Chapter 3: Credit Cards and Consumer Loans

Chapter 3: Credit Cards and Consumer Loans In these tough economic times, credit card debt has escalated to a new high. Lenders like American Express, Citigroup, Discover, Capital One etc. are tightening their standards thereby making it more difficult to get a new credit card and they are also cutting credit limits. Here, we will learn how mathematics is used in credit cards and how to use credit cards wisely. Mostly we use a credit card to make a large purchase. When we use a credit card, we are actually receiving loan which we have to eventually pay back. Generally, this loan payment is done in installments. There are two types of loans used for installment purchases: Closed-End Credit and Open-End Credit. A. Closed-End Credit This is the first type of consumer credit installment plan. Closed-End Credit is a loan where you make equal payments at regular intervals for some specific period of time. It is also called an installment loan. It is a traditional way of paying off loans. Mostly these loan payments are made monthly. An installment is the name given to each of these loan payments. The size of the payment is determined by the amount of the purchase and also the interest rate charged on the loan. The interest charged on the loan is called the finance charge. Method to compute the payments on an installment loan: Add-On Interest Method In this method, we use the simple interest formula to calculate the finance charge on the loan. Then the interest is added to the amount borrowed so that both the interest and the amount borrowed are paid back over the time period for which the loan was taken. The formula for the payment (installment) is given by P I Installment n where P = loan amount (present value), n = total number of payments I = amount of interest due on the loan (finance charge), 1

Example 1 Suppose that you take an add-on loan for $1000 to buy a washer/dryer at an annual rate of interest of 10% to be repaid in monthly installments over four years. How much will your monthly payments be? Solution: P = 1000, n = 4 x 12 = 48 months We calculate the interest using the simple interest formula: I = Prt = 1000 x.1 x 4 = $ 400. Thus, the monthly payment is = Example 2 1000 400 1400 48 48 $ 29.16. Suppose that you take an add-on loan for $28,000 at an annual rate of interest of 3.5% to be repaid in yearly installments over thirty years. How much will be your yearly payments? Solution: P = 28000, n = 30 years We calculate the interest using the simple interest formula: I = Prt = 28000 x.035 x 30 = $ 29400. Thus, the yearly payment is = Example 3 28000 29400 57400 30 30 $ 1913.33. Suppose that you take an add-on loan for $4,000 at an annual rate of interest of 5.45% to be repaid in semi-annual (six-month) installments over two years. How much will your semi-annual payments be? Solution: P = 4000, n = 4 semi-annual installments We calculate the interest using the simple interest formula: I = Prt = 4000 x.0545 x 2 = $ 436. Thus, the semi-annual payment is = 4000 436 4 4436 4 $ 1109. 2

Annual Percentage Rate (APR): In the add-on interest method you are not keeping the entire borrowed amount for the entire time of the loan period, which makes us wonder if the annual interest rate told to us is actually correct. Let us take a simple example to understand what a true APR is. Suppose you borrow $3000 to be repaid in three yearly installments using an add-on interest rate of 10%. What is your true interest rate? Using the add-on method, we can find the amount of interest by using the formula I = Prt = 3000 x.10 x 3 = $ 900. Thus, the amount to be paid back in three yearly installments is 3000 + 900 = $3900. Therefore, each payment is 3900 / 3 = $ 1300, of which 1000 is being paid on the principal and $300 is for the interest. Hence, for the first year you borrowed $ 3000 and paid an interest of $ 300. Then using the simple interest formula, the interest rate was 10% ( 300 = 3000 x r x 1), which is the same as the annual interest rate. Now in the second year, you make a payment of $1300, of which $ 1000 goes to reduce the principal and $ 300 is interest. Keep in mind that you have already paid back $1000 of the loan. Thus, you now have only $2000 to pay back and so you have paid $300 interest on a $2000 loan. Let us calculate the interest rate we have paid in the second year. Use simple interest formula: 300 = 2000 x r x 1. We find interest rate to be r =.15 or 15%. So in reality the interest rate on the loan for the second year is 15%. Now in the third year, you make the final payment of $1300. For this last year, you have paid $300 interest on a $1000 loan. Let us calculate the interest rate we have paid in the third year. Use simple interest formula: 300 = 1000 x r x 1. We find interest rate to be r =.3 or 30%!. So in reality the interest rate on the loan for the third year is 30%. Question: What is the true interest rate for this loan? The true interest rate we are looking for is called the Annual Percentage Rate. We will denote it by a. We see that the Interest for the first year + interest for the second year + interest for the third year = 900 Using the simple interest formula we can write this as 3000 x a x 1 + 2000 x a x 1 + 1000 x a x 1 = 900 6000 a = 900 a = 900 / 6000 =.15 Thus, the annual percentage rate is 15%. 3

