Suppose that you decide to borrow $15,000
for a new car. You can select one of the following loans, each requiring regular monthly payments.
Installment Loan A: three-year loan at 5.5%
Installment Loan B: five-year loan at 5.2%
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) Installment Loan A:
Plug the given values into the payment formula:
P = 15,000
r = 5.5/100 = 0.055 (always convert to decimal)
n = 12 (monthly payments in a year)
t = 3 years
The formula for the monthly payment (PMT) is:
PMT = [P * (r/n)] / [1 - (1 + r/n)^-nt]
PMT = [15000 * (0.055/12)] / [1 - (1 + 0.055/12)^-(12*3)]
PMT = [15000 * 0.00458333] / [1 - (1.004583)^-36]
PMT = 68.75 / [1 - 0.77951567]
PMT = 68.75 / 0.22048433
PMT = $311.97
You need to pay $311.97 per month for a three-year loan at 5.5%.
b) Installment Loan B:
Plug the given values into the payment formula:
P = 15,000
r = 5.2/100 = 0.052 (always convert to decimal)
n = 12 (monthly payments in a year)
t = 5 years
The formula for the monthly payment (PMT) is:
PMT = [P * (r/n)] / [1 - (1 + r/n)^-nt]
PMT = [15000 * (0.052/12)] / [1 - (1 + 0.052/12)^-(12*5)]
PMT = [15000 * (0.00433333)] / [1 - (1.004333)^-60]
PMT = 65.00 / [1 - 0.610772]
PMT = 65.00 / 0.389228
PMT = $167.11
You need to pay $167.11 per month for a five-year loan at 5.2%.
c) Comparison between Loans A & B:
Loan A has higher monthly payments ($311.97) but you pay off the loan faster (3 years), while Loan B has lower monthly payments ($167.11) but the loan term is longer (5 years). It depends on your monthly budget to afford the installments and how quickly you want to pay off the loan.
To complete parts (a) through (c), we'll use the given information about the loans and the formula PMT, which stands for the regular monthly payment.
(a) Calculate the regular monthly payment for Installment Loan A:
Loan amount: $15,000
Interest rate: 5.5%
Loan term: 3 years
Using the PMT formula:
PMT = P * (r/n) / (1 - (1 + r/n)^(-nt))
Here, P = 15,000 (loan amount), r = 5.5% (interest rate), n = 12 (number of payments per year), and t = 3 (loan term in years).
PMT_A = 15,000 * (0.055/12) / (1 - (1 + 0.055/12)^(-12*3))
Now, we can use a calculator to find PMT_A.
(b) Calculate the regular monthly payment for Installment Loan B:
Loan amount: $15,000
Interest rate: 5.2%
Loan term: 5 years
Using the same PMT formula as before:
PMT_B = 15,000 * (0.052/12) / (1 - (1 + 0.052/12)^(-12*5))
Again, use a calculator to find PMT_B.
(c) Compare the regular monthly payments of Installment Loan A and Installment Loan B.
Compare PMT_A and PMT_B calculated in parts (a) and (b), respectively. The loan with the lower regular monthly payment will require a lower payment each month, while the loan with the higher regular monthly payment will likely be paid off earlier.
Therefore, by comparing PMT_A and PMT_B, you can determine which loan requires a lower regular monthly payment.