Question

Can you help me with this question?

Compare the monthly payments and total loan costs for the following pairs of loan options. Assume that both loans are fixed rate and have the same closing costs. You need a $40,000 loan. Option 1: a 30-year loan at an APR of 8.15%. Option 2: a 15-year loan at an APR of 7.75% 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 amount paid for each option. The total payment for option 1 is $ The total payment for option 2 is (Use the answers from the previous step. Round to the nearest cent as needed.) Compare the two options to determine which is the better economic option. Which appears to be the better option?

Answer 1

this can be solved using present value of annuity formula

present value of annuity = PMT[1 - (1+r)^-n / r]

where,PMT = monthly payments

r = rate of interest

n = number of periods

monthly payments:

option 1:

here interest rate = 8.15%

since monthly payments = 8.15 / 12 = 0.67917%

n = 30 x 12 = 360

40,000 = PMT[1 - (1.0067917)^{-12 x 30} / 0.0067917]

PMT = 40,000 / 134.3637884

monthly payments = $297.70

Option 2:

here interest rate = 7.75%

since monthly payments = 7.75 / 12 =0.645833%

n = 15 x 12 = 180

  40,000 = PMT[1 - (1.00645833)^-180 / 0.00645833]

PMT = 40,000 / 106.2388

monthly payments = $376.51

Total payment for option 1 = 360 x 297.70 = $107,172

Total Payment for Option 2 = 180 x 376.51 = $67,771.85

when we compare both the options Option 2 is economically better because it has low interest rate even though Option 1 seems to be better option because of low monthly payments we end up being pay more interest.