§8.5, Installment Loans, Amortization, and Credit Cards

Learning Targets Compute the monthly payment and interest costs for a mortgage. Prepare a partial loan amortization schedule. Compute payments and interest for other kinds of installment loans. Find the interest, balance due, and the minimum monthly payment for credit card loans.

VOCABULARY A mortgage is a long-term loan for the purpose of buying a home. The down payment is the portion of the sale price of the home that the buyer initially pays to the seller. The amount of the mortgage is the difference between the sale price and the down payment. Some companies, called mortgage brokers, offer to find you a mortgage lender willing to make you a loan. Fixed-rate mortgages have the same monthly payment during the entire time of the loan. Variable-rate mortgages known as adjustable-rate mortgages (ARMs), have payment amounts that change from time to time depending on changes in the interest rate.

Computations Involved with Buying a Home Most lending institutions require the buyer to pay one or more points at the time of closing—that is, the time at which the mortgage begins. A point is a one-time charge that equals 1% of the loan amount. For example, two points means that the buyer must pay 2% of the loan amount at closing. A document, called the Truth-in-Lending Disclosure Statement, shows the buyer the APR for the mortgage. In addition, lending institutions can require that part of the monthly payment be deposited into an escrow account, an account used by the lender to pay real estate taxes and insurance.

Computation Involved with Buying a Home Loan Payment Formula for Fixed Installment Loans The regular payment amount, PMT, required to repay a loan of P dollars paid n times per year over t years at an annual rate r is given by

Example 1: Computing the Monthly Payment and Interest Costs for a Mortgage The price of a home is $195,000. The bank requires a 10% down payment and two points at the time of closing. The cost of the home is financed with a 30-year fixed rate mortgage at 7.5%. Find the required down payment. Find the amount of the mortgage. How much must be paid for the two points at closing? Find the monthly payment (excluding escrowed taxes and insurance). Find the total interest paid over 30 years.

Example 1: Computing the Monthly Payment and Interest Costs for a Mortgage Solution: The required down payment is 10% of $195,000 or 0.10  $195,000 = $19,500. The amount of the mortgage is the difference between the price of the home and the down payment.

Example 1: Computing the Monthly Payment and Interest Costs for a Mortgage To find the cost of two points on a mortgage of $175,500, find 2% of $175,500. 0.02  $175,000 = $3510 The down payment ($19,500) is paid to the seller and the cost of two points ($3510) is paid to the lending institution.

Example 1: Computing the Monthly Payment and Interest Costs for a Mortgage We need to find the monthly mortgage payment for $175,500 at 7.5% for 30 years. We use the loan payment formula for installment loans. The monthly mortgage payment for principal and interest is approximately $1227.00.

Example 1: Computing the Monthly Payment and Interest Costs for a Mortgage e. The total cost of interest over 30 years is equal to the difference between the total of all monthly payments and the amount of the mortgage. The total of all monthly payments is equal to the amount of the monthly payment multiplied by the number of payments. We found the amount of each monthly payment in (d): $1227. The number of payments is equal to the number of months in a year, 12, multiplied by the number of years in the mortgage, 30: 12  30 = 360. Thus, the total of all monthly payments = $1227  360.

Example 1: Computing the Monthly Payment and Interest Costs for a Mortgage Now we calculate the interest over 30 years. The total interest paid over 30 years is approximately $266,220.

Loan Amortization Schedules When a loan is paid off through a series of regular payments, it is said to be amortized, which literally means “killed off.” Although each payment is the same, with each successive payment the interest portion decreases and the principal portion increases. A document showing important information about the status of the mortgage is called a loan amortization schedule.

Example 2: Preparing a Loan Amortization Schedule Prepare a loan amortization schedule for the first two months of the mortgage loan shown in the following table:

Example 2: Preparing a Loan Amortization Schedule Solution: We begin with payment number 1. Interest for the month = Prt = $130,000  0.095  1/12 ≈ $1029.17 Principal payment = Monthly payment  Interest payment = $1357.50  $1029.17 = $328.33 Balance of loan = Principal balance  Principal payment = $130,000  $328.33 = $129,671.67

Example 2: Preparing a Loan Amortization Schedule Now, starting with a loan balance of $129,671.67, we repeat these computations for the second month. Interest for the month = Prt = $129,671.67  0.095  1/12 = $1026.57 Principal payment = Monthly payment – Interest payment = $1357.50 – $1026.57 = $330.93 Balance of loan = Principal balance – Principal payment = $129,671.67 – $330.93 = $129,340.74

Example 2: Preparing a Loan Amortization Schedule

Example 3: Comparing Car Loans You decide to take a $20,000 loan for a new car. You can select one of the following loans, each requiring regular monthly payments: Installment Loan A: 3-year loan at 7%. Installment Loan A: 5-year loan at 9%. Find the monthly payments and the total interest for Loan A. Find the monthly payments and the total interest for Loan B. Compare the monthly payments and total interest for the two loans.

Example 3: Comparing Car Loans Solution: We first determine monthly payments and total interest for Loan A. The monthly payments are approximately $618.

Example 3: Comparing Car Loans Now we calculate the interest paid over 3 years or 36 months. The interest paid over 3 years is approximately $2248.

Example 3: Comparing Car Loans Next, we determine monthly payment and total interest for Loan B. The monthly payments are approximately $415.

Example 3: Comparing Car Loans Now we calculate the interest over 5 years, or 60 months. The total interest paid over 5 years is approximately $4900.

Example 3: Comparing Car Loans c. The monthly payments and total interest for the two loans is given in the table below.