Compare the monthly payment and total payment for the following pairs of loan options. Assume that both loans are fixed rate and have the same closing costs. (Round all answers to the nearest cent as needed.)
You need a $120,000 loan.
Option 1: a 30-year loan at an APR of 8%.
Option 2: a 15-year loan at an APR of 7%.
a) What is the monthly payment for option 1?
b) What is the monthly payment for option 2?
c) What is the total amount paid for option 1?
d) What is the total amount paid for option 2?
e) If the borrower can afford the higher monthly payments over the entire term of the loan, which appears to be the better option?
To compare the monthly payment and total payment for the loan options, we can use the following formulas:
Monthly Payment Formula:
\(M = P \cdot \frac{r \cdot (1 + r)^n}{(1 + r)^n - 1}\)
Total Payment Formula:
\(T = M \cdot n\)
Where:
M is the monthly payment,
P is the loan amount,
r is the monthly interest rate (APR divided by 12),
n is the total number of payments (number of years multiplied by 12).
Let's calculate the values step by step.
a) Monthly payment for option 1:
Given:
Loan amount (P) = $120,000
APR = 8%
Number of years (n) = 30
To calculate the monthly interest rate (r), we divide the APR by 12 and convert it to a decimal: \(r = \frac{0.08}{12}\).
Now substituting the values into the Monthly Payment Formula:
\(M = 120,000 \cdot \frac{\frac{0.08}{12} \cdot (1 + \frac{0.08}{12})^{30}}{(1 + \frac{0.08}{12})^{30} - 1}\)
Using a calculator or spreadsheet software, the monthly payment for option 1 is approximately $879.26.
b) Monthly payment for option 2:
Given:
Loan amount (P) = $120,000
APR = 7%
Number of years (n) = 15
To calculate the monthly interest rate (r), we divide the APR by 12 and convert it to a decimal: \(r = \frac{0.07}{12}\).
Now substituting the values into the Monthly Payment Formula:
\(M = 120,000 \cdot \frac{\frac{0.07}{12} \cdot (1 + \frac{0.07}{12})^{15}}{(1 + \frac{0.07}{12})^{15} - 1}\)
Using a calculator or spreadsheet software, the monthly payment for option 2 is approximately $1,073.64.
c) Total amount paid for option 1:
To calculate the total payment, we multiply the monthly payment by the total number of payments: \(T = M \cdot n\).
Substituting the values:
\(T = 879.26 \cdot (30 \cdot 12)\)
Using a calculator or spreadsheet software, the total amount paid for option 1 is approximately $316,536.00.
d) Total amount paid for option 2:
Substituting the values:
\(T = 1,073.64 \cdot (15 \cdot 12)\)
Using a calculator or spreadsheet software, the total amount paid for option 2 is approximately $193,854.40.
e) If the borrower can afford the higher monthly payments over the entire term of the loan, the better option would be option 2. Although option 2 has a higher monthly payment, the total amount paid is significantly less than option 1. This means the borrower would save money in the long run by choosing option 2, the 15-year loan with a lower APR. However, it's important to consider other factors such as the borrower's financial situation and long-term goals before making a decision.