How does money grow when interest compounds, and how do regular savings build an investment?
Apply the compound interest formula and model annuities with regular contributions to find future values of investments.
How to use the compound interest formula with different compounding periods, find the future value of an investment, and model an annuity that grows through regular contributions.
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What this dot point is asking
You must apply the compound interest formula with correct period rates, find future values, and model an annuity that grows through regular deposits.
The compound interest formula
Compound interest is calculated on the growing balance, so earlier interest itself earns interest.
Here is the principal (starting amount), is the final amount, is the interest rate per compounding period (as a decimal) and is the number of compounding periods.
The key step is converting an annual rate to the period rate and the years to the number of periods.
Why compounding frequency matters
For the same annual rate, more frequent compounding gives slightly more growth, because interest is added to the balance sooner and starts earning interest itself. Daily beats monthly beats annual, though the differences are modest at typical rates.
Annuities: investing with regular contributions
An annuity is an investment that receives equal payments at regular intervals, with compound interest applied each period. The balance grows from two sources: interest on what is already there, plus each new contribution.
You can model an annuity step by step with a recurrence. If is the balance after periods, is the period rate and is the regular deposit made each period:
Reading the result in context
For investments, the future value answers "how much will I have?" and the interest earned is (or final balance minus total deposits for an annuity). Examiners often ask you to compare two options or to find how long an investment takes to reach a target.
Exam-style practice questions
Practice questions written in the style of SACE Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
2021 SACE Stage 22 marksEilidh contributes approximately $1960 per quarter to a superannuation fund earning 5.85% per annum, compounded quarterly. Calculate the expected balance in the account after 43 years of working in this job.Show worked answer →
This is a future value of an annuity with regular contributions. Use the financial solver (or annuity formula) with:
Quarterly rate = 5.85% / 4 = 1.4625% per quarter.
Number of periods n = 43 times 4 = 172 quarters.
Payment PMT = 1960 per quarter, present value = 0.
Future value FV = PMT times [((1 + i)^n - 1) / i]
= 1960 times [((1.014625)^172 - 1) / 0.014625]
which gives approximately 1.08 million).
Award 1 mark for the correct rate per period and number of periods, and 1 mark for the future value of roughly 1960" payment.
2022 SACE Stage 22 marksArjun invests 50 into the account each week. Show that the balance of the account will be approximately $30 100 after 3 years.Show worked answer →
This combines a compounding lump sum with a regular-deposit annuity. Use:
Weekly rate = 3.1% / 52 = 0.0596% per week.
Number of periods n = 3 times 52 = 156 weeks.
Present value PV = 20 000, payment PMT = 50 per week.
Growth of the lump sum: 20 000 times (1 + 0.031/52)^156 = approximately 21 950.
Growth of the deposits: 50 times [((1 + 0.031/52)^156 - 1) / (0.031/52)] = approximately 8160.
Total = 21 950 + 8160 = approximately $30 100, as required.
Award 1 mark for setting up both components with the correct weekly rate and 156 periods, and 1 mark for combining them to show the result is about $30 100.
2019 SACE Stage 22 marksLewis saves 32 000. Show that it will take Lewis approximately 62 fortnights to save $32 000.Show worked answer →
This is a future value of an annuity; solve for the number of periods n.
Fortnightly rate i = 3.2% / 26 = 0.12308% per fortnight = 0.0012308.
Set FV = 32 000, PMT = 500.
32 000 = 500 times [((1 + i)^n - 1) / i].
Using the financial solver (or rearranging and solving for n), n is approximately 61.8, so n is about 62 fortnights.
A quick check: 62 times 500 = 31 000 of deposits, plus about 1000 of interest, gives roughly 32 000, confirming the answer. Award 1 mark for the correct annuity setup with the fortnightly rate and 1 mark for solving to show n is approximately 62 fortnights.