Mathematics Advanced • Year 12 • Module 7 • Lesson 16
Comparing Investment Products
Build fluency with net returns, after-tax returns and fee-adjusted future values.
1. Quick recall
Answer each question in the space provided. 1 mark each
Q1.1 Complete the formulas:
Net return: rnet = ____________________
After-tax return: rafter-tax = ____________________
Future value (single deposit): FV = ____________________
Q1.2 A managed fund returns 7.0% p.a. gross, charges 1.2% in fees and the investor pays 32.5% marginal tax on the distribution.
rnet (after fees) = ____________
rafter-tax (on the net) = ____________
Q1.3 In one sentence, state why two products advertised at the same headline rate can deliver very different final balances.
2. Worked example — balanced fund vs term deposit
Follow every line. Each step has a short reason.
Problem. An investor has $30,000. Product A: term deposit at 4.8% p.a., no fees. Product B: balanced fund at 6.5% p.a. gross, 0.8% fees. Find the future value of each after 15 years. (Compound annually.)
Step 1 — Find net return for each product.
A: rnet = 4.8% − 0% = 4.8%
B: rnet = 6.5% − 0.8% = 5.7%
Reason: only the return that actually reaches the investor compounds.
Step 2 — Apply FV = PV(1 + rnet)ⁿ for each.
A: FV = 30,000 × (1.048)¹⁵ = 30,000 × 2.0094 = $60,281.94
B: FV = 30,000 × (1.057)¹⁵ = 30,000 × 2.30773 = $69,232.00
Step 3 — Compare.
Difference = 69,232.00 − 60,281.94 = $8,950.06 in favour of B
Conclusion. Product B wins by approximately $8,950 (about 14.8% more) — but only because the 0.9 percentage-point net-return advantage was preserved after fees.
3. Faded example — fill in the missing steps
Two products on $20,000 over 10 years. Product X: 5.2% p.a., no fees. Product Y: 7.0% p.a. gross, 1.2% fees. 4 marks
Step 1 — Find net returns:
rX = ______________ rY = ______________
Step 2 — Substitute into FV = PV(1 + rnet)ⁿ:
FVX = 20,000 × (1 + ______)¹⁰ = 20,000 × ______ = $______________
FVY = 20,000 × (1 + ______)¹⁰ = 20,000 × ______ = $______________
Step 3 — Difference:
FVY − FVX = $______________
Conclusion. Product ______ wins by $______________ even though its headline rate is 1.8 pp higher and its fees are higher.
4. Graduated practice — net returns and future values
Show the substitution and the final amount (to nearest dollar) for each. Compound annually unless stated.
Foundation — single-step net return or FV (4 questions)
| Q | Scenario | Working & answer |
|---|---|---|
| 4.1 1 | Gross 8% p.a., fees 1.5%. State rnet. | |
| 4.2 1 | Gross 6% p.a., investor tax rate 32.5%. State rafter-tax (treat fees as 0%). | |
| 4.3 1 | $50,000 at net 4.5% p.a. for 20 years — find FV. | |
| 4.4 1 | $50,000 at net 7.5% p.a. for 20 years — find FV. |
Standard — typical HSC difficulty (6 questions)
Show working in the space below each part — at least one substitution line and one evaluation line.
4.5 Reproduce the lesson's headline comparison: $50,000 over 20 years in (a) the term deposit at 4.5% net, (b) the managed fund at 6.0% net, (c) the growth portfolio at 7.5% net. Quote each FV to the nearest hundred dollars. 3 marks
4.6 A managed fund has gross 7% and currently 1% fees. Recompute the FV on a $50,000, 20-year investment if fees rise to 2%. State the dollar cost of the extra 1% fee. 2 marks
4.7 A super product returns 8.0% p.a. gross with 0.6% admin fees. The investor's tax on earnings is 15%. Compute the effective rate the investor actually accrues, then find the FV of $25,000 over 30 years. 3 marks
4.8 $100,000 is invested for 25 years. Compare a 6.5% growth portfolio (net) with an 8.0% growth portfolio (net). State the difference in FV in dollars and as a percentage of the lower FV. 2 marks
4.9 A 60-year-old has $50,000 to invest for 5 years. Compare the term deposit at 4.5% net against the growth portfolio at 7.5% net. State both FVs and one sentence on why the dollar gap is small at this horizon. 2 marks
4.10 A 25-year-old has $50,000 to invest for 40 years in the growth portfolio at 7.5% net. State the FV and the multiple by which the initial capital has grown. 2 marks
Extension — combine concepts (2 questions)
4.11 A managed fund has gross 7% p.a. Find the break-even fee level at which its 20-year FV on $50,000 ties the term deposit's 4.5% net FV on the same deposit. State the fee to one decimal place. 3 marks
4.12 An investor compares two 10-year products on $40,000: Product P (4.0% net, no tax), Product Q (6.5% gross, 0.5% fees, 30% tax on the post-fee return). Which gives the larger FV? Justify with the FV figures. 3 marks
5. Self-check the easy 3
Tick the first three once you have checked the method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1 — Formulas
rnet = rgross − rfees. rafter-tax = rgross × (1 − t). FV = PV(1 + rnet)ⁿ.
