Credit Card Analysis
Stuart Boersma, Central Washington University
Based on material in "Case Studies for Quantitative Reasoning: A Casebook of Media Articles" by Bernard L. Madison, Stuart Boersma, Caren L. Diefenderfer, and Shannon W Dingman.
Summary
In this example, students are asked to obtain a credit card disclosure statement, identify the various interest rates and payment rules, and keep track of a revolving monthly balance under several payment scenarios. While modeling several months to a year of credit card statements is easily accomplished with pencil, paper, and a basic calculator, this example is also well-suited for analysis with a spreadsheet.
Topics
Mathematics Grade Level
College Introductory
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Learning Goals
Learning goals include:
- Careful reading of financial rules and regulations,
- Calculating with interest,
- Calculating recursively, and
- Calculating a revolving monthly balance.
Context for Use
Before beginning this assignment, students should be familiar with the notion of compound interest. This example could be used in a course as a final assessment on compound interest.
Description and Teaching Materials
In a letter from 1993 (Seattle Times), a young adult describes the debt he accumulated as a result of not understanding the finances of buying products with credit cards. In particular, he states his confusion about required minimum payments. A more recent New York Times article also refers to the large amount of interest that is charged to someone who only makes the required minimum payments. In order to fully understand these truths, work through the following calculations.
- Obtain a current credit card offer and read through the terms. Identify the following information (you may have to dig into the "fine print"):
a. APR (Annual Percentage Rate): Note that there are often different categories (payment, transfer) and rates (introductory and regular). These rates can also be fixed or variable. Determine the regular payment APR for your card. If the rate is listed as variable, indicate how this rate is determined. Does your card have a Default APR? If so, what is it and when does it get used?
b. Grace Period: Most credit cards have a grace period on new purchases. State whether your card has a grace period and when it applies.
c. Does your card have an annual fee?
d. How is the minimum payment determined?
e. What happens if you miss one or more payments?
- Credit cards allow the holder to purchase an item and take it home to enjoy while paying for the item at a later date. Purchasers may pay for the item in full when they receive their first statement. For companies that offer "grace periods," no additional interest is charged. However, often people use credit cards when they plan to pay for an item over several months or years. If one does not pay off the entire credit card balance, interest will be charged on ALL purchases, and the interest owed is then calculated from the date of purchase (not the date the bill is received). Thus, there is no grace period at all.
Suppose you have your eye on a nice big screen TV that costs $2,000. You decide to use your new card to make the purchase. Assume that you do not plan to pay the debt off in full at the first billing, so you will be subjected to interest charges each month.
a. Using the regular purchase APR, determine the monthly interest rate for your card.
b. If your card has a grace period, when you receive your first bill (say, in one month) there should be no interest charge. If you make the required minimum payment, how much will you have to pay the first month? What is the remaining balance (amount you owe) on the card?
c. Assuming you only make the required minimum payment, your second bill will have an interest charge. Assuming you are charged one month of interest on the remaining balance, how much will you be charged in interest on your second bill?
d. Your new minimum payment will be computed by using the starting balance plus the interest charged. What is the minimum payment required on your second bill?
e. Assuming that you again just make the minimum payment, what percent of your payment goes towards interest and what percent goes towards reducing your original balance of $2,000?
f. When your third bill comes, what is the new balance and minimum payment?
- Suppose you keep making the minimum payment for one year.
a. After one year, what is the current balance on the credit card?
b. After one year, how much have you paid the credit card company?
c. Of all the money you paid the credit card company, what percent went towards reducing the original balance of $2,000, and what percent went towards interest payments?
d. If you keep making minimum payments, how long do you think it will take to pay off the purchase of the TV? An estimate based on your work so far will suffice.
- Use a spreadsheet, such as Excel, to keep track of the following quantities:
- Statement number (first, second, etc.)
- Interest charged that month
- Current balance
- Minimum Payment required
- Total amount of interest paid to date
How long will it take to pay off the original purchase price of $2,000? How much interest would you have to pay for this privilege?
How would the scenario change if you could afford to pay twice the minimum payment every month?
What happens if, for some reason, you miss your payments twice during the first year?
Is it worth "shopping around" for a credit card with a lower APR? Explain when this would be advantageous and why.
