Suppose you purchased a house exactly $5$ years ago at a cost of $$450,000.$ Assume that you were able to make a down payment of $20\%$ of the purchase price, and you negotiated a conventional $25-$year mortgage for the remaining balance. The fixed rate on the mortgage was $8\mathrm{.}25\%,$ and banks are required to compound interest semi$-$annually.

$($i$)$ What is your monthly payment

$($ii$)$ Suppose mortgage rates fell over the five years such that now, rates on $20-$year mortgages are presently $5\mathrm{.}85\%.$ What is your remaining mortgage balance immediately prior to refinancing? How much have you paid in interest, and how much have you paid in principal over the last $5$ years? Calculate your new monthly payments if you do indeed refinance.

$($iii$)$ Suppose you decide to continue to pay your initial monthly payments instead of the newly calculated payments. How long will it take you to pay off your mortgage?