Treasury bill discount-rate convention to calculate the bill’s dollar price from face value, discount rate, and days to maturity.
It then converts that price into a coupon-equivalent yield using the bill’s price, face value, and time to maturity. The example accounts for settlement one business day after the observation date, which affects the remaining maturity. This gives a practical calculation path for interpreting a quote, but the post is tied to a particular displayed value and settlement assumption; users should verify the applicable Treasury convention, maturity, and settlement date for other quotes.
Key ideas
- Treasury bills have no coupon, so their quotes are interpreted using bill-specific discount-rate conventions.
- The discount rate and days to maturity determine the bill’s price relative to face value.
- The price can then be converted to a coupon-equivalent yield.
- Settlement timing changes the remaining days to maturity used in the calculation.
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Full text
# How is the price of US T-Bill's calculated here on CNBC?
# How is the price of US T-Bill's calculated here on CNBC?
https://www.cnbc.com/quotes/US6M
Here, the price is given (as of the time of asking) as 5.285, and it is not clear to me how the yield and price here are exactly related. It is clear for me however, how it is done for T-Bonds, using the following formula
$$ \begin{align} P &= \begin{matrix} \left(\frac{C}{1+i}+\frac{C}{(1+i)^2}+ ... +\frac{C}{(1+i)^N}\right) + \frac{M}{(1+i)^N} \end{matrix}\\ &= \begin{matrix} \left(\sum_{n=1}^N\frac{C}{(1+i)^n}\right) + \frac{M}{(1+i)^N} \end{matrix}\\ &= \begin{matrix} C\left(\frac{1-(1+i)^{-N}}{i}\right)+M(1+i)^{-N} \end{matrix} \end{align} $$
as given on https://en.wikipedia.org/wiki/Bond_valuation page on Wikipedia. I cannot see how to use this formula for T-Bills where coupon is zero.
## Answer by oronimbus (score 3, accepted)
https://quant.stackexchange.com/a/76224
T-Bill's follow the discount rate methodology outlined by the U.S. Treasury in their document Price, Yield and Rate Calculations for a Treasury Bill here. What CNBC is displaying is the converted yield, i.e. you use the Discount Rate of 5.28% and then calculate the dollar price of the Bill
`=100*(1-0.0528*182/360)`
which gives approx 97.3. Next convert the price using the "Coupon Equivalent Yield" formula to get to the value displayed (5.515%):
`=(100-97.3306667)/97.3306667*366/182*100`
Note that 182 is the days to maturity (accounting for T+1 settlement from today).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.