Suppose you take out a 45-year $100,000 mortgage with an APR of 6%. You make payments for 2 years (24 monthly payments) and then consider refinancing the original loan. The new loan would have a term of 20 years, have an APR of 5.9%, and be in the amount of the unpaid balance on the original loan. (The amount you borrow on the new loan would be used to pay off the balance on the original loan.) The administrative cost of taking out the second loan would be $2500. Use the information to complete parts (a) through (d) below
a. What are the monthly payments on the original loan?
b. A short calculation shows that the unpaid balance on the original loan after 2 years is $99,077.21, which would become the amount of the second loan. What would the monthly payments be on the second loan?
c. What would be the total amount you would pay if you continued with the original 45-year loan without refinancing?
d. What would be the total amount you would pay with the refinancing?
for first part:
i = .06/12 = .005
n = 45(12) = 540
let the payment be p
p(1 - 1.005^-540)/.005 = 100000
I get p = $536.2846
Now, you are doing this for 24 months
outstanding balance = 100000(1.005)^24 - 536.2846(1.005^24 - 1)/.005
= $99,077.21 , as stated in the problem
Plus the admin fee of 2500 gives you a balance of 101577.21
MORE THAN WHAT YOU STARTED WITH 2 YEARS AGO !!!!
new i = .00491666...
new n = 20(12) = 240
new payment --- q
q(1 - 1.00491666..^-240)/.00491666... = 101577.21
solve for q
c) and d) are invalid questions, since it asks you to add up values of money which are not in the same time spot.
However, if in your course the instructor would accept 540*$536.2846, go ahead and do it. In terms of actuarial mathematics it would be a bogus result.
To solve this problem, we can use the formula for calculating monthly mortgage payments.
a. The formula for calculating monthly mortgage payments is:
M = P [ i(1 + i)^n ] / [ (1 + i)^n - 1]
where:
M = monthly payment
P = principal (loan amount)
i = monthly interest rate
n = number of monthly payments
In this case, for the original loan:
P = $100,000
APR = 6%, so the monthly interest rate (i) = 6% / 12 = 0.005
n = 45 years * 12 months/year = 540 monthly payments
Plug these values into the formula:
M = 100,000 [ 0.005(1 + 0.005)^540 ] / [ (1 + 0.005)^540 - 1]
Using a calculator, we can calculate M, which will give us the monthly payments on the original loan.
b. The unpaid balance on the original loan after 2 years is $99,077.21, which becomes the amount of the second loan.
To calculate the monthly payments on the second loan, we can use the same formula as in part (a), but with different values of P and n.
P = unpaid balance on the original loan after 2 years = $99,077.21
APR = 5.9%, so the monthly interest rate (i) = 5.9% / 12 = 0.00492
n = 20 years * 12 months/year = 240 monthly payments
Plug these values into the formula:
M = 99,077.21 [ 0.00492(1 + 0.00492)^240 ] / [ (1 + 0.00492)^240 - 1]
Using a calculator, we can calculate M, which will give us the monthly payments on the second loan.
c. To calculate the total amount paid if the original loan is continued without refinancing, we multiply the monthly payment calculated in part (a) by the total number of payments (n = 540).
Total amount paid = Monthly payment * Number of payments
d. To calculate the total amount paid with refinancing, we multiply the monthly payment calculated in part (b) by the total number of payments (n = 240) and add the administrative cost of $2500.
Total amount paid = (Monthly payment * Number of payments) + Administrative cost