So far we have looked at loans for large amounts that are usually paid off over long periods. Credit cards are used by consumers to take loans exactly equal to the cost of a purchase, generally for a small amount (less than $\$10000$$10000), on a more frequent basis. The expectation is that the money borrowed will be repaid in the near future, so interest rates on credit cards tend to be much higher than those with a longer repayment period.
Calculating interest
A $\$5000000$$5000000 home loan may have an interest rate of $4%$4% per annum to be paid off over $25$25 years, whereas money borrowed using a credit card to purchase a $\$50$$50 shirt may have an interest rate of $15%$15% per annum. The credit card provider expects you to pay off this small loan in the same month that the purchase is made.
Generally, interest on credit cards will be charged daily. So if an interest rate is given per annum, we'll have to make sure to calculate the equivalent daily interest rate. To do this we use the same method we used for varying compounding periods as in 5.03. We assume that there are $365$365 days in a regular year, so we divide the annual interest rate by $365$365 to find the equivalent daily interest rate.
Interest-free periods
The high interest rate applied to credit card purchases may make customers think twice about taking these small loans, so in order to encourage customers to use their credit cards many banks offer an interest-free period on purchases. As long as any loans are paid off within that time, no interest will be charged, so the customer will only have to pay back the original purchase amount.
It's important to note that interest-free periods, typically $55$55 days, always begin on the first day of the month, not the date of any particular transaction. This means that a purchase on the $30$30th of the month will only have $25$25 days in the interest-free period, with a new interest-free period beginning on the $1$1st of the following month.
In this diagram, a customer pays for a meal in January on their credit card, and repays it in February. They also pay a phone bill in February with their credit card, and repay it in March:
The meal was repaid within the $55$55 day interest-free period that began on $1$1 January, and the phone bill was repaid within the $55$55 day interest-free period that began on $1$1 February, so no interest accrues for either purchase.
In this next example, the customer pays for both a meal and their phone bill in January, and repays them both in February:
Here the customer did not repay the full amount they borrowed during January by the end of the interest-free period. This means that they will be charged interest, an the amount they are charged will depend on the terms and conditions defined by their bank. They may even be charged interest from the date of purchase, as though the interest-free period never occurred, even though the customer paid the amount they borrowed for the phone bill only a few days after the end of the interest-free period, they could be charged interest on both purchases for the entire month!
Exploration
Steve has a credit card that charges $18%$18% interest per annum. Using the card, he makes two purchases of $\$150$$150 and $\$300$$300 on $10$10 March and $20$20 March respectively. If the card has a $55$55 day interest-free period, how much interest will he accrue:
a. if he repays $\$450$$450 onto his card by the end of March?
b. if he fully repays his card on $15$15 June?
c. if he repays $\$250$$250 on $30$30 March, and the remainder of the balance on $7$7 May?
In general, there are three situations we could find ourselves in:
- The loans are repaid in full during the interest-free period,
- None of the loan is repaid during the interest-free period,
- Some, but not all of the loan is repaid during the interest-free period.
The important date in these calculations is $55$55 days after $1$1 March, which is $24$24 April. This is when the March interest-free period expires. We also need to remember to subtract the original purchase amount after we do our calculations, since this question is asking for the interest charges only.
a. Both purchases were made during March, so paying off the credit card at the end of the month will definitely fall inside the interest-free period. This means he will pay no interest.
b. The $15$15 June falls well outside the interest-free period, so Steve's loans will accrue interest from their date of purchase. The first purchase is repaid $98$98 days after it was made, and the second is repaid $88$88 days after.
Credit cards are a type of reducing balance loan, so we can use the future value formula to calculate interest.
| Purchase 1 | ||
| Interest | $=$= | $FV$FV $-$− Repayment |
| $=$= | $PV\left(1+r\right)^n$PV(1+r)n $-$− Repayment | |
| $=$= | $150\left(1+\frac{18}{36500}\right)^{98}-150$150(1+1836500)98−150 | |
| $=$= | $157.43-150$157.43−150 | |
| $=$= | $\$7.43$$7.43 | |
| Purchase 2 | ||
| Interest | $=$= | $FV$FV $-$− Repayment |
| $=$= | $PV\left(1+r\right)^n$PV(1+r)n $-$− Repayment | |
| $=$= | $300\left(1+\frac{18}{36500}\right)^{88}-300$300(1+1836500)88−300 | |
| $=$= | $313.30-300$313.30−300 | |
| $=$= | $\$13.30$$13.30 | |
The total amount of interest paid on the credit card is $\$20.73$$20.73.
c. $\$250$$250 is paid off within the interest-free period, so at the conclusion of this period there is still $\$200$$200 remaining to be repaid. This amount is repaid $49$49 days after the purchase date, and if we make the assumption that interest is calculated only on the remaining balance at the end of the interest-free period, then:
| Interest | $=$= | $FV$FV $-$− Repayment |
| $=$= | $PV\left(1+r\right)^n$PV(1+r)n $-$− Repayment | |
| $=$= | $200\left(1+\frac{18}{36500}\right)^{49}-200$200(1+1836500)49−200 | |
| $=$= | $204.89-200$204.89−200 | |
| $=$= | $\$4.89$$4.89 |
The total amount of interest paid on the credit card is $\$4.89$$4.89.
The assumption made in part (c) will not always be true. In fact, almost every bank takes a different approach to calculating interest when interest-free periods are involved.
Credit card statements
Apart from the additional interest accrued by making loans with a credit card, a consumer should also expect to pay additional fees if they do not repay a certain minimum amount each month. If a credit card is used for anything other than making purchases, such as withdrawing cash from an ATM, then extra charges (fees) may also apply.
At the end of each month a consumer will receive a credit card statement which summarises their purchases and repayments during that month. This statement should also contain the minimum repayment expected of the consumer, as well as any relevant payment due dates. Here is an example:
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Practice questions
Question 1
Question 2
Xanthe has a credit card that charges $22%$22% interest p.a. Using the card, she buys a new pair of hiking boots for $\$250$$250 on 1 October, a rainproof jacket for $\$180$$180 on 20 October and a backpack for $\$75$$75 on 25 November. She pays off her credit card on 10 December. The card has a $30$30 day interest-free period that is forfeited completely if the balance is not repaid before it.
How much interest will Xanthe accrue for these purchases?