Question

Prof. Finance decides to buy a 2019 Mazda 6 Signature Edition. After paying a down payment and taxes, Prof. Finance can finance the rest of the purchase price with a loan of $24,000 for 60 months at a special finance rate offered by Mazda of 1.9% APR compounded monthly.

• He finds out that Mazda has a second offer of $1000 cash back (rebate) in place of the special 1.9% finance rate offered with the cash back being an additional down payment. Prof. Finance finds he can get 2.9% APR financing online for 60 months if he takes the $1000 cash back offer. Answer the following questions.

1. What is the effective annual rate for each loan?

2. What would be the monthly car loan payment under the Mazda’s 1.9% APR financing offer (assume a 60-month loan term)?

3. What would be the monthly car loan payment under the Mazda’s $1000 cash back offer and the 2.9% APR pre-approved financing (assume a 60-month loan term)?

4. At what APR would Prof. Finance be indifferent between the two offers? In other words, what APR (assuming a 60-month loan term) for the $1000 cash back offer would have the same monthly payment with the 1.9% APR financing offer?

5. Let’s assume you go with the offer in question #3. Construct an amortization schedule for the loan for all 60 monthly payments (see section 5-18 of the textbook). What is your loan balance after 36 months?

6. The local Mazda dealer has found a special 2.49% APR loan rate for 60 months from Citibank that Prof. Finance qualifies for if he elects the $1000 cash back option. Prof. Finance says that’s great! What would be the monthly payment under this loan?

7. Prof. Finance is more than happy with the 2.49% APR and $1000 cash back offer but wants a monthly payment of $375 (assume a 60-month loan term) and realizes he will have to put more money down. How much additional money will Prof. Finance have to put down in order to achieve his target monthly payment? Note: original loan amount with cash back was $23,000.

Answer 1

1]

EAR = (1 + (APR / compounding frequency)^{compounding frequency} - 1

Here, the compounding frequency is 12 as the payments are monthly

EAR of 1.9% loan = (1 + (1.9%/12))¹² - 1 = 1.917%

EAR of 2.9% loan = (1 + (2.9%/12))¹² - 1 = 2.939%

2]

Monthly payment is calculated using PMT function in Excel :

rate = 1.9% / 12 (converting annual rate into monthly rate)

nper = 60 (60 months is the loan term)

pv = 24000 (loan amount)

PMT is calculated to be $419.62

3]

Monthly payment is calculated using PMT function in Excel :

rate = 2.9% / 12 (converting annual rate into monthly rate)

nper = 60 (60 months is the loan term)

pv = 23000 (With the additional down payment, loan amount is reduced by $1000)

PMT is calculated to be $412.26

4]

APR for 2.9% loan is calculated using RATE function in Excel :

nper = 60

pmt = -419.62 (monthly payment with 1.9% APR. This is entered as a negative number because it is a cash outflow)

pv = 23000 (loan amount)

RATE is calculated to be 0.3014%. This is the monthly rate. To get annual rate, we multiply by 12. Annual rate is 3.62%

The APR should be 3.62%