Formula to approximate the Annual Percentage Rate for an add-on interest loan is 2nr APR n 1 Where r annual interest rate, n total number of payments. In 1968, the Truth-in Lending Act was passed by the Congress. This act requires all lenders to state the true annual interest rate, which is called the APR and is based on the actual amount owed. This regulation enables you to compare interest rates before you sign a contract, which must state the APR even if you haven t asked for it. Example 4: Consider a 2007 Blazer with a price of $28,505 that is advertised at a monthly payment of $631 for 60 months. What is the APR (to the nearest tenth of a percent)? We know A = P + I; I = 631(60) 28505 = 9355. Then we use the simple interest formula ( I = Prt) to find r: r = I / Pt = 9355 / 28505 ( 5) =.0656 ( Note that 60 months = 5 years) 2nr 2(60)(.0656) APR.1291 12.91% n 1 61 Example 5: Sara purchases a sofa set for a price of $2500 with a loan that advertises a 9% simple interest rate to be repaid in three equal monthly payments. What is the APR (to the nearest tenth of a percent)? 2nr APR n 1 2(3)(.09) 4.135 13.5% 4

Problems: 1. Use the add-on method for determining the installments for the following. The amount of the loan, the annual interest rate, the term of the loan is given, and the installment plan. a. $879, 12%, 2 years, monthly b. $ 6390, 7.80%, 5 years, monthly c. $900, 10%, 6 years, semi-annually d. $ 790, 5.40%, 3 years, every six months e. $1320, 13.5%, 6 months, every two months f. $7890, 6.5%, 60 months, yearly g. $ 800, 12.77%, 3 years, monthly h. $ 5400, 11.05%, 8 years, quarterly i. $ 600, 4.5%, 2 years, quarterly 2. Find the APR on the add-on loan amount using the given number of payments and annual interest rate. a. n = 36, r = 6.4% b. n = 48, r = 4.8% c. n = 42, r = 7% d. n = 30, r = 15.48 % 3. Ann took out a 24-month add-on loan for the amount $2000 to go to China at an interest rate of 8%. What is her APR (round to the nearest tenth)? 4. What is the APR of a 36-month add-on loan with an interest rate of 8.2%? 5. Trey took out a $25, 000 loans to remodel his house and will repay it by making 60 monthly payments of $485. Find his APR on the remodeling loan. 6. Carrie took an add-on loan for $1500 to buy a new laptop and will repay it by making 24 monthly payments of $71.25. What is Carrie s APR on the loan? 5

7. Decide which has better APR if you want to repay a $ 5000 loan. Assume that you are making monthly payments. a. An add-on interest loan at 8.4% for 3 years b. 24 payments of $ 230 ( hint: You have to find the interest rate first.) 8. Decide which has better APR if you want to repay a $ 5000 loan. Assume that you are making monthly payments. a. An add-on interest loan at 8.4% for 1 years b. 24 payments of $ 240 ( hint: You have to find the interest rate first.) 9. Decide which has better APR if you want to repay a $ 5000 loan. Assume that you are making monthly payments. a. An add-on interest loan at 7.2% for 2 years b. 36 payments of $ 165 ( hint: You have to find the interest rate first.) 10. Jamie 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. If Jamie pays off the boat in 48 months. a. What are his monthly payments? b. If he makes a down payment of $2000, how much will be his monthly payments? c. Estimate the APR. 11. Kelly buys a computer for $2400. If Kelly agrees to repay the balance in 24 equally monthly payments at an annual simple interest rate of 10%. a. What are her monthly payments? b. If she makes a down payment of $360, how much will be her monthly payments? c. Estimate the APR. 6

B. Open-End Credit This is a second type of consumer credit. It is also called revolving credit or a credit card loan. Examples of this type of credit card loan are MasterCard, American Express, Discover etc. This type of loan allows for purchases or cash advances up to a specified maximum line of credit and has a flexible repayment schedule. There are several ways a credit card company charges finance charges. We will look at two methods. I. Unpaid Balance Method In the Unpaid Balance Method, the finance charges are applied on the previous month s balance. Most of the credit cards issue monthly bills. The due date on the bill is usually 1 month after the billing date (the date the bill is prepared and sent to the customer). If you pay the bill by the due date, then no finance charges are applied; otherwise a finance charge is added to the next bill. This method uses the simple interest formula to calculate the finance charge on the balances you owe to the credit card company. The formula for the amount P you will owe is P = previous month s balance + finance charge + purchases made payments returns Example 6 Suppose you have a credit card which has an annual rate of interest of 18% and at the beginning of the last month you had an unpaid balance of $144. Since then you returned the winter boots you had bought for $75.00, made a purchase of a dell desktop for $500, and sent in a payment of $200. What is the balance you owe for this month? What is your finance charge for the next month? Solution: First we will calculate the finance charge on previous month balance: I = 144 x.18 x ( ) = $ 2.16 Now we will find the balance you owe the credit card company for this month. P = 144 + 2.16 + 500 75 200 = $ 371.16 To find the finance charge for the next month we apply the simple interest formula on the new amount ($ 371.16) which you owe for this month: I = Prt = 371.16 x.18 x ( ) = $ 5.57. It is easy to use to credit cards for purchases and to only pay the minimum payment stated on the credit card bill. This is dangerous as it will lead to an increase in what you owe and sometimes it gets out of hand. Let us look at an example. 7