Teaching Notes and Tips
General Notes
This example can create many questions from students. It is important to be able to perform some sort of triage on the questions. For instance, it may not be worthwhile to make sure every student is computing the balance in exactly the same manner (is interest added before or after the minimum payment is calculated? Does "balance after one year" mean the amount before or after the payment is made?). What is important is that students understand that every monthly cycle (other than perhaps the first one) should contain a) a starting balance, b) an interest charge, c) a minimum payment calculation, and d) a payment. Whether payments are credited at the end of one cycle or the beginning of another cycle is not of great importance.
Rounding is another cause for discrepancy. In the following calculations, interest rates were rounded to the nearest hundredth of a percent and dollar amounts were rounded to the nearest cent.
Since answers can vary tremendously with different interest rates, it would help to have a spreadsheet available to quickly check to see if students appear to be "in the ballpark" on these calculations.
Answers to question 6 can also vary quite a bit depending on which months students decide to "miss". This is an exercise in assumption making. Students should be able to assume which statements will be missed. The instructor needs to check 1) any late/missed payment fees are added appropriately, 2) the default APR is introduced at the right time (if applicable), and 3) the APR is reset to the original after the correct amount of time.
Sample Solution
- Obtain a current credit card offer and read through the terms. Identify the following information (you may have to dig into the "fine print"):
The following pertains to the Capital One Platinum Prestige card as found at CapitalOne.com on July 23, 2009.
a. APR (Annual Percentage Rate): Note that there are often different categories (payment, transfer) and rates (introductory and regular). These rates can also be fixed or variable. Determine the regular payment APR for your card. If the rate is listed as variable, indicate how this rate is determined. Does your card have a Default APR? If so, what is it and when does it get used?
Regular payment APR is 11.9%. This is a variable rate determined by adding 8.65% to the Prime rate.
The default APR is 29.4%. This is also a variable rate determined by adding 26.15% to the Prime rate. The default APR is used if payment is received late twice within any 12 billing periods (months).
b. Grace Period: Most credit cards have a grace period on new purchases. State whether your card has a grace period and when it applies.
This card has a grace period of 25 days from the date of the periodic statement on new purchases. It only applies if the balance is paid in full by the due date.
c. Does your card have an annual fee?
No.
d. How is the minimum payment determined?
If the balance is less than $15, the minimum payment equals the balance. Otherwise, the minimum payment will be the greater of $15 or 1% of the balance plus interest and late payment fees.
e. What happens if you miss one or more payments?
If a single payment is missed (hence late) there is a late payment fee based on the amount of the balance: A balance between $0 and $99.99 results in a $15 fee, balance of $100 - $249.99 results in a $29 fee, balance of $250 or more results in a $39 fee. If payment is missed (or received late) twice within any 12 month period, the default APR goes into effect. Also, there is a $39 returned payment fee.
- Credit cards allow the holder to purchase an item and take it home to enjoy while paying for the item at a later date. Purchasers may pay for the item in full when they receive their first statement. For companies that offer "grace periods," no additional interest is charged. However, often people use credit cards when they plan to pay for an item over several months or years. If one does not pay off the entire credit card balance, interest will be charged on ALL purchases, and the interest owed is then calculated from the date of purchase (not the date the bill is received). Thus, there is no grace period at all.
Suppose you have your eye on a nice big screen TV that costs $2,000. You decide to use your new card to make the purchase. Assume that you do not plan to pay the debt off in full at the first billing, so you will be subjected to interest charges each month.
a. Using the regular purchase APR, determine the monthly interest rate for your card.
Dividing the regular APR by 12 yields a monthly interest rate of 0.99%.
b. If your card has a grace period, when you receive your first bill (say, in one month) there should be no interest charge. If you make the required minimum payment, how much will you have to pay the first month? What is the remaining balance (amount you owe) on the card?
The minimum payment will end up being 1% of the $2000 balance or $20. If we pay $20, then there is a remaining balance of $1980.
c. Assuming you only make the required minimum payment, your second bill will have an interest charge. Assuming you are charged one month of interest on the remaining balance, how much will you be charged in interest on your second bill?
We will be charged 0.99% of $1980 or $19.60.
d. Your new minimum payment will be computed by using the starting balance plus the interest charged. What is the minimum payment required on your second bill?
Rounding to the nearest $.01 results in a minimum payment of $20 + $19.6 = $39.6.
e. Assuming that you again just make the minimum payment, what percent of your payment goes towards interest and what percent goes towards reducing your original balance of $2,000?
$19.60/$39.60 = 49% goes towards the interest while 51% of our payment goes towards reducing the original balance.
f. When your third bill comes, what is the new balance and minimum payment?
