$5.(\backslash (\backslash$mathbf$\{20\}\backslash )$ points in total$)$ A $\backslash (6-\backslash )$ month zero$-$coupon bond with a face value of $\backslash (\backslash$$ $100\backslash )$ is currently selling for $\backslash (\backslash$$ $98\backslash ).$ In addition, a $1-$year bond with a face value of $\backslash (\backslash$$ $100\backslash ),$ offering a $\backslash (5\backslash \%\backslash )$ annual coupon rate with semi$-$anntal payments, is priced at $\backslash (\backslash$$ $100\backslash ).$

a$)$ Calculate the $6-$month and $1-$year continuously compounded zero rates. $(10$ points$)$

b$)$ Determine the forward rate between the $6-$month and $1-$year periods. $(10$ points$)$