What is the YTM with Fractional Period on US Treasury Note/Bond?

If you are purchasing a T-Note/T-Bond in middle of a coupon payment period, the present value expression in term of YTM (Yield To Maturity) needs to be adjusted in 2 places:

1. The investment (also called PV: Present Value) of the T-Note/T-Bond needs to include a non-zero accrued interest:

PV = Present Value = Par*Price/100 + (Accrued Interest)

2. The first period needs to be considered as a fractional period:

f = 1 - Days/Period (first period fraction)
Days = Accrued days (days since the last payment date)
Period = Days in the first period

After applying these 2 adjustments, our present value expression becomes:

$\mathrm{PV}={\displaystyle \frac{\mathrm{Par}}{{(1+0.5\times \mathrm{YTM})}^{n+f}}}+{\displaystyle \sum _{i=0}^n}{\displaystyle \frac{\mathrm{Coupon}}{{(1+0.5\times \mathrm{YTM})}^{i+f}}}$
$\mathrm{PV}=\text{Present Value (Investment)}$
$\mathrm{PV}=\text{Par*Price/100 + (Accrued Interest)}$
$\mathrm{Par}=\text{Par value (Face value)}$
$n=\text{Number of full coupon periods}$
$f=\text{1 - Days/Period (first period fraction)}$
$\mathrm{Days}=\text{Accrued days (days since the last payment date)}$
$\mathrm{Period}=\text{Days in the first period}$

Let's look at the same example from the previous tutorial, where you purchased $1,000.00 of the 10-year T-Note 91282CLW9 at its 1st reopening auction and settled on December 16, 2024:

Release Date: December 11, 2024 (the 1st reopening auction date)
Term and Type of Security: 9-Year 11-Month Note
CUSIP Number: 91282CLW9
Series: F-2034

Interest Rate: 4-1/4%
High Yield: 4.235% (the highest accepted YTM)
Price: 100.114150 (the lowest accepted price)
Accrued Interest per $1,000: $3.63950 

Median Yield: 4.190% (50% accepted at or blow this YTM)
Low Yield: 4.080% (5% accepted at or blow this YTM)

Issue Date: December 16, 2024 (the settlement date)
Maturity Date: November 15, 2034
Original Issue Date: November 15, 2024
Dated Date: November 15, 2024 

Applying our present value expression on this example, we have:

$\mathrm{PV}={\displaystyle \frac{\mathrm{Par}}{{(1+0.5\times \mathrm{YTM})}^{n+f}}}+{\displaystyle \sum _{i=0}^n}{\displaystyle \frac{\mathrm{Coupon}}{{(1+0.5\times \mathrm{YTM})}^{i+f}}}$
$\mathrm{ }={\displaystyle \frac{\mathrm{1000.0}}{{(1+0.5\times \mathrm{0.04235})}^{\mathrm{19}+\mathrm{0.82872928176796}}}}+{\displaystyle \sum _{i=0}^{\mathrm{19}}}{\displaystyle \frac{\mathrm{21.25}}{{(1+0.5\times \mathrm{0.04235})}^{i+\mathrm{0.82872928176796}}}}$
$\mathrm{ }=1004.8121653921$
$\mathrm{Price}=(\mathrm{PV}-\mathrm{Accrued})\times {\displaystyle \frac{\mathrm{100.0}}{\mathrm{Par}}}=(\mathrm{1004.8121653921}-3.6395)\times \mathrm{0.1}=100.11726653921$

The output does not match well with the price of 100.114150 reported in the auction result as shown below with other intermediary values:

Price = 100.114150

Interest Rate = 4.25% (Coupon Rate)
High Yield: 4.235% (the highest accepted YTM)
Accrued = 3.6395 (Accrued Interest)
Par = 1000.0 (Par value)
Days = 31 ("2024-12-16" - "2024-11-15")
Period = 181 ("2025-05-15" - "2024-11-15")
f = 1 - Days/Period = 1 - 31/181 = 0.82872928176796 (first period fraction)
Coupon = 21.25 (Coupon value = 0.5*Par*(Interest Rate)

This discrepancy is actually a result of the lower precision in the YTM value. Let's to try to increase it by 1 decimal digit from 4.235% to 4.2354%, and calculate the price again:

$\mathrm{PV}={\displaystyle \frac{\mathrm{1000.0}}{{(1+0.5\times \mathrm{0.04235})}^{\mathrm{19}+\mathrm{0.82872928176796}}}}+{\displaystyle \sum _{i=0}^{\mathrm{19}}}{\displaystyle \frac{\mathrm{21.25}}{{(1+0.5\times \mathrm{0.04235})}^{i+\mathrm{0.82872928176796}}}}$
$\mathrm{ }=1004.7800283763$
$\mathrm{Price}=(\mathrm{1004.7800283763}-3.6395)\times \mathrm{0.1}=100.11405283763$

The output now matches better with the price of 100.114150 reported in the auction result.

You can also verify the above results using our conversion tools:

• Click this YTM to Price conversion link, "T-Note 91282CLW9 settled on 2024-12-16 with a YTM of 4.235%", you will see the purchase price of $100.1173 displayed.
• Click this Price to YTM conversion link, "T-Note 91282CLW9 settled on 2024-12-16 with a purchase price of $100.114150", you will see the YTM of 4.2354% 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:

• PRICE function, by Microsoft.
• U.S. Treasury Securities - Price and Yield Calculations, by Larry Ng.

⇒ One-Period YTM on US T-Note/T-Bond

⇐ Accrued Interest of US Treasury Note/Bond

⇑ Performance Measurements of US Treasury Note/Bond

⇑⇑ US Treasury Securities