One type of mortgage is the adjustable rate mortgage $($ARM$).$ With an ARM, the interest rate

changes periodically as determined by a measure of current interest rates, called an index. The

most common index in the United States, called the $1-$year CMT $($constant$-$maturity Treasury

index has ranged from $1\%$ to about $6\mathrm{.}3\%).$ The interest rate on the ARM is reset by adding a fixed

percent called the margin to the index percent. For instance, if the current value of the CMT index

is $3\%$ and the margin is $2\mathrm{.}7\%,$ then the adjusted interest rate for the mortgage would be $5\mathrm{.}7\%.$ The

margin usually remains fixed for the duration of the loan.

Of the many types of ARMs, we will consider the $5/1$ ARM. With this type of mortgage, the

interest rate is fixed for the first five years, and then is readjusted each year depending on the value

of the index.

$1.)$ Consider a $$200,0005/1$ ARM that has a $2\mathrm{.}7\%$ margin, is based on the CMT index, and

has $30-$year duration. Suppose that the interest rate is initially $5\mathrm{.}7\%$ and the value of the

CMT index is $4\mathrm{.}5\%$ five years later when the rate adjusts. Assume monthly compounding.

a$.)$ Calculate the monthly payment for the first $5$ years.

b$.)$ Calculate the unpaid balance at the end of the first $5$ years.

c$.)$ Calculate the monthly payment for the $6$ th

year.