10.5 Buying a House With a Mortgage 637 Adjustable-Rate Mortgages A mortgage in which the interest rate is not fixed for the entire length of the loan is called an adjustable-rate mortgage (ARM) or a variable-rate mortgage . Generally, an ARM rate is fixed for an initial period of time, called the initial rate period . Thereafter, the rate may go up or down based on movements in the interest rate market. Most ARMs have an initial rate period of 5 or 7 years. The initial rate is usually lower than the rate for conventional mortgages, thus making the loan attractive to buyers. However, after the initial rate period, the rate may rise and cause the monthly mortgage payments to also rise. Typically, after the initial rate period, the ARM rate is adjusted once a year. In the United States, the interest rates of ARMs are often tied to one-year U.S. Treasury bills. The stated rate on the Treasury bill is called the base rate of the ARM. A lender then adds on a certain percentage called the add-on rate or margin . For example, if the current one-year Treasury bill rate is 0.22% and the addon rate is 3.0%, then the ARM rate would be 3.22%. Should the one-year Treasury bill rate rise or fall, then so, too, would the ARM rate. Example 5 An Adjustable Rate Mortgage The Ghiselins purchased a condominium for $275,000 with a down payment of $115,000. They obtained a 15-year adjustable rate mortgage with the following terms. The interest rate is based on the one-year Treasury bill rate, which currently is 0.5%, and the add-on rate, which is 3.0%. The initial rate period is 5 years, and thereafter the interest rate is adjusted once a year and a new monthly mortgage payment is calculated. a) Determine the Ghiselins’ initial ARM rate. b) Determine the Ghiselins’ initial monthly payment for principal and interest. c) If, after the 5-year initial rate period, the rate of the one-year Treasury bill rises to 1.5%, determine the Ghiselins’ new ARM rate. Solution a) The ARM rate is the sum of the one-year Treasury bill rate, 0.5%, and the add-on rate, 3.0%. Thus, the Ghiselins’ initial ARM rate is 0.5% 3.0%, + or 3.5%. b) We will use Table 10.4. The amount of the loan is $275,000 $115,000, − or $160,000. Divide the amount of the loan by $1000 to get $160,000/$1000, or 160. Now, looking at Table 10.4 with r 3.5% = for 15 years, we determine the value 7.14883. 160 $7.14883 $1143.81 × ≈ Thus, the initial monthly payment for principal and interest is $1143.81. Note that we could have also used the principal and interest payment formula to calculate the monthly payment for principal and interest. c) The sum of the new one-year Treasury bill rate, 1.5%, and the add-on rate, 3.0%,is 4.5%. Thus, the Ghiselins’ new ARM rate is 4.5%. 7 Now try Exercise 27 MATHEMATICS TODAY Other Types of Mortgages When purchasing a home, consumers may have options other than conventional mortgages and adjustable-rate mortgages. Federal Housing Authority (FHA) and Veterans Affairs (VA) mortgages are commonly used by qualified buyers who are purchasing their first home. Both of these mortgages assist the buyer by requiring a smaller down payment than a conventional mortgage. Both of these mortgages also are guaranteed by the federal government, thus providing incentive to lenders to extend the loan to the consumer. Other benefits are provided by FHA and VA mortgages. More information about FHA, VA, and other types of mortgages can be found online at Home.Treasury.gov. Why This Is Important When purchasing a home, knowing about the various mortgage options can help you make an informed decision. To prevent rapid increases in interest rates of adjustable rate mortgages, some banks have a rate cap. A rate cap limits the maximum amount the interest rate may change. A periodic rate cap limits the amount the interest rate may increase in any one period. For example, your mortgage could provide that your rate can go up only 1% per year. An aggregate rate cap limits the interest rate increase and decrease over the entire life of the loan. If the initial interest rate is 6% and the aggregate rate cap is 2%, the interest rate could go no higher than 8% and no lower than 4% over the life of the mortgage. Conventional mortgages and adjustable-rate mortgages are not the only methods of financing the purchase of a home. The Mathematics Today box, left, discusses some other types of mortgages that are available to consumers. More information about other types of mortgages can be found online at Home.Treasury.gov. Instructor Resources for Section 10.5 in MyLab Math • Objective-Level Videos 10.5 • Interactive Concept Video: Amortization Schedules • Interactive Concept Video: Regular Monthly Mortgage Payments • Animation: Mortgage Calculator • PowerPoint Lecture Slides 10.5 • MyLab Exercises and Assignments 10.5 Imagenet/Shutterstock
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