Math in Our World Section 8.4 Installment Buying

Learning Objectives Find amount financed, total installment price, and finance charge for a fixed installment loan. Use a table to find APR for a loan. Compute unearned interest and payoff amount for a loan paid off early. Compute credit card finance charges using the unpaid balance method. Compute credit card finance charges using the average daily balance method.

Installment Buying Installment buying is when an item is purchased and the buyer pays for it by making periodic partial payments, or installments.

Fixed Installment Loans A fixed installment loan is a loan that is repaid in equal payments. Sometimes the buyer will pay part of the cost at the time of purchase. This is known as a down payment.

Fixed Installment Loans The amount financed is the amount a borrower will pay interest on. Amount financed = Price of item – Down payment The total installment price is the total amount of money the buyer will ultimately pay. Total installment price = Sum of all payments + Down pmt The finance charge is the interest charged for borrowing the amount financed. Finance charge = Total installment price – Price of item

EXAMPLE 1 Calculating Information About a Car Loan Cat bought a 2-year old Santa Fe for $12,260. Her down payment was $3,000, and she will have to pay $231.50 for 48 months. Find the amount financed, the total installment price, and the finance charge. SOLUTION Using the formulas previously shown: Amount financed = Cash price – Down payment = $12,260 – $3,000 = $9,260

EXAMPLE 1 Calculating Information About a Car Loan SOLUTION Since she paid $231.50 for 48 months and her down payment was $3,000, Total installment price = Total of monthly payments + Down pmt = (48 x $231.50) + $3,000 = $14,112.00 Now we can find the finance charge: Finance charge = Total installment price – Cash price = $14,112.00 – $12,260.00 = $1,852.00 The amount financed was $9,260.00; the total installment price was $14,112.00, and the finance charge was $1,852.00.

EXAMPLE 2 Computing a Monthly Payment After a big promotion, a young couple bought $9,000 worth of furniture. The down payment was $1,000. The balance was financed for 3 years at 8% simple interest per year. (a) Find the amount financed. (b) Find the finance charge (interest). (c) Find the total installment price. (d) Find the monthly payment.

EXAMPLE 2 Computing a Monthly Payment SOLUTION (a) Amount financed = Price of item – Down payment = $9,000 – $1,000 = $8,000 (b) To find the finance charge, we use the simple interest formula: I = Prt = $8,000 x 0.08 x 3 = $1,920 (c) In this case, the total installment price is simply the cost of the furniture plus the finance charge: Total installment price = $9,000 + $1,920 = $10,920

EXAMPLE 2 Computing a Monthly Payment SOLUTION (d) To calculate the monthly payment, divide the amount financed plus the finance charge ($8,000 + $1,920) by the number of payments: Monthly payment = $9,920 ÷ 36 = $275.56 In summary, the amount financed is $8,000, the finance charge is $1,920, the total installment price is $10,920, and the monthly payment is $275.56.

Annual Percentage Rate (APR) Lenders are required by law to disclose an annual percentage rate, or APR, that reflects the true interest charged. This allows consumers to compare loans with different terms. This is a partial APR table. See Text for a more complete table.

Using the APR Table Step 1 Find the finance charge per $100 borrowed using the formula Step 2 Find the row in the table marked with the number of payments and move to the right until you find the amount closest to the number from Step 1. Step 3 The APR (to the nearest half percent) is at the top of the corresponding column.

EXAMPLE 3 Finding APR Burk Carter purchased a color laser printer for $600.00. He made a down payment of $50.00 and financed the rest for 2 years with a monthly payment of $24.75. Find the APR. SOLUTION Find the finance charge per $100.00. The total amount he will pay is $24.75 per month x 24 payments, or $594.00. Since he financed $550.00, the finance charge is $594.00 – $550.00 = $44.

EXAMPLE 3 Finding APR SOLUTION Step 2 Find the row for 24 payments and move across the row until you find the number closest to $8.00. In this case, it is exactly $8.00. Step 3 Move to the top of the column to get the APR. It is 7.5%.

Unearned Interest One way to save money on a fixed installment loan is to pay it off early. This will allow a buyer to avoid paying the entire finance charge. The amount of the finance charge that is saved when a loan is paid off early is called unearned interest. There are two methods for calculating unearned interest, the actuarial method and the rule of 78.

Actuarial Method where u = unearned interest k = number of payments remaining, excluding the current one R = monthly payment h = finance charge per $100 for a loan with the same APR and k monthly payments

EXAMPLE 4 Using the Actuarial Method Our friend Burk from Example 3 decides to use part of his tax refund to pay off the full amount of his laser printer with his 12th payment. Find the unearned interest and the payoff amount. SOLUTION To use the formula for the actuarial method, we’ll need values for k, R, and h. Half of the original 24 payments will remain, so k = 12. From Example 3, the monthly payment is $24.75 and the APR is 7.5%.

