$(20$ points$)$ A bank has $$100$ million worth of $5-$year Treasury bonds with a $6$ percent annual coupon selling at par, and $$80$ million worth of liabilities with a duration of $0\mathrm{.}4$ years. The bank wants to hedge its duration gap with a swap. The swap is for $5$ years, and payments are swapped at the end of each year. The floating rate will reset at the end of each year. The current $1-$year floating rate $=4\mathrm{.}5\%$ and the fixed rate also equals $4\mathrm{.}5\%.1.$ Find the amount of the swap to effectively hedge against the interest rate risk on the value of the bank's equity. $2.$ Based on your answer to the first question, does the bank need to buy or sell the swap? Provide a qualitative explanation for how this position helps the bank hedge its interest rate risk.