Example 8 Suppose Tom has a credit card debt of $ 4000 which has an annual rate of interest 18%. The minimum payment for the month of June is $60. Suppose Tom pays back only the minimum required payment. What is the balance Tom has to pay back at the end of June? The finance charges are computed using the unpaid balance method. Solution: Tom s interest is I = Prt = (4000)(.18)(1/12) = $60. Thus at the end of June he will owe P = 4000 + 60 60 = $4000! Tom has made no progress in paying down his credit card debt! 8

Problems: 1. Given last month s balance, the annual interest rates, and any other transactions, find the amount you owe and the finance charge to be paid on this month s credit card, using the unpaid balance method. a. Last month balance $ 5000; rate 21%; returned TV $1000, payment $600 b. Last month balance $ 800; rate 16%; bought sandals $100, payment $300 c. Last month balance $ 500; rate 19%; bought calculator $82, returned jeans $30.00, payment $180 d. Last month balance $ 1300; rate 20%; bought socks $10, bought iron $30, returned lamp $45.00, returned luggage $145.00, payment $580 2. Tracy bought personal computer for $600.00 using her credit card which has 18% annual interest rate. The unpaid balance on her credit card at the beginning of last month was $ 300. Since then, she has purchased her statistics book for $ 70.00 and sent in payment of $ 200. Using the Unpaid balance method, what is her credit card bill this month? What is her finance charge next month? 3. Compute the finance charge for the Month of August ( 31 days), when the previous month s (July) balance was $ 280 and the following transaction took place in the month of August. Assume an annual interest rate of 21%. Date Transaction August 5 Made payment of $75 August 10 Charged $ 130 for trekking boots August 18 Charged $ 40 for gas August 23 Charged $30.34 for restaurant meal 4. Compute the finance charge for the Month of November ( 30 days), when the previous month s balance was $ 212 and the following transaction took place in the month of November. Assume an annual interest rate of 21%. Date Nov. 9 Transaction Charged $ 40 for DVD Nov. 15 Charged $ 34 for gas Nov. 21 Made payment of $175 Nov. 23 Charged $190.98 for groceries 9

II. Average Daily Balance Method In reality, many credit card companies use the average daily balance method to calculate the interest on the balance you have. In this method, the finance charge is based on the balance in the account for each day of the billing period. It is calculated by dividing the sum of the total amounts owed each day by the number of days in the billing period. Example 8: sum of the total amounts owed each day Average Daily Balance number of days in the billing period Suppose an unpaid bill of $315 had a due date of April 10. A purchase of $28 was made on April 12, and $123 was charged on April 24. A payment of $50 was made on April 15. The next billing date is May 10. The interest on the average daily balance is 18%. Find the finance charge on the May 10 bill. Solution: First we will create a table showing the unpaid balance for each purchase, the number of days the balance is owed, and the product of these numbers. A negative sign in the payments or purchases column of the table indicates that a payment was made. Date Payments or Purchases Balance each Day No. of days until Balance Changes Unpaid Balance Times Number of Days April 10-11 $ 315 2 $ 630 April 12-14 $ 28 $ 343 3 $ 1029 April 15-23 -$ 50 $ 293 9 $ 2637 April 24-May 9 $ 123 $ 416 16 $ 6656 Total $ 10,952 10,952 Average daily balance = $365. 07 30 The finance charge is given by I = Prt = 365.07 (.18)(30/365) $ 5.40 10

Problems: 1. Calculate the finance charge for a credit card that has given average daily balance and interest rate. Assume a 30 day billing period. a. Average daily balance = $ 118.72; annual interest rate = 15% b. Average daily balance = $ 391.64; annual interest rate = 21% c. Average daily balance = $ 2034.91; annual interest rate = 18% d. Average daily balance = $ 546.45; annual interest rate = 21% 2. Use the average daily balance method to compute the finance charge on the credit card account for the stated month. The starting balance and transactions on the account for the month are given. Assume an annual interest rate of 21%. a. Compute the finance charge for the month of August (31 days), when the previous month s (July) balance was $280 and the following transaction took place in the month of August. Date Transaction August 5 Made payment of $75 August 10 Charged $130 for trekking boots August 18 Charged $40 for gas August 23 Charged $30.34 for restaurant meal b. Compute the finance charge for the month of November (30 days), when the previous month s balance was $212 and the following transaction took place in the month of November. Date Nov. 9 Transaction Charged $40 for DVD Nov. 15 Charged $34 for gas Nov. 21 Made payment of $175 Nov. 23 Charged $190.98 for groceries 3. A credit card account had a $244 balance on March 5. A purchase of $152 was made on March 12, and a payment of $100 was made on March 28. Find the average daily balance and the finance charge if the billing date is April 5. Assume an annual interest rate of 21%. 11