I have worked and reworked this problem, but I cannot seem to figure it out. Can anyone help?
Jeffries & Sons is borrowing $95,000 for four years at an APR of 7.05 percent. The principal is to be repaid in equal annual payments over the life of the loan with interest paid annually. Payments will be made at the end of each year. What is the total payment due for Year 3 of this loan?
Thank you in advance!
To find the total payment due for Year 3 of the loan, we need to calculate the amount of principal repaid, as well as the interest paid.
Step 1: Calculate the principal repaid in Year 3:
The loan is to be repaid in equal annual payments over 4 years, so the principal repaid each year will be $95,000 divided by 4 years, which is $23,750.
Step 2: Calculate the interest paid in Year 3:
The interest rate is 7.05 percent, and interest is paid annually. So, the interest paid in Year 3 will be 7.05 percent of the remaining balance after Year 2's payment.
To calculate the remaining balance after Year 2's payment, subtract the principal repaid in Year 2 ($23,750) from the original loan amount ($95,000):
Remaining balance after Year 2 = $95,000 - $23,750 = $71,250
Now, calculate the interest paid in Year 3 by multiplying the remaining balance after Year 2 by the interest rate:
Interest paid in Year 3 = 7.05 percent of $71,250 = 0.0705 × $71,250 = $5,020.62 (rounding to two decimal places)
Step 3: Calculate the total payment due in Year 3:
The total payment due in Year 3 will be the sum of the principal repaid in Year 3 and the interest paid in Year 3.
Total payment due in Year 3 = Principal repaid in Year 3 + Interest paid in Year 3
Total payment due in Year 3 = $23,750 + $5,020.62 = $28,770.62
Therefore, the total payment due for Year 3 of the loan is $28,770.62.
To find the total payment due for Year 3 of the loan, we'll need to calculate the annual payment amount first. This can be done using the formula for calculating the equal annual payment for an amortizing loan.
The formula for calculating the equal annual payment is:
PMT = [P x i(1 + i)^n] / [(1 + i)^n - 1]
Where:
PMT = Payment
P = Principal amount borrowed
i = Interest rate per period
n = Total number of periods
In this case, the principal (P) is $95,000, the interest rate (i) is 0.0705 (7.05% expressed as a decimal), and the total number of periods (n) is 4 years.
Using this formula, let's calculate the annual payment amount first:
PMT = [95,000 x 0.0705(1 + 0.0705)^4] / [(1 + 0.0705)^4 - 1]
PMT = [95,000 x 0.0705(1.0705)^4] / [(1.0705)^4 - 1]
PMT = [95,000 x 0.0705(1.316035673)] / [(1.316035673) - 1]
PMT = (95,000 x 0.0924) / (0.316035673)
PMT = 8,748 / 0.316035673
PMT ≈ $27,678.23
Now that we have the annual payment amount, we can determine the total payment due for Year 3. Each year, the principal and interest components of the payment change as the remaining balance decreases. To calculate the payment for Year 3, we need to calculate the interest paid in Year 3 and subtract it from the total payment amount.
To find the interest paid in Year 3, multiply the outstanding balance at the beginning of Year 3 by the annual interest rate:
Interest Paid in Year 3 = Outstanding Balance at the Beginning of Year 3 x Annual Interest Rate
To find the outstanding balance at the beginning of Year 3, we need to subtract the principal paid in Years 1 and 2 from the original loan amount. In each year, the principal paid equals the annual payment minus the interest paid.
Let's calculate the outstanding balance at the beginning of Year 3 and the interest paid in Year 3:
First, let's find the principal paid in Years 1 and 2:
Principal Paid in Years 1 and 2 = Annual Payment - Interest Paid in Year 1 - Interest Paid in Year 2
The interest paid in Year 1 can be calculated as:
Interest Paid in Year 1 = Outstanding Balance at the Beginning of Year 1 x Annual Interest Rate
Similarly, the interest paid in Year 2 can be calculated as:
Interest Paid in Year 2 = Outstanding Balance at the Beginning of Year 2 x Annual Interest Rate
The outstanding balance at the beginning of Year 1 is the original loan amount, and for Year 2, we subtract the principal paid in Year 1 from the original loan amount.
Let's calculate the principal paid in Years 1 and 2:
Principal Paid in Year 1 = Annual Payment - (Outstanding Balance at the Beginning of Year 1 x Annual Interest Rate)
Principal Paid in Year 2 = Annual Payment - (Outstanding Balance at the Beginning of Year 2 x Annual Interest Rate)
To find the outstanding balance at the beginning of Year 2, we subtract the principal paid in Year 1 from the original loan amount:
Outstanding Balance at the Beginning of Year 2 = Original Loan Amount - Principal Paid in Year 1
Now we can calculate the principal paid in Year 2 and the outstanding balance at the beginning of Year 2.
Finally, the outstanding balance at the beginning of Year 3 can be found by subtracting the principal paid in Year 2 from the outstanding balance at the beginning of Year 2.
Once we have the outstanding balance at the beginning of Year 3, we can calculate the interest paid in Year 3 by multiplying it by the annual interest rate.
The total payment due for Year 3 is the sum of the principal paid in Year 3 and the interest paid in Year 3.
To summarize the steps:
1. Use the formula for calculating the equal annual payment to find the annual payment amount.
2. Calculate the interest paid in Year 1 and Year 2.
3. Calculate the principal paid in Year 1 and Year 2.
4. Calculate the outstanding balance at the beginning of Year 2.
5. Calculate the principal paid in Year 3.
6. Calculate the outstanding balance at the beginning of Year 3.
7. Calculate the interest paid in Year 3.
8. Calculate the total payment due for Year 3 by adding the principal paid in Year 3 and the interest paid in Year 3.
Good luck with your calculations.