Adjustable rate mortgage—approximating payments: 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:
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 $325,000 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 4.5% APR. Calculate your monthly payment.
b. Suppose you earn $6000 per month. What percentage of your income is going toward your house payment?
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 $325,000. (This is not as unreasonable an assumption as it may appear. The correct calculation) What is your monthly payment now? Be careful: The term of the loan is now 28 years, not 30 years.
d. Using the assumptions of part c, what percentage of your income is going to your house payment now?