To find out how much Derek must give the bank after 16 months of payments, we need to perform some calculations using a finance calculator or a financial formula method.
Here's how to break down the information given:
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Loan Amount (PV): This is the current value of the loan, which is $43,312.00.
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Interest Rate (i/y): This is the annual interest rate divided by the number of periods (monthly payments). The annual interest rate of 6.11% needs to be converted to a monthly rate: \[ i/y = \frac{6.11%}{12} = \frac{0.0611}{12} \approx 0.00509167 \text{ (as a decimal)} \]
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Total Number of Payments (N): The total number of payments for a 6-year loan, making monthly payments, is: \[ N = 6 \times 12 = 72 \text{ months} \]
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Monthly Payment (PMT): We will calculate the monthly payment using the loan amount, interest rate, and total number of payments. The formula to find PMT is given by the loan payment formula: \[ PMT = \frac{PV \cdot (i/y)}{1 - (1 + i/y)^{-N}} \] Plugging in the values: \[ PMT = \frac{43312 \times 0.00509167}{1 - (1 + 0.00509167)^{-72}} \approx \frac{220.0006566}{1 - (1.00509167)^{-72}} \] Calculate
(1 + 0.00509167)^{-72}and complete the calculation to find PMT. -
Payments Made: After 16 months, the amount Derek has paid is: \[ \text{Total Payments Made} = PMT \times 16 \]
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Remaining Balance Calculation: After 16 months of payments, we need to find the remaining balance (the amount he owes). This can be calculated with the remaining number of payments left (56 in this case) and the interest accrued until that point. The formula used would be slightly different to credit the amount already paid: \[ \text{Remaining Balance} = PV \times (1 + i/y)^{t} - PMT \times \frac{(1 + i/y)^{t} - 1}{i/y} \] where \( t \) is the number of payments made (16 months).
Finally, enter these values into a financial calculator:
- N: 72 (total number of payments for the loan)
- PV: -43312.00 (loan amount is negative because it's money owed)
- i/y: 0.00509167 (monthly interest rate)
- PMT: Calculate to get the monthly payment
- FV: 0 (the goal is to settle the loan to zero)
Then use the remaining balance calculation to find out how much Derek must give the bank after 16 payments. Remember to input the proper amount for PMT once calculated.
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