Calculate the price of a 29-year 11-month Treasury bond
U.S. Treasury notes or bonds pay a coupon every six month and return the face value at maturity. The price of a N-year Treasury bond equals to the present value of all future cash flows: \begin{equation}P=\frac{\frac{r}{2} * F}{1+ \frac{y}{2}} + \frac{\frac{r}{2} * F}{(1+ \frac{y}{2})^2} +\cdots + \frac{\frac{r}{2} * F}{(1+ \frac{y}{2})^{2N}} + \frac{ F}{(1+ \frac{y}{2})^{2N}}\,,\tag{1}\end{equation}
For the recent 30-Year Treasury bond, the issue date is Aug. 15, 2024 and the maturity date is Aug. 15, 2054. Submitting the coupon rate r=4.250\% and the yield to maturity y=4.314\% into Eq. (1) or (2), we can compute the bond price as P=98.928757 for the face value F=100. The computed price agrees with the price as listed in the official Treasury document, proving the correctness of Eq. (1) and (2).
The issue occurs in the calculation of a recent 29-year 11-month Treasury bond with the issue date Sept. 16, 2024 and the maturity date Aug. 15, 2054. The coupon rate r=4.250\% and the yield to maturity y=4.015\%. For the face value F=100, the clean price of this bond of is 104.064869 and the accrued interest is 0.369565, as listed in the official Treasury document.
To verify the price, we first calculate the price on the first coupon payment date Feb. 15, 2025 by treating the bond as a 29.5-year Treasury bond. In this case, we can directly apply Eq. (1) or (2) with N=29.5 and compute the price 104.041343. We then continue to backward this price as well as the coupon paid on Feb. 15, 2025 back to the issue date Sept. 16, 2024: \begin{equation} \frac{104.041343 + \frac{r}{2}*F} {\left(1 + \frac{y}{2}\right)^{152/184}}=104.437385\,,\tag{3}\end{equation}
Note that the price in Eq. (3) can also be calculated by first computing the price on Aug. 15, 2024 as 104.076997 by treating the bond as a 30-year Treasury bond, and then forward the price 104.076997 from Aug. 15, 2024 to Sept. 16, 2024: \begin{equation}104.076997 * (1 + y/2)^{32/184}=104.437385\,,\end{equation}
By definition, the accrued interest is \begin{equation}\frac{r}{2}*F*\frac{32}{184}=0.369565,\tag{4} \end{equation}
The final clean price is \begin{equation}104.437385-0.369565=104.067820\,,\tag{5}\end{equation}
If we use some online bond price calculator, the result agrees with my calculation as shown in the screenshot:
So where does the difference 0.003 come from?
It turns out the Treasury department uses the one-order approximation when backwarding the price from Feb. 15, 2025 to Sept. 16, 2024. Instead of using Eq. (3), their calculation is \begin{equation} \frac{104.041343 + \frac{r}{2}*F} {1 + \frac{y}{2}*\frac{152}{184}}=104.434434\,,\tag{6}\end{equation}
The question is: does the first order approximation in Eq. (6) make sense?
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