To calculate the payment for a loan, you can use the formula for the monthly payment of a loan:
P = (Pv * r) / (1 - (1 + r)^(-n))
Where:
P = monthly payment
Pv = present value of the loan (loan amount)
r = monthly interest rate
n = total number of payments
First, let's calculate the monthly interest rate (r) by converting the annual interest rate (9%) to a monthly rate.
r = annual interest rate / 12 = 9% / 12 = 0.0075
Next, let's calculate the total number of payments (n) for a loan term of 7 years with monthly compounding:
n = total number of years * 12 = 7 * 12 = 84
Now, we can substitute the values into the formula to calculate the monthly payment (P):
P = (50,000 * 0.0075) / (1 - (1 + 0.0075)^(-84))
By evaluating this formula, the monthly payment (P) for a $50,000 loan with a 9% interest rate compounded monthly for 7 years is approximately $814.57.
To calculate the balance of the loan at the end of the first year, we can use the formula for the unpaid balance of a loan after a certain period. The formula is:
Unpaid Balance = Pv * (1 + r)^n - (P * [(1 + r)^n - 1]) / r
Where:
Unpaid Balance = remaining balance at the end of the specified period
Pv = present value of the loan (loan amount)
r = monthly interest rate
n = number of payments made
For the end of the first year, n = 1. Substituting the values into the formula, we have:
Unpaid Balance = 50,000 * (1 + 0.0075)^1 - (814.57 * [(1 + 0.0075)^1 - 1]) / 0.0075
Evaluating this expression will give you the remaining balance at the end of the first year.