Question

You are borrowing $6,400 now, which you will pay back at the end of 13 years. You are considering two separate interest rates:
a. A fixed 7.6% annual interest rate.
b. A variable interest of 3.9% annual interest for the first 5 years; 5.6% annual interest rate for next 2 years; and i% interest rate for the remaining years.
Assuming you desire to pay the least amount of money back at the end of 13 years, what value of i makes you indifferent between options a and b? With both options, you make exactly one payment at the end of 13 years. Enter your answer as a percentage between 0 and 100

Answer 1

Option a) Fixed annual interest rate @ 7.6%

Amount borrowed: $6400

Total Interest payable at the end of 13 Years: 6400*7.6%*13= $ 6323.20.......................(1)

Total Amount Payable at the end of 13 Years (Principal plus Interest): $12723.20

Option b) Variable Interest of 3.9% p.a. for first 5 years; 5.6% p.a. for next 2 years and "i%" for next 6 years

Amount Borrowed; $ 6400

Interest payable at the end of 5 Years @ 3.9% p.a. : 6400*3.9%*5= $ 1248

Interest payable for the next 2 Years @ 5.9% p.a.: 6400*5.6%*2= $ 716.80

Total Interest payable at the end of 7 Years = $(1248+716.80)=$ 1964.80

Now let i% per annum interest is charged for the next 6 Years, then Interest payable for the next 6 years = 6400*(i/100)*6= $ 384i.

Total Interest payable at the end of 13 years under option b above= $ (1964.80 + 384i).....................(2)

For calculating the value of i , Comparing the above two equations (1) and (2)

.i.e 1964.80+384i=6323.20

This implies, 384i=4358.40

This implies, i= 11.35% per annum (p.a.)

Now as per the requirement of question, since we are required to be indifferent between the two available options and need to payback the least amount at the end of 13 Years: The variable rate of interest per annum for the remaining 6 Years should be less than 11.35%.