Why Treasury Bill Discount Quotes Differ from Continuous-Yield Pricing
Summary
The note explains why a Treasury bill’s quoted discount rate does not reproduce its market price when inserted into a continuous-compounding formula. The displayed bill price is generated under a quote convention based on a simple discount calculation, while continuous yield uses exponential discounting; the two rates are not interchangeable.
The response emphasizes that the traded price is the amount paid, whereas yield is derived from that price under a chosen convention. Converting yield back into a theoretical price requires matching the convention, including how time to maturity is measured. Calendar-day counts, trading-day counts, and year fractions can lead to different results. The example illustrates the issue but does not prescribe one universal convention: bond and bill quoting practices vary, and market-specific price details may not be captured by a generic yield calculation. Continuous rates therefore require an explicit day-count and time basis.
Key ideas
- A bill discount quote and a continuously compounded yield use different pricing conventions.
- Market price is the traded amount; yield is calculated from that price using an agreed convention.
- Reconstructing price from a yield requires matching the instrument’s day-count and time basis.
- Calendar days, trading days, and year fractions can produce different discount factors.
Tags
Full text
# How to use exp(-r*t) to calculate tbill price
# How to use exp(-r*t) to calculate tbill price
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
```
## Answer by D Stanley (score 3, accepted)
https://quant.stackexchange.com/a/79101
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.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.