What is the difference between Discount Rate and BEY (Bond Equivalent Yield) on long US Treasury Bills?

For long US Treasury Bills (T-Bills) that are maturing in more than one half-year, the discount rate is defined in the same form as short T-Bills. However, the BEY (Bond Equivalent Yield) is defined in a more complex form, because of the interest compounding for the first half-year as shown in previous tutorials.

The discount rate on a long T-Bill is defined as the annualized rate of the discount you get off the face value when you purchase the T-Bill.

$P=F\times (1-d\times {\displaystyle \frac{\mathrm{Days}}{\mathrm{Year}}})$
$P=\text{Purchase price}$
$F=\text{Face value}$
$d=\text{Discount Rate}$
$\mathrm{Days}=\text{Number of days to maturity}$
$\mathrm{Year}=360=\text{Constant number of days in a year}$

The BEY (Bond Equivalent Yield), also called investment/interest rate or equivalent coupon rate, on a long T-Bill is defined as a compounding interest form

$F=P\times (1+i\times {\displaystyle \frac{{\displaystyle \frac{\mathrm{Year}}{2}}}{\mathrm{Year}}})\times (1+i\times {\displaystyle \frac{\mathrm{Days}-{\displaystyle \frac{\mathrm{Year}}{2}}}{\mathrm{Year}}})$
$P=\text{Purchase price}$
$F=\text{Face value}$
$i=\text{BEY (Bond Equivalent Yield)}$
$\mathrm{Days}=\text{Days to maturity}$
$\mathrm{Year}=\text{365 or 366}=\text{Actual number of days in the year}$

Similar to case of shot T-Bills, we can also merge their defining formulas into a single equation:

$(1-d\times {\displaystyle \frac{\mathrm{Days}}{360}})\times (1+i\times \mathrm{0.5})\times (1+i\times ({\displaystyle \frac{\mathrm{Days}}{\mathrm{Year}}}-0.5))=1$
$(1-d\times {\displaystyle \frac{\mathrm{Days}}{360}})\times (1+i\times {\displaystyle \frac{\mathrm{Days}}{\mathrm{Year}}}+i^2\times (0.5{\displaystyle \frac{\mathrm{Days}}{\mathrm{Year}}}-0.25))=1$

We can then easily rewrite it as an conversion formula from BEY (Bond Equivalent Yield) to discount rate:

$d={\displaystyle \frac{360}{\mathrm{Days}}}\times (1-{\displaystyle \frac{1}{(1+i\times {\displaystyle \frac{\mathrm{Days}}{\mathrm{Year}}}+i^2\times (0.5\times {\displaystyle \frac{\mathrm{Days}}{\mathrm{Year}}}-0.25))}})$

We can also rewrite it as a quadratic equation on the BEY:

$(0.5\times {\displaystyle \frac{\mathrm{Days}}{\mathrm{Year}}}-0.25)\times i^2+{\displaystyle \frac{\mathrm{Days}}{\mathrm{Year}}}\times i+1-{\displaystyle \frac{\mathrm{360}}{(360-d\times \mathrm{Days})}}=0$

Then we can use the quadratic formula for the conversion from discount rate to BEY:

$a{i}^2+bi+c=0$
$i={\displaystyle \frac{-b+\sqrt{b^2-4ac}}{2a}}$
$a=0.5\times {\displaystyle \frac{\mathrm{Days}}{\mathrm{Year}}}-0.25$
$b={\displaystyle \frac{\mathrm{Days}}{\mathrm{Year}}}$
$c=1-{\displaystyle \frac{360}{(360-d\times \mathrm{Days})}}$

Here is an example of 52-week T-Bill auctioned by The US Department of the Treasury:

Term and Type of Security: 364-Day Bill 
CUSIP: 912797QD

High Rate: 3.820% (Discount rate)
Investment Rate: 3.989% (BEY - Bond Equivalent Yield)

Issue Date: April 17, 2025 
Maturity Date: April 16, 2026 

We can use this example to verify the conversion formulas between discount rate and BEY (Bond Equivalent Yield):

$a=0.5\times {\displaystyle \frac{\mathrm{Days}}{\mathrm{Year}}}-0.25=0.5\times {\displaystyle \frac{364}{365}}-0.25=0.2486301369863$
$b={\displaystyle \frac{\mathrm{Days}}{\mathrm{Year}}}={\displaystyle \frac{364}{365}}=0.9972602739726$
$c=1-{\displaystyle \frac{360}{(360-d\times \mathrm{Days})}}=1-{\displaystyle \frac{360}{(360-0.03820\times 364)}}=-0.040176224159474$
$i={\displaystyle \frac{-b+\sqrt{b^2-4ac}}{2a}}$
$={\displaystyle \frac{-0.9972602739726+\sqrt{{0.9972602739726}^2+4\times 0.2486301369863\times 0.040176224159474}}{2\times 0.2486301369863}}$
$=0.039889890421657=\mathrm{3.989\%}$

⇒ YTM (Yield To Maturity) on US T-Bill

⇐ Discount Rate vs. BEY on Short US T-Bill

⇑ Performance Measurements of US Treasury Bills

⇑⇑ US Treasury Securities