What is YTM (Yield To Maturity) of US Treasury Note/Bond?

Yield on US Treasury Note/Bond (T-Note/T-Bond) refers to the equivalent compounding interest rate for all revenues generated during the entire investment term. YTM (Yield To Maturity) refers to the yield of a T-Note/T-Bond, if you hold it until its maturity date.

To derive the relation between YTM and the purchase price, we can start with a simple investment case that generates a single revenue on maturity. The revenue includes the return of investment and interest compounded annually. In this case, the relation between investment and YTM can expressed as the "Future Value" formula:

$\mathrm{FV}=\mathrm{PV}\times {(1+\mathrm{YTM})}^N$
$N=\text{Number of years}$
$\mathrm{PV}=\text{Present Value (Investment)}$
$\mathrm{FV}=\text{Future Value (Revenue)}$
$\mathrm{YTM}=\text{Yield To Maturity}$

If the interest is compounded semiannually, as in the case of T-Note/T-Bond, the relation between investment and YTM should be adjusted as:

$\mathrm{FV}=\mathrm{PV}\times {(1+0.5\times \mathrm{YTM})}^n$
$n=\text{Number of interest periods}$
$\mathrm{PV}=\text{Present Value (Investment)}$
$\mathrm{FV}=\text{Future Value (Revenue)}$
$\mathrm{YTM}=\text{Yield To Maturity}$

If we consider all coupons generated from the T-Note/T-Bond and its par value (face value) as individual revenues and apply the above relation on them, we have a series of future value expressions:

$\mathrm{Coupon}={\mathrm{PV}}_1\times {(1+0.5\times \mathrm{YTM})}^1$
$\mathrm{Coupon}={\mathrm{PV}}_2\times {(1+0.5\times \mathrm{YTM})}^2$
$\text{...}$
$\mathrm{Coupon}={\mathrm{PV}}_n\times {(1+0.5\times \mathrm{YTM})}^n$
$\mathrm{Par}={\mathrm{PV}}_p\times {(1+0.5\times \mathrm{YTM})}^n$

If we reverse positions of future values and present values, we have a series of present value expressions:

${\mathrm{PV}}_1={\displaystyle \frac{\mathrm{Coupon}}{{(1+0.5\times \mathrm{YTM})}^1}}$
${\mathrm{PV}}_2={\displaystyle \frac{\mathrm{Coupon}}{{(1+0.5\times \mathrm{YTM})}^2}}$
$\text{...}$
${\mathrm{PV}}_n={\displaystyle \frac{\mathrm{Coupon}}{{(1+0.5\times \mathrm{YTM})}^n}}$
${\mathrm{PV}}_p={\displaystyle \frac{\mathrm{Par}}{{(1+0.5\times \mathrm{YTM})}^n}}$

Finally, if we add those expressions up, we have the total present value expression, which defines the YTM for a T-Note/T-Bond:

$\mathrm{PV}={\displaystyle \frac{\mathrm{Par}}{{(1+0.5\times \mathrm{YTM})}^n}}+{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{\mathrm{Coupon}}{{(1+0.5\times \mathrm{YTM})}^i}}$
$\mathrm{PV}=\text{Present Value (Investment)}$
$\mathrm{Par}=\text{Par value (Face value)}$
$n=\text{Number of coupon periods}$

Here is an example of 2-year T-Note sold through US treasury auction with a YTM (Yield To Maturity) of 3.795%:

Term and Type of Security: 2-Year Note
CUSIP Number: 91282CMY4
Series: BA-2027 

Interest Rate: 3-3/4% (Coupon rate: 3.75%)
High Yield: 3.795% (the highest accepted YTM)
Price: 99.914113 (the lowest accepted price)
Accrued Interest per $1,000: None

Issue Date: 2025-04-30
Maturity Date: 2027-04-30

We can use this example to verify the YTM (Yield To Maturity) definition:

$\mathrm{PV}={\displaystyle \frac{\mathrm{Par}}{{(1+0.5\times \mathrm{YTM})}^n}}+{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{\mathrm{Coupon}}{{(1+0.5\times \mathrm{YTM})}^i}}$
$\mathrm{ }={\displaystyle \frac{\mathrm{100.0}}{{(1+0.5\times \mathrm{0.03795})}^4}}+{\displaystyle \sum _{i=1}^4}{\displaystyle \frac{\mathrm{1.875}}{{(1+0.5\times \mathrm{0.03795})}^i}}$
$\mathrm{ }=99.914112573572=99.914113$

The output matches well with the auction result:

PV = 99.914113 (Price as Present Value)
YTM = 0.03795 (High Yield as YTM (Yield To Maturity))
Par = 100.0 (Par value is an assumption)
Coupon = 1.875 (Coupon value = 0.5*Par*(Interest Rate)
n = 4 (There are 4 coupon periods in 2 years)

You can also use our price calculator to validate the result. Click this link "T-Note 91282CMY4 settled on 2025-04-30 with a YTM of 3.795", you will see the purchase price of $99.9141 displayed.

Related topics:

YTM (Yield To Maturity) to Price Converter for T-Notes

Price to YTM (Yield To Maturity) Converter for T-Notes

YTM (Yield To Maturity) to Price Converter for T-Bonds

Price to YTM (Yield To Maturity) Converter for T-Bonds

References:

• Bond Yield Formula, by educba.com.
• Future Value, by Wikipedia.

⇒ High/Median/Low Yield of US Treasury Note/Bond

⇐ Price Discount/Premium of US Treasury Note/Bond

⇑ Performance Measurements of US Treasury Note/Bond

⇑⇑ US Treasury Securities