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I wonder why : $1 - \left(\frac{4.91\% \times 358}{360}\right) = 95.1172778 $

and why $\exp\left(-4.91\% \times \frac{358}{360}\right)$ does not give 95.1172778

T Bill Description :

B 0 04/17/25 ( 912797KS5 )

Discount           4.91000   
Settle             04/24/24
Price              95.1172778   
Issue              04/18/2024
Days to Maturity   358  
Maturity           04/17/2025

enter image description here

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    $\begingroup$ I think you're just confused by the price quote convention used which is not the exact same value as the continuous yield. $\endgroup$
    – D Stanley
    Commented Apr 23 at 15:40
  • $\begingroup$ @DStanley Then which rate should I use to retrieve the correct price using exp(-rt) ? Thanks $\endgroup$
    – TourEiffel
    Commented Apr 23 at 15:42
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    $\begingroup$ The price is the price. The quoted annualized yield will be different than the calculated continuous yield. $\endgroup$
    – D Stanley
    Commented Apr 23 at 15:43
  • $\begingroup$ @DStanley since Tbill is 0 coupon I except the discount factor to be the price so exp(-rt) should work ? Can you please clarify what is the "Discount Rate" and it's difference with UST Treasury Convention Yield ? Mathematical answer is more than welcome. $\endgroup$
    – TourEiffel
    Commented Apr 23 at 15:46
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    $\begingroup$ Apply the correct formula for prices of Bills: P = 100 * (1 - 0.0491 * 359 /360). Assuming a continuously compounded rate is equal to the discount rate on a bill is an error, as explained. $\endgroup$
    – Attack68
    Commented Apr 23 at 16:10

1 Answer 1

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It's just a quote convention that likely comes from tradition before computers were prevalent. Calculating a "yield" from simple addition and multiplication/division is easier to do without computers than using the actual continuous yield or compounding, which requires exponential functions.

When looking at data feeds like Bloomberg, it's important to remember that the price is the price - that's what you actually trade the security for. The yield is calculated from that price, and may be different depending on what conventions you use.

One can use yields to calculate an equivalent price for a security, but that doesn't necessarily mean that you can trade for that price. Bonds often trade with idiosyncrasies that are not always captured by the yield curve or other market data.

In order to calculate the actual continuous yield (e.g. r in exp(-r*t)) you have to decide how to calculate t which is where different conventions come into play. In academics, t is generally given and is a fraction or multiple of a year (e.g. 1 or 0.5 or 2). In reality, what is t for a bond that matures in 358 calendar days? Is it 358/365? should you only consider trading days instead of calendar days? This is where different quoting conventions come into play.

The various yields you see in Bloomberg (and the "discount" which is a common convention for zero-coupon bonds) are all based on different conventions, none of which can be directly used as a continuous yield.

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