$2.$ Information about Bonds $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ is given below. Construct a portfolio of these two bonds to match the value and duration of a target bond with $\backslash (\backslash$$ $100\backslash )$ face value $20-$year $\backslash (8\backslash \%\backslash )$ coupon bond currently trading at par. $-$ Bond A: $1$ year zero coupon bond. Price $\backslash (=\backslash$$ $94\mathrm{.}26\backslash )$ for $\backslash (\backslash$$ $100\backslash )$ face value. $-$ Bond B: $30$ year zero coupon bond. Price $\backslash (\backslash$$ $16\mathrm{.}97\backslash )$ for $\backslash (\backslash$$ $100\backslash )$ face value. a$)$ What is the duration of the target bond you are matching? b$)$ If you have $\backslash (\backslash$mathbf$\{\backslash$$ $100\}\backslash )$ total to invest in bonds $\backslash (\backslash$mathbf$\{$A$\}\backslash )$ and $\backslash ($ B $\backslash ),$ how much would you invest in each so that your portfolio will have the same duration as the target bond? $($Hint solve for weight in $\backslash ($ A $\backslash )$ and weight in $\backslash ($ B $\backslash ),$ remember that these two weights must add to $1).$