The following information relates to Questions $11$ and $12$

The investment manager for a UK defined$-$benefit pension scheme is considering two bonds about to be issued by a large life insurance company. The first is a $30-$year, $4\%$ semiannual coupon payment bond. The second is a $100-$year, $4\%$ semiannual coupon payment "century" bond. Both bonds are expected to trade at par value at issuance. $($NOTE: Use $6$ decimal places in your calculator when you calculate!$)$

$11.$ Calculate the approximate modified duration and approximate convexity for the $30-$year bond using a $5bp(0\mathrm{.}05\%)$ increase and decrease in the annual yield$-$to$-$maturity $($YTM$).$

$12.$ Calculate the approximate modified duration and approximate convexity for the $100-$year bond $($the "century" bond$)$ using a $5bp(0\mathrm{.}05\%)$ increase and decrease in the annual yield$-$tomaturity $($YTM$).$