Extra repayments, frequency changes and offsets | Year 12

01 / LEARN

Understand the idea.

Extra repayments change later interest

An extra principal repayment lowers the balance from its payment date onward. Later interest is then calculated on a smaller debt. Start the remaining recurrence from the reduced balance; the eventual balance difference includes both the extra payment and the interest it saves.

For a fair frequency comparison, specify the annual payment budget and rate conversion. An effective annual rate $e$ gives $i = {(1 + e)}^{1 / f} - 1$ for $f$ equal periods. Half a monthly payment every fortnight produces 26 half-payments, equivalent to 13 monthly payments per year.

An offset reduces the interest base

An offset is a separate asset that reduces the debt used to calculate interest. In this model the charge is $i max (B - O, 0)$ , where $B$ is the loan balance and $O$ is the offset balance. Interest cannot become negative.

With an end-of-period repayment, use $B_{k + 1} = B_{k} + i max (B_{k} - O, 0) - R$ . State when offset deposits or withdrawals occur. Comparing an offset with a lump-sum repayment requires net debt as well as the displayed loan balance, since offset funds remain separately owned.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Build the model

AUD 10,000 is owed at 0.5% monthly, with AUD 500 repayments. Compare the balance after 12 months with and without an extra AUD 1,000 paid immediately at time 0.

Worked solution

1
Calculate the balance with no extra repayment.
$\begin{aligned} B_{12} & = 10000 {(1.005)}^{12} \\ - 500 \frac{{(1.005)}^{12} - 1}{0.005} \\ \approx 4448.996932 \end{aligned}$
The original debt receives twelve months of interest and the same twelve month-end payments.
2
Reduce the opening principal for the immediate extra payment.
$P_{extra} = 10000 - 1000 = 9000$
A payment at time zero reduces the balance before any of the twelve interest charges.
3
Calculate the second balance with the same monthly schedule.
$\begin{aligned} B_{12, extra} & = 9000 {(1.005)}^{12} \\ - 500 \frac{{(1.005)}^{12} - 1}{0.005} \\ \approx 3387.319120 \end{aligned}$
Only the starting principal changes; the rate and regular repayments remain identical.
4
Compare the remaining debts.
$\begin{aligned} Δ B & = 1000 {(1.005)}^{12} \\ \approx AUD 1061.68 \end{aligned}$
The balances round to AUD 4,449.00 and AUD 3,387.32; their difference includes AUD 1,000 paid early and about AUD 61.68 of saved interest.

Answer

No extra: AUD 4,449.00; extra: AUD 3,387.32; balance difference AUD 1,061.68.

EXAMPLE 02

Use the model to make a decision

A loan balance is AUD 30,000, an offset holds AUD 5,000, and the monthly rate is 0.5%. Find the month’s interest.

Worked solution

1
Find the balance exposed to interest.
$max (30000 - 5000, 0) = 25000$
The offset reduces the interest base without changing the displayed AUD 30,000 loan balance.
2
Apply the monthly interest rate.
$I = 25000 \times 0.005 = AUD 125.00$
The stated 0.5% is already a monthly rate.
3
Distinguish the offset asset from repaid principal.
$B = 30000, O = 5000$
This interest calculation does not transfer the offset funds into the loan or make a principal repayment.

Answer

AUD 125.00

03 / YOUR TURN

Try it yourself.

Check your understanding

Find the fortnightly rate equivalent to an effective annual rate of 6%.

Show answer & reasoning

0.2244%

Let $i$ be the fortnightly rate. There are 26 model periods, so ${(1 + i)}^{26} = 1.06$ . Thus $i = {1.06}^{1 / 26} - 1 \approx 0.002243625$ , or $0.2244 %$ per fortnight.

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

Read the questions online

A monthly repayment is AUD 800. Compare the annual cash paid with AUD 400 paid every fortnight.
3 marks
Find the fortnightly rate equivalent to an effective annual rate of 6%.
3 marks
A balance is AUD 20,000 with an offset of AUD 3,000. Interest is 0.4% monthly and the month-end repayment is AUD 600. Find the new loan balance.
3 marks
A debt of AUD 1,500 has an offset of AUD 2,000. Under a model with no negative interest and an AUD 200 month-end repayment, find the new debt.
3 marks
A no-interest loan of AUD 6,000 is repaid at AUD 300 monthly. How many months are saved by instead paying AUD 400 monthly?
3 marks
Explain why an offset balance of AUD 4,000 is not the same accounting entry as a permanent AUD 4,000 principal repayment.
3 marks

Read the worked solutions

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

A monthly repayment is AUD 800. Compare the annual cash paid with AUD 400 paid every fortnight.
3 marks
Monthly total is $12 \times 800 = 9600$ .
Fortnightly total is $26 \times 400 = 10400$ .
Monthly: AUD 9,600.00; fortnightly: AUD 10,400.00; extra annual outflow AUD 800.00.
Find the fortnightly rate equivalent to an effective annual rate of 6%.
3 marks
There are 26 model periods per year.
$i = {1.06}^{\frac{1}{26}} - 1$ .
0.2244%
A balance is AUD 20,000 with an offset of AUD 3,000. Interest is 0.4% monthly and the month-end repayment is AUD 600. Find the new loan balance.
3 marks
Interest is $(20000 - 3000) \times 0.004 = 68$ .
Loan balance is $20000 + 68 - 600 = 19468$ .
The AUD 3,000 remains in the offset; it is not subtracted from the debt again.
AUD 19,468.00
A debt of AUD 1,500 has an offset of AUD 2,000. Under a model with no negative interest and an AUD 200 month-end repayment, find the new debt.
3 marks
The interest base is $max (1500 - 2000, 0) = 0$ .
New debt is $1500 - 200 = 1300$ .
AUD 1,300.00
A no-interest loan of AUD 6,000 is repaid at AUD 300 monthly. How many months are saved by instead paying AUD 400 monthly?
3 marks
At AUD 300, $\frac{6000}{300} = 20$ months.
At AUD 400, $\frac{6000}{400} = 15$ months.
Difference is 5 months.
5 months
Explain why an offset balance of AUD 4,000 is not the same accounting entry as a permanent AUD 4,000 principal repayment.
3 marks
Both may initially reduce the interest base by AUD 4,000.
Compare net positions and access to cash, not only the loan’s displayed balance.
The offset leaves both the loan debt and an accessible savings asset; a principal repayment reduces the debt and uses the cash.

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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 ↗

In this lesson

Understand the idea.

Extra repayments change later interest

An offset reduces the interest base

See the method in action.

Build the model

Calculate the balance with no extra repayment.

Reduce the opening principal for the immediate extra payment.

Calculate the second balance with the same monthly schedule.

Compare the remaining debts.

Use the model to make a decision

Find the balance exposed to interest.

Apply the monthly interest rate.

Distinguish the offset asset from repaid principal.

Try it yourself.

Check your understanding

Keep practising