9.3. Consumer Loans Objectives

Transcription

1 9.3 Consumer Loans Objectives 1. Determine payments for an add-on loan. 2. Compute finance charges on a credit card using the unpaid balance method. 3. Use the average daily balance method to compute credit card charges. 4. Compare credit card finance charge methods. Debt is a bottomless sea. Thomas Carlyle In the spring of 2008, publications such as Fortune and Money magazines featured articles highlighting a record U.S. credit card debt of $915 billion, and financial institutions such as Citigroup, American Express, and Bank of America were, to quote Fortune magazine, strapping on their Kevlar vests, anticipating an impending financial explosion. From ancient times to the present, the Bible, William Shakespeare, Benjamin Franklin, and others have warned about the dangers of unbridled credit. In this section, we will

2 9.3 y Consumer Loans 415 explain the mathematics of credit cards and show you how you can use credit wisely, to avoid drowning in Carlyle s bottomless sea. Imagine that you have just signed the lease for your first apartment and now all you have to do is furnish it. If you buy living room furniture for $1,100, which you pay for in payments, you are taking out an installment loan. Loans having a fixed number of payments are called closedended credit agreements (or installment loans). Each payment is called an installment. The size of your payments is determined by the amount of your purchase and also by the interest rate that the seller is charging. The interest charged on a loan is often called a finance charge. KEY POINT The add-on interest method is a simple way to compute payments on an installment loan. The Add-On Interest Method We use the simple interest formula from Section 9.2 to calculate the finance charge for an installment loan. To determine the payments for an installment loan, we add the simple interest due on the loan to the loan amount and then divide this sum by the number of monthly payments. FORMULA FOR DETERMINING THE MONTHLY PAYMENT OF AN INSTALLMENT LOAN monthly payment = P I n + where P is the amount of the loan, I is the amount of interest due on the loan, and n is the number of monthly payments., This method is sometimes called the add-on interest method because we are adding on the interest due on the loan before determining the payments. Quiz Yourself 11 Suppose that you take an installment loan for $360 for 1 year at an annual interest rate of 21%. What are your monthly EXAMPLE 1 Determining Payments for an Add-On Interest Loan A new pair of Bose speakers for your home theater system costs $720. If you take out an addon loan for 2 years at an annual interest rate of 18%, what will be your monthly SOLUTION: We first use the simple interest formula to calculate the interest: I = Prt = 720(0.18)2 = Next, we add the interest to the purchase price: = To find the monthly payments, we divide this amount by 24: Monthly payments are therefore $ Now try Exercises 5 to 12. ] 11 = In Example 1, the annual interest rate of 18% is quite misleading. If we think about it, the purchase price was $720, so it would be fair to say that you are paying off $720/24 = $30 of the loan amount each month; the other $10.80 is interest. When you reach the last month, although you only owe $30 on the purchase, you are still paying $10.80 in interest. Simple arithmetic shows that 10.80/30 = So in a certain sense, the interest rate for the last month of the loan is actually 36%. Because you are paying 36% interest for one month, this is equivalent to an annual interest rate of 12 * 36% = 432%. What we want to point out here is that although simple interest is easy to compute, as you pay off the loan amount, the actual interest you are paying on the outstanding balance is higher than the stated interest rate.

