Question

4) The effective annual interest rate is 6%. A 30 year loan is repaid as follows (payments starting at the end of the first year):

For the first 10 years, interest only.

For the second 10 years, each payment is twice the interest due in that period.

For the final 10 years, level payments of X per year.

Find the outstanding balance at the end of each 10 year period, and find X. (Optional: do it without using a spreadsheet.)

Answer 1

1

Outstanding payment at the end of first 10 years = Actual loan amount

because only interest has been paid in first 10 years so loan amount is still due.

2

Outstanding payment at the end of second 10 years

Payment each year for 10 year = twice the interest due in that period

Interest due = 6% of outstanding amount

Payment = 12% (6% * 2) of outstanding amout

Principal payment = 6% of principal (12% total - 6% interest = 6% principal)

if loan amount is reducing by 6% every year for 10 years then amount outstanding after 10 years will be =

= Loan amount * (1-reducing rate)^¹⁰

Let the loan amount be 100

Outstanding amount after second 10 year = 100 * (1-6%)^¹⁰ = 53.86

Outstanding amount at the end of second 10 years = 53.86% of original loan

3

Outstanding payment at the end of third 10 years

Since the loan is repaid in 30 years, the loan amount outstanding after 30 years = 0.

4

For the final 10 years, level payments of X per year, X = ?

Equal installments (X) = {P * R * (1+R)^ⁿ} / {(1+R)^ⁿ - 1}

P = remaining outstanding amount = 53.86

R = 6%

N = 10 years

X = (53.86 * 6% * (1.06^¹⁰)) / ((1.06^¹⁰) - 1) = 7.32

For an initial loan amount of 100, X is 7.32

Thank you.