Superannuation models | Year 12

01 / LEARN

Understand the idea.

Separate contribution flows from investment growth

Separate three quantities: salary, the contribution calculated from salary, and the investment return earned on the account. If the stated deduction fraction is $t$ , a gross contribution $C$ adds only $C (1 - t)$ to the balance.

Follow the supplied timing. When returns are credited before year-end contributions and fees, use $B_{k} = B_{k - 1} (1 + i) + C_{k} (1 - t) - F_{k}$ . The new contribution earns no return during that completed year. Use only the deduction and contribution assumptions given in the question.

B_{k} = B_{k - 1} (1 + i) + C_{k} (1 - t) - F_{k}

Changing contributions and uncertain returns

For a constant net contribution and fixed year-end fee, combine them as one cash flow and accumulate it with an ordinary-annuity factor. A fee deducted from each contribution reduces the amount that can earn returns in later years.

If salary, contributions, fees or returns change, make one row per year and carry the closing balance forward. A single constant-rate annuity would impose different assumptions. Compare scenarios on the same timing and remember that a projected constant return is a modelling assumption.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Build the model

For this model, salary is AUD 50,000, the annual contribution is 10% of salary, and 15% is deducted from contributions. Find the net annual contribution.

Worked solution

1
Calculate the gross contribution from salary.
$C = 50000 \times 0.10 = 5000$
The 10% contribution is a percentage of salary, not of the account balance.
2
Calculate the stated contribution deduction.
$D = 5000 \times 0.15 = 750$
The 15% deduction applies to the gross contribution only.
3
Find the amount actually entering the account.
$C_{net} = 5000 - 750 = AUD 4250.00$
Only the retained contribution is available for future investment growth.

Answer

AUD 4,250.00

EXAMPLE 02

Use the model to make a decision

Start with AUD 20,000. A net AUD 3,000 is added at each year end for 5 years. Returns are 4% annually and a fixed AUD 100 year-end fee is charged. Find the final balance.

Worked solution

1
Combine the end-of-year contribution and fee.
$R = 3000 - 100 = 2900$
Both cash flows occur at the same year end, so their net addition is AUD 2,900.
2
Grow the starting balance for five years.
$20000 {(1.04)}^{5} = 24333.058048$
The initial fund receives five annual returns.
3
Accumulate the five net additions.
$2900 \frac{{(1.04)}^{5} - 1}{0.04} = 15707.335424$
Equal year-end additions form an ordinary savings annuity.
4
Add the two final-date components.
$\begin{aligned} B_{5} & = 24333.058048 + 15707.335424 \\ = 40040.393472 \approx AUD 40040.39 \end{aligned}$
Both components represent money in the fund immediately after year five’s contribution and fee.

Answer

AUD 40,040.39

03 / YOUR TURN

Try it yourself.

Check your understanding

A fund starts at AUD 8,000, earns 5%, then receives a net AUD 2,200 contribution and pays a fixed AUD 90 fee at year end. Find its new balance.

Show answer & reasoning

AUD 10,510.00

First credit interest on the opening fund: $8000 (1.05) = 8400$ . Then apply the year-end cash flows: $8400 + 2200 - 90 = 10510$ . The closing balance is AUD 10,510.00.

Keep practising

6 worksheet questions · 17 marks

Open worksheet Worked solutions

Read the questions online

A model has an annual gross contribution of AUD 6,400 and a stated contribution deduction of 12%. Find the net contribution.
3 marks
A fund starts at AUD 8,000, earns 5%, then receives a net AUD 2,200 contribution and pays a fixed AUD 90 fee at year end. Find its new balance.
3 marks
A constant net contribution is AUD 2,500 at each year end. At 4%, starting from zero, find the balance after 12 years.
3 marks
A balance of AUD 10,000 earns 5% in year 1 and $- 3 %$ in year 2. Add AUD 2,000 at each year end. Find the final balance.
3 marks
Compare fixed year-end fees of AUD 80 and AUD 150 over 10 years at 4%, with all other cash flows identical. Find the difference in final balances.
3 marks
Explain why a projected balance based on 6% every year is not a guaranteed result.
2 marks

Read the worked solutions

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

A model has an annual gross contribution of AUD 6,400 and a stated contribution deduction of 12%. Find the net contribution.
3 marks
Net is $6400 (1 - 0.12)$ .
$6400 \times 0.88 = 5632$ .
AUD 5,632.00
A fund starts at AUD 8,000, earns 5%, then receives a net AUD 2,200 contribution and pays a fixed AUD 90 fee at year end. Find its new balance.
3 marks
Interest is $8000 \times 0.05 = 400$ .
Balance is $8000 + 400 + 2200 - 90 = 10510$ .
AUD 10,510.00
A constant net contribution is AUD 2,500 at each year end. At 4%, starting from zero, find the balance after 12 years.
3 marks
This is an ordinary annuity with $R = 2500, i = 0.04$ and $n = 12$ .
Use $\frac{R ({(1 + i)}^{n} - 1)}{i}$ .
AUD 37,564.51
A balance of AUD 10,000 earns 5% in year 1 and $- 3 %$ in year 2. Add AUD 2,000 at each year end. Find the final balance.
3 marks
After year 1: $10000 \times 1.05 + 2000 = 12500$ .
After year 2: $12500 \times 0.97 + 2000 = 14125$ .
AUD 14,125.00
Compare fixed year-end fees of AUD 80 and AUD 150 over 10 years at 4%, with all other cash flows identical. Find the difference in final balances.
3 marks
The fee difference is AUD 70 each year.
Its accumulated effect is the future value of ten end-of-year AUD 70 payments.
AUD 840.43
Explain why a projected balance based on 6% every year is not a guaranteed result.
2 marks
Identify the assumed quantities.
A valid calculation only describes what follows if those assumptions hold.
The return is an assumed constant; actual yearly returns, contributions, fees and timing may differ.

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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.

Separate contribution flows from investment growth

Changing contributions and uncertain returns

See the method in action.

Build the model

Calculate the gross contribution from salary.

Calculate the stated contribution deduction.

Find the amount actually entering the account.

Use the model to make a decision

Combine the end-of-year contribution and fee.

Grow the starting balance for five years.

Accumulate the five net additions.

Add the two final-date components.

Try it yourself.

Check your understanding

Keep practising