3 416 CHAPTER 9 y Consumer Mathematics When you use your credit card to pay for gas at a gas station, you are using open-ended credit. With open-ended credit, the calculation of finance charges can be more complicated than with closed-ended credit. Although you may be making monthly payments on your loan, you may also be increasing the loan by making further purchases. There are several ways that credit card companies compute finance charges. We will look at two methods and compare them at the end of this section. You will see that if you understand the method being used to compute your finance charges, you can use this information to reduce the cost of borrowing money. KEY POINT The unpaid balance method computes finance charges on the balance at the end of the previous month. The Unpaid Balance Method The first method that we will discuss for computing finance charges is called the unpaid balance method. With this method, the interest is based on the previous month s balance. THE UNPAID BALANCE METHOD FOR COMPUTING THE FINANCE CHARGE ON A CREDIT CARD LOAN This method also uses the simple interest formula I = Prt; however, P = previous month s balance + finance charge + purchases made - returns - payments. t = 1 12 The variable r is the annual interest rate, and. EXAMPLE 2 Using the Unpaid Balance Method for Finding Finance Charges Assume that the annual interest rate on your credit card is 18% and your unpaid balance at the beginning of last month was $600. Since then, you purchased ski boots for $130 and sent in a payment of $170. a) Using the unpaid balance method, what is your credit card bill this month? b) What is your finance charge next month? Quiz Yourself 12 Assume that the annual interest rate on your credit card is 21%. Your outstanding balance last month was $300. Since then, you have charged a purchase for $84 and made a payment of $100. What is the outstanding balance on your card at the end of this month? What is next month s finance charge on this balance? SOLUTION: a) We will list the items that we need to know to compute this month s balance. Previous month s balance: $600 1 Finance charge on last month s balance; $ $9 12 Purchases made: $130 annual interest rate Time is 1 month. Returns: $0 Payment: $170 Therefore, you owe Previous month s balance + finance charge + purchases made - returns - payments = = $569. b) The finance charge for next month will be $569 * 0.18 * a b = $8.54 Now try Exercises 21 to 26. ] 12 Note that you can use the unpaid balance method to your advantage by making a large purchase early in the billing period and then paying it off just before the billing date. This is not to the credit card company s advantage because you can use the credit card company s money for free for almost a whole month. It is easy to use credit cards for purchases and so tempting to pay only the minimum payment that appears on your credit card bill that you may find your debt increasing even

4 9.3 y Consumer Loans 417 though you are sending in a payment every month. Example 3 illustrates how credit card debt can get out of hand. EXAMPLE 3 Paying Off a Credit Card Debt Assume that you want to pay off your credit card debt of $6,589 by making the minimum payment of $100 a month. What will your balance be at the end of 1 month? Assume that the annual interest rate on your card is 18% and that the credit card company is using the unpaid balance method to compute your finance charges. SOLUTION: When you send in your $100, it is credited to your account, so the next month you have an unpaid balance of 6, = $6,489. The annual interest rate is 18 18%, so your monthly interest rate is 12 = 1.5%. Therefore, at the end of the month you still owe 6,489 + (0.015)(6,489) = 6, = $6, balance interest Thus, your $100 payment has reduced your debt by only 6,589-6, = $2.66. ] Example 3 illustrates how difficult it is to pay off a large credit card bill. The best practice is to pay off as much of your outstanding balance as you can to avoid paying a large amount of interest. Although it is possible to borrow cash from one company to pay off debts at another, doing this is not reducing the size of your debt. Instead, the debt may still grow at a rate of 18% or more per year. When you use credit, always look at the annual interest rate. In some cases, if you read the fine print in the agreement, you will find rates as high as 24% or 25%. Also, be very careful when taking a cash advance on your credit card because the interest rate is often much higher than the rate you are charged for making purchases on the card. KEY POINT The average daily balance method computes finance charges based on the balance in the account for each day of the month. The Average Daily Balance Method A more complicated method for determining the finance charge on a credit card is called the average daily balance method, which is one of the most common methods used by credit card companies. With this method, the balance is the average of all daily balances for the previous month. THE AVERAGE DAILY BALANCE METHOD FOR COMPUTING THE FINANCE CHARGE ON A CREDIT CARD LOAN 1. Add the outstanding balance for your account for each day of the month. 2. Divide the total in step 1 by the number of days in the month to find the average daily balance. 3. To find the finance charge, use the formula I = Prt, where P is the average daily balance found in step 2, r is the annual interest rate, and t is the number of days in the month divided by 365. EXAMPLE 4 Using the Average Daily Balance Method for Finding Finance Charges Suppose that you begin the month of September (which has 30 days) with a credit card balance of $240. Assume that your card has an annual interest rate of 18% and that during September the following adjustments are made on your account: September 11: A payment of $60 is credited to your account. September 18: You charge $24 for itune downloads. September 23: You charge $12 for gasoline.

