A homeowner installed a new swimming pool for $8500. He paid 8% down and then paid 36 monthly payments of $243.07. Determine the APR of the loan to the nearest one-half of a percent.
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A homeowner installed a new swimming pool for $8500. He paid 8% down and then paid 36 monthly payments of $243.07. Determine the APR of the loan to the nearest one-half of a percent.
Po =(1-0.08) * 8500 = $7820.
P = 243.07/mo. * 36mo. = $8750.52
P = Po + Po*r*t = 8750.52
7820 + 7820*r*3 = 8750.52
7820 + 23,460r = 8750.52
23,460r = 8750.52-7820 = 930.52
r = 3.97% or 4.0% = APR.
To determine the APR (Annual Percentage Rate) of the loan, we need to calculate the interest paid over the course of the loan.
First, let's calculate the total cost of the pool by adding the down payment to the total of the monthly payments:
Down payment = $8,500 x 8% = $680
Total cost = $8,500 + $680 = $9,180
Now, let's determine the total interest paid over the loan period. Since the monthly payments are fixed at $243.07, we can multiply this by the number of payments to get the total amount paid:
Total payments = $243.07 x 36 = $8,752.52
Total interest paid = Total payments - Loan amount = $8,752.52 - $9,180 = -$427.48
Since the total interest paid is negative, it indicates that the borrower paid back less than the original loan amount. This can happen due to rounding errors or other factors. In this case, it means there is no actual interest paid over the course of the loan.
To calculate the APR, we need to find the interest rate that, when applied to the loan amount, would result in the total payments and the total interest paid.
Let's use an iterative approach to find the APR. We'll start with a guess and adjust it until we get the desired total payments and total interest paid.
1. Set an initial guess for the APR (e.g., 5%).
2. Calculate the monthly interest rate by dividing the APR by 12 (months) and converting it to a decimal.
Monthly interest rate = (APR / 12) / 100
3. Use the monthly interest rate to calculate the loan balance after each payment:
Starting balance = Loan amount
For each payment:
Interest payment = Starting balance x Monthly interest rate
Principal payment = Monthly payment - Interest payment
Ending balance = Starting balance - Principal payment
Set the starting balance for the next payment as the ending balance.
4. Repeat step 3 for all 36 payments.
5. Check if the final ending balance is close to zero.
- If yes, the guessed APR is close to the actual APR.
- If no, adjust the initial guess and repeat the process from step 2.
By repeating these steps with different APR guesses, you can find the APR that results in a final ending balance close to zero. You can use an Excel spreadsheet or a financial calculator to automate the process.
Once you find the APR, you can round it to the nearest one-half of a percent based on the calculation result.
Note: In this case, since the total interest paid is negative, it implies that the APR is effectively zero or very close to it, as the homeowner paid back less than the original loan amount.