An FI is planning the purchase of a $$5$ million loan to raise the existing average duration of its

assets from $3\mathrm{.}5$ years to $5$ years. It currently has total assets worth $$20$ million comprising $$5$

million in cash $(0$ duration$)$ and $$15$ million in loans. All the loans are fairly priced.

$($a$)$ Assuming it uses the cash to purchase the loan, should it purchase the loan if its duration is

seven years? $($Hint to students: Recall what we learned in Lecture $4$ on Duration Gap Model.

Duration of a portfolio of assets or liabilities is calculated by summing the proportion weighted

duration of each asset.$)$

$($b$)$ What asset duration loans should it purchase in order to raise its average duration to five years?