5 418 CHAPTER 9 y Consumer Mathematics Use the average daily balance method to compute the finance charge that will appear on your October credit card statement. SOLUTION: To answer this question, we must first find the average daily balance for September. The easiest way to calculate the balance is to keep a day-by-day record of what you owe the credit card company for each day in September, as we do in Table 9.3. Day Balance Number of Days * Balance 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 $ * 240 = 2,400 11, 12, 13, 14, 15, 16, 17 $180 7 * 180 = 1,260 18, 19, 20, 21, 22 $204 5 * 204 = 1,020 23, 24, 25, 26, 27, 28, 29, 30 $216 8 * 216 = 1,728 TABLE 9.3 Daily balances for September. Quiz Yourself 13 Recalculate the average daily balance in Example 4, except now assume you bought the itunes downloads on September 3 instead of September 18. Make a table similar to Table 9.3. The average daily balance is therefore (10 * 240) + (7 * 180) + (5 * 204) + (8 * 216) 30 2, , , ,728 = 30 = 6, = We next apply the simple interest formula, where P = $213.60, r = 0.18, and t = * Thus, I = Prt = 213.6(0.18)A B = Your finance charge on the October statement will be $3.16. Now try Exercises 27 to 30. ] 13 As you will see in Example 5, the amount of finance charges you pay on a loan will vary depending on the method used to compute the charges. Math in Your Life Will You Be Part of Generation Broke? You ve heard of Generation X and Generation Y but have you ever heard of Generation Broke? Article after article, from Time Magazine to USA Today, warns of impending financial doom for young adults. Depending on which study you read, the average graduating college senior now owes between $5,000 and $20,000 in student loans and another $3,000 to $4,000 in credit card debt. According to a public policy group in New York called Demos,... young adults are doing everything society tells them to do...[but] they can t get ahead because of the debt they went into to get the degree and get the good job. Demos sees two trends fueling this rise in debt among younger Americans: increasing college costs and aggressive credit card marketing on college campuses. In a report in the Christian Science Monitor, one student laments not having had a class in personal financial management, If I had that type of class, I wouldn t have gotten into such a credit mess. * t = because we are using the credit card for 30 days out of a 365-day year. To read the full report, Generation Broke: The Growth of Debt Among Younger Americans, refer to

6 9.3 y Consumer Loans 419 Credit and Interest* Credit cards were not widely used in the United States until the 1950s when cards such as Diners Club, Carte Blanche, and American Express made the use of plastic money more popular. Today, Americans charge about $1 trillion per year on their cards. Credit is not a modern idea. Surprisingly, there are ancient Sumerian documents dating back to about 3000 BC that show the regular use of credit in borrowing grain and metal. Interest on these loans was often in the 20% to 30% range similar to the 18% to 21% charged on many of today s credit cards. As the use of credit increased, so did its HISTORICAL HIGHLIGHT misuse. Many societies wrote laws to prevent its abuse particularly the charging of unfairly high interest rates, which is called usury. Credit and interest can appear in many diverse forms. The Ifugao tribe of the Philippines charges 100% on a loan. If rice is borrowed, then at the next harvest, the loan must be paid in double. In Vancouver, Canada, the Kwakiutl have a system of credit based on blankets. The rules of interest state that if five blankets are borrowed, in 6 months they become seven. In Northern Siberia, loans are made in reindeer, usually at a 100% interest rate. Comparing Financing Methods EXAMPLE 5 Comparing Methods for Finding Finance Charges Suppose that you begin the month of May (which has 31 days) with a credit card balance of $500. The annual interest rate is 21%. On May 11, you use your credit card to pay for a $400 car repair, and on May 29, you make a payment of $500. Calculate the finance charge that will appear on the statement for next month using the two methods we have discussed. SOLUTION: Method P r t Finance Charge = I = Prt Unpaid balance Average daily balance last month s balance + finance charge - payment + charge for car repair = = * * * = 22, = % % (408.75)(0.21)a 1 12 b = $7.15 (722.58)(0.21)a b = $12.89 With the unpaid balance method, the finance charge is $7.15; with the average daily balance method, the finance charge is $ ] Example 5 shows that the exact same charges on two different credit cards having the same annual interest rate can result in very different finance charges. If you understand the method your credit card company is using, you may be able to schedule your purchases and payments to minimize your finance charges. In deciding how to use credit, you must consider many other issues that we have not discussed in this section. Some credit card companies charge an annual fee; others return part of your interest payments. For some credit cards, there is a grace period. If you reduce your balance to zero during this grace period, then you pay no finance *This Historical Highlight is based on S. Homer and R. Sylla, A History of Interest Rates, 3rd ed. (Piscataway, NJ: Rutgers University Press, 1991), pp

