Suppose that you decide to borrow $16 comma 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
%
Use PMT equals StartStartFraction Upper P left parenthesis StartFraction r Over n EndFraction right parenthesis OverOver left bracket 1 minus left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript negative nt right bracket EndEndFraction
to complete parts (a) through (c) below.
a) To calculate the monthly payment for each loan, we can use the PMT formula mentioned above.
For Loan A, the interest rate is 5.9% and the loan period is three years (36 months). Plugging these values into the formula, we have:
PMT = $16,000 * (0.059/12) / (1 - (1 + 0.059/12)^-36)
Calculating this expression, we find that the monthly payment for Loan A is approximately $486.46.
For Loan B, the interest rate is 5.8% and the loan period is five years (60 months). Plugging these values into the formula, we have:
PMT = $16,000 * (0.058/12) / (1 - (1 + 0.058/12)^-60)
Calculating this expression, we find that the monthly payment for Loan B is approximately $308.95.
b) To determine the total amount paid over the duration of the loan, we need to multiply the monthly payment by the total number of months.
For Loan A, the total amount paid over three years is $486.46 * 36 = $17,511.36.
For Loan B, the total amount paid over five years is $308.95 * 60 = $18,537.
c) To decide which loan option is better, we need to consider both the monthly payment and the total amount paid.
In terms of the monthly payment, Loan B has a lower monthly payment of $308.95 compared to Loan A's $486.46. This may be more affordable for some borrowers.
However, when considering the total amount paid, Loan A actually ends up being cheaper at $17,511.36 compared to Loan B's $18,537. This is because Loan A has a shorter loan period and therefore less interest built up over time.
Ultimately, the decision between Loan A and Loan B depends on the individual's financial situation and priorities. If the borrower wants to minimize the total amount paid, Loan A may be the better option. If having a lower monthly payment is more important, Loan B may be preferable.
(a) Calculate the monthly payment for Loan A.
To calculate the monthly payment for Loan A, we need to use the formula:
PMT = P * (r/n) / (1 - (1 + r/n)^(-nt))
Where:
PMT = Monthly payment
P = Principal amount borrowed = $16,000
r = Annual interest rate = 5.9% = 0.059 (as a decimal)
n = Number of payments per year = 12 (since it's a monthly payment)
t = Number of years = 3
Plugging in the values, we get:
PMT = 16000 * (0.059/12) / (1 - (1 + (0.059/12))^(-3*12))
Calculating this expression, the monthly payment for Loan A is approximately $485.49.
(b) Calculate the monthly payment for Loan B.
To calculate the monthly payment for Loan B, we use the same formula as above, with the values:
P = Principal amount borrowed = $16,000
r = Annual interest rate = 5.8% = 0.058 (as a decimal)
n = Number of payments per year = 12 (since it's a monthly payment)
t = Number of years = 5
Plugging in these values, we get:
PMT = 16000 * (0.058/12) / (1 - (1 + (0.058/12))^(-5*12))
Calculating this expression, the monthly payment for Loan B is approximately $310.15.
(c) Which loan option has the lower monthly payment?
Comparing the monthly payments for Loan A and Loan B, we see that Loan B has the lower monthly payment. So, Loan B is the better option if you are looking for a lower monthly payment.