Mathematics Standard • Year 12 • Module 7 • Lesson 10
Investment Strategies — Past-Paper Style
Practise HSC Mathematics Standard 2-style writing on investment strategies — three short-answer questions and one longer scenario with marking criteria.
1. Short-answer questions
1.1 A term deposit pays 5.5% p.a. compounded annually. Inflation is 2.8% p.a.
(a) State the real return rate.
(b) Calculate the purchasing power of $10,000 invested for 4 years, in today's dollars. 2 marks Band 3
1.2 A share portfolio returned 10%, −4%, 8%, 12% and −1% over 5 years.
(a) Calculate the arithmetic mean annual return.
(b) Using that mean, project the FV of $25,000 invested for the 5-year period at the mean rate compounded annually.
(c) State in one sentence why this projection may overstate the actual outcome. 3 marks Band 3-4
1.3 $25,000 is invested for 6 years. Compare a 5% p.a. term deposit (guaranteed) with shares at expected 8% p.a. but a possible 25% loss in year 1.
(a) Calculate the term-deposit FV.
(b) Calculate the shares' best-case FV (no losses).
(c) Calculate the shares' FV if year 1 loses 25% and years 2-6 each return 8%.
(d) Briefly state which option a risk-averse investor with a 6-year horizon should take, and why. 4 marks Band 4
2. Extended response
2.1 A 32-year-old couple has $40,000 to deploy over 10 years. Inflation is forecast at 2.5% p.a.
Option A: Voluntary super contribution earning expected 7% p.a., with super tax of 15% applied to earnings (use after-tax effective rate ≈ 7% × 0.85 = 5.95%).
Option B: Pay down their mortgage at 5.4% p.a. (treat as a guaranteed 5.4% return).
Option C: Savings account at 4% p.a.
(a) Calculate the nominal FV of the $40,000 under Option A after 10 years.
(b) Calculate the nominal FV (or equivalent saving) under Option B after 10 years.
(c) Calculate the nominal FV under Option C after 10 years.
(d) State the real return rate for each option.
(e) Recommend the best option for the couple's long-term wealth, naming the option and the dollar difference vs Option C, and state one realistic trade-off (e.g. liquidity, super-access age, market risk). 7 marks Band 5-6
Explicit marking criteria
Part (a) — 1 mark
• 1 mark — correct FV using 40,000 × (1.0595)¹⁰.
Part (b) — 1 mark
• 1 mark — correct FV using 40,000 × (1.054)¹⁰.
Part (c) — 1 mark
• 1 mark — correct FV using 40,000 × (1.04)¹⁰.
Part (d) — 1 mark
• 1 mark — all three real returns correctly stated (Real = nominal/effective − inflation).
Part (e) — 3 marks
• 1 mark — comparison sentence stating all three FVs on the same basis.
• 1 mark — recommendation names the chosen option AND quotes the dollar difference vs Option C.
• 1 mark — one realistic trade-off named (e.g. super preservation age, market risk on the 7% expectation, lost liquidity if money is in mortgage).
Your response:
Stuck on (e)? After ranking the three FVs, the conclusion must name the option, quote the dollar gap vs Option C, AND mention one realistic trade-off to land all three marks.How did this worksheet feel?
What I'll revisit before next class:
1.1 — Real return and purchasing power (2 marks)
(a) Sample response. Real return = 5.5 − 2.8 = 2.7% p.a.
(b) Sample response. PP = 10,000 × (1.027)⁴ ≈ 10,000 × 1.1124 ≈ $11,124 in today's dollars.
Marking notes. (a) 1 mark — correct real return. (b) 1 mark — uses (1 + real)ⁿ not (1 + nominal)ⁿ. A common error is computing nominal FV; that scores 0/1 for part (b).
1.2 — Mean return and projection (3 marks)
(a) Sample response. Mean = (10 − 4 + 8 + 12 − 1) / 5 = 25 / 5 = 5% p.a.
(b) Sample response. FV = 25,000 × (1.05)⁵ ≈ 25,000 × 1.2763 ≈ $31,907.
(c) Sample response. The arithmetic mean hides the volatility — actual sequence-of-returns effects (especially big losses early on) typically give a slightly lower realised compound result.
Marking notes. (a) 1 mark — correct sum/5. (b) 1 mark — correct FV using mean. (c) 1 mark — explanation references volatility / sequence of returns or similar.
1.3 — TD vs shares (4 marks)
(a) Sample response. FV_TD = 25,000 × (1.05)⁶ ≈ 25,000 × 1.3401 ≈ $33,503.
(b) Sample response. Best case: 25,000 × (1.08)⁶ ≈ 25,000 × 1.5869 ≈ $39,672.
(c) Sample response. Yr 1 loss case: 25,000 × 0.75 × (1.08)⁵ ≈ 18,750 × 1.4693 ≈ $27,549.
(d) Sample response. A risk-averse investor should take the term deposit: $33,503 guaranteed is more than the loss-case shares value ($27,549), and they avoid the chance of being below the original $25,000.
Marking notes. (a)-(c) 1 mark each — correct FV with substitution shown. (d) 1 mark — recommendation links risk-aversion to the calculated worst-case figure.
2.1 — Couple's $40,000 over 10 years (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Option A — super at after-tax 5.95%.
FV_A = 40,000 × (1.0595)¹⁰ ≈ 40,000 × 1.7825 ≈ $71,302. [1 mark — correct FV.]
(b) Option B — mortgage paydown at 5.4%.
FV_B = 40,000 × (1.054)¹⁰ ≈ 40,000 × 1.6919 ≈ $67,675. [1 mark — correct FV.]
(c) Option C — savings at 4%.
FV_C = 40,000 × (1.04)¹⁰ ≈ 40,000 × 1.4802 ≈ $59,210. [1 mark — correct FV.]
(d) Real return rates.
A: 5.95 − 2.5 = 3.45%; B: 5.4 − 2.5 = 2.9%; C: 4 − 2.5 = 1.5%. [1 mark — all three real returns correct.]
(e) Comparison and recommendation.
Nominal FV ranking: A = $71,302, B = $67,675, C = $59,210. [1 mark — comparison on the same FV basis.]
Conclusion: the couple should take Option A (super contribution) — at expected 5.95% after-tax, the $40,000 grows to $71,302, which is $12,092 more than Option C's $59,210. [1 mark — names Option A and quotes the dollar difference vs Option C.] The trade-off is that super is locked until preservation age (currently around 60), so this money cannot be used for emergencies or major purchases in the next 10 years. [1 mark — one realistic trade-off named.]
Total: 7/7.
Band descriptors for marker.
Band 3: Computes one or two FVs correctly; real returns missing. ≈ 2 marks.
Band 4: All three FVs and all three real returns correct, but no recommendation sentence. ≈ 4 marks.
Band 5: Full numerical solution plus recommendation that names Option A and the dollar difference vs Option C, but no trade-off discussion. ≈ 6 marks.
Band 6: Complete: three FVs, three real returns, recommendation with dollar saving, AND one realistic trade-off sentence. 7/7.