Suppose that you decide to borrow $16,000
for a new car. You can select one of the following loans, each requiring regular monthly payments.
Installment Loan A: three-year loan at 5.9%
Installment Loan B: five-year loan at 5.8%
To find out the monthly payments for each loan option, we can use the loan amortization formula:
Monthly Payment = (Loan Amount * Monthly Interest Rate) / (1 - (1 + Monthly Interest Rate)^(-Number of Payments))
For Installment Loan A:
Loan Amount = $16,000
Annual Interest Rate = 5.9%
Monthly Interest Rate = 5.9% / 12 = 0.4917%
Number of Payments = 3 years * 12 months/year = 36
Monthly Payment (Installment Loan A) = (16000 * 0.4917%) / (1 - (1 + 0.4917%)^(-36))
For Installment Loan B:
Loan Amount = $16,000
Annual Interest Rate = 5.8%
Monthly Interest Rate = 5.8% / 12 = 0.4833%
Number of Payments = 5 years * 12 months/year = 60
Monthly Payment (Installment Loan B) = (16000 * 0.4833%) / (1 - (1 + 0.4833%)^(-60))
Now, we can calculate the monthly payments for each loan option:
Monthly Payment (Installment Loan A) = ($79.12)
Monthly Payment (Installment Loan B) = ($71.31)
Therefore, the monthly payment for Installment Loan A is $79.12, while the monthly payment for Installment Loan B is $71.31.
To compare the two loans, we'll calculate the monthly payments and the total amount paid for each loan.
Loan A: Three-year loan at 5.9%
Step 1: Convert the interest rate to a decimal by dividing it by 100.
5.9% = 0.059
Step 2: Determine the number of months for the loan.
Number of months = 3 years * 12 months/year = 36 months
Step 3: Use the formula for monthly payment calculation:
Monthly payment = (Principal * Monthly Interest Rate) / (1 - (1 + Monthly Interest Rate)^(-Number of months))
Monthly Interest Rate = Annual Interest Rate / 12
Monthly Interest Rate = 0.059 / 12 = 0.004917
Monthly payment = (16000 * 0.004917) / (1 - (1 + 0.004917)^(-36))
Monthly payment ≈ $480.97
Step 4: Calculate the total amount paid over the 3-year loan term.
Total amount paid = Monthly payment * Number of months
Total amount paid ≈ $480.97 * 36 = $17,313.12
Loan B: Five-year loan at 5.8%
Step 1: Convert the interest rate to a decimal.
5.8% = 0.058
Step 2: Determine the number of months for the loan.
Number of months = 5 years * 12 months/year = 60 months
Step 3: Use the formula mentioned earlier to find the monthly payment.
Monthly Interest Rate = 0.058 / 12 = 0.004833
Monthly payment = (16000 * 0.004833) / (1 - (1 + 0.004833)^(-60))
Monthly payment ≈ $315.50
Step 4: Calculate the total amount paid over the 5-year loan term.
Total amount paid = Monthly payment * Number of months
Total amount paid ≈ $315.50 * 60 = $18,930.00
Therefore, Loan A: three-year loan at 5.9% has a monthly payment of approximately $480.97 and a total amount paid of approximately $17,313.12.
Loan B: five-year loan at 5.8% has a monthly payment of approximately $315.50 and a total amount paid of approximately $18,930.00.