7 420 CHAPTER 9 y Consumer Mathematics charges. A credit card may have a low introductory rate that changes to a much higher rate at a later time. One common enticement is that you can make a purchase and pay no interest payments until several months later. With such deals, however, you must be careful. Often, if you do not pay off the purchase completely at the end of the interest-free period, then all the interest that would have accumulated is added to your balance. It is difficult to understand all the pros and cons of the many different types of credit contracts. However, if you read credit agreements carefully and remember the principles that you learned in this section, you will be an intelligent consumer who will be able to use credit wisely. Exercises 9.3 Looking Back* These exercises follow the general outline of the topics presented in this section and will give you a good overview of the material that you have just studied. 1. Describe briefly how we calculated the payments for the add-on interest loan in Example In Example 2, we computed the finance charge from the previous month using the expression Prt. What did each variable represent? 3. What does Table 9.3 show in Example 4? 4. In reference to the Math in Your Life on Generation Broke, what do you think are some of the causes of this phenomenon? Sharpening Your Skills In Exercises 5 8, compute the monthly payments for each add-on interest loan. The amount of the loan, the annual interest rate, and the term of the loan are given. 5. $900; 12%; 2 years 6. $840; 10%; 3 years 7. $1,360; 8%; 4 years 8. $1,710; 9%; 3 years 9. Paying off a computer. Luis took out an add-on interest loan for $1,280 to buy a new laptop computer. The loan will be paid back in 2 years and the annual interest rate is 9.5%. How much interest will he pay? What are his monthly 10. Paying off furniture. Mandy bought furniture costing $1,460 for her new apartment. To pay for it, her bank gave her a 5-year add-on interest loan at an annual interest rate of 10.4%. How much interest will she pay? What are her monthly 11. Financing equipment. Angela s bank gave her a 4-year add-on interest loan for $6,480 to pay for new equipment for her antiques restoration business. The annual interest rate is 11.65%. How much interest will she pay? What are her monthly *Before doing these exercises, you may find it useful to review the note How to Succeed at Mathematics on page xix. 12. Paying for a sculpture. Mikeal purchased an antique sculpture from a gallery for $1,320. The gallery offered him a 3-year add-on loan at an annual rate of 9.75%. How much interest will he pay? What are his monthly In Exercises 13 16, use the add-on method for determining interest on the loan. Determine the annual interest rate during the last month of the loan. 13. $900; 12%; 2 years 14. $840; 10%; 3 years 15. $1,360; 8%; 4 years 16. $1,710; 9%; 4 years Applying What You ve Learned 17. Financing a boat. Ben is buying a new boat for $11,000. The dealer is charging him an annual interest rate of 9.2% and is using the add-on method to compute his monthly payments. a. If Ben pays off the boat in 48 months, what are his monthly b. If he makes a down payment of $2,000, how much will this reduce his monthly c. If he wants to have monthly payments of $200, how large should his down payment be? 18. Financing a swimming pool. Mr. Phelps is buying a new swimming pool for $14,000. The dealer is charging him an annual interest rate of 8.5% and is using the add-on method to compute his monthly payments. a. If Mr. Phelps pays off the pool in 48 months, what are his monthly b. If he makes a down payment of $3,000, how much will this reduce his monthly c. If he wants to have monthly payments of $250, how large should his down payment be? 19. Financing rare coins. Anna is buying $15,000 worth of rare coins as an investment. The dealer is charging her an annual interest rate of 9.6% and is using the add-on method to compute her monthly payments.

