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 the fixed monthly payment on a loan. This formula is:
PMT = (P * r * (1 + r)^n) / ((1 + r)^n - 1)
Where:
- PMT is the monthly payment
- P is the principal amount (loan amount)
- r is the monthly interest rate
- n is the number of monthly payments
First, let's calculate the monthly interest rate. The annual interest rate is 10%, so we need to convert it to a monthly rate by dividing it by 12 (since there are 12 months in a year):
Monthly interest rate = 10% / 12 = 0.008333 (approximately)
Next, let's substitute the values into the formula:
P = $35,000
r = 0.008333
n = 5 years * 12 months/year = 60 months
PMT = ($35,000 * 0.008333 * (1 + 0.008333)^60) / ((1 + 0.008333)^60 - 1)
Using a proper mathematical calculator or a spreadsheet, the monthly loan payment can be calculated.
Now, let's calculate the loan's effective annual interest rate (EAR). The EAR takes into account any compounding effects of interest throughout the year. To calculate the EAR, we can use the following formula:
EAR = (1 + r/n)^n - 1
Where:
- r is the nominal interest rate (monthly rate)
- n is the number of compounding periods per year
Substituting the values, we get:
r = 0.008333
n = 12 (since payments are monthly)
EAR = (1 + 0.008333/12)^12 - 1
Again, using a calculator or spreadsheet, you can find the loan's effective annual interest rate (EAR).