Finanical Mathematics Question

Question

Jimmy borrows $80,000 at 5.4%/a compounded monthly for a period of 30 years to build a treehouse. He is guaranteed that interest rate for 10 years and makes monthly payments of $449.22. After 10 years, the interest rate drops by 0.6%/a and he continues with the regular payments for another 10 years, at which point the rate increases by 1.2%. If he continues to pay the same monthly payment, how much will the time to pay off the loan change?

Answer 1

First of all, I used a computational software to help me. So i will try to explain, but the rough calculations were done outside this text.

Since the payments are made monthly, we need to consider the interest monthly. To do this we'll use the formula:

(1+r_y) = (1+r_m)¹²

Where r_m and r_y are the monthly and yearly interest rates respectively. Applying this formula, we get:

5.4%/a → 0.44%/m

0.6%/a → 0.05%/m

1.2%/a → 0.10%/m

In the month zero, he will own 80.000.

In the month one, he will own 80000* (1+r_m) - P, where P is the monthly payment.

In the month two, he will own (80000* (1+r_m) - P)*(1+r_m) - P and so long, and so forth.

Realize that in the month 121 the rate will change, the same as in the month 241.

Doing the calculations, we get he will pay his debt at the month 270. Therefore the time to pay the loan will change in 90 months, which is 7.5 years.