equipment cost $20,000 and is financed over a period of five years at an interest rate of 12%...What is the monthly payment...What is the loan balance at the end of four years and how much interest will have been paid on the loan after five years?
To calculate the monthly payment, loan balance at the end of four years, and the total interest paid on the loan after five years, we need to use the formula for calculating loan payments, loan balance, and total interest paid.
1. Monthly Payment:
The formula to calculate the monthly payment is given by the following formula:
M = P * (r * (1+r)^n) / ((1+r)^n - 1)
Where:
M = Monthly payment
P = Principal amount (equipment cost)
r = Monthly interest rate (annual interest rate divided by 12)
n = Number of monthly payments (duration of the loan in years multiplied by 12)
In this case:
P = $20,000
r = 12% / 100 / 12 = 0.01
n = 5 * 12 = 60
Now, we can calculate the monthly payment:
M = $20,000 * (0.01 * (1+0.01)^60) / ((1+0.01)^60 - 1)
By plugging the values into the formula, we can solve for M.
2. Loan Balance at the end of four years:
To calculate the loan balance at the end of four years, we need to find the remaining principal amount. We can use the loan amortization formula:
B = P * (1+r)^n - ((1+r)^p - 1) / r
Where:
B = Remaining balance
P = Principal amount (equipment cost)
r = Monthly interest rate
n = Number of monthly payments (duration of the loan in years multiplied by 12)
p = Number of monthly payments made (4 years multiplied by 12)
By plugging in the values, we can find the loan balance.
3. Total Interest Paid:
To calculate the total interest paid on the loan after five years, we need to deduct the principal amount from the total payments made. The total payments can be calculated by multiplying the monthly payment by the number of monthly payments:
Total Payments = M * n
Then, the total interest paid can be determined by subtracting the principal amount from the total payments.
Let's calculate these values one by one.
To calculate the monthly payment, we can use the formula for a fixed-rate loan:
Monthly Payment = P * r * (1 + r)^n / ((1 + r)^n - 1)
where P is the principal loan amount, r is the monthly interest rate, and n is the total number of payments.
Step 1: Convert the annual interest rate to a monthly interest rate.
The monthly interest rate is calculated by dividing the annual interest rate by 12.
r = 12% / 12 = 0.01.
Step 2: Determine the total number of payments.
The loan is financed over a period of five years, which is equivalent to 60 months.
n = 5 years * 12 months/year = 60 months.
Step 3: Calculate the monthly payment.
P = $20,000.
Monthly Payment = $20,000 * 0.01 * (1 + 0.01)^60 / ((1 + 0.01)^60 - 1).
Using a calculator, the monthly payment is approximately $448.29.
To calculate the loan balance at the end of four years, we need to determine the number of remaining payments.
Step 1: Calculate the number of remaining payments.
The loan is financed for five years, but at the end of four years, there will be one year (12 months) remaining.
Remaining payments = 60 months - 48 months = 12 months.
Step 2: Calculate the loan balance.
To calculate the loan balance, we can use the formula:
Loan Balance = P * ((1 + r)^n - (1 + r)^p) / ((1 + r)^n - 1)
where P is the principal loan amount, r is the monthly interest rate, n is the total number of payments, and p is the number of remaining payments.
Loan Balance = $20,000 * ((1 + 0.01)^60 - (1 + 0.01)^12) / ((1 + 0.01)^60 - 1).
Using a calculator, the loan balance at the end of four years is approximately $8,051.51.
To calculate the total interest paid on the loan after five years, we can subtract the original principal loan amount from the total amount paid.
Step 1: Calculate the total amount paid.
The total amount paid is the monthly payment multiplied by the number of payments.
Total Amount Paid = Monthly Payment * Total Number of Payments.
Total Amount Paid = $448.29 * 60 = $26,897.40.
Step 2: Calculate the total interest paid.
Total Interest Paid = Total Amount Paid - Principal Loan Amount.
Total Interest Paid = $26,897.40 - $20,000 = $6,897.40.
Therefore, the loan balance at the end of four years is approximately $8,051.51, and the total interest paid on the loan after five years is $6,897.40.