Comparing investments | Year 11

01 / LEARN

Understand the idea.

Create a fair comparison

A fair comparison needs a common starting amount, a common final date and clearly specified cash flows. Build a separate calculation for each account, using its own periodic rate and period count. Include the stated fees before comparing final balances.

A larger nominal percentage does not settle the comparison when compounding frequencies differ. Work out the actual growth factors over the same time. Use unrounded balances to identify the winner, then round the monetary results for reporting.

Test a conclusion

A break-even comparison asks when two final-value expressions are equal. Write both in terms of the same unknown principal, set them equal and solve. The difference between their expressions also shows which account is larger on either side of that principal.

Treat the conclusion as conditional on the supplied rules. A small rate or fee change can reverse a close result, and access, risk or changing rates may lie outside the model. Equal time horizons are necessary when comparing interest dollars.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example

For 3 years, compare AUD 8000 at 4.4% annual compounding with 4.3% nominal compounded quarterly. No fees.

Worked solution

1
Calculate the annual-compounding account.
$A = 8000 {(1.044)}^{3} \approx 9103.145472$
Three years at annual compounding gives three 4.4% growth periods.
2
Set up the quarterly-compounding account.
$i_{B} = \frac{0.043}{4} = 0.01075, n_{B} = 4 (3) = 12$
The nominal annual quote must be converted to a quarterly rate and twelve quarters.
3
Calculate the quarterly account's balance.
$B = 8000 {(1.01075)}^{12} \approx 9095.257248$
Both balances now represent the same starting AUD 8000 after the same three years.
4
Compare using the unrounded balances.
$\begin{aligned} A & \approx AUD 9103.15 \\ B & \approx AUD 9095.26 \\ A - B & \approx AUD 7.89 \end{aligned}$
The annual account finishes higher; using full precision prevents rounding from deciding a close comparison.

Answer

Annual account: AUD 9,103.15; quarterly account: AUD 9,095.26. The annual account is higher by AUD 7.89.

EXAMPLE 02

Worked example

For one year, A pays 5% simple interest and no fee. B pays 5.5% simple interest and charges AUD 25. For what principal do their final amounts match?

Worked solution

1
Express both final amounts in terms of the principal.
$A = 1.05 P, B = 1.055 P - 25$
Each pays one year of simple interest, while B also deducts its fixed fee.
2
Set the final balances equal.
$1.05 P = 1.055 P - 25$
Break-even means neither account finishes with more money.
3
Collect the principal terms and solve.
$0.005 P = 25 ⟹ P = \frac{25}{0.005} = AUD 5000.00$
The extra 0.5 percentage points of interest must exactly cover the AUD 25 fee.
4
Check the shared balance.
$1.05 (5000) = 1.055 (5000) - 25 = AUD 5250$
Both rules produce the same final amount at the calculated principal.

Answer

AUD 5,000.00

03 / YOUR TURN

Try it yourself.

Check your understanding

An extra 0.4 percentage points of annual simple interest must cover AUD 20 fees. What principal breaks even?

Show answer & reasoning

AUD 5,000.00

An extra 0.4 percentage points means a decimal rate difference of $0.004$ . Break-even requires $0.004 P = 20$ , so $P = \frac{20}{0.004} = AUD 5000.00$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Compare AUD 3000 for 2 years at 5% annual compound interest and 4.8% nominal compounded monthly. No fees.
2 marks
A AUD 2500 account earns 4% simple interest for one year and charges AUD 12. Find its final amount.
2 marks
For one year, A pays 3% simple interest/no fee; B pays 4% and charges AUD 18. Find the break-even starting amount.
2 marks
A model predicts A exceeds B by AUD 4 after 2 years. Explain why a possible AUD 10 closing fee matters.
2 marks
One account pays a fixed 4% for 2 years; another pays 5% only for the first year. Can a 2 year final comparison be determined?
2 marks
Compare 10% total growth over 2 years with 12% total growth over 3 years using annualised rates.
2 marks

Read the worked solutions

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

Compare AUD 3000 for 2 years at 5% annual compound interest and 4.8% nominal compounded monthly. No fees.
2 marks
Evaluate $3000 {(1.05)}^{2}$ and $3000 {(1.004)}^{24}$ .
Annual: AUD 3,307.50; monthly: AUD 3,301.64.
A AUD 2500 account earns 4% simple interest for one year and charges AUD 12. Find its final amount.
2 marks
$2500 + 100 - 12 = 2588$ .
AUD 2,588.00
For one year, A pays 3% simple interest/no fee; B pays 4% and charges AUD 18. Find the break-even starting amount.
2 marks
$1.03 P = 1.04 P - 18$ .
$P = \frac{18}{0.01} = 1800$ .
AUD 1,800.00
A model predicts A exceeds B by AUD 4 after 2 years. Explain why a possible AUD 10 closing fee matters.
2 marks
If only A incurs the extra 10,its advantage becomes $4 - 10 = - 6$ .
It can reverse the ranking.
One account pays a fixed 4% for 2 years; another pays 5% only for the first year. Can a 2 year final comparison be determined?
2 marks
A first-year promotional rate does not specify both years.
No; the second-year rate or balance rule is missing.
Compare 10% total growth over 2 years with 12% total growth over 3 years using annualised rates.
2 marks
Annual factors are ${1.10}^{\frac{1}{2}}$ and ${1.12}^{\frac{1}{3}}$ .
The two-year option has the higher annualised rate.
Two-year option: 4.881%; three-year option: 3.850%.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: investing and borrowing. Supplied educational models cover interest, investment comparisons, shares and credit; they are not recommendations of real products or statements of personal tax liability.

Official curriculum source ↗

In this lesson

Understand the idea.

Create a fair comparison

Test a conclusion

See the method in action.

Worked example

Calculate the annual-compounding account.

Set up the quarterly-compounding account.

Calculate the quarterly account's balance.

Compare using the unrounded balances.

Worked example

Express both final amounts in terms of the principal.

Set the final balances equal.

Collect the principal terms and solve.

Check the shared balance.

Try it yourself.

Check your understanding

Keep practising