Mathematics Advanced • Year 12 • Module 7 • Lesson 12
Superannuation Modelling
Apply the super model to fee-comparison, salary-growth and retirement-design scenarios.
Problem 1 — Two friends, two fees (Think First scenario)
Two friends both start work at age 25 on a $70,000 salary with 11.5% employer contribution. The fund earns 7% p.a. They work to 65 (n = 40 years). Friend A pays 1.5% fees; Friend B pays 0.5% fees. Assume A0 = $0 and constant salary.
Set up: What are we solving for?
(i) State C and the two net returns rnet for Friend A and Friend B. 1 mark
(ii) Use the closed form to find each friend's balance at age 65. 3 marks
(iii) Express the fee-cost in two ways: dollar gap (A balance − A balance for friend with lower fee), and as a percentage of Friend B's balance. Comment on whether the impact is small, medium, or massive. 3 marks
Stuck? Revisit lesson § Fee Impact table.Problem 2 — Choosing between two funds
A 40-year-old has A0 = $80,000 in super. They are 25 years from retirement on a $90,000 salary with 11.5% contributions. They are deciding between two funds.
Fund P: Return 7.5% p.a., fees 1.5% p.a.
Fund Q: Return 6.5% p.a., fees 0.3% p.a.
Set up: What are we solving for?
(i) Calculate rnet for each fund. 1 mark
(ii) Project A25 for each fund using the closed form. 3 marks
(iii) State which fund the saver should choose and explain in 2 sentences why higher headline returns do not always win once fees are considered. 2 marks
Problem 3 — The effect of a voluntary top-up
A 35-year-old has $60,000 in super. Salary $85,000. Employer contributes 11.5%. Fund return 7%, fees 0.8%. They consider adding a voluntary $3,000 per year on top of the employer contribution.
Set up: What are we solving for?
(i) Compute the new total annual contribution C with the top-up included. 1 mark
(ii) Project A30 with and without the $3,000 top-up. 3 marks
(iii) Find the dollar difference at age 65 and express it as a return on the extra $90,000 contributed over 30 years. 2 marks
Problem 4 — Two-stage career
An employee earns $50,000 from age 25 to 35, then $80,000 from age 35 to 65. The employer contributes 11.5%. The fund earns rnet = 6%. A0 = $0 at age 25.
Set up: What are we solving for?
(i) State the two annual contributions C1 (ages 25–35) and C2 (ages 35–65). 1 mark
(ii) Find A10 using the closed form with C = C1. This is the starting balance for the second stage. 2 marks
(iii) Use A10 as the new A0 and project 30 more years at C = C2 to find the balance at age 65. 3 marks
Stuck? When C changes, you cannot use one closed form for the whole career — split into stages and chain them.Problem 5 — Designing for a $1.5 million target
A 25-year-old has $10,000 in super and wants $1.5 million at age 65 (40 years). Their salary is $70,000 and the employer contributes 11.5%. They can choose a fund with rnet = 5.5% (high-fee) or rnet = 6.5% (low-fee).
Set up: What are we solving for?
(i) Calculate the projected A40 under each fund using the closed form. 2 marks
(ii) If neither fund alone reaches $1.5 million, what additional annual voluntary contribution Cextra would the low-fee fund need? (Use the annuity formula to solve for Cextra from the gap.) 3 marks
(iii) Write a one-sentence recommendation to the 25-year-old: which fund, and what extra contribution? 2 marks
Stuck? Use gap = Cextra × [(1+rnet)n − 1] ÷ rnet, then solve for Cextra.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Two friends
Set up. Project each balance with the closed form and compare.
(i) C = 70,000 × 0.115 = $8,050. Friend A: rnet = 0.07 − 0.015 = 0.055. Friend B: rnet = 0.07 − 0.005 = 0.065.
(ii) Friend A: (1.055)⁴⁰ ≈ 8.5133. A40 = 0 + 8,050 × (7.5133 ÷ 0.055) = 8,050 × 136.605 ≈ $1,099,673. Friend B: (1.065)⁴⁰ ≈ 12.4161. A40 = 8,050 × (11.4161 ÷ 0.065) = 8,050 × 175.633 ≈ $1,413,847.
