How does a loan balance fall as repayments are made, and how is interest split out?
Model reducing-balance loans with recurrence relations and amortisation tables.
Reducing-balance loans, recurrence relations, amortisation tables splitting interest from principal, and total interest paid in TCE Mathematics Applications.
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What this dot point is asking
A reducing-balance loan charges interest only on the amount still owing. Because the balance falls with every repayment, the interest charged each period also falls, so a larger slice of each fixed repayment goes toward the principal as time passes.
The recurrence relation
The balance after each period is found by adding interest then subtracting the repayment.
The amortisation table
An amortisation table records, for each period, the interest charged, the principal repaid, and the new balance. It makes the inner workings of each repayment visible.
Total interest paid
The total interest over the life of the loan is the sum of all repayments minus the amount originally borrowed. This is often the figure that surprises borrowers, because over many years the interest can rival the principal.
The final repayment
The last repayment is usually smaller than the rest, because a full fixed repayment would overshoot and leave a negative balance. The final payment is exactly the outstanding balance plus its final period of interest, clearing the loan to zero.
A complete answer uses with as the per-period rate, splits at least one repayment into interest and principal, and where asked finds total interest as total repayments minus the amount borrowed.
Exam-style practice questions
Practice questions written in the style of TASC exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
2024 TASC General Mathematics2 marksAlex is trying to calculate how much they will still owe on their home loan after 5 years. They repay $900 each month, interest is 7% p.a. compounding monthly and the loan is set to last 20 years. Their calculation uses the present-value-of-annuity formula with n = 5 x 12 in the exponent. What common error have they made?Show worked answer →
(2 marks) The formula Alex used finds the present value of only the payments made over 5 years, not the balance still owing after 5 years.
To find the amount outstanding after 5 years you must use the number of payments that are still left, which is the remaining 15 years: n = 15 x 12 = 180, not 5 x 12 = 60. By substituting 60 they have valued the wrong set of payments. The correct balance owing is the present value of the 180 remaining repayments.
2023 TASC General Mathematics6 marksUse the interest factor table for monthly compound interest on a loan. a) Find the amount borrowed over a 10-year loan period if monthly repayments of 200000 taken over a 20-year period at 5% p.a. compounded monthly. c) How long would it take to repay 425.75 were made and the interest rate was 4% p.a. compounded monthly?Show worked answer →
a) (2 marks) Amount borrowed = repayment x factor, where the factor is read from the table for i = 0.005 (6%/12) and n = 120. This gives amount borrowed = $150000.
b) (2 marks) Rearrange to repayment = loan / factor, using i = 5%/12 and n = 240. The factor gives a monthly repayment of about $1319.91.
c) (2 marks) Here repayment = loan / factor, so factor = 5000 / 425.75 = 11.744. Find this factor in the i = 4%/12 column of the table; it corresponds to n = 12, so the loan is repaid in 12 months (1 year).