QLDGeneral MathematicsSyllabus dot point

Topic 1: Loans, investments and annuities - how do regular contributions build, and regular withdrawals draw down, an invested balance?

Model an annuity investment (regular deposits earning compound interest) and an annuity that pays a regular income (drawing down a lump sum) using first-order recurrence relations, compute the future value of contributions and the duration a payout annuity lasts, and apply this to superannuation

A focused answer to the QCE General Mathematics Unit 4 dot point on annuities and superannuation. Covers annuity investments with regular deposits, payout annuities that draw down a balance, the future-value recurrence, working out how long a payout lasts, and superannuation applications, with arithmetic-verified worked examples for IA3 and the external assessment.

Generated by Claude Opus 4.76 min answerUpdated 2026-05-29

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking

QCAA wants you to model two faces of an annuity: building wealth by making regular deposits into an account that earns compound interest (an annuity investment, the engine of superannuation), and drawing it back down later as a regular income (a payout annuity). Both run on the same recurrence as a loan, but with the payment sign chosen to suit deposits or withdrawals. You compute future values, work out how long a payout annuity lasts, and connect it to superannuation. This is a distinct slice of Unit 4 Topic 1, beyond reducing-balance loans, and is common in IA3 and the external assessment.

The answer

The shared recurrence

Every annuity uses the period recurrence

V_{n+1} = V_n (1 + r) + D,

where $r$ is the interest rate per compounding period and $D$ is the regular payment. For an annuity investment the deposit is added, so $D$ is positive. For a payout annuity the withdrawal is subtracted, so $D$ is negative. The rate per period is the annual rate divided by the number of periods per year.

Annuity investments and superannuation

In an annuity investment you start with a balance (possibly zero) and add a fixed deposit $D$ each period while the balance earns interest. The balance grows from both the interest and the new contributions, so it accelerates over time. Superannuation is exactly this model: regular contributions from wages are invested and compound for decades, which is why starting early matters so much.

The future value is the balance after the final deposit. You usually read it from the recurrence on technology, stepping period by period or using a financial solver.

Payout annuities

In a payout annuity you begin with a lump sum (such as a superannuation balance at retirement) and withdraw a fixed income $D$ each period while the remainder keeps earning interest. The balance falls because the withdrawal usually exceeds the interest earned. The annuity is exhausted when the balance first reaches zero, and a key question is how many periods the income can last.

How long a payout lasts

To find the duration, run the recurrence with the negative withdrawal until the balance reaches zero or below. The number of full periods before that point is how long the income runs at the stated level; the final period may be a smaller top-up payment. If the withdrawal is smaller than the interest earned, the balance grows instead and the annuity lasts indefinitely.

Worked example

Growth of a superannuation contribution over two periods

An account starts at $\$ 10,000 $and earns$ 6 $percent per year compounding monthly. A deposit of$ \ $500$ is made at the end of each month. Find the balance after two months.

Step 1: Monthly rate.

r = \frac{0.06}{12} = 0.005.

Step 2: Month 1. Grow the balance then add the deposit.

V_1 = 10000 \times 1.005 + 500 = 10050 + 500 = 10550.

Verify: $10000 \times 1.005 = 10050$ and $10050 + 500 = 10550$ .

Step 3: Month 2.

V_2 = 10550 \times 1.005 + 500.

Step 4: Evaluate.

10550 \times 1.005 = 10602.75, \qquad 10602.75 + 500 = 11102.75.

Verify: $10550 \times 0.005 = 52.75$ , so $10550 + 52.75 = 10602.75$ , then $+500 = 11102.75$ . The balance after two months is $\$ 11,102.75$, made of interest plus the two deposits.

Common traps

Using the wrong sign for the payment. Deposits are added and withdrawals are subtracted. A payout annuity with a positive $D$ grows instead of drawing down.

Adding the deposit before applying interest, when the question says end of period. The standard convention is grow the balance by the interest first, then add the end-of-period deposit. Doing it in the wrong order shifts every figure.

Treating superannuation as simple interest: Superannuation compounds, so each period's interest is earned on the whole growing balance, not just the original contributions.
Stopping a payout annuity one period too late: The income runs at full level only while the balance stays positive; once it would go negative, the final payment is a smaller top-up. Counting that as a full period overstates the duration.
Forgetting to convert the annual rate: Monthly compounding needs the annual rate divided by $12$ before use; leaving it annual inflates every balance.

