Comparison rates and choosing between loan models | Year 12

01 / LEARN

The idea, step by step

Build a like-for-like comparison

A lower quoted interest rate may come with fees or a longer term. Start with the amount actually needed, repayment frequency, term, rate convention, upfront fees and ongoing fees. A lower monthly payment alone does not establish a lower total cost.

When an upfront fee is paid separately, include it in total cash outflow. If it is financed, increase the loan principal before calculating repayments. Keep the two cases distinct and state the convention used.

A rate summarises a particular cash-flow model

For an educational comparison-rate model, let the borrower receive net advance P−F at time 0 and make equal periodic outflows R+f. Solve $P - F = (R + f) a_{n} (i)$ numerically, then annualise according to the required convention.

A published comparison rate uses stated assumptions and does not capture every possible cost or borrower choice. For a question, use the supplied inclusions and exclusions. Explain how the answer changes with term, extra repayments or fees instead of treating one rate as a universal recommendation.

P - F = (R + f) \frac{1 - {(1 + i)}^{- n}}{i}

A common trap

Choosing the smallest repayment without checking the term and fees.

Instead: Compare the total outflow over the relevant term and report any differences in loan conditions.

02 / WORKED EXAMPLES

See the thinking

Example 1

Build the model

Two no-fee loans each advance AUD 10,000. A requires 24 payments of AUD 445; B requires 36 payments of AUD 310. Compare total payments.

A total is $24 \times 445 = 10680$ .
B total is $36 \times 310 = 11160$ .
B has a smaller payment but a larger total cost.

A: AUD 10,680.00; B: AUD 11,160.00. A costs AUD 480.00 less in total.

Example 2

Use the model to make a decision

A loan of AUD 8,000 has 24 repayments of AUD 360, an upfront fee of AUD 120 paid separately, and a monthly fee of AUD 5. Find total cash paid and the amount above the principal.

Repayments is $24 \times 360 = 8640$ .
Add upfront 120 and monthly fees $24 \times 5 = 120$ .
Total is 8880; subtract 8000 for the excess cost.

Total AUD 8,880.00; above principal AUD 880.00.

Check your understanding

A fee of AUD 300 is financed into an AUD 12,000 loan. What principal should be used in the repayment formula?

Show answer

AUD 12,300.00

The lender advances enough to cover the cash need and the financed fee. Use principal $12000 + 300 = 12300$ .

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Offer A has total repayments AUD 12,300 and an upfront fee AUD 200. Offer B has total repayments AUD 12,450 and no fees. Both advance the same amount over the same term. Compare stated total costs.
3 marks
A fee of AUD 300 is financed into an AUD 12,000 loan. What principal should be used in the repayment formula?
3 marks
A loan advertises 5% interest with a comparison rate of 6%. Does this prove every borrower pays exactly 6% interest? Explain.
3 marks
A fee-free loan advances AUD 2,000 and requires 12 monthly payments of AUD 175. Find the monthly rate in this cash-flow model and its effective annual rate.
3 marks
For an educational model, a loan’s net advance is AUD 5,000 after fees. There are 10 equal annual payments of AUD 700. Write the equation defining its annual cost rate $i$ .
3 marks
A borrower plans to repay after one year, but an advertised comparison rate assumes five years. State two costs to check before choosing between offers.
3 marks

Read the worked solutions

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

Offer A has total repayments AUD 12,300 and an upfront fee AUD 200. Offer B has total repayments AUD 12,450 and no fees. Both advance the same amount over the same term. Compare stated total costs.
3 marks
Include A’s upfront fee.
Compare 12500 with 12450.
A: AUD 12,500.00; B: AUD 12,450.00; B is AUD 50.00 lower.
A fee of AUD 300 is financed into an AUD 12,000 loan. What principal should be used in the repayment formula?
3 marks
The lender advances enough to cover the cash need and the financed fee.
Use principal $12000 + 300 = 12300$ .
AUD 12,300.00
A loan advertises 5% interest with a comparison rate of 6%. Does this prove every borrower pays exactly 6% interest? Explain.
3 marks
Separate the quoted interest rate from the summary comparison measure.
Different amounts, terms or excluded costs can change the effective cost.
No. The comparison rate summarises specified interest and fee assumptions; it is not the contractual interest rate for every borrower.
A fee-free loan advances AUD 2,000 and requires 12 monthly payments of AUD 175. Find the monthly rate in this cash-flow model and its effective annual rate.
3 marks
Solve $2000 = 175 (1 - {(1 + i)}^{-} 12) / i$ numerically.
The monthly rate is 0.007587.
Annualise with ${(1 + i)}^{12} - 1$ .
Monthly 0.7587%; effective annual 9.4943%.
For an educational model, a loan’s net advance is AUD 5,000 after fees. There are 10 equal annual payments of AUD 700. Write the equation defining its annual cost rate $i$ .
3 marks
Match the net advance to the present value of every required outflow.
Use annual periods because the payments are annual.
$5000 = 700 \frac{1 - {(1 + i)}^{- 10}}{i}$
A borrower plans to repay after one year, but an advertised comparison rate assumes five years. State two costs to check before choosing between offers.
3 marks
The advertised term does not match the planned borrowing period.
Recalculate the relevant cash flows using the actual scenario rather than assuming the advertised ordering persists.
Check upfront/ongoing fees over the actual one-year term and any early-repayment or exit costs stated in the offers.

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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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