Question $1-(20$ points$)$

An investment fund has outstanding liabilities totalling $2$ million due in

$2$ years, $4$ million due in $4$ years, $9$ million due in $9$ years, and $12$ million due

in $12$ years. This fund holds two investments, labelled as $x$ and $Y.$

Investment $x$ provides an annual income of $1$ million for the next five

years, without any capital repayment. Investment $Y$ is a zero coupon bond

that pays a lump sum of $R$ at the end of $n$ years $($where $n$ may not be an

integer$).$ The annual effective interest rate is $5\%.$

Determine values for $n$ and $R$ that ensure the fund is adequately protected

against minor fluctuations in the interest rate. $.$