Assume the car can be purchased for 0% down for 60 months (in lieu of rebate).
A car with a sticker price of $42,900 with factory and dealer rebates of $5,100
(a) Find the monthly payment if financed for 60 months at 0% APR. (Round your answer to the nearest cent.)
$
(b) Find the monthly payment if financed at 2.5% add-on interest for 60 months. (Round your answer to the nearest cent.)
$
(c) Use the APR approximation formula to find the APR for part (b). (Round your answer to one decimal place.)
%
(d) State whether the 0% APR or the 2.5% add-on rate should be preferred.
0% APR
2.5% add-on rate
a. (42,900-5100)/60 =
b. Po = 42900 - 5100 = $37,800.
P = Po*r*t/(1-(1+r)^-t)
r = (2.5%/12)/100* = 0.00208 = Monthly %
rate expressed as a decimal.
t = 60 Months.
P = (37,800*0.00208*60)/(1-1.00208^-60)
= $40,251.02
Monthly Payments = P/t
To find the answers to each question, you need to use the formula for calculating monthly payments on a loan:
Monthly Payment = [Loan Amount / (1 - (1 + interest rate)^(-number of months))]
Now let's solve each part of the question step by step:
(a) To find the monthly payment at 0% APR, we need to calculate the loan amount. The sticker price minus the rebates will give us the loan amount:
Loan Amount = Sticker Price - Rebates
= $42,900 - $5,100
= $37,800
Substituting this value into the formula:
Monthly Payment = [$37,800 / (1 - (1 + 0)^(-60))]
= [$37,800 / (1 - 1)]
= [$37,800 / 0]
Since dividing by zero is undefined, there is no monthly payment at 0% APR.
Thus, the answer is $0.
(b) To find the monthly payment at 2.5% add-on interest, we first need to calculate the amount being financed. In this case, the rebates are not applied, so the financed amount is equal to the sticker price:
Financed Amount = Sticker Price
= $42,900
Next, we calculate the interest amount:
Interest = Financed Amount * (interest rate / 100)
= $42,900 * (2.5 / 100)
= $42,900 * 0.025
= $1,072.50
Therefore, the Total Loan Amount = Financed Amount + Interest
= $42,900 + $1,072.50
= $44,972.50
Substituting this value into the formula:
Monthly Payment = [$44,972.50 / (1 - (1 + 2.5/100)^(-60))]
= [$44,972.50 / (1 - (1 + 0.025)^(-60))]
= [$44,972.50 / (1 - 1.62889424326)]
= [$44,972.50 / (-0.62889424326)]
≈ $71,593.25
Therefore, the monthly payment at 2.5% add-on interest for 60 months is approximately $71,593.25.
(c) To find the APR using the APR approximation formula, we can rewrite the formula for calculating monthly payments as:
Loan Amount = Monthly Payment * [(1 - (1 + interest rate)^(-number of months)) / interest rate]
Rearranging this formula to solve for the interest rate:
interest rate = [(1 - (Loan Amount / Monthly Payment))^(1 / -number of months)] - 1
Substituting the values from part (b):
interest rate = [(1 - ($44,972.50 / $71,593.25))^(1 / -60)] - 1
= [(1 - 0.62889424326)^(1 / -60)] - 1
Using a calculator, we can find that the interest rate ≈ -0.0125.
Multiply by 100 to convert the rate to a percentage:
interest rate ≈ -0.0125 * 100
≈ -1.25%
Therefore, the APR for the loan with a 2.5% add-on rate is approximately -1.25%.
(d) Based on the given options, the 0% APR should be preferred over the 2.5% add-on rate since 0% interest means no interest is charged on the loan, while 2.5% add-on rate means you will be paying additional interest on top of the loan amount.