Q.366. Compound interest - equatted monthly intalments of loan

Question 366.
A woman borrows $50 000 in order to buy a house. Compound interest at the rate of 12% per annum in charge on the loan. She agrees to pay back the loan in 25 equal instalments at yearly intervals, the first repayment being made exactly one year after the loan is taken out. Calculate the value of each instalment.

Answer 366.
Let the equal yearly instalments = $ x

Loan outstanding at the end of year 1 = $ 50000 * (1.12) - x
Loan outstanding at the end of year 2
= $ [50000 * (1.12) - x]*(1.12) - x
= $ 50000 * (1.12)^2 - x * (1 + 1.12)

Loan outstanding at the end of year 3
= $ [50000 * (1.12)^2 - x * (1 + 1.12)] * (1.12) - x
= $ 50000 * (1.12)^3 - x [1 + 1.12 + (1.12)^2]

Loan outstanding at the end of year 25
= $ 50000 * (1.12)^25 - x [1 + (1.12) + ... + (1.12)^24]

Loan outstanding at the end of year 25 should be zero
=> x * [1 + 1.12 + (1.12)^2 + ... + (1.12)^24] = 50000 * (1.12)^25
=> x * [(1.12)^25 - 1] / (1.12 - 1) = 850003.220332
=> x * (17.00006 - 1) = (0.12) * (850003.220332)
=> 16.00006 x = 102000.386
=> x = 6375
=> yearly equal instalments
= $ 6375 each year.

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