Question

A 20-year loan of 150,000 is negotiated with the borrower agreeing to repay principal and interest at 5%. A level payment of 9,000 will apply during the first ten years and a higher level payment will apply during the remaining ten years. Each time the lender receives a payment from the borrower, he will deposit the portion representing the principal into a sinking fund with an annual effective interest rate of 4%. (Assume that the interest portion remains level throughout these 20 years and assume that all but the interest portion is deposited into the sinking fund.) This scheme will replace the lender’s capital.

What is the higher payment (rounded to the nearest dollar) that applies during the years 11-20?

Answer 1

Interest portion = 5% x 150,000 = $ 7,500

Hence, principal portion in the first 10 years payment = $ 9,000 - 7,500 = $ 1,500

Let A be the higher payment that applies during the years 11 to 20. The principal portion will be A - 7,500.

Hence, FV of 10 annuities of $ 1,500 + FV of 10 annuities of (A - 7,500) = 150,000

FV of 10 annuities at the end of 10th year = 1,500 / r x [(1 + r)ⁿ - 1] = 1,500 / 4% x [(1 + 4%)¹⁰ - 1] = $  18,009

FV of these 10 annuities at the end of 20 years = $  18,009 x (1 + r)¹⁰ = 18,009 x 1.04¹⁰ =  26,658

FV of 10 annuities of (A - 7,500) starting from year 11 and ending at the end of year 20 = (A - 7,500) / r x [(1 + r)ⁿ - 1] = (A - 7,500) / 4% x [(1 + 4%)¹⁰ - 1] = 12.01 x (A - 7,500)

Hence, 18,009 + 12.01 x (A - 7,500) = 150,000

Hence, A = (150,000 - 18,009) / 12.01 + 7,500 = $ 18,493.64 = $ 18,494

HEnce, the higher payment (rounded to the nearest dollar) that applies during the years 11-20 = A = $ 18,494