# Suppose that you have decided to purchase a house for $400,000 using an adjustable-rate... ## Question: Suppose that you have decided to purchase a house for$400,000 using an adjustable-rate mortgagewith the terms provided below. Loan-to-value ratio: 90%Index rate: one-year Treasury yield (currently 3.00%)Margin: 250 basis pointsAmortization: 15 years with monthly payments and compoundingAnnual cap: 1.5 percentage pointsLifetime cap: 5 percentage pointsAdjustment period: AnnuallyTeaser Rate 2.50%

What is the monthly payment during the first year of the loan?

a. $2133.73 b.$2400.44

c. $2667.16 d.$3268.33

e. None of the above

An adjustable rate mortgage is a mortgage whose interest rate changes periodically over the term of a loan. Depending on the conditions of a loan, each interest rate period will stay in effect for a stated length of time, after which it will reset to reflect any new market conditions. Adjustable rate mortgages are often based off of a benchmark or index. Their rates are usually marginally higher than the index rate because they will include an additional margin. When there are changes in the benchmark or index rate, the interest rate of an adjustable rate mortgage will change to reflect that change at its next adjustment period.

There are rate caps that limit how much the interest rate can change over a given time period (i.e. a year or the lifetime of the loan). Because the interest rate will fluctuate over the lifespan of the loan, the mortgage payment amount will change accordingly. Adjustable rate mortgages can result in lower total interest expenses, but they do not provide the payment predictability of fixed rate mortgages.

A mortgage payment amount is determined by the principal loan amount, the rate of financing, and the number of payment/compounding periods. We can gather all of this data from the question. The remainder of the information dictates the terms of the adjustable rate mortgage, but does not influence the mortgage payment amount in the first year.

We can calculate a mortgage payment amount using the following formula:

{eq}M = P ((r(1+r)^n) / ((1+r)^n - 1)) {/eq}

Where:

 M = Mortgage payment amount P = Principal amount r = Interest rate (expressed as a monthly rate) n = Total number of payments

We must first determine our principal amount (P), our monthly interest rate (r), and our total number of payments (n).

The principal amount is the total amount of the loan. Considering a 90% loan-to-value ratio on a $400,000 home, we are able to determine the principal amount. P = 400,000(0.90) = 360,000 =$360,000.00

To determine our monthly interest rate, we can simply divide the annual teaser interest rate by 12.

r = 0.025/12

= 0.00208333

To determine our total number of payments, we multiply the number of years the loan will be in place by the number of payments that will be made each year.

n = 15 x 12

= 180

We can now enter this data into our payment formula.

M = 360,000 ((0.00208333(1+0.00208333)^180) / ((1+0.00208333)^180 - 1))

= 2400.4412

= $2,400.44  The monthly payment during the first year is b.$2400.44