Mathematics Advanced • Year 12 • Module 7 • Lesson 12

Superannuation Modelling

Practise HSC-style writing on super projections and an extended response on the impact of fees.

Master · Past-Paper Style

1. Short-answer questions

1.1 A worker has $30,000 in super, salary $85,000, employer contribution 11.5%, fund return 7% p.a., fees 1.2% p.a.
(a) Find the annual contribution C and the net return r_net.
(b) Using the closed form, project the balance after 30 years to the nearest dollar. 3 marks Band 3

1.2 Two super funds both quote a 7% gross return. Fund A's fees are 1.5% p.a.; Fund B's fees are 0.5% p.a. A 25-year-old with A₀ = $0, salary $70,000 and 11.5% employer contribution invests for 40 years.
(a) Calculate the projected balance in each fund.
(b) State the dollar gap and express it as a percentage of Fund B's balance. 3 marks Band 3-4

1.3 A super recurrence is given by A_n+1 = 1.058A_n + 9,200 with A₀ = $20,000.
(a) Iterate to find A₁ and A₂.
(b) Verify A₂ using the closed form and state the difference (in dollars or cents) due to rounding.
(c) Interpret the values of r_net = 0.058 and C = $9,200: what salary and which combination of return/fees could have produced these numbers? 4 marks Band 4

Stuck on 1.3(c)? Solve C = salary × 0.115 for the salary, then pick any (return, fee) pair that differs by 5.8%.

2. Extended response

2.1 A 25-year-old has $0 in super. Salary $80,000. Employer contribution 11.5%. Three funds are available, all advertised as "7% p.a. growth", but they charge different fees.

Fund Low: 7% return, 0.5% fees.

Fund Mid: 7% return, 1.0% fees.

Fund High: 7% return, 1.5% fees.

(a) Calculate each fund's net return r_net and the projected balance at age 65 (40 years), showing the closed-form substitution.
(b) Rank the funds from highest to lowest balance, and quantify the dollar penalty of each 0.5% fee step (Low→Mid, Mid→High).
(c) Explain in 2-3 sentences why a small annual fee difference results in a six-figure final difference, referencing how fees enter the recurrence relation and compound over time. 8 marks Band 5-6

Explicit marking criteria

Part (a) — 4 marks

• 1 mark — correct C = 80,000 × 0.115 = $9,200 stated explicitly.

• 1 mark — correct three r_net values (0.065, 0.060, 0.055).

• 1 mark — at least two projected balances correct to the nearest thousand dollars.

• 1 mark — all three projected balances correct to the nearest thousand dollars.

Part (b) — 2 marks

• 1 mark — correct ranking Low > Mid > High.

• 1 mark — correct dollar gaps for both steps.

Part (c) — 2 marks

• 1 mark — explicitly states that fees compound against the balance just as returns compound for it (or equivalent phrasing).

• 1 mark — links the recurrence A_n+1 = (1 + r − f)A_n + C to the long term: small differences in r − f raise (1 + r − f)ⁿ by a large factor over decades.

Your response:

Stuck on (c)? Reference the lesson's "Hidden Cost of Fees" table: a 1% fee gap can cost over $600k over 40 years.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — sample responses + marking notes

1.1 — $30,000 + 11.5% on $85,000, return 7%, fees 1.2%, n = 30 (3 marks)

Sample response. (a) C = 85,000 × 0.115 = $9,775. r_net = 0.07 − 0.012 = 0.058. (b) (1.058)³⁰ ≈ 5.404. A₃₀ = 30,000(5.404) + 9,775 × (4.404 ÷ 0.058) = 162,120 + 9,775 × 75.931 = 162,120 + 742,222 ≈ $904,342.

Marking notes. 1 mark — correct C and r_net. 1 mark — correct substitution into closed form. 1 mark — correct numerical answer (accept ±$2,000 for rounding). A bald final answer with no working scores 1/3.

