Problem $3($Required$,35$ marks$)$ An investor has $\backslash (\backslash$$ $15000\backslash )$ initially and would like to invest the capital for $6$ years. The investor would like to invest capital into the following $3$ bonds: $-$ Bond A$-3-$year coupon bond with face value $\backslash ($ F$\_\{$A$\}=100\backslash )$ and annual coupon rate $4\backslash \%$ payable annually. $-$ Bond $\backslash (\backslash$mathrm$\{$B$\}-8\backslash )$ year zero coupon bond with face value $\backslash ($ F$\_\{$B$\}=100\backslash )-$ Bond C$-12$ year zero coupon bond with face value $\backslash ($ F$\_\{$C$\}=100\backslash )$ The term structure assumes to be flat and the annual effective interest rate is $\backslash ($ i$\_\{0\}=\backslash )5\backslash \%.$ Suppose that the investor wishes to achieve an annual return not less than $5\backslash \%,$ i$.$e$.\backslash ($ i$\_\{$I R R$\}\backslash$geq $5\backslash \%\backslash ),$ regarding of the future interest rate movement. $($a$)$ Describe the best bond portfolio for the investor. Provide full justification to your answer. $(*$Note: Please express your proposed portfolio in the following way: Invest $\backslash$$ $($exact amount$)$ into bond A Invest $\backslash$$ $($exact amount$)$ into bond B Invest $\backslash$$ $($exact amount$)$ into bond $\backslash ($ C $\backslash ))($b$)$ Using second$-$order approximation, estimate the internal rate of return $($IRR$)$ of the bond investment if the interest rate is raised to $\backslash (7\mathrm{.}5\backslash \%\backslash )$ in future. $($c$)$ Suppose that the interest rate is raised to $\backslash ($ i$=6\backslash \%\backslash )$ after $1$ year and the investor wishes to rebalance the portfolio that can maintain the effectiveness of the bond investment strategy, determining the best balancing portfolio. Provide full justification to your answer.