(iii) Gap = 1,413,847 − 1,099,673 ≈ $314,000. As a percentage of B's balance: 314,000 ÷ 1,413,847 ≈ 22%. The impact is massive — a 1% lower fee saves nearly a quarter of the entire retirement balance over a working life.
Problem 2 — Two funds for a 40-year-old
Set up. Project each fund's A25 and choose the larger.
(i) Fund P: rnet = 0.075 − 0.015 = 0.060. Fund Q: rnet = 0.065 − 0.003 = 0.062.
(ii) C = 90,000 × 0.115 = $10,350. Fund P: (1.060)²⁵ ≈ 4.2919. A25 = 80,000(4.2919) + 10,350 × (3.2919 ÷ 0.060) = 343,352 + 10,350 × 54.865 = 343,352 + 567,853 ≈ $911,205. Fund Q: (1.062)²⁵ ≈ 4.5028. A25 = 80,000(4.5028) + 10,350 × (3.5028 ÷ 0.062) = 360,224 + 10,350 × 56.497 = 360,224 + 584,744 ≈ $944,968.
(iii) Choose Fund Q (by ~$34,000). Headline 7.5% looks better than 6.5%, but once the 1.2% fee gap is applied, Fund Q's net 6.2% beats Fund P's net 6.0%, and the difference compounds over 25 years.
Problem 3 — Voluntary top-up
Set up. Project A30 with and without the top-up to find the marginal benefit.
(i) Employer C = 85,000 × 0.115 = $9,775. With top-up, C = 9,775 + 3,000 = $12,775.
(ii) rnet = 0.062. (1.062)³⁰ ≈ 6.099. Without top-up: A30 = 60,000(6.099) + 9,775 × (5.099 ÷ 0.062) = 365,940 + 9,775 × 82.241 = 365,940 + 803,906 ≈ $1,169,846. With top-up: A30 = 60,000(6.099) + 12,775 × 82.241 = 365,940 + 1,050,629 ≈ $1,416,569.
(iii) Difference ≈ $246,723. Extra contributed = 3,000 × 30 = $90,000. Return on the extra contributions = 246,723 ÷ 90,000 ≈ 274% over 30 years — each extra dollar contributed earns roughly $2.74 in additional balance because it compounds at net 6.2% for an average of ~15 years.
Problem 4 — Two-stage career
Set up. Project the first 10 years with the lower contribution, then chain into 30 years with the higher contribution.
(i) C1 = 50,000 × 0.115 = $5,750. C2 = 80,000 × 0.115 = $9,200.
(ii) (1.06)¹⁰ ≈ 1.7908. A10 = 0 + 5,750 × (0.7908 ÷ 0.06) = 5,750 × 13.181 ≈ $75,789.
(iii) Use A10 = $75,789 as new A0. (1.06)³⁰ ≈ 5.7435. A40 = 75,789(5.7435) + 9,200 × (4.7435 ÷ 0.06) = 435,393 + 9,200 × 79.058 = 435,393 + 727,338 ≈ $1,162,731 at age 65.
Problem 5 — $1.5 million target
Set up. Project under each fund, then size the voluntary top-up needed.
(i) C = 70,000 × 0.115 = $8,050. High-fee (rnet = 0.055): (1.055)⁴⁰ ≈ 8.5133. A40 = 10,000(8.5133) + 8,050 × 136.605 = 85,133 + 1,099,668 ≈ $1,184,801. Low-fee (rnet = 0.065): (1.065)⁴⁰ ≈ 12.4161. A40 = 10,000(12.4161) + 8,050 × 175.633 = 124,161 + 1,413,846 ≈ $1,538,007.
(ii) The low-fee fund already exceeds the $1.5 million target by ~$38,000, so no extra contribution is required. If we did need to close a gap, e.g. for the high-fee fund (gap = 1,500,000 − 1,184,801 ≈ $315,199), Cextra = 315,199 ÷ 136.605 ≈ $2,308/year would be required for the high-fee path.
(iii) Choose the low-fee fund; no voluntary top-up is needed at this salary if the target is $1.5 million. Switching from high-fee to low-fee was worth a $2,300/year voluntary contribution and still has the advantage of guaranteed access without committing the cash.