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 You need a $\$ 100,000$ loan.
Option 1: a 30-year loan at an APR of $8 \%$.
Option 2. a 15 -year loan at an APR of $7 \%$
Find the monthly payment for each option
The monthly payment for option 1 is $\$$
The monthly payment for option 2 is $\$$
(Do not round until the final answer. Then round to the nearest cent as needed)
Find the total payment for each option
The total payment for option 1 is $\$$
The total payment for option 2 is's
(Round to the nearest cent as needed)
Compare the two options. Which appears to be the better option?
A. Option 1 will always be the better option
B. Option 1 is the better option, but only if the borrower plans to stay in the same home for the entire term of the loan.
C. Option 2 is the better option, but only if the borrower can afford the higher monthly payments over the entire term of the loan
D. Option 2 will always be the better option.
Answer
\(\boxed{\text{Therefore, Option 2 is the better option, but only if the borrower can afford the higher monthly payments over the entire term of the loan.}}\)
Step 1 ：Calculate the monthly payment for each option using the formula \(P = \frac{{r*PV}}{{1 - (1 + r)^{-n}}}\), where P is the monthly payment, r is the monthly interest rate (annual rate / 12), PV is the present value, or the amount of the loan, and n is the number of payments (years * 12).
Step 2 ：For Option 1: \(r = \frac{{8\%}}{{12}} = 0.00666667\), \(PV = \$100,000\), and \(n = 30 * 12 = 360\).
Step 3 ：Substitute these values into the formula to get \(P = \frac{{0.00666667 * 100,000}}{{1 - (1 + 0.00666667)^{-360}}} = \$733.76\).
Step 4 ：For Option 2: \(r = \frac{{7\%}}{{12}} = 0.00583333\), \(PV = \$100,000\), and \(n = 15 * 12 = 180\).
Step 5 ：Substitute these values into the formula to get \(P = \frac{{0.00583333 * 100,000}}{{1 - (1 + 0.00583333)^{-180}}} = \$899.33\).
Step 6 ：Calculate the total payment for each option, which is the monthly payment times the number of payments.
Step 7 ：For Option 1: \(\$733.76 * 360 = \$264,153.60\).
Step 8 ：For Option 2: \(\$899.33 * 180 = \$161,679.40\).
Step 9 ：Compare the total payments for the two options. While the monthly payment for Option 2 is higher, the total payment is significantly lower.
Step 10 ：\(\boxed{\text{Therefore, Option 2 is the better option, but only if the borrower can afford the higher monthly payments over the entire term of the loan.}}\)
