An adjustable rate mortgage, or ARM, is a mortgage whose interest rate varies over the life of the loan. The interest rate is often tied

in some fashion to the prime rate, which may go up or down. One advantage of an ARM is that it usually has an initial rate that is

lower than that of a fixed rate mortgage. In the summer of $2007,$ defaults on home mortgages led to a crisis in the U$.$S$.$ economy. At

least part of the blame was placed on ARMs. This exercise illustrates the difficulties that many homeowners faced during this period.

We make use of the following formula for the monthly payment.

$M=\frac{Pr(1+r{)}^t}{(1+r{)}^t-1}$

Here, $M$ is the monthly payment, in dollars, $P$ is the amount borrowed, in dollars, $t$ is the term of the loan, in months, and $r$ is the

monthly interest rate as a decimal, with $r=$ APR$/12.$ In this exercise, round $r$ to five decimal places.

Suppose you purchased a home in $2005,$ securing a mortgage of $$385,000$ with a $30-$year ARM.

$($a$)$ In $2005$ interest rates were at historical lows. Suppose that at the time of the loan the rate for your ARM was $4\mathrm{.}5\%$ APR.

Calculate your monthly payment. $($Round your answer to two decimal places.$)$

$M=$

$($b$)$ Suppose you earn $$7000$ per month. What percentage of your income is going toward your house payment? $($Round your

answer to the nearest percent.$)$

$\%$

$($c$)$ Suppose that after $24$ payments your ARM rate adjusted to $7\%$ APR. We will assume that after $24$ months your loan

balance is still $$385,000.($This is not as unreasonable an assumption as it may appear. The correct calculation is shown in a

later exercise.$)$ What is your monthly payment now? Be careful: The term of the loan is now $28$ years, not $30$ years. $($Round

your answer to two decimal places.$)$

$M=$

$($d$)$ Using the assumptions of part $($c$),$ what percentage of your income is going to your house payment now? $($Round your

answer to the nearest percent.$)$

$\%$