Q1.2 — Managed fund net and after-tax
rnet = 7.0 − 1.2 = 5.8%. rafter-tax = 5.8 × (1 − 0.325) = 3.915%.
Q1.3 — Same headline, different finals
The headline (gross) rate hides fees, tax and compounding frequency. Only the net return — after every cost the investor actually pays — compounds through (1 + r)ⁿ, so two products at the same gross rate can deliver wildly different FVs.
Q3 — Faded example: Product X vs Y on $20,000, 10 years
rX = 5.2%; rY = 7.0 − 1.2 = 5.8%. FVX = 20,000 × (1.052)¹⁰ = 20,000 × 1.6585 = $33,170. FVY = 20,000 × (1.058)¹⁰ = 20,000 × 1.7586 = $35,172. Difference = $2,002 in favour of Y; Y wins despite paying 1.2% in fees because its post-fee return is still higher.
Q4.1 — Net return
rnet = 8 − 1.5 = 6.5%.
Q4.2 — After-tax return
rafter-tax = 6 × (1 − 0.325) = 6 × 0.675 = 4.05%.
Q4.3 — $50,000 at 4.5% for 20 years
FV = 50,000 × (1.045)²⁰ = 50,000 × 2.41171 = $120,586 (lesson rounds to $120,300).
Q4.4 — $50,000 at 7.5% for 20 years
FV = 50,000 × (1.075)²⁰ = 50,000 × 4.24785 = $212,392 (lesson rounds to $212,200). About 76% more than the term deposit.
Q4.5 — Headline three-way comparison
(a) Term deposit: FV ≈ $120,300. (b) Managed fund at 6.0% net: FV = 50,000 × (1.06)²⁰ = 50,000 × 3.2071 = $160,356. (c) Growth portfolio: FV ≈ $212,200.
Q4.6 — Fee shock on managed fund
Original net = 6.0%: FV = $160,356. New net = 5.0%: FV = 50,000 × (1.05)²⁰ = 50,000 × 2.6533 = $132,665. The extra 1% in fees costs $27,691 over 20 years on a $50,000 deposit.
Q4.7 — Super product with tax
rnet = 8.0 − 0.6 = 7.4%. Effective rate to investor = 7.4 × (1 − 0.15) = 6.29%. FV = 25,000 × (1.0629)³⁰ = 25,000 × 6.241 = $156,025.
Q4.8 — 6.5% vs 8.0% on $100,000 over 25 years
6.5%: FV = 100,000 × (1.065)²⁵ = 100,000 × 4.8277 = $482,770. 8.0%: FV = 100,000 × (1.08)²⁵ = 100,000 × 6.8485 = $684,850. Difference = $202,080, or about 41.9% of the lower FV — a small rate gap compounds into a six-figure dollar gap.
Q4.9 — Short horizon for the 60-year-old
Term deposit: FV = 50,000 × (1.045)⁵ = 50,000 × 1.2462 = $62,310. Growth portfolio: FV = 50,000 × (1.075)⁵ = 50,000 × 1.4356 = $71,782. The dollar gap is only $9,472 because 5 years is too short for compounding to do heavy lifting — and the growth option could lose 20% in a downturn before the investor needs the money.
Q4.10 — Long horizon for the 25-year-old
FV = 50,000 × (1.075)⁴⁰ = 50,000 × 18.044 = $902,180 — about 18 times the initial capital. (Lesson quotes ≈ $871,000 from the comparator's slightly different rounding.)
Q4.11 — Break-even fee on the managed fund
We want (1 + rnet)²⁰ = (1.045)²⁰, i.e. rnet = 4.5%. So fees = gross − net = 7.0 − 4.5 = 2.5% p.a. Any fee above 2.5% causes the 7.0% gross fund to lose to a 4.5% term deposit over 20 years.
Q4.12 — Tax-adjusted product comparison
P (4.0% net, no tax): FV = 40,000 × (1.04)¹⁰ = 40,000 × 1.4802 = $59,210. Q: rnet = 6.5 − 0.5 = 6.0%; after 30% tax this becomes 6.0 × 0.70 = 4.20%. FV = 40,000 × (1.042)¹⁰ = 40,000 × 1.5083 = $60,331. Q wins by $1,121 — but the gap is small because tax eats most of the gross-return advantage.