a small business loan in the amount of 50,000 need to interest rate of 9% that compounds monthly for 7 years. What is the payment for 7 years, need formula for this. And what is the balance of the loan at the end of the 1st year? Show the formula used for each variable and the unpaid balance at end of 1st year
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.