EXAMPLE 4 Using the Actuarial Method SOLUTION Using the APR Table, we find the row for 12 payments and the column for 7.5%; the intersection shows $4.11, so h = $4.11. Substituting k = 12, R = 24.75 and h = 4.11:

EXAMPLE 4 Using the Actuarial Method SOLUTION The unearned interest is $11.72. The payoff amount is the amount remaining on the loan minus unearned interest. At this point, Burk has made 11 payments, so there would be 13 remaining if he were not paying the loan off early. Payoff amount = (13 x $24.75) – $11.72 = $310.03 With a payment of $310.03, Burk is the proud owner of a laser printer.

The Rule of 78 where u = unearned interest f = finance charge k = number of remaining monthly payments n = original number of payments

EXAMPLE 5 Using the Rule of 78 A $5,000 car loan is to be paid off in 36 monthly installments of $172. The borrower decides to pay off the loan after 24 payments have been made. Find the amount of interest saved by paying the loan off early. Use the rule of 78. SOLUTION Find the finance charge (i.e. total interest). $172 x 36 = $6,192 ($172 x 36 payments) $6,192 – $5,000 = $1,192 (Total payments – Amount financed)

EXAMPLE 5 Using the Rule of 78 SOLUTION Substitute into the formula using f = $1,192, n = 36, and k = 36 – 24 = 12. By paying off the loan a year early, the borrower saved $139.60.

Open-Ended Credit Open-ended credit has no fixed number of payments or payoff date. By far the most common example of this is credit cards. With the unpaid balance method, interest is charged only on the balance from the previous month.

EXAMPLE 6 Computing a Credit Card Finance Charge For the month of April, Elliot had an unpaid balance of $356.75 at the beginning of the month and made purchases of $436.50. A payment of $200.00 was made during the month. The interest on the unpaid balance is 1.8% per month. Find the finance charge and the balance on May 1. SOLUTION Step 1 Find the finance charge on the unpaid balance using the simple interest formula with rate 1.8%. (r = 0.018) I = Prt = $356.75 x 0.018 x 1 (1 month, so t = 1) = $6.42 (rounded)

EXAMPLE 6 Computing a Credit Card Finance Charge SOLUTION The finance charge is $6.42. Step 2 To the unpaid balance, add the finance charge and the purchases for the month; then subtract the payment to get the new balance. New balance = $356.75 + $6.42 + $436.50 – $200 = $599.67 The new balance as of May 1 is $599.67.

Average Daily Balance Method When using the average daily balance method, the balance for each day of the month is used to compute an average daily balance, and interest is computed on that average.

Average Daily Balance Method Procedure for the ADB Method Step 1 Find the balance as of each transaction. Step 2 Find the number of days for each balance. Step 3 Multiply the balances by the number of days and find the sum. Step 4 Divide the sum by the number of days in the month. Step 5 Find the finance charge (multiply the average daily balance by the monthly rate). Step 6 Find the new balance (add the finance charge to the balance as of the last transaction).

EXAMPLE 7 Computing a Credit Card Finance Charge Betty’s credit card statement showed the following transactions during the month of August. August 1 Previous balance $165.50 August 7 Purchases 59.95 August 12 Purchases 23.75 August 18 Payment 75.00 August 24 Purchases 107.43 Find the average daily balance, the finance charge for the month, and the new balance on September 1. The interest rate is 1.5% per month on the average daily balance.

EXAMPLE 7 Computing a Credit Card Finance Charge SOLUTION Step 1 Find the balance as of each transaction. August 1 $165.50 August 7 $165.50 + $59.95 = $225.45 August 12 $225.45 + $23.75 = $249.20 August 18 $249.20 + $75.00 = $174.20 August 24 $174.20 + $107.43 = $281.63 Step 2 Find the number of days for each balance. Date Balance Days Calculations August 1 $165.50 6 (7 – 1 = 6) August 7 $225.45 5 (12 – 7 = 5) August 12 $249.20 6 (18 – 12 = 6) August 18 $174.20 6 (24 – 18 = 6) August 24 $281.63 8 (31 – 24 + 1 = 8)

EXAMPLE 7 Computing a Credit Card Finance Charge SOLUTION Step 3 Multiply each balance by the number of days, and add these products. Date Balance Days Calculations August 1 $165.50 6 $165.50(6) = $993.00 August 7 $225.45 5 $225.45(5) = $1,127.25 August 12 $249.20 6 $249.20(6) = $1,495.20 August 18 $174.20 6 $174.20(6) = $1,045.20 August 24 $281.63 8 $281.63(8) = $2,253.04 31 $6,913.69 Step 4 Divide the total by the number of days in the month to get the average daily balance. Average daily balance = $6,913.69/31 ≈ $223.02

EXAMPLE 7 Computing a Credit Card Finance Charge SOLUTION Step 5 Find the finance charge. Multiply the average daily balance by the rate, which is 1.5%, or 0.015. Finance charge = $223.02 x 0.015 ≈ $3.35. Step 6 Find the new balance. Add the finance charge to the balance as of the last transaction. New balance: $281.63 + $3.35 = $284.98 The average daily balance is $223.02. The finance charge is $3.35, and the new balance is $284.98.