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

For short US Treasury Bills (T-Bills) that are maturing in less than or equal to one half-year, both discount rate and Discount Rate and BEY (Bond Equivalent Yield) are defined in simple forms as shown in previous tutorials.

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

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

The BEY (Bond Equivalent Yield). also called investment/interest rate or equivalent coupon rate), on a short T-Bill is defined as the annualized rate of interest (the price discount) you earn on your investment (the price you paid for the T-Bill).

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

As you can see from their definitions, the main differences between the discount rate and the BEY (Bond Equivalent Yield) on short T-Bills are:

Numerator - Both the discount rate and the BEY are using same numerator: the difference between the face value and the purchase price.

But the discount rate considers the difference as a discount from the face value. While the BEY considers the difference as an interest earned from the purchase price.

Denominator - The discount rate and the BEY are using different denominators. The discount rate uses the face value as the denominator. While the BEY uses the purchase price as the denominator.

Since the BEY uses a lower number (purchase price is typically lower than the face value) as the denominator, its value is typically higher than the discount rate value.

Annulization - The discount rate and the interest rate are using different day count methods. The discount rate uses 360-day year. While the BEY uses actual-day year.

To compare their values, we can merge their defining formulas into a single equation:

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

We can then rewrite it as conversion formulas:

$d={\displaystyle \frac{i\times 360}{\mathrm{Year}+i\times \mathrm{Days}}}$
$i={\displaystyle \frac{d\times \mathrm{Year}}{\mathrm{360}-d\times \mathrm{Days}}}$

To validate the above formulas, let's take an example of 26-week T-Bill bought through TreasuryDirect.gov with a discount rate of 4.170% and an BEY of 4.319%:

CUSIP: 912797NW3
Type: 26-Week Bill 
Par Amount: $10,000.00 (Face value) 
Purchase Price: $9,789.18

Investment/Interest Rate: 4.319% (BEY - Bond Equivalent Yield)
Yield: 4.170% (Discount rate)
Price per $100: 97.891833

Discount: $210.82
 
Issue Date: 2024-12-26
Maturity Date: 2025-06-26

Here is the validation result of conversion from BEY to discount rate:

$d={\displaystyle \frac{i\times 360}{\mathrm{Year}+i\times \mathrm{Days}}}={\displaystyle \frac{\mathrm{0.04319}\times 360}{\mathrm{365}+\mathrm{0.04319}\times \mathrm{182}}}$
$\mathrm{ }=0.041700305245462=\text{4.170\%}$

Here is the validation result of conversion from discount rate to BEY:

$i={\displaystyle \frac{d\times \mathrm{Year}}{\mathrm{360}-d\times \mathrm{Days}}}={\displaystyle \frac{\mathrm{0.04170}\times 365}{\mathrm{360}-\mathrm{0.04170}\times \mathrm{182}}}$
$\mathrm{ }=0.043189677041496=\text{4.319\%}$

Related topics:

BEY to Discount Rate Converter for T-Bills

Discount Rate to BEY Converter for T-Bills

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

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

⇑ Performance Measurements of US Treasury Bills

⇑⇑ US Treasury Securities