Reducing-Balance Loans
Master reducing-balance loans for NSW Year 12 Mathematics Standard 2. Interest is charged each period on the outstanding balance, so the loan behaves like compound interest with regular repayments — you set the working out in an amortisation table of opening balance, interest, repayment and closing balance.
You will build the table period by period, read the monthly repayment, total interest and amount owing from a per-\(\$1000\) table, and work out how an extra or lump-sum repayment repays the loan sooner with less interest — core loan skills for Standard 2 financial mathematics.
Theory
Reducing-balance loans are part of the loans topic in Year 12 Standard 2 (NSW). Interest is charged each period on the outstanding balance, so the loan is compound interest with regular repayments. This guide shows how to build an amortisation table, find the monthly repayment and total interest from a per-\(\$1000\) table, and work out the effect of extra repayments.
A reducing-balance loan charges interest each period on the outstanding balance — the amount still owing — not on the original amount borrowed. Each period you add that interest and then subtract the repayment, so the balance falls a little more each time.
Because the interest is worked out on the balance, it is really compound interest with regular repayments. The calculation is set out period by period in an amortisation table with columns for the opening balance, interest, repayment and closing balance.
In this Year 12 Standard 2 (NSW) topic you build the table for up to four periods, read the monthly repayment, total interest and amount owing from a per-\(\$1000\) table, and work out how an extra or lump-sum repayment shortens the loan.
For a balance \(B\), monthly interest rate \(r\) and fixed repayment \(R\), each period:
From a table of the monthly repayment per \(\$1000\) borrowed:
How to work through a reducing-balance loan
- Opening balance. Start the period with the amount still owing (the previous closing balance, or the original loan for period 1).
- Interest. Multiply the monthly rate by the opening balance and round to the nearest cent.
- Closing balance. Add the interest and subtract the repayment: \(B_{\text{closing}} = B_{\text{opening}} + \text{Interest} - R\).
- Carry forward and repeat. Use the closing balance as the next opening balance. For an extra or lump-sum payment, subtract it as well in that period.
For example, an \(\$16000\) loan at \(1.5\%\) per month with \(\$600\) repayments:
| Month | Opening ($) | Interest ($) | Repayment ($) | Closing ($) |
|---|---|---|---|---|
| 1 | 16000.00 | 240.00 | 600.00 | 15640.00 |
| 2 | 15640.00 | 234.60 | 600.00 | 15274.60 |
Each month: interest \(=0.01\times\) opening; closing \(=\) opening \(+\) interest \(-\) repayment, carried forward.
| Month | Opening ($) | Interest ($) | Repayment ($) | Closing ($) |
|---|---|---|---|---|
| 1 | 9000.00 | 90.00 | 500.00 | 8590.00 |
| 2 | 8590.00 | 85.90 | 500.00 | 8175.90 |
| 3 | 8175.90 | 81.76 | 500.00 | 7757.66 |
| \(\text{month 1}\) | \(\) | \(0.01\times9000 = 90,\ \ 9000+90-500=8590\) |
| \(\text{month 2}\) | \(\) | \(0.01\times8590 = 85.90,\ \ 8590+85.90-500=8175.90\) |
| \(\text{month 3}\) | \(\) | \(0.01\times8175.90 \approx 81.76,\ \ =7757.66\) |
After three months the balance owing is \(\$7757.66\).
Add one month's interest to the balance, then subtract the repayment.
| \(\text{interest}\) | \(=\) | \(0.009\times18000 = 162\) |
| \(\text{balance}\) | \(=\) | \(18000 + 162 - 700\) |
| \(\) | \(=\) | \(17462\) |
The balance owing is \(\$17462\).
Multiply the table value by the number of thousands, then compare total paid with the amount borrowed.
| Term | 6.0% p.a. | 7.0% p.a. |
|---|---|---|
| 10 years | 11.10 | 11.61 |
| 20 years | 7.16 | 7.75 |
| 25 years | 6.44 | 7.07 |
| \(\text{repayment}\) | \(=\) | \(240\times 6.44 = 1545.60\) |
| \(\text{total paid}\) | \(=\) | \(1545.60\times 12\times 25 = 463680\) |
| \(\text{interest}\) | \(=\) | \(463680 - 240000 = 223680\) |
Priya repays \(\$1545.60\) per month and pays \(\$223680\) in interest.
Complete month 1, then in month 2 subtract the larger \(\$2400\) repayment.
| \(\text{month 1}\) | \(\) | \(0.01\times25000 = 250,\ \ 25000+250-900=24350\) |
| \(\text{month 2 interest}\) | \(=\) | \(0.01\times24350 = 243.50\) |
| \(\text{with extra}\) | \(=\) | \(24350+243.50-2400 = 22193.50\) |
| \(\text{without extra}\) | \(=\) | \(24350+243.50-900 = 23693.50\) |
The extra \(\$1500\) leaves the balance \(\$1500\) lower at \(\$22193.50\), so less interest is charged from then on and the loan is repaid sooner.
Common pitfalls
Frequently asked questions
What is a reducing-balance loan?
It is a loan where interest is charged each period on the outstanding balance — the amount you still owe — rather than on the original amount borrowed. Each period you add the interest and subtract your repayment, so the balance and the interest both fall over time. It is compound interest with regular repayments.
How do you calculate the interest each month?
Multiply the monthly interest rate by that month's opening balance, then round to the nearest cent. For example, at 1 percent per month on an opening balance of $9000 the interest is 0.01 times 9000, which is $90. Next month the interest is charged on the new, smaller balance.
How do you make an amortisation table?
Use columns for the month, opening balance, interest, repayment and closing balance. Each row: interest equals the rate times the opening balance; closing balance equals opening plus interest minus the repayment. Carry that closing balance down as the next row's opening balance and repeat for each period.
Why does the balance fall faster near the end of the loan?
The repayment is fixed, but as the balance drops less of it is needed for interest, so more goes towards the principal. That means the amount owing falls by a larger amount each period, and the balance curve drops more steeply the closer you get to paying it off.
How do extra or lump-sum repayments help?
An extra or lump-sum repayment reduces the balance immediately. Because interest is charged on the balance, every future period's interest is then smaller, so more of each normal repayment pays off the principal. The loan is repaid sooner and the total interest paid is less.
How do you find the monthly repayment and total interest from a per-$1000 table?
Divide the loan by 1000 and multiply by the table value for that rate and term to get the monthly repayment. Multiply the repayment by the number of months to get the total paid, then subtract the amount borrowed to get the total interest. For example, $240000 at 6.0% over 25 years gives 240 times 6.44, which is $1545.60 a month.