What is BEY (Bond Equivalent Yield) of Long US Treasury Short Bill?

BEY (Bond Equivalent Yield), or Coupon Equivalent Yield, of Long US Treasury Bill (T-Bill) is the equivalent interest rate if you consider the price discount as the compounding (semiannually) interest and the purchase price as the investment capital.

Long T-Bills are T-Bills that mature in more than one half-year, and less than or equal to 52 weeks.

Since US Treasury securities are paying interests semiannually, the equivalent interest needs to be compounded every 6 months. So the compounding interest formula provides in a very easy-to-understand form to define the BEY (Bond Equivalent Yield):

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

The above formula says that the face value is the result of the purchase price being applied by 2 interest factors:

$\text{First half-year interest factor}:(1+i\times {\displaystyle \frac{{\displaystyle \frac{\mathrm{Year}}{2}}}{\mathrm{Year}}})$
$\text{Remaining-term interest factor}:(1+i\times {\displaystyle \frac{\mathrm{Days}-{\displaystyle \frac{\mathrm{Year}}{2}}}{\mathrm{Year}}})$

In order to calculate the BEY (Bond Equivalent Yield), we need transform the above compounding interest formula into a quadratic equation. Let's simplify the half-year interest factor a little bit first:

$F=P\times (1+0.5\times i)\times (1+i\times ({\displaystyle \frac{\mathrm{Days}}{\mathrm{Year}}}-0.5))$

Then we can transform it into a quadratic equation with multiplication operator omitted:

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

Now we can use the quadratic formula to calculate BEY (Bond Equivalent Yield):

$a{i}^2+bi+c=0$
$i={\displaystyle \frac{-b+\sqrt{b^2-4ac}}{2a}}$
$a=0.5{\displaystyle \frac{\mathrm{Days}}{\mathrm{Year}}}-0.25$
$b={\displaystyle \frac{\mathrm{Days}}{\mathrm{Year}}}$
$c=1-{\displaystyle \frac{F}{P}}$

Here is an example of 52-week T-Bill auctioned by The US Department of the Treasury with a BEY (Bond Equivalent Yield) of 3.989%:

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

Price: 96.137556
Investment Rate: 3.989% (BEY - Bond Equivalent Yield)

Issue Date: 2025-04-17 
Maturity Date: 2026-04-16 

We can use this example to verify the BEY (Bond Equivalent Yield) formula:

$a=0.5{\displaystyle \frac{\mathrm{Days}}{\mathrm{Year}}}-0.25=0.5{\displaystyle \frac{364}{365}}-0.25=0.2486301369863$
$b={\displaystyle \frac{\mathrm{Days}}{\mathrm{Year}}}={\displaystyle \frac{364}{365}}=0.9972602739726$
$c=1-{\displaystyle \frac{F}{P}}=1-{\displaystyle \frac{100.00}{96.137556}}=-0.040176224159474$
$i={\displaystyle \frac{-b+\sqrt{b^2-4ac}}{2a}}$
$={\displaystyle \frac{-0.9972602739726+\sqrt{{0.9972602739726}^2+4\times 0.2486301369863\times 0.040176224159474}}{2\times 0.2486301369863}}$
$=0.039889890421657=\mathrm{3.989\%}$

We have also provided a BEY (Bond Equivalent Yield) online calculator for you to use. Click this link "T-Bill 912797QD settled on 2025-04-17 with a price of 96.137556", you will see the BEY (Bond Equivalent Yield) of 3.9890% displayed.

You can also find the BEY (Bond Equivalent Yield), also called Coupon Equivalent Yield, formula for long T-Bills in the "Price, Yield and Rate Calculations for a Treasury Bill" document provided by treasurydirect.gov.

Related topics:

BEY (Bond Equivalent Yield) to Price Converter for T-Bills

Price to BEY (Bond Equivalent Yield) Converter for T-Bills

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

⇐ BEY (Bond Equivalent Yield) of Short US Treasury Bill

⇑ Performance Measurements of US Treasury Bills

⇑⇑ US Treasury Securities