Reducing-balance loans | Year 12

01 / LEARN

Understand the idea.

Interest is charged before the repayment

For an end-of-period repayment, calculate interest on the opening debt, add it, and then subtract the payment: $B_{k + 1} = B_{k} (1 + i) - R$ . The part of the payment beyond interest reduces principal, making later interest charges smaller.

This recurrence assumes a constant rate, equal period lengths, no fees and payments at period ends. Retain unrounded balances unless the question specifies cent-by-cent rounding. A beginning-of-period repayment requires a different order because it reduces the interest base first.

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

Choose a repayment that reaches zero

A fully amortising loan equals the present value of its repayments. For $i > 0$ , solving $P = R \frac{1 - {(1 + i)}^{- n}}{i}$ gives the fixed payment needed to reach zero after $n$ payments. At zero interest, simply use $R = \frac{P}{n}$ .

Check affordability of repayment within the model before finding a payoff term. For a positive-rate loan, $R = P i$ leaves the initial debt unchanged, while $R < P i$ makes it grow. A fixed payment must exceed the initial interest to clear the debt.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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.

Worked solution

1
Calculate the first month’s interest.
$I_{1} = 10000 \times 0.01 = 100$
The first interest charge uses the opening AUD 10,000 balance.
2
Subtract the first repayment after interest.
$B_{1} = 10000 + 100 - 500 = AUD 9600.00$
AUD 100 covers interest, so AUD 400 of the payment reduces principal.
3
Calculate interest on the smaller second opening balance.
$I_{2} = 9600 \times 0.01 = 96$
The second charge uses AUD 9,600 rather than the original principal.
4
Find the second closing balance.
$B_{2} = 9600 + 96 - 500 = AUD 9196.00$
The second payment reduces principal by AUD 404 after paying that month’s interest.

Answer

AUD 9,600.00; AUD 9,196.00.

EXAMPLE 02

Use the model to make a decision

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

Worked solution

1
Write the present-value equation for five repayments.
$20000 = R \frac{1 - {(1.06)}^{- 5}}{0.06}$
There are five year-end repayments, each discounted to the borrowing date.
2
Solve for the annual repayment.
$\begin{aligned} R & = \frac{20000 \times 0.06}{1 - {(1.06)}^{- 5}} \\ \approx 4747.928009 \end{aligned}$
The repayment must cover the initial principal and all modelled interest over five years.
3
Verify exhaustion using the exact repayment.
$B_{5} = 20000 {(1.06)}^{5} - R \frac{{(1.06)}^{5} - 1}{0.06} = 0$
Substituting the unrounded expression for R confirms that the fifth payment clears the loan.
4
Report the repayment to cents.
$R \approx AUD 4747.93$
The exact model repayment is rounded for display; a cents-only schedule can require a small final adjustment.

Answer

AUD 4,747.93

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

AUD 400.00

At zero interest every payment reduces principal directly. Solve $24 R = 9600$ to obtain $R = 400$ , so the equal monthly payment is AUD 400.00.

Keep practising

6 worksheet questions · 17 marks

Open worksheet 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, $P = n R$ .
$R = \frac{9600}{24} = 400$ .
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 $i = \frac{0.06}{12} = 0.005, n = 36$ .
$R = \frac{12000 \times 0.005}{1 - {1.005}^{- 36}}$ .
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.

Checking saved progress…

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 ↗

In this lesson

Understand the idea.

Interest is charged before the repayment

Choose a repayment that reaches zero

See the method in action.

Build the model

Calculate the first month’s interest.

Subtract the first repayment after interest.

Calculate interest on the smaller second opening balance.

Find the second closing balance.

Use the model to make a decision

Write the present-value equation for five repayments.

Solve for the annual repayment.

Verify exhaustion using the exact repayment.

Report the repayment to cents.

Try it yourself.

Check your understanding

Keep practising