you have contacted a number of dearlerships to determine the best interest rate on a new car loan, dealship quoted a 5 year, 10% loan in the amount of $35,000 that will require monthly payments. What is the monthly loan payment. And what will the loan effective annual interest rate (EAR)?
To calculate the monthly loan payment, we can use the formula for calculating the monthly payment for a fixed-rate loan:
M = P * (r * (1 + r)^n) / ((1 + r)^n - 1)
Where:
- M is the monthly payment
- P is the loan amount ($35,000 in this case)
- r is the monthly interest rate (annual interest rate divided by 12)
- n is the total number of monthly payments (5 years = 5 * 12 = 60 months)
Let's calculate the monthly payment first:
r = 10% / 100 / 12 = 0.008333 (monthly interest rate)
n = 5 * 12 = 60 (total number of payments)
M = 35,000 * (0.008333 * (1 + 0.008333)^60) / ((1 + 0.008333)^60 - 1)
M ≈ $744.82 (rounded to the nearest cent)
Therefore, the monthly loan payment will be approximately $744.82.
To calculate the loan's effective annual interest rate (EAR), we need to use the formula:
EAR = (1 + r)^m - 1
Where:
- EAR is the effective annual interest rate
- r is the monthly interest rate
- m is the number of compounding periods per year (in this case, m = 12 since it's compounded monthly)
Using the given monthly interest rate of 0.008333, let's calculate the EAR:
EAR = (1 + 0.008333)^12 - 1
EAR ≈ 10.4714% (rounded to the nearest hundredth)
Therefore, the loan's effective annual interest rate (EAR) is approximately 10.47%.