Exam-style practice questions

Practice questions written in the style of QCAA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

2021 QCAA5 marksJo contributes

2500 per quarter to an annuity earning 3.6% p.a. compounding quarterly. At the end of 4 years, Jo makes a one-off extra contribution of

10 000 and continues with the regular quarterly contributions. Determine the value of the annuity at the end of 6 years, to the nearest dollar.

Show worked answer →

Treat the two streams separately and add them.

Step 1 - parameters (1 mark): quarterly rate i = 3.6% / 4 = 0.9% = 0.009. The regular contributions of $2500 run for the full 6 years, so n = 6 x 4 = 24 quarters.

Step 2 - future value of the regular contributions (1 mark): A = M x ((1 + i)^n - 1) / i = 2500 x (1.009^24 - 1) / 0.009 = 66 639.94.

Step 3 - grow the one-off $10 000 (1 mark). It is added at the end of year 4, so it compounds for the remaining 2 years, n = 2 x 4 = 8 quarters: A = P(1 + i)^n = 10 000 x 1.009^8 = 10 743.09.

Step 4 - add the two values (1 mark): 66 639.94 + 10 743.09 = 77 383.03.

Step 5 - round and state (1 mark): the annuity is worth about $77 383 at the end of 6 years.

The key is the timing: the regular deposits accumulate over all 24 quarters, but the lump sum only grows for the 8 quarters between when it is added and the end.

2023 QCAA7 marksThe table shows the average superannuation account balance for workers of various ages in two industries. The coefficient of determination, R^2, for age versus account balance is 0.95 for industry A and 0.96 for industry B. Age (years) with balances Industry A / Industry B: 22: 7500/8100; 32: 42 000/60 000; 42: 98 000/120 000; 52: 160 000/210 000; 62: 290 000/360 000; 72: 400 000/480 000. 40-year-old Leigh works in the industry for which age explains a higher percentage of the balance variation. Tony is 10 years older than Leigh and works in the other industry. Use linear models to predict the difference in current superannuation account balances for Leigh and Tony.

Show worked answer →

Step 1 - assign each person to an industry (1 mark). A higher R^2 means age explains more of the variation, so Leigh (40) is in industry B (R^2 = 0.96) and Tony (50) is in industry A.

Step 2 - fit the line for industry B (1 mark). Least-squares regression of balance on age for B gives about y = 9570x - 243 440.

Step 3 - fit the line for industry A (1 mark): y = 7910x - 205 520.

Step 4 - predict Leigh's balance (1 mark): industry B at x = 40, y = 9570 x 40 - 243 440 = 139 360.

Step 5 - predict Tony's balance (1 mark): industry A at x = 50, y = 7910 x 50 - 205 520 = 189 980.

Step 6 - find the difference (1 mark): 189 980 - 139 360 = 50 620.

Step 7 - communicate clearly (1 mark): Tony's predicted balance is about $50 620 more than Leigh's. The final mark rewards a logical layout that makes clear which line belongs to which person.

2023 QCAA5 marksFive years ago, a retiree invested

100 000 in a compound interest account earning 3.8% p.a. compounding monthly. They now intend to use the balance of the account to begin a perpetuity that will return 4% p.a. compounding annually and pay them

6000 each year. Provide advice to the retiree about whether their compound interest investment is large enough to finance the perpetuity.

Show worked answer →

Step 1 - grow the investment (1 mark). Monthly rate i = 3.8% / 12, periods n = 5 x 12 = 60: balance = 100 000 x (1 + 0.038/12)^60.

Step 2 - evaluate the balance (1 mark): = $120 888.66.

Step 3 - principal a perpetuity needs (1 mark). A perpetuity pays interest forever without touching the principal, so the principal must satisfy payment = principal x rate: principal = 6000 / 0.04.

Step 4 - evaluate that principal (1 mark): = $150 000.

Step 5 - compare and advise (1 mark): the accumulated balance $120 888.66 is less than the$ 150 000 required, so the investment is not large enough. The retiree cannot fund a $6000 perpetuity at 4% p.a.; they would need to save more or accept a smaller annual payment.

The perpetuity condition payment = principal x rate is the heart of the question; the compound-interest step just supplies the principal you then test.