5. Numerical exercise. In the file BondQuotes-PS01-2022-23.csv posted on the website you will find quotes for US Treasury bonds published by the Wall Street Journal in on September 23rd, 2022. All bonds chosen mature on February 15th, in different years, starting from 2023 until 2050. Use the software R for the following calculations. If you need help for question (d), please wait for the TA section where more detailed instructions will be provided. (a) What days of each year do these bond make their semi-annual payments? (b) What was the day of the last semi-annual payment for each bond? (c) What is the accrued interest and the dirty price for each bond? (take the clean price to be the mid point between the bid and the ask price.) (d) Assume that at the current time t, which is September 23, 2022, the spot rate is given by a third order polynomial: rt,u = a1u + a2u2 + a3u3,

\begin{tabular}{|r|r|r|r|} \hline Maturity & Coupon & \multicolumn{2}{|l|}{ Bid } \\ \hline 2/15/23 & 2 & 99.13 & 99.134 \\ \hline 2/15/24 & 2.75 & 97.296 & 97.302 \\ \hline 2/15/25 & 2 & 94.296 & 94.302 \\ \hline 2/15/26 & 6 & 105.194 & 105.2 \\ \hline 2/15/27 & 2.25 & 92.206 & 92.212 \\ \hline 2/15/28 & 2.75 & 94.006 & 94.012 \\ \hline 2/15/29 & 2.625 & 92.21 & 92.214 \\ \hline 2/15/30 & 1.5 & 85.13 & 85.14 \\ \hline 2/15/31 & 1.125 & 81.17 & 81.18 \\ \hline 2/15/32 & 1.875 & 85.156 & 85.166 \\ \hline 2/15/36 & 4.5 & 109.256 & 109.266 \\ \hline 2/15/37 & 4.75 & 112.214 & 112.224 \\ \hline 2/15/39 & 3.5 & 97.08 & 97.09 \\ \hline 2/15/41 & 4.75 & 112.176 & 112.196 \\ \hline 2/15/43 & 3.125 & 88.152 & 88.172 \\ \hline 2/15/45 & 2.5 & 78.17 & 78.19 \\ \hline 2/15/47 & 3 & 86.022 & 86.042 \\ \hline 2/15/49 & 3 & 87.264 & 87.284 \\ \hline 2/15/51 & 1.875 & 68.09 & 68.11 \\ \hline 2/15/52 & 2.25 & 74.266 & 74.286 \\ \hline \end{tabular} \begin{tabular}{|r|r|r|r|} \hline Maturity & Coupon & \multicolumn{2}{|l|}{ Bid } \\ \hline 2/15/23 & 2 & 99.13 & 99.134 \\ \hline 2/15/24 & 2.75 & 97.296 & 97.302 \\ \hline 2/15/25 & 2 & 94.296 & 94.302 \\ \hline 2/15/26 & 6 & 105.194 & 105.2 \\ \hline 2/15/27 & 2.25 & 92.206 & 92.212 \\ \hline 2/15/28 & 2.75 & 94.006 & 94.012 \\ \hline 2/15/29 & 2.625 & 92.21 & 92.214 \\ \hline 2/15/30 & 1.5 & 85.13 & 85.14 \\ \hline 2/15/31 & 1.125 & 81.17 & 81.18 \\ \hline 2/15/32 & 1.875 & 85.156 & 85.166 \\ \hline 2/15/36 & 4.5 & 109.256 & 109.266 \\ \hline 2/15/37 & 4.75 & 112.214 & 112.224 \\ \hline 2/15/39 & 3.5 & 97.08 & 97.09 \\ \hline 2/15/41 & 4.75 & 112.176 & 112.196 \\ \hline 2/15/43 & 3.125 & 88.152 & 88.172 \\ \hline 2/15/45 & 2.5 & 78.17 & 78.19 \\ \hline 2/15/47 & 3 & 86.022 & 86.042 \\ \hline 2/15/49 & 3 & 87.264 & 87.284 \\ \hline 2/15/51 & 1.875 & 68.09 & 68.11 \\ \hline 2/15/52 & 2.25 & 74.266 & 74.286 \\ \hline \end{tabular}