Refer to the following for Exercises 31-35, about credit card payments. Many credit cards use a similar formula for the minimum payment, which is the new balance (if less than $25), or else the greatest of $25 or 1% of the new balance (excluding interest and late fees), plus the interest billed, rounded down to the nearest dollar. Any late fees are then added on to this calculated amount. Moreover, when any interest is due, there is a minimum charge of $1.50.
Question 22.63
33. The purpose of such a complicated formula for the minimum payment on a credit card is to avoid the situation of a customer who makes just the minimum payment but nevertheless falls farther and farther behind. For example, formerly some banks set the minimum payment at balance due or else the larger of $10 or 2% of the total new balance (including interest). However, for a high enough interest rate, paying 2% of the balance due will not cover the interest, so the balance actually would increase (this is called negative amortization). How high would the APR have to be to make this happen? (Hint: It’s not just )
33.
Let be the APR, with the old balance (after the preceding payment) and the new balance (after addition of interest for this period), and let the bill be for 30 days. The daily interest rate is , and we have . The interest is . If the interest is greater than 0.02, a payment of 2% of will not keep up with the interest due. Solving
gives first , then , and , so that , yielding finally . Using a 31-day month gives .