8 9.3 y Exercises 421 a. If Anna pays off the coins in 36 months, what are her monthly b. If she makes a down payment of $3,000, how much will this reduce her monthly c. If she wishes to have monthly payments of $300, how large should her down payment be? 20. Financing a musical instrument. Walt is buying a music synthesizer for his rock band for $6,500. The music store is charging him an annual interest rate of 8.5% and is using the add-on method to compute his monthly payments. a. If Walt pays off the synthesizer in 24 months, what are his monthly b. If he makes a down payment of $1,500, how much will this reduce his monthly c. If he wants to have monthly payments of $150, how large should his down payment be? In Exercises 21 26, use the unpaid balance method to find the finance charge on the credit card account. Last month s balance, the payment, the annual interest rate, and any other transactions are given. 21. Computing a finance charge. Last month s balance, $475; payment, $225; interest rate, 18%; bought ski jacket, $180; returned camera, $ Computing a finance charge. Last month s balance, $510; payment, $360; interest rate, 21%; bought exercise equipment, $470; bought fish tank, $ Computing a finance charge. Last month s balance, $640; payment: $320; interest rate, 16.5%; bought dog, $140; bought pet supplies, $35; paid veterinarian bill, $ Computing a finance charge. Month: August (31 days); previous month s balance: $280 Date Transaction August 5 Made payment of $75 August 15 Charged $135 for hiking boots August 21 Charged $16 for gasoline August 24 Charged $26 for restaurant meal 28. Computing a finance charge. Month: October (31 days); previous month s balance: $190 Date Transaction October 9 Charged $35 for a book October 11 Charged $20 for gasoline October 20 Made payment of $110 October 26 Charged $13 for lunch 29. Computing a finance charge. Month: April (30 days); previous month s balance: $240 Date Transaction April 3 Charged $135 for a coat April 13 Made payment of $150 April 23 Charged $30 for DVDs April 28 Charged $28 for groceries 30. Computing a finance charge. Month: June (30 days); previous month s balance: $350 Date Transaction June 9 Made payment of $200 June 15 Charged $15 for gasoline June 20 Charged $180 for skis June 26 Made payment of $ Computing a finance charge. Last month s balance, $340; payment, $180; interest rate, 17.5%; bought coat, $210; bought hat, $28; returned boots, $ Computing a finance charge. Last month s balance, $460; payment, $300; interest rate, 18.8%; bought plane ticket, $140; bought luggage, $135; paid hotel bill, $ Computing a finance charge. Last month s balance, $700; payment, $480; interest rate, 21%; bought ring, $210; bought theater tickets, $142; returned vase, $128 In Exercises 27 30, use the average daily balance method to compute the finance charge on the credit card account for the previous month. The starting balance and transactions on the account for the month are given. Assume an annual interest rate of 21% in each case. In Exercises 31 34, redo the specified exercise using the unpaid balance method to calculate the finance charges. 31. Exercise Exercise Exercise Exercise Comparing financing methods. Mayesha purchased a largescreen TV for $1,000 and can pay it off in 10 months with an add-on interest loan at an annual rate of 10.5%, or she can use her credit card that has an annual rate of 18%. If she uses her credit card, she will pay $100 per month (beginning next month) plus the finance charges for the month. Assume that Mayesha s credit card company is using the unpaid balance method to compute her finance charges and that she is making no other transactions on her credit card. Which option will have the smaller total finance charges on her loan? 36. Comparing financing methods. Repeat Exercise 35, but now assume that Mayesha purchased an entertainment center for $2,000, the rate for the add-on loan is 9.6%, and she is paying off the loan in 20 months.

9 422 CHAPTER 9 y Consumer Mathematics 37. Accumulated interest. A Home Depot advertises 0% financing for 3 months for purchases made before the new year. The fine print in the advertisement states that if the purchase is not paid off within 3 months, the purchaser must pay interest that has accumulated at a monthly rate of 1.75%. Assume that you buy a refrigerator for $1,150 and make no payments during the next 3 months. How much interest has accumulated on your purchase during this time? 38. Accumulated interest. Repeat Exercise 37, but now assume that the purchase is for $1,450 and the annual interest rate is 24%. Communicating Mathematics 39. Why is an add-on interest loan called by that name? 40. What was the point of our discussion in the paragraph following Example 1? 41. How can you use the unpaid balance method to your advantage when borrowing? 42. How would you explain informally the difference between computing finance charges with the unpaid balance method and the average daily method to a classmate? Using Technology to Investigate Mathematics 43. Your instructor has an Excel spreadsheet that will carry out the computations for the average daily balance method. Use this spreadsheet to duplicate the computations in Example 4 and also to solve some of the exercises.* 44. Use the Web to research the terms of some actual credit cards. See if you can determine how the finance charges are computed with those cards. What other terms are in those agreements that we have not discussed in this section? For Extra Credit 45. In Example 5, we found that the average daily balance method gave the highest finance charges. Explain why this happened. 46. Make up transactions on a hypothetical credit card so that the unpaid balance method will give you lower finance charges than the average daily balance method does. Assume an annual interest rate of 18%. 47. Make up transactions on a hypothetical credit card so that the average daily balance method will give you lower finance charges than the unpaid balance method does. Assume an annual interest rate of 21%. 48. In our discussions about credit, we have ignored the fact that whatever money is not used to pay off a loan can be invested. Assume that you can earn 5% on any money that you do not use for paying off a loan. However, any money you earn as interest is subject to federal, state, and local taxes. Assume that these taxes total 20%. Discuss how that might affect your decision to pay off your credit card debt.