Answer :
To solve this problem, we'll break it down into the individual parts provided in the question. We'll use the given parameters and formulas related to loan amortization to determine the answers.
Given parameters:
- Debt Principal ([tex]\( P \)[/tex]): \[tex]$14,000 - Annual Interest Rate (\( R \)): 9% or 0.09 - Repayment Period: 8 years - Payment Interval: 1 month (there are 12 payment intervals in a year) - Outstanding Principal After: 6th payment ### (a) The Size of the Periodic Payment First, we need to convert the annual interest rate to a monthly interest rate, since payments are made monthly. \[ \text{Monthly Interest Rate} = \frac{0.09}{12} = 0.0075 \] Next, we calculate the total number of payments over the 8-year period: \[ \text{Total Payments} = 8 \text{ years} \times 12 \text{ months per year} = 96 \text{ payments} \] We use the annuity formula to determine the size of the periodic payment (\( PMT \)): \[ PMT = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \] Where: - \( P \) is the principal amount - \( r \) is the monthly interest rate - \( n \) is the total number of payments Substituting the values: \[ PMT = \frac{14000 \cdot 0.0075 \cdot (1 + 0.0075)^{96}}{(1 + 0.0075)^{96} - 1} \] After performing the calculations, the periodic payment (\( PMT \)) is: \[ PMT \approx \$[/tex]205.10
\]
### (b) The Outstanding Principal at the Time Indicated (After 6 Payments)
To find the outstanding principal after 6 payments, we use the formula for the remaining balance:
[tex]\[ \text{Principal Remaining} = P \cdot \frac{(1 + r)^n - (1 + r)^k}{(1 + r)^n - 1} \][/tex]
Where:
- [tex]\( k \)[/tex] is the number of payments made (6 in this case)
Substituting the values:
[tex]\[ \text{Principal Remaining} = 14000 \cdot \frac{(1 + 0.0075)^{96} - (1 + 0.0075)^6}{(1 + 0.0075)^{96} - 1} \][/tex]
After performing the calculations, the outstanding principal after 6 payments is:
[tex]\[ \text{Principal Remaining} \approx \$13,388.01 \][/tex]
### (c) The Interest Paid by the Payment Following the Time Indicated
The interest part of the next (7th) payment is calculated by multiplying the remaining principal by the monthly interest rate:
[tex]\[ \text{Interest Paid} = \text{Principal Remaining} \times \text{Monthly Interest Rate} \][/tex]
Substituting the values:
[tex]\[ \text{Interest Paid} = 13388.01 \times 0.0075 \][/tex]
After performing the calculations, the interest paid in the 7th payment is:
[tex]\[ \text{Interest Paid} \approx \$100.41 \][/tex]
### (d) The Principal Repaid by the Payment Following the Time Indicated
The principal repaid by the next (7th) payment is the difference between the total periodic payment and the interest part of the payment:
[tex]\[ \text{Principal Repaid} = PMT - \text{Interest Paid} \][/tex]
Substituting the values:
[tex]\[ \text{Principal Repaid} = 205.10 - 100.41 \][/tex]
After performing the calculations, the principal repaid in the 7th payment is:
[tex]\[ \text{Principal Repaid} \approx \$104.69 \][/tex]
### Summary
(a) The size of the periodic payment is:
[tex]\[ \$205.10 \][/tex]
(b) The outstanding principal after 6 payments is:
[tex]\[ \$13,388.01 \][/tex]
(c) The interest paid by the 7th payment is:
[tex]\[ \$100.41 \][/tex]
(d) The principal repaid by the 7th payment is:
[tex]\[ \$104.69 \][/tex]
These values give a comprehensive breakdown of the loan's amortization at the specified intervals.
Given parameters:
- Debt Principal ([tex]\( P \)[/tex]): \[tex]$14,000 - Annual Interest Rate (\( R \)): 9% or 0.09 - Repayment Period: 8 years - Payment Interval: 1 month (there are 12 payment intervals in a year) - Outstanding Principal After: 6th payment ### (a) The Size of the Periodic Payment First, we need to convert the annual interest rate to a monthly interest rate, since payments are made monthly. \[ \text{Monthly Interest Rate} = \frac{0.09}{12} = 0.0075 \] Next, we calculate the total number of payments over the 8-year period: \[ \text{Total Payments} = 8 \text{ years} \times 12 \text{ months per year} = 96 \text{ payments} \] We use the annuity formula to determine the size of the periodic payment (\( PMT \)): \[ PMT = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \] Where: - \( P \) is the principal amount - \( r \) is the monthly interest rate - \( n \) is the total number of payments Substituting the values: \[ PMT = \frac{14000 \cdot 0.0075 \cdot (1 + 0.0075)^{96}}{(1 + 0.0075)^{96} - 1} \] After performing the calculations, the periodic payment (\( PMT \)) is: \[ PMT \approx \$[/tex]205.10
\]
### (b) The Outstanding Principal at the Time Indicated (After 6 Payments)
To find the outstanding principal after 6 payments, we use the formula for the remaining balance:
[tex]\[ \text{Principal Remaining} = P \cdot \frac{(1 + r)^n - (1 + r)^k}{(1 + r)^n - 1} \][/tex]
Where:
- [tex]\( k \)[/tex] is the number of payments made (6 in this case)
Substituting the values:
[tex]\[ \text{Principal Remaining} = 14000 \cdot \frac{(1 + 0.0075)^{96} - (1 + 0.0075)^6}{(1 + 0.0075)^{96} - 1} \][/tex]
After performing the calculations, the outstanding principal after 6 payments is:
[tex]\[ \text{Principal Remaining} \approx \$13,388.01 \][/tex]
### (c) The Interest Paid by the Payment Following the Time Indicated
The interest part of the next (7th) payment is calculated by multiplying the remaining principal by the monthly interest rate:
[tex]\[ \text{Interest Paid} = \text{Principal Remaining} \times \text{Monthly Interest Rate} \][/tex]
Substituting the values:
[tex]\[ \text{Interest Paid} = 13388.01 \times 0.0075 \][/tex]
After performing the calculations, the interest paid in the 7th payment is:
[tex]\[ \text{Interest Paid} \approx \$100.41 \][/tex]
### (d) The Principal Repaid by the Payment Following the Time Indicated
The principal repaid by the next (7th) payment is the difference between the total periodic payment and the interest part of the payment:
[tex]\[ \text{Principal Repaid} = PMT - \text{Interest Paid} \][/tex]
Substituting the values:
[tex]\[ \text{Principal Repaid} = 205.10 - 100.41 \][/tex]
After performing the calculations, the principal repaid in the 7th payment is:
[tex]\[ \text{Principal Repaid} \approx \$104.69 \][/tex]
### Summary
(a) The size of the periodic payment is:
[tex]\[ \$205.10 \][/tex]
(b) The outstanding principal after 6 payments is:
[tex]\[ \$13,388.01 \][/tex]
(c) The interest paid by the 7th payment is:
[tex]\[ \$100.41 \][/tex]
(d) The principal repaid by the 7th payment is:
[tex]\[ \$104.69 \][/tex]
These values give a comprehensive breakdown of the loan's amortization at the specified intervals.
