Chapter 1.1, Problem 31E

Reminder Round all answers to two decimal places unless otherwise indicated.

Note Some of the formulas below use the special number $e$, which was presented in the Prologue.

Adjustable Rate Mortgage-Exact Payments This is a continuation of Exercise $30$. In Exercise $30$, we only approximated the increased payment when the rate for an ARM increases: We assumed that over the first $24$ months, you accrue $0$ equity in your home. (Your equity in a home is the total you have paid toward the principal.)We address that point here. We make use of the equity formula

$E=P\times \frac{{\left(1+r\right)}^k-1}{{\left(1+r\right)}^t-1}$.

Here $E$ is the equity, in dollars, after $k$ monthly payments. The quantities $P,t,$ and $r$ are defined as in Exercise $30$. Round $r$ to five decimal places.

We assume as in Exercise $30$ that you borrowed $\$325,000$ at an initial APR of $4.5\%$ with a term of $30$ years.

a. What equity have you accrued after $24$ months?

b. When your rate adjusts to $7\%$ after $24$ months, the new amount borrowed is $\$325,000$ less your equity. The term is now $28$ years. What is your new monthly payment?

30. 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 same 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 $2007$, 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:

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

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 $r=\text{APR/12}$. 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 a historical lows. Suppose that at the time of the loan, the rate for your ARM was a $4.5\%$ 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 is shown in Exercise 31.) 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?