Reducing-balance loans | Year 12

01 / LEARN

The idea, step by step

Interest is charged before the repayment

In a reducing-balance model, each period begins with an outstanding balance. Add interest on that balance and then subtract the repayment. If the repayment exceeds the interest, the debt falls; as it falls, the next period’s interest also falls.

The model here uses a fixed periodic rate and equal end-of-period repayments, no fees and no intermediate cent rounding. Real schedules may use different conventions; use the convention specified in a question.

B_{k + 1} = B_{k} (1 + i) - R

Choose a repayment that reaches zero

A loan is the present value of its future repayments. Solving $P = R a_{n}$ gives $R = P i / (1 - {(1 + i)}^{- n})$ . If the rate is zero, use $R = P / n$ .

A payment equal to the initial period interest leaves the debt unchanged. A smaller payment makes the debt grow. Before using a logarithmic payoff formula, check that $R$ > $P i$ so the fixed-payment loan can actually be repaid.

A common trap

Subtracting the repayment before calculating interest when the model specifies an end-of-period payment.

Instead: Calculate interest on the opening balance first, then subtract the payment.

02 / WORKED EXAMPLES

See the thinking

Example 1

Build the model

A loan starts at AUD 10,000, with 1% monthly interest and AUD 500 month-end repayments. Find the first two closing balances.

Month 1: $10000 \times 1.01 - 500 = 9600$ .
Month 2: $9600 \times 1.01 - 500 = 9196$ .

AUD 9,600.00; AUD 9,196.00.

Example 2

Use the model to make a decision

Find the annual end repayment that clears AUD 20,000 over 5 years at 6% annually.

Use $P = R a_{5} (0.06)$ .
$R = 20000 \times 0.06 / (1 - {1.06}^{- 5})$ .
Check the balance after five exact repayments is zero.

AUD 4,747.93

Check your understanding

AUD 9,600 is repaid in 24 equal month-end payments at zero interest. Find the payment.

Show answer

AUD 400.00

At zero interest, $P = n R$ . $R = 9600 / 24 = 400$ .

03 / WORKSHEET

Your turn

6 questions · 17 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

On AUD 6,000 at 0.5% monthly interest, what is the first month’s interest?
3 marks
AUD 9,600 is repaid in 24 equal month-end payments at zero interest. Find the payment.
3 marks
A balance is AUD 5,000 at 1% monthly interest. Explain what happens if each month’s repayment is AUD 40.
3 marks
Find the monthly repayment for AUD 12,000 over 36 months at 6% nominal annually, compounded monthly.
3 marks
A debt of AUD 8,000 has a monthly interest rate of 0.75%. What repayment leaves the balance unchanged after that month?
3 marks
For a loan balance B and periodic rate $i$ , explain why a fixed repayment can reduce the debt faster in later periods.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

On AUD 6,000 at 0.5% monthly interest, what is the first month’s interest?
3 marks
Interest is calculated on the opening debt.
$6000 \times 0.005 = 30$ .
AUD 30.00
AUD 9,600 is repaid in 24 equal month-end payments at zero interest. Find the payment.
3 marks
At zero interest, $P = n R$ .
$R = 9600 / 24 = 400$ .
AUD 400.00
A balance is AUD 5,000 at 1% monthly interest. Explain what happens if each month’s repayment is AUD 40.
3 marks
First interest is $5000 \times 0.01 = 50$ .
The AUD 40 payment is less than interest, so $5000 + 50 - 40 = 5010$ .
Future interest grows as the balance grows.
The balance grows; first closing balance is AUD 5,010.
Find the monthly repayment for AUD 12,000 over 36 months at 6% nominal annually, compounded monthly.
3 marks
Use $i = 0.06 / 12 = 0.005, n = 36$ .
$R = 12000 \times 0.005 / (1 - {1.005}^{- 36})$ .
AUD 365.06
A debt of AUD 8,000 has a monthly interest rate of 0.75%. What repayment leaves the balance unchanged after that month?
3 marks
A payment equal to interest leaves principal unchanged.
$8000 \times 0.0075 = 60$ .
AUD 60.00
For a loan balance B and periodic rate $i$ , explain why a fixed repayment can reduce the debt faster in later periods.
2 marks
Separate each payment into interest and principal reduction.
Principal reduction is $R$ −B $i$ ; this increases as B decreases.
As B falls, the interest B $i$ falls, so more of the same repayment reduces principal.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 General Mathematics financial models: investments and annuities, superannuation, income streams and borrowing. Original practice uses supplied hypothetical rates, taxes and fees; statutory or commercial conditions are not inferred.

Official curriculum source ↗

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