620 CHAPTER 10 Consumer Mathematics In Example 5, if Tino made all 48 payments, his finance charge would be $1576. If he decides to repay the loan after making 30 payments, must he pay the total finance charge? The answer is no. By paying off his loan early, Tino is not obligated to pay the entire finance charge. The amount of the reduction of the finance charge from paying off a loan early is called the unearned interest . We will learn about the most common way of calculating unearned interest, the actuarial method . Now we give the formula for determining the unearned interest using the actuarial method. Example 5 Financing a Restored Car Tino wants to purchase a classic 1966 Ford Mustang that sells for $9800. To purchase the car, he takes out a loan. He does not recall the APR of the loan but remembers that there are 48 payments of $237. If he did not make a down payment on the car, determine the APR. Solution First determine the finance charge by subtracting the cash price of the car from the total amount paid. Finance charge (237 48) 9800 11,376 9800 $1576 = × − = − = Now divide the finance charge by the amount of the loan and multiply this quotient by 100. 1576 9800 100 16.08 × ≈ Next find 48 payments in the left column of Table 10.2. Move to the right until you find the value that is closest to 16.08. The value closest to 16.08 is 16.0587 (circled in green). At the top of the column is the APR of 7.5%. Thus, the APR for Tino’s loan is approximately 7.5%. 7 Now try Exercise 25 MATHEMATICS TODAY Credit CARD Act of 2009 The Credit Card Accountability Responsibility and Disclosure (CARD) Act of 2009 amended the Truth in Lending Act to establish fair practices among credit card companies and their customers. As a result of the CARD Act, credit card companies must give customers advance notice of changes to interest rates and fees and allow customers the option of canceling their card before the changes take effect. The act also requires companies to inform their customers of the following: how long it will take to pay off the balance if they continue to make the minimum monthly payments; what monthly payment is necessary to pay off the balance in three years; and the exact late fees the customer will be charged if a monthly payment is late. The act has many other features, including regulations regarding the maximum interest rates and the maximum fees that companies can charge their customers. Why This Is Important Credit card use is a large part of financial literacy. If you have a credit card, it is very important to understand the rules regarding payments, interest rates, and fees. Reading your monthly statement will help you understand these rules and thus make better financial decisions. Actuarial Method for Unearned Interest u n P V V 100 = ⋅ ⋅ + where u is the unearned interest, n is the number of remaining monthly payments (excluding the current payment), P is the monthly payment, and V is the value from the APR table that corresponds to the annual percentage rate for the number of remaining payments (excluding the current payment). Example 6 illustrates the actuarial method for calculating unearned interest when an installment loan is paid off early. Example 6 Using the Actuarial Method In Example 5, we determined the APR of Tino’s loan to be about 7.5%. Instead of making his 30th payment of his 48-payment loan, Tino wishes to pay his remaining balance and terminate the loan. a) Use the actuarial method to determine how much interest Tino will save (the unearned interest, u ) by repaying the loan early. b) What is the total amount due to pay off the loan early on the day he makes his final payment? Theethawat Bootmata/ Shutterstock
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