You plan to retire in exactly $20$ years. Your goal is to create a fund that will allow you to receive N$$400000$ at the end of each year for the $30$ years between retirement and death $($you have estimated your death according to life mortality tables used by insurance companies$).$ You know that you will be able to earn $11\%$ per year during the $30-$year retirement period.

$($a$)$ How large a fund will you need when you retire in $20$ years in order to provide the $30-$year N$$400000$ per year retirement annuity?

$($b$)$ How much would you need to provide today as a single amount to the fund as calculated in $($a$)$ if you earn only $9\%$ per year during the $20$ years before retirement?