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
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
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
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 ≈ 8.513309; annuity factor ≈ 136.605614. A40 = 8,050 × 136.605614 ≈ $1,099,675. Friend B: (1.065)⁴⁰ ≈ 12.416075; annuity factor ≈ 175.631916. A40 = 8,050 × 175.631916 ≈ $1,413,837.
(iii) Gap = 1,413,837 − 1,099,675 ≈ $314,162. As a percentage of B's balance: 314,162 ÷ 1,413,837 ≈ 22.2%. In this fixed-return model, the 1 percentage-point fee difference produces a substantial difference in the projected balance over 40 years.
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 (rnet = 0.060): (1.060)²⁵ ≈ 4.291871; annuity factor ≈ 54.864512. A25 = 80,000(4.291871) + 10,350(54.864512) ≈ $911,197. Fund Q (rnet = 0.062): (1.062)²⁵ ≈ 4.498968; annuity factor ≈ 56.434970. A25 = 80,000(4.498968) + 10,350(56.434970) ≈ $944,019.
(iii) Under the stated fixed-return model, choose Fund Q (by about $32,822). Its higher net rate, 6.2% compared with Fund P's 6.0%, produces the larger projected balance 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 ≈ 6.077647; annuity factor ≈ 81.897539. Without top-up: A30 = 60,000(6.077647) + 9,775(81.897539) ≈ 364,659 + 800,548 ≈ $1,165,207. With top-up: A30 = 60,000(6.077647) + 12,775(81.897539) ≈ 364,659 + 1,046,241 ≈ $1,410,900.
(iii) The added contributions increase the projected final balance by about $245,693. The extra deposits total $3,000 × 30 = $90,000, so the increase is about 2.73 times those deposits; the net gain above the deposits is about $155,693, or a 173% return on the $90,000, under this fixed-return, end-of-year contribution model.
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 ≈ 1.790848; annuity factor ≈ 13.180795. A10 = 5,750 × 13.180795 ≈ $75,789.57.
(iii) Use the unrounded A10 as the new A0. (1.06)³⁰ ≈ 5.743491; annuity factor ≈ 79.058186. A40 = 75,789.57(5.743491) + 9,200(79.058186) ≈ 435,297 + 727,335 ≈ $1,162,632 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 ≈ 8.513309; annuity factor ≈ 136.605614. A40 = 10,000(8.513309) + 8,050(136.605614) ≈ 85,133 + 1,099,675 ≈ $1,184,808. Low-fee (rnet = 0.065): (1.065)⁴⁰ ≈ 12.416075; annuity factor ≈ 175.631916. A40 = 10,000(12.416075) + 8,050(175.631916) ≈ 124,161 + 1,413,837 ≈ $1,537,998.
(ii) The low-fee fund exceeds the $1.5 million target by about $37,998, so no extra contribution is required under this model. For comparison, the high-fee fund's shortfall is about $315,192; dividing by its 40-year annuity factor 136.605614 gives a top-up of about $2,307.63/year.
(iii) Under the stated fixed-return model, choose the low-fee fund; it reaches the $1.5 million target without a voluntary top-up. This projection does not guarantee any future balance.