Northbrook Bank purchases a four$-$year cap for a fee of $3$ percent of notional principal valued at $$100$ million, with an interest rate ceiling of $9$ percent and LIBOR as the index representing the market interest rate. Assume that LIBOR is expected to be $8$ percent, $10$ percent, $12$ percent, and $13$ percent $($respectively$)$ at the end of each of the next four years.

$($A$).$ Determine the initial fee paid, as well as the expected payments to be received by Northbrook if LIBOR moves as forecasted.

$($b$)$ Determine the dollar amount to be received $($or paid$)$ by the seller of the interest rate cap based on the assumed forecasts of LIBOR.