$5.$ Consider a $30-$year bond with a face value of $\backslash (\backslash$$ $1,000\backslash )$ and a coupon rate of $\backslash (8\mathrm{.}8\backslash \%\backslash )($annual coupon payments$).$ The bond currently has a YTM of $\backslash (8\mathrm{.}1\backslash \%\backslash ).$ a$)$ What is this bond's duration and modified duration? b$)$ What is this bond's convexity? c$)$ Assume the YTM increases from $\backslash (8\mathrm{.}1\backslash \%\backslash )$ to $\backslash (9\mathrm{.}1\backslash \%\backslash )($a change of $\backslash (+1\backslash \%\backslash )).$ Estimate the percentage change in price of this bond based on this change in yield and using convexity and duration.