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} which can be simplified as \begin{equation}\frac{P}{F}= \frac{r}{y}+\frac{1-\frac{r}{y}}{(1+ \frac{y}{2})^{2N}}\,.\tag{2}\end{equation} where $P$ is the bond price and $F$ is the face value of the bond. $r$ is the coupon rate (or the interest rate in Treasury documents) and $y$ is the yield to maturity. Eq. (1) and (2) are computed in a semi-annual basis: there are $2N$ periods for a $N$-year bond.

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} where $152$ is the number of days from Sept. 16, 2024 to Feb. 15, 2025 and $184$ is the number of days from Aug. 15, 2024 to the first coupon payment date Feb. 15, 2025. 

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} where $32$ is the number of days from Aug. 15, 2024 to Sept. 16, 2024 and $184$ is the number of dates from Aug. 15, 2024 to the first coupon payment date Feb. 15, 2025.

By definition, the accrued interest is \begin{equation}\frac{r}{2}*F*\frac{32}{184}=0.369565,\tag{4} \end{equation} which agrees with the given value, showing the correctness of day counting.

The final clean price is \begin{equation}104.437385-0.369565=104.067820\,,\tag{5}\end{equation} which differs $0.003$ from the given clean price $104.064869$! 

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} and the final clean price is $104.434434-0.369565= 104.064869$, which matches the given price perfectly.

The question is: does the first order approximation in Eq. (6) make sense?









Location: 美国加利福尼亚帕罗奥图

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