Case Study Comparing Car Loans$-7\mathrm{.}5\%$

After reading consumer car guides and receiving advice from family and friends, Aysha has chosen the new car she would like to purchase. She now wants to research her financing options to choose the best way to pay for the car.

Aysha knows that with taxes, license, delivery, and dealer preparation fees, her car will cost She has saved toward the purchase price but must borrow the rest. She has narrowed her financing choices to three options: dealer financing, credit union financing, and bank financing.

$1.$ The car dealer has offered $48-$month financing at $4\mathrm{.}5\%$ compounded monthly.

$2.$ The credit union has offered $36-$month financing at $5\%$ compounded quarterly. It has also offered $48-$month financing at $5\mathrm{.}3\%$ compounded quarterly.

$3.$ The bank has offered $36-$month financing at $4\mathrm{.}8\%$ compounded semi$-$annually. It has also offered $48-$month financing at $5\mathrm{.}1\%$ compounded semi$-$annually.

Aysha desires the financing option that offers the best interest rate. However, she also wants to explore the financing options that allow her to pay off her car loan more quickly.

Questions

$1.$ Aysha wants to compare the $48-$month car loan options offered by the car dealer, the credit union, and the bank.

a$.$ What is the effective annual rate of interest for each $48-$month option?

b$.$ How much interest will Aysha save by choosing the best option compared to the worst option?

$2.$ Suppose Aysha wants to pay off her car loan within three years.

a$.$ What is the effective annual rate of interest for both of the $36-$month options?

b$.$ How much interest will Aysha save by choosing the better option?

$3.$ If you wanted to get a car loan today, what are the current rates of interest for $36-$month and $48-$month terms? Are car dealers offering better interest rates than banks or credit unions? If so$,$ why?