You buy a US bond. The bond has a nominal value of USD $100,000$ and bears $4\%$ interest. The bond bears interest payments and they are paid every year. The bond's principal will be paid in one payment after $12$ years. The next interest due date is exactly $1$ year. The yield at which you buy the bond is $5\mathrm{.}8\%$

a$)$ At what price do you buy the bond?

b$)$ What will be the price of the bonds in $3$ years if we assume that they will be sold at the yield $($YTM$)$ of $3\mathrm{.}7\%?$