A five-year loan of $25,000 at 7.2% compounded quarterly requires quarterly payments. (Round your answer to 2 decimal places.)
a. Calculate the interest component of Payment 10.
Interest component $
b. Calculate the principal component of Payment 13.
Principal component $
c. Calculate the total interest in Payments 5 to 10 inclusive.
Total interest $
d. Calculate the principal paid in Year 4.
Principal paid $
first step is to find the quarterly payment.
Did you do that ?
Using pv formula?
yes
To answer these questions, we need to understand the formula for calculating the quarterly payment of a loan:
Payment = Principal component + Interest component
We can find the interest component and principal component using the following formulas:
Interest component = Principal balance * Quarterly interest rate
Principal component = Payment - Interest component
Let's calculate each of the requested values step by step:
a. Calculate the interest component of Payment 10:
First, we need to find the principal balance of Payment 10. To do this, we need to calculate the principal balance after Payment 9, as each payment reduces the principal balance.
Principal balance after Payment 9 = Principal balance - Principal component of Payment 9
Since this is the 10th payment, we need to calculate the principal balance after 9 payments. We can use the formula for the principal balance after n payments:
Principal balance after n payments = Principal balance * (1 + Quarterly interest rate)^n - Payment * ((1 + Quarterly interest rate)^n - 1) / Quarterly interest rate
In this case, n = 9. Let's calculate the principal balance after Payment 9:
Principal balance after Payment 9 = $25,000 * (1 + 0.072/4)^9 - Payment * ((1 + 0.072/4)^9 - 1) / (0.072/4)
Now, we can calculate the interest component of Payment 10 using the formula:
Interest component = Principal balance after Payment 9 * Quarterly interest rate
b. Calculate the principal component of Payment 13:
First, we need to find the principal balance of Payment 13. To do this, we can use the formula mentioned earlier:
Principal balance after n payments = Principal balance * (1 + Quarterly interest rate)^n - Payment * ((1 + Quarterly interest rate)^n - 1) / Quarterly interest rate
In this case, n = 13. Let's calculate the principal balance after Payment 13:
Principal balance after Payment 13 = $25,000 * (1 + 0.072/4)^13 - Payment * ((1 + 0.072/4)^13 - 1) / (0.072/4)
Now, we can calculate the principal component of Payment 13 using the formula:
Principal component = Payment - Interest component
c. Calculate the total interest in Payments 5 to 10 inclusive:
First, we need to find the principal balance after Payment 4, as each payment reduces the principal balance.
Principal balance after Payment 4 = Principal balance - Principal component of Payment 4
Since we want to calculate the total interest from Payment 5 to 10, we need to calculate the principal balance after 4 payments and 1 to 10 payments. We can use the formula mentioned earlier to calculate the principal balance after n payments.
Next, we need to find the interest component of each payment from Payment 5 to 10 and sum them up to find the total interest.
d. Calculate the principal paid in Year 4:
To calculate the principal paid in Year 4, we need to find the principal balance after 3 years (12 payments) and subtract it from the original loan amount.
Principal balance after 3 years = Principal balance * (1 + Quarterly interest rate)^(12 * 3) - Payment * ((1 + Quarterly interest rate)^(12 * 3) - 1) / (Quarterly interest rate)
Principal paid in Year 4 = Loan amount - Principal balance after 3 years
By following these steps and performing the required calculations, you should be able to find the answers to all the questions.