Question
anton took out a 20 year loan for $90,000 at an APR of 10.5%, compounded monthly. approximately what would be the total cost of his loan if he paid it off 12 years early?
A. $215,649.60
B. $86,259.84
C. $73,401.85
D. $159,661.69
A. $215,649.60
B. $86,259.84
C. $73,401.85
D. $159,661.69
Answers
GPT-4o mini
To find the total cost of Anton's loan if he paid it off 12 years early, we need to determine the monthly payment amount first and then calculate the total amount paid over the time he actually keeps the loan.
### Step 1: Calculate the Monthly Payment
Given:
- Loan amount (principal, \( P \)) = $90,000
- Annual interest rate (APR) = 10.5%
- Monthly interest rate (\( r \)) = 10.5% / 12 = 0.875% = 0.00875
- Total number of payments for a 20-year loan (\( n \)) = 20 * 12 = 240 months
The formula for the monthly payment (M) on an amortizing loan is:
\[
M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
Substituting the values into the formula:
\[
M = 90000 \frac{0.00875(1 + 0.00875)^{240}}{(1 + 0.00875)^{240} - 1}
\]
Calculating \( (1 + 0.00875)^{240} \):
\[
(1 + 0.00875)^{240} \approx 6.6724
\]
Now substituting back:
\[
M = 90000 \frac{0.00875 \cdot 6.6724}{6.6724 - 1}
\]
\[
M = 90000 \frac{0.0584}{5.6724}
\]
\[
M = 90000 \cdot 0.01031 \approx 926.28
\]
### Step 2: Calculate Total Payments Made
If Anton pays off the loan 12 years early, he will pay for 8 years instead, which means:
Total payments for 8 years:
\[
\text{Total Payments} = M \times \text{number of payments made}
\]
\[
\text{Number of payments over 8 years} = 8 \times 12 = 96
\]
\[
\text{Total Payments} \approx 926.28 \times 96 \approx 88,092.88
\]
### Check Total Payments
To check for correctness:
Given that \( M \approx 926.28 \):
\[
\text{Total cost of the loan paid off in 12 years} \approx 926.28 \times 96 \approx 88,092.88
\]
### Conclusion of Total Cost of the Loan
- Therefore, the possible options provided do not match with the calculated total cost. Based on this calculation, we are estimating approximately $88,092.88 which is closest to \(B. 86,259.84\).
So the closest answer from provided options would be:
**B. $86,259.84**
If you want a more precise calculation, you can use financial calculators or software to arrive at the correct monthly payment and total. Otherwise, the approximation gives us a clear direction.
### Step 1: Calculate the Monthly Payment
Given:
- Loan amount (principal, \( P \)) = $90,000
- Annual interest rate (APR) = 10.5%
- Monthly interest rate (\( r \)) = 10.5% / 12 = 0.875% = 0.00875
- Total number of payments for a 20-year loan (\( n \)) = 20 * 12 = 240 months
The formula for the monthly payment (M) on an amortizing loan is:
\[
M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
Substituting the values into the formula:
\[
M = 90000 \frac{0.00875(1 + 0.00875)^{240}}{(1 + 0.00875)^{240} - 1}
\]
Calculating \( (1 + 0.00875)^{240} \):
\[
(1 + 0.00875)^{240} \approx 6.6724
\]
Now substituting back:
\[
M = 90000 \frac{0.00875 \cdot 6.6724}{6.6724 - 1}
\]
\[
M = 90000 \frac{0.0584}{5.6724}
\]
\[
M = 90000 \cdot 0.01031 \approx 926.28
\]
### Step 2: Calculate Total Payments Made
If Anton pays off the loan 12 years early, he will pay for 8 years instead, which means:
Total payments for 8 years:
\[
\text{Total Payments} = M \times \text{number of payments made}
\]
\[
\text{Number of payments over 8 years} = 8 \times 12 = 96
\]
\[
\text{Total Payments} \approx 926.28 \times 96 \approx 88,092.88
\]
### Check Total Payments
To check for correctness:
Given that \( M \approx 926.28 \):
\[
\text{Total cost of the loan paid off in 12 years} \approx 926.28 \times 96 \approx 88,092.88
\]
### Conclusion of Total Cost of the Loan
- Therefore, the possible options provided do not match with the calculated total cost. Based on this calculation, we are estimating approximately $88,092.88 which is closest to \(B. 86,259.84\).
So the closest answer from provided options would be:
**B. $86,259.84**
If you want a more precise calculation, you can use financial calculators or software to arrive at the correct monthly payment and total. Otherwise, the approximation gives us a clear direction.