Question

Kyle Parker of Concord, New Hampshire, has been shopping for a new car for several weeks. He has negotiated a price of $33,000 on a model that carries a choice of a $2,500 rebate or dealer financing at 2 percent APR. The dealer loan would require a $1,000 down payment and a monthly payment of $561 for 60 months. Kyle has also arranged for a loan from his bank with a 6 percent APR. Use the Run the Numbers worksheet to advise Kyle about whether he should use the dealer financing or take the rebate and use the financing from the bank. Round your answer to two decimal places.

Adjusted APR (dealer financing):

Kyle should use either the financing from the bank or the dealer finance?

Answer 1

Adjusted APR (dealer financing):

Option 1: Suppose Kyle Parker goes for dealer financing at 2 percent APR. The dealer loan would require a $1,000 down payment and a monthly payment of $561 for 60 months.

Loan amount = Car price – down payment = $33,000 - $1,000 = $32,000

We can use PV of an Annuity formula to calculate the monthly payment of car loan

PV = PMT * [1-(1+i) ^-n)]/i

Where PV of loan = $32,000

PMT = Monthly payment =?

n = N = number of payments = 60 months

i = I/Y = interest rate per year = 2%, therefore monthly interest rate is 2%/12 = 0.16% per month

Therefore,

$32,000 = PMT* [1- (1+0.16%) ^-60]/0.16%

= $560.89 or $561

Monthly payment of car loan is $561

Option 2: Suppose Kyle Parker take a $2,500 rebate from dealer and go for bank finance

Loan amount after rebate = Car price – rebate = $33,000 - $2,500 = $30,500

We can use PV of an Annuity formula to calculate the monthly payment of car loan

PV = PMT * [1-(1+i) ^-n)]/i

Where PV of loan = $30,500

PMT = Monthly payment =?

n = N = number of payments = 60 months

i = I/Y = interest rate per year = 6%, therefore monthly interest rate is 6%/12 = 0.5% per month

Therefore,

$30,500 = PMT* [1- (1+0.5%) ^-60]/0.5%

= $589.65 or $590

Monthly payment of bank loan is $590

Kyle should use either the financing from the bank or the dealer finance?

The difference between monthly payments from both options

= Monthly payment of bank loan - Monthly payment of dealer finance

= $589.65 - $560.89 =$28.76

Now calculate the present value of this difference in monthly payments for 60 months

We can use PV of an Annuity formula

PV = PMT * [1-(1+i) ^-n)]/i

Where PV of loan =?

PMT = Monthly payment = $28.76

n = N = number of payments = 60 months

i = I/Y = interest rate per year = 6%, therefore monthly interest rate is 6%/12 = 0.5% per month

Therefore,

PV = $28.76* [1- (1+0.5%) ^-60]/0.5%

= $1,487.74

This amount is more than the down payment of $1,000 in dealer financing, therefore Kyle should use the dealer financing because it is cheaper.