A couple takes out a $$1,500,000$ mortgage amortized by monthly payments for $25$ years. Interest is $9\mathrm{.}5\%/$year compounded semi$-$annually for the first $5$ years and could change at the end of that time. No penalty is charged for full or partial payment of the mortgage after those $5$ years.

$($a$)$ Calculate the regular monthly payment and the reduced, final mortgage payment assuming the $9\mathrm{.}5\%$ rate continues for the entire $25$ years.

$($b$)$ What is the outstanding balance after

$($i$)2$ years;

$($ii$)5$ years?

$($c$)$ During the $5-$year period, there is a penalty of $6$ months$$ interest on any principal repaid early. After $2$ years, interest rates on $3-$year mortgages fall to $8\%/$year compounded semi$-$annually. Calculate the new monthly payment if the loan is refinanced assuming the $8\%$ rate continues for the entire remaining $23$ years. If the couple believes it can earn interest on their savings account at a rate of $3\%/$year compounded semi$-$annually for the next three years, would it pay to refinance