Comparison rates and choosing between loan models | Year 12

01 / LEARN

Understand the idea.

Build a like-for-like comparison

Compare loans from the borrower’s actual cash flows: the money received, payment amounts, dates, term and all included fees. A smaller instalment can result from spreading repayments over more periods, so it does not by itself establish a lower total cost.

An upfront fee paid separately is an additional cash outflow. A financed fee instead increases the principal used to calculate repayments. Count it once according to its treatment; adding a financed fee again after calculating repayments would double-count it.

A rate summarises a particular cash-flow model

An educational comparison-rate model finds the interest rate matching the borrower’s net advance to the present value of required outflows. If the net receipt is $P - F$ and each periodic outflow is $R + f$ , solve $P - F = (R + f) \frac{1 - {(1 + i)}^{- n}}{i}$ .

Find the periodic rate numerically, then annualise using the convention requested. Such a rate summarises the specified amount, term, fees and payment assumptions. Changing those assumptions can change the comparison, so retain the underlying cash-flow totals when explaining the result.

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

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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.

Worked solution

1
Add all payments for loan A.
$T_{A} = 24 \times 445 = AUD 10680.00$
Loan A has twenty-four payments and no additional fees.
2
Add all payments for loan B.
$T_{B} = 36 \times 310 = AUD 11160.00$
B’s smaller individual payments continue for twelve more periods.
3
Compare the total future outflows.
$T_{B} - T_{A} = 11160 - 10680 = AUD 480.00$
Both loans advance the same AUD 10,000, so A costs AUD 480 less in total under the stated schedules.

Answer

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

EXAMPLE 02

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.

Worked solution

1
Find the total scheduled loan repayments.
$24 \times 360 = 8640$
These twenty-four payments exclude the separately stated monthly fee.
2
Add the upfront fee and all monthly fees.
$\begin{aligned} F_{monthly} & = 24 \times 5 = 120 \\ T & = 8640 + 120 + 120 = AUD 8880.00 \end{aligned}$
The upfront fee is paid separately once, while the monthly fee is paid twenty-four times.
3
Separate the amount above the cash principal.
$T - 8000 = 8880 - 8000 = AUD 880.00$
The excess includes interest and fees; the full AUD 880 is not all loan interest.

Answer

Total AUD 8,880.00; above principal AUD 880.00.

03 / YOUR TURN

Try it yourself.

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 & reasoning

AUD 12,300.00

The financed fee becomes part of the opening debt: $P = 12000 + 300 = 12300$ . Use AUD 12,300.00 in the repayment formula, so the repayments also service the financed fee.

Keep practising

6 worksheet questions · 18 marks

Open worksheet 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 $2000 = \frac{175 (1 - {(1 + i)}^{- 12})}{i}$ 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 ↗

In this lesson

Understand the idea.

Build a like-for-like comparison

A rate summarises a particular cash-flow model

See the method in action.

Build the model

Add all payments for loan A.

Add all payments for loan B.

Compare the total future outflows.

Use the model to make a decision

Find the total scheduled loan repayments.

Add the upfront fee and all monthly fees.

Separate the amount above the cash principal.

Try it yourself.

Check your understanding

Keep practising