Mathematics Standard • Year 12 • Module 7 • Lesson 10
Investment Strategies — Problem Set
Apply investment-strategy reasoning to realistic Australian decisions — mortgage vs super vs savings, inflation, risk tolerance, time horizon and diversification.
Problem 1 — A retiree's $300,000 income decision
Renee has $300,000 to invest at retirement. She is choosing between a term deposit at 4.8% p.a. compounded annually, and a 60/40 share/bond portfolio with expected return 7% but a possible 15% loss in any one year. Her time horizon is 3 years (she will draw on the funds for living costs). Inflation is 2.5% p.a.
Set up: What are we solving for?
(i) Calculate the nominal FV of the term deposit after 3 years. 1 mark
(ii) Calculate the expected nominal FV of the share/bond portfolio after 3 years, and also the value if year 1 loses 15% then years 2 and 3 return 7%. 2 marks
(iii) Recommend which option Renee should take and explain in one sentence why time horizon and risk tolerance are decisive here. 2 marks
Stuck? Revisit lesson § Time Horizon — short horizons favour safer assets.Problem 2 — A couple's $40,000 windfall
A couple receive $40,000 and consider three uses over 10 years. Inflation is 2.5% p.a.
Option A: Pay down their mortgage at 5.4% p.a. (treated as a guaranteed 5.4% saving).
Option B: Voluntary super contribution at expected 7% p.a., with super earnings taxed at 15% (use after-tax effective return ≈ 7% × 0.85 = 5.95%).
Option C: Savings account at 4% p.a.
Set up: What are we solving for?
(i) Calculate the nominal value after 10 years under each option. 3 marks
(ii) Calculate the real return rate for each option (real = effective − inflation). 1 mark
(iii) Recommend the best option for the couple's long-term wealth, naming the option and stating the dollar difference vs Option C, with one sentence on a key trade-off (e.g. liquidity, access age). 2 marks
Stuck? Revisit lesson § Comparing Investments — real returns and § Activity 2 Q2 — mortgage as a guaranteed return.Problem 3 — Diversifying a $100,000 portfolio
Han has $100,000. He considers two portfolios:
Portfolio X: 100% shares, expected 8% return, possible 30% loss in a bad year.
Portfolio Y: 50% shares, 30% bonds, 20% property, expected 6.5% return, possible 12% loss in a bad year.
Set up: What are we solving for?
(i) Calculate the expected nominal FV of each portfolio after 5 years (compounded annually at the expected rate). 2 marks
(ii) Calculate the worst-case value of each portfolio after a single bad year (no recovery, just the loss applied to the original $100,000). 2 marks
(iii) In one or two sentences, explain why a diversified portfolio "sacrifices some return for significantly less risk", referring to your numbers from (i) and (ii). 2 marks
Stuck? Revisit lesson § Diversification — the 100% shares vs balanced portfolio example.Problem 4 — A savings account vs inflation
Mia keeps $50,000 in a transaction account paying 1.5% p.a. interest. Inflation runs at 3.2% p.a. over the next 5 years.
Set up: What are we solving for?
(i) Calculate the nominal FV after 5 years. 1 mark
(ii) Calculate the real return rate, and use it to find Mia's purchasing power after 5 years in today's dollars. 2 marks
(iii) Comment in one sentence on what this calculation tells Mia about leaving large sums in low-interest accounts during high-inflation periods. 2 marks
Stuck? Revisit lesson § Comparing Investments — Real return = Nominal − Inflation.Problem 5 — A 28-year-old's superannuation choice
Caleb (28) can choose a "growth" super option (expected 7% p.a., possible bad-year loss 15%) or a "balanced" super option (expected 5.5% p.a., possible bad-year loss 6%). His balance is $40,000 and his time horizon is 37 years (to age 65).
Set up: What are we solving for?
