Question

You are considering signing up for a new "cash back" credit card. By using this credit card, the card company will give you a cash payment at the end of each month equal to 4% of the value of your credit card bill during that month (e.g., if you spend $100 on your credit card in January, the card company will pay you $4 in cash at the end of the month). In order to sign up for this credit card, you will have to pay an annual fee at the beginning of the year. Suppose you reliably spend $1,000 on your credit card each month. If you can receive . 75% monthly compound interest on your savings, what is the largest annual fee that you'd be willing to pay in order to enroll in this credit card?

A. $309.41
B. $457.40
C. $500.30
D. $3094.11
E. $4573.97

Answer 1

To determine the largest annual fee that you'd be willing to pay for the cash back credit card, calculate the present value of the cash back payments using the formula P = A / (1 + r)^n, and compare it to the annual fee. For this scenario, the largest annual fee is approximately $309.41.

Step-by-step explanation:

To determine the largest annual fee that you'd be willing to pay in order to enroll in this credit card, we need to calculate the present value of the cash back payments and compare it to the cost of the annual fee. The present value can be calculated using the formula P = A / (1 + r)^n, where P is the present value, A is the future value (cash back payment), r is the monthly interest rate, and n is the number of months.

In this case, the future value (cash back payment) is 4% of the credit card bill each month, which is $40 ($1,000 * 4% = $40). The monthly interest rate can be calculated by dividing the annual interest rate (0.75%) by 12, which is approximately 0.0625%. Plugging these values into the formula, we get P = $40 / (1 + 0.0625%)^12, which is approximately $309.41. Therefore, the largest annual fee that you'd be willing to pay is $309.41.