I am purchasing a car for $8,500. I have a loan for the purchase with a 9.2% annual interest rate compounded continuously and the loan will run for three years. How much do I pay per month?
500
To determine the monthly payment on a loan, you can use the formula for continuous compounding:
A = P * e^(rt)
Where:
A = the final amount (loan balance) after time t
P = the principal amount (initial loan balance)
e = the base of the natural logarithm (approximately 2.71828...)
r = the annual interest rate (expressed as a decimal)
t = the time period (in years)
In this case, the principal amount P is $8,500, the annual interest rate r is 9.2% (or 0.092 as a decimal), and the time period t is 3 years.
Now let's calculate the monthly payment step by step:
Step 1: Convert the annual interest rate to a monthly interest rate. Since there are 12 months in a year, divide the annual interest rate by 12:
monthly interest rate = 0.092 / 12 = 0.00766
Step 2: Calculate the total amount owed after 3 years using the continuous compounding formula:
A = P * e^(rt)
A = 8500 * e^(0.00766 * 3)
A ≈ $10,056.59
Step 3: Divide the total amount owed by the number of months in the loan period (3 years * 12 months/year) to calculate the monthly payment:
Monthly payment = 10056.59 / (3 * 12) ≈ $279.35
Therefore, the monthly payment on the loan will be approximately $279.35.