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.