(i) Calculate the expected nominal FV of each option after 37 years (use FV = P(1 + r)ⁿ). 2 marks
(ii) Calculate the difference in FV in dollars. 1 mark
(iii) Recommend an option for Caleb and explain in one or two sentences why a young investor can usually tolerate the higher short-run risk of the growth option. 2 marks
Stuck? Revisit lesson § Time Horizon — long horizons allow higher-risk strategies.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Retiree's 3-year horizon
Set up. Compute the FVs and compare on a 3-year basis.
(i) FV_TD = 300,000 × (1.048)³ ≈ 300,000 × 1.1510 ≈ $345,310.
(ii) Expected FV (shares/bonds) = 300,000 × (1.07)³ ≈ 300,000 × 1.2250 ≈ $367,503. With a 15% year-1 loss: 300,000 × 0.85 × (1.07)² ≈ 255,000 × 1.1449 ≈ $291,949 — below the original $300,000.
(iii) Recommend the term deposit: with only 3 years before drawing on the money, a bad year for shares could leave Renee with less than she started, and the term deposit's guaranteed $345,310 covers her cost-of-living needs.
Problem 2 — $40,000 over 10 years
Set up. Compute FV for each effective rate; compare real returns and dollars.
(i) A: 40,000 × (1.054)¹⁰ ≈ 40,000 × 1.6919 ≈ $67,675. B: 40,000 × (1.0595)¹⁰ ≈ 40,000 × 1.7825 ≈ $71,302. C: 40,000 × (1.04)¹⁰ ≈ 40,000 × 1.4802 ≈ $59,210.
(ii) Real returns: A = 5.4 − 2.5 = 2.9%; B = 5.95 − 2.5 = 3.45%; C = 4 − 2.5 = 1.5%.
(iii) Recommend Option B (super): $71,302 − $59,210 = $12,092 more than Option C over 10 years. Trade-off: super is locked until preservation age, so the money is not accessible for emergencies — Option A (mortgage paydown) is a guaranteed return with no super access restriction.
Problem 3 — Diversification
Set up. FVs at expected rates; worst-year values from a single 30% / 12% loss.
(i) Portfolio X: 100,000 × (1.08)⁵ ≈ 100,000 × 1.4693 ≈ $146,930. Portfolio Y: 100,000 × (1.065)⁵ ≈ 100,000 × 1.3701 ≈ $137,010.
(ii) X worst case: 100,000 × 0.70 = $70,000. Y worst case: 100,000 × 0.88 = $88,000.
(iii) Portfolio Y gives up about $9,900 in expected FV but caps Han's worst-case loss at $12,000 instead of $30,000 — i.e. he trades roughly $10k of expected upside for $18k of downside protection in a bad year. That smaller worst-case risk is the value of diversification.
Problem 4 — Inflation erosion
Set up. Compute nominal FV; subtract inflation to find real rate; compute today's-dollar value.
(i) FV_nominal = 50,000 × (1.015)⁵ ≈ 50,000 × 1.0773 ≈ $53,864.
(ii) Real return = 1.5 − 3.2 = −1.7% p.a. PP = 50,000 × (1 − 0.017)⁵ ≈ 50,000 × (0.983)⁵ ≈ 50,000 × 0.9178 ≈ $45,890 in today's dollars.
(iii) Even though the nominal balance grows by about $3,900, Mia's purchasing power shrinks by about $4,100 — leaving cash in a low-rate account during high inflation is effectively losing money in real terms.
Problem 5 — 37-year super horizon
Set up. Compound at each expected rate for 37 years; compare dollars.
(i) Growth: 40,000 × (1.07)³⁷ ≈ 40,000 × 12.224 ≈ $488,968. Balanced: 40,000 × (1.055)³⁷ ≈ 40,000 × 7.318 ≈ $292,720.
(ii) Difference = 488,968 − 292,720 = $196,248 more under Growth.
(iii) Recommend the Growth option: with 37 years until retirement, even a bad year that loses 15% has nearly four decades to recover, and the compounding gap from a higher average return is enormous (almost $200k extra).