After paying $39.60 and receiving a $19.40 charge in interest, the new balance is $1979.40. This results in a minimum payment of $39.19.
- Suppose you keep making the minimum payment for one year.
a. After one year, what is the current balance on the credit card?
The balance on the twelfth statement is $1806.60
b. After one year, how much have you paid the credit card company?
After making the minimum payment on the twelfth statement, you will have paid the credit card company $434.22. After making this payment, but before any interest from the next month is added, your new balance would be $1770.82.
c. Of all the money you paid the credit card company, what percent went towards reducing the original balance of $2,000, and what percent went towards interest payments?
Since the balance on the credit card is $1770.82, this means that $229.18 of payments went towards the original purchase of $2000. This represents about 53% of the total payments. The rest, 47%, went to paying interest.
d. If you keep making minimum payments, how long do you think it will take to pay off the purchase of the TV? An estimate based on your work so far will suffice.
One could argue that the balance is reduced by about $190 a year. At this rate it would take over 10 years to pay off the purchase.
- Use a spreadsheet (Excel 2007 (.xlsx) 60kB Nov6 09), such as Excel, to keep track of the following quantities:
- Statement number (first, second, etc.)
- Interest charged that month
- Current balance
- Minimum Payment required
- Total amount of interest paid to date
How long will it take to pay off the original purchase price of $2,000? How much interest would you have to pay for this privilege?
It took about 166 months, nearly 14 years! Nearly $1,500 in interest (75% of the original purchase price) was paid over this time period.
How would the scenario change if you could afford to pay twice the minimum payment every month?
If one were to pay twice the minimum payment, the balance on the twelfth statement would be $1458.20, a nice improvement over the earlier $1806.60. After 62 months, or just over five years the account would paid off. This is an extremely significant time savings over the 14 years! This scenario results in only about $514 (about a quarter of the original purchase price) going towards interest.
What happens if, for some reason, you miss your payments twice during the first year?
Consider the scenario where we are paying twice the minimum payment each month. We'll assume we miss payments 5 and 7. Therefore no payments will be made during those months (but interest charges will continue to accrue), two $39 fees will be charged on statements 6 and 8, and starting with statement 8 the APR will be increased to the default interest rate of 29.4%. The interest rate will lower back to the original APR after 12 on-time payments, beginning with statement 19. With these assumptions, the spreadsheet (Excel 2007 (.xlsx) 52kB Nov6 09) shows the resulting figures. During the two months when the $39 fee was assessed, it was added to the monthly payment.
This increases the amount of time to pay off the account by about three months. The total amount paid is increased significantly: $2789.42 (an increase of about $275).
Is it worth "shopping around" for a credit card with a lower APR? Explain when this would be advantageous and why.
Of course, it is worth finding a card that has better interest rate than this one. As we saw, if one makes twice the minimum payment each month after a year the balance will be about $1458. If one could reduce the annual interest rate by, say, 2.5%, then the balance after a year would be $1488, a $50 savings. With this lower interest rate the original purchase would be paid off about one month earlier and saving roughly $90 in interest charges. Clearly it is always better to have a lower interest rate. However, if the lower rate comes at some other expense, say an annual fee, then it may not be worth it. For example, if the above interest reduction came with a card that had a $100 annual fee, it would not be worth switching over.
Assessment
Assessment for this example varies greatly depending on how much class time one wishes to devote to this project, the mathematical level of the students, and the available technology.
Assessment Plan 1 (Beginners): All students should complete item 1 before coming to class. Working in groups and encouraging peer-to-peer instruction, students should work through the calculations associated with item 2. Some instructor guidance on organization may prove helpful. Items 1 and 2 can then be assigned for individual homework and collected the following class day. On the second day of the activity, the instructor may wish to show the class how a spreadsheet can be used to quickly repeat the series of calculations from item 2. Using a spreadsheet projected in front of the entire class, the instructor can perform the necessary calculations in order to answer questions posed in items 3 - 7.
Assessment Plan 2 (Intermediate/Advanced): The seven study questions can be used as the basis for a longer written project. Students should be required to create spreadsheets to analyze the various payment situations, the effects of missed payments and the effects of different interest rates.
References and Resources
- A variety of credit card disclosure statements may be found at Bank of America.
- An example of an Excel Spreadsheet (Excel 2007 (.xlsx) 60kB Nov6 09) that can be used to duplicate the types of calculations associated with this example.
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