1.2 — Fund A vs Fund B at 7%/1.5% vs 7%/0.5%, 40 years (3 marks)

Sample response. C = 70,000 × 0.115 = $8,050. (a) Fund A (r_net = 0.055): A₄₀ = 8,050 × [(1.055)⁴⁰ − 1] ÷ 0.055 = 8,050 × 136.605 ≈ $1,099,673. Fund B (r_net = 0.065): A₄₀ = 8,050 × [(1.065)⁴⁰ − 1] ÷ 0.065 = 8,050 × 175.633 ≈ $1,413,847. (b) Gap ≈ $314,174, about 22% of Fund B's balance.

Marking notes. 1 mark — correct C. 1 mark — both projections within ±$5,000. 1 mark — correct gap and percentage. Students who use the wrong (gross) rate score 0/2 on the calculation.

1.3 — Recurrence A_n+1 = 1.058A_n + 9,200, A₀ = 20,000 (4 marks)

Sample response. (a) A₁ = 1.058(20,000) + 9,200 = 21,160 + 9,200 = $30,360. A₂ = 1.058(30,360) + 9,200 = 32,120.88 + 9,200 = $41,320.88. (b) Closed form: A₂ = 20,000(1.058)² + 9,200 × ((1.058)² − 1) ÷ 0.058 = 20,000(1.119364) + 9,200 × (0.119364 ÷ 0.058) = 22,387.28 + 18,934.91 = $41,322.19. Difference ≈ $1.31 — pure rounding error. (c) C = 9,200 = salary × 0.115 implies salary = $80,000. r_net = 5.8% could come from return 7%, fees 1.2%; or return 6.5%, fees 0.7% — many pairs work, since only the difference matters.

Marking notes. 1 mark — correct A₁ and A₂. 1 mark — correct closed-form A₂. 1 mark — explicit acknowledgement of the rounding gap (no need to compute exactly). 1 mark — correct salary $80,000 and a valid return/fee pair.

2.1 — Three funds at 7% with different fees, 40 years (8 marks): sample Band-6 response

Sample Band-6 response.

C = 80,000 × 0.115 = $9,200 per year. The three net returns are:

Fund Low: r_net = 0.07 − 0.005 = 0.065

Fund Mid: r_net = 0.07 − 0.010 = 0.060

Fund High: r_net = 0.07 − 0.015 = 0.055

[2 marks: C correct, all three r_net correct]

(a) Projected balances at age 65 (n = 40, A₀ = 0).

Fund Low: A₄₀ = 9,200 × [(1.065)⁴⁰ − 1] ÷ 0.065 = 9,200 × 175.633 ≈ $1,615,824.

Fund Mid: A₄₀ = 9,200 × [(1.060)⁴⁰ − 1] ÷ 0.060 = 9,200 × 154.762 ≈ $1,423,810.

Fund High: A₄₀ = 9,200 × [(1.055)⁴⁰ − 1] ÷ 0.055 = 9,200 × 136.605 ≈ $1,256,766.

[2 marks: all three balances correct]

(b) Ranking and penalties. Low > Mid > High. Low → Mid penalty = 1,615,824 − 1,423,810 ≈ $192,000. Mid → High penalty = 1,423,810 − 1,256,766 ≈ $167,000. [2 marks]

(c) Why a 0.5% fee step costs six figures. Each year's fee reduces the growth factor in the recurrence A_n+1 = (1 + r − f)A_n + C, so fees compound against the balance just as returns compound for it. Over 40 years, a 0.5% gap in r_net raises the growth factor (1 + r_net)⁴⁰ by a factor of roughly 1.13–1.14, multiplying through the entire balance. Because the balance reaches seven figures, a 13% multiplicative change is worth roughly $170,000 — for the cost of choosing the next-cheaper fund. [2 marks]

Total: 8/8.

Band descriptors for marker.

Band 3: Two of three balances correct, ranking present but no quantification of fee penalty. ≈ 4 marks.

Band 4: All three balances correct, ranking with one penalty quantified. Part (c) restates "fees reduce return" without compound reasoning. ≈ 5-6 marks.

Band 5: Correct calculations and ranking, both penalty values, structural explanation of compounding in (c). ≈ 6-7 marks.

Band 6: Full calculations to within a few thousand dollars, ranking with both penalties, and part (c) explicitly references the recurrence form and the compounding multiplier (1 + r − f)ⁿ. 8/8.