Consider an amortized loan of $10,000 ( with 10.0% interest compounded monthly) paid back over two yrs in equal, monthly installments. For each of the first two months, calculate three things: the interest paid for the month, the amound of the principal paid off and the balance at the end of the month.
To calculate the interest paid for each month, the amount of principal paid off, and the balance at the end of the month for an amortized loan, we can follow these steps:
Step 1: Calculate the monthly interest rate
Since the annual interest rate is 10.0%, we need to convert it to a monthly rate. Divide the annual rate by 12 (the number of months in a year) and express it as a decimal:
Monthly interest rate = (10.0% / 12) / 100 = 0.00833333
Step 2: Calculate the monthly payment
To determine the monthly installment, we need to use a loan amortization formula. In this case, we can calculate it using the "loan repayment" formula, which is:
Monthly Payment = (Principal * Monthly interest rate) / (1 - (1 + Monthly interest rate)^(-Number of months))
Plugging in the values:
Principal = $10,000
Monthly interest rate = 0.00833333
Number of months = 2 years x 12 months/year = 24 months
Monthly Payment = (10,000 * 0.00833333) / (1 - (1 + 0.00833333)^(-24))
Monthly Payment = $446.25 (rounded to two decimal places)
Step 3: Calculate for month 1:
Interest paid for the month:
For the first month, multiply the balance at the beginning of the month (which is the principal) by the monthly interest rate:
Interest for month 1 = Principal * Monthly interest rate = $10,000 * 0.00833333 = $83.33
Principal paid off for the month:
Subtract the interest paid from the monthly payment:
Principal paid for month 1 = Monthly Payment - Interest for month 1 = $446.25 - $83.33 = $362.92 (rounded to two decimal places)
Balance at the end of the month:
Subtract the principal paid off for the month from the beginning balance (principal) to get the balance at the end of the month:
Balance at the end of month 1 = Principal - Principal paid for month 1 = $10,000 - $362.92 = $9,637.08 (rounded to two decimal places)
Step 4: Calculate for month 2:
Interest paid for the month:
For the second month, multiply the balance at the beginning of the month (which is the remaining balance from month 1) by the monthly interest rate:
Interest for month 2 = Balance at the end of month 1 * Monthly interest rate = $9,637.08 * 0.00833333 = $80.31 (rounded to two decimal places)
Principal paid off for the month:
Subtract the interest paid from the monthly payment:
Principal paid for month 2 = Monthly Payment - Interest for month 2 = $446.25 - $80.31 = $365.94 (rounded to two decimal places)
Balance at the end of the month:
Subtract the principal paid off for the month from the beginning balance (remaining balance from month 1) to get the balance at the end of the month:
Balance at the end of month 2 = Balance at the end of month 1 - Principal paid for month 2 = $9,637.08 - $365.94 = $9,271.14 (rounded to two decimal places)
Therefore, for the first two months:
Month 1:
- Interest paid: $83.33
- Principal paid off: $362.92
- Balance at the end of the month: $9,637.08
Month 2:
- Interest paid: $80.31
- Principal paid off: $365.94
- Balance at the end of the month: $9,271.14