Mathematics Standard • Year 12 • Module 7 • Lesson 10
Investment Strategies — Skill Drill
Build fluency comparing investment options: real return = nominal − inflation, future value at a given rate, mean annual return, and risk-vs-return ranking.
1. Quick recall
Answer each question in the space provided. 1 mark each
Q1.1 Complete the real-return formula.
Real return = Nominal return − ____________
Q1.2 Rank these investments from lowest typical risk to highest: shares, savings account, term deposit, property, cryptocurrency.
1. ____________ 2. ____________ 3. ____________ 4. ____________ 5. ____________
Q1.3 Fill in the time-horizon rule of thumb.
Short term (0-3 years): _________________________________________________
Long term (10+ years): _________________________________________________
2. Worked example — real return and future value
Follow each line of working. Every step has a reason on the right.
Problem. $20,000 is invested at 4.5% p.a. compounded annually for 8 years while inflation runs at 2.5% p.a. Find the real return rate, the nominal future value and the purchasing power in today's dollars.
Step 1 — Real return = Nominal − Inflation.
Real = 4.5% − 2.5% = 2.0%
Reason: inflation erodes purchasing power, so the "real" gain is what your money buys above the rising cost of goods.
Step 2 — Nominal future value FV = P(1 + r)ⁿ.
FV = 20,000 × (1.045)⁸ ≈ 20,000 × 1.4221 ≈ $28,442
Reason: compound interest on a single lump sum at the nominal rate over 8 years.
Step 3 — Purchasing power = P × (1 + real)ⁿ.
PP = 20,000 × (1.02)⁸ ≈ 20,000 × 1.1717 ≈ $23,434
Reason: re-express the nominal $28,442 in today's prices using the real return.
Conclusion. Real return 2.0% p.a.; nominal FV $28,442; today's-dollar value $23,434.
3. Faded example — average annual return of a share portfolio
A share portfolio returned 12%, −5%, 8%, 15% and −2% over 5 years. Find the arithmetic mean annual return and then use it to project the future value of $10,000 over 5 years at the mean rate. Fill in each blank. 4 marks
Step 1 — Sum of the annual returns:
Sum = 12 + (−5) + 8 + 15 + (−2) = ____________ %
Step 2 — Mean annual return:
Mean = ____________ / 5 = ____________ % = ____________ (decimal)
Step 3 — Projected FV at the mean rate, compounded annually:
FV = 10,000 × (1 + ____________)⁵ ≈ $ ____________
Comment. The mean return hides the year-to-year volatility — actual sequence-of-returns may give a different result.
4. Graduated practice — investment calculations
Show your working below each part. Round dollars to 2 dp.
Foundation — single-step substitution (4 questions)
| Q | Problem | Answer |
|---|---|---|
| 4.1 1 | Calculate the real return for a 5% p.a. term deposit when inflation is 3%. | |
| 4.2 1 | Calculate the real return for 8% p.a. shares when inflation is 2.5%. | |
| 4.3 1 | $15,000 at 4% p.a. compounded annually for 5 years. Find the nominal FV. | |
| 4.4 1 | A portfolio returns 6%, 8%, 4%, 10%. Find the arithmetic mean annual return. |
Standard — typical HSC difficulty (6 questions)
Show formulas before substituting and label final answers with units.
4.5 $30,000 is invested for 10 years at 6% p.a. compounded annually with inflation at 2% p.a. Find (a) nominal FV, (b) real return rate, (c) purchasing power in today's dollars. 3 marks
4.6 Repeat 4.5 with a 4% p.a. rate (everything else the same). State which of the two investments gives the higher real return. 2 marks
4.7 A 100%-shares portfolio averages 8% return but loses 30% in a bad year. A diversified portfolio (50% shares, 30% bonds, 20% property) averages 6.5% but only loses 12% in a bad year. State (a) the difference in average return and (b) the difference in worst-year loss. Which is more suited to a 1-year horizon? 2 marks
4.8 Calculate the real FV (in today's dollars) of $25,000 invested for 6 years at 5% p.a. with inflation 2.8% p.a. 2 marks
4.9 A retiree splits $200,000 as 70% term deposit at 4.5%, 30% shares at expected 8%. Find (a) expected dollar return after 1 year, (b) expected combined rate (weighted average). 2 marks
4.10 Inflation is forecast at 3% p.a. A savings account pays 2.5% p.a. State whether real purchasing power grows, shrinks or stays the same, and explain in one sentence. 2 marks
Extension — risk-adjusted comparison (2 questions)
4.11 $20,000 for 8 years. Option A: 4.5% term deposit. Option B: expected 7.5% shares but with possible 20% loss in year 1. With inflation 2.5%, find (a) Option A real FV in today's dollars, (b) Option B nominal FV in the best case, (c) Option B nominal FV if year 1 loses 20% then averages 7.5% for the remaining 7 years. 3 marks
4.12 Paying down a mortgage at 5.4% gives a "guaranteed return" of 5.4%. Investing the same money in shares has expected 7%. Inflation is 2.5% and the marginal tax rate on share dividends is 30%. (a) After-tax share return, (b) compare to the guaranteed mortgage return, (c) state in one sentence which has the better risk-adjusted return for a risk-averse investor. 3 marks
5. Self-check the easy 3
Tick the first three once you've checked your method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1 — Real-return formula
Real return = Nominal return − Inflation rate.
Q1.2 — Risk ranking (low to high)
1. Savings account 2. Term deposit 3. Property 4. Shares 5. Cryptocurrency.
Q1.3 — Time horizon rule of thumb
Short term (0-3 years): safety first — savings or term deposits. Long term (10+ years): growth focused — mostly shares and property.
Q3 — Faded example (mean annual return)
Sum = 12 − 5 + 8 + 15 − 2 = 28%. Mean = 28 / 5 = 5.6% = 0.056. FV = 10,000 × (1.056)⁵ ≈ 10,000 × 1.3134 ≈ $13,134.
Q4.1 — Real return
5% − 3% = 2% p.a.
Q4.2 — Real return
8% − 2.5% = 5.5% p.a.
Q4.3 — Nominal FV
FV = 15,000 × (1.04)⁵ ≈ 15,000 × 1.2167 ≈ $18,250.
Q4.4 — Mean annual return
Mean = (6 + 8 + 4 + 10) / 4 = 28 / 4 = 7% p.a.
Q4.5 — $30,000 at 6% for 10 yr, inflation 2%
(a) FV = 30,000 × (1.06)¹⁰ ≈ 30,000 × 1.7908 ≈ $53,725. (b) Real = 6 − 2 = 4% p.a. (c) PP = 30,000 × (1.04)¹⁰ ≈ 30,000 × 1.4802 ≈ $44,407 in today's dollars.
Q4.6 — Same scenario at 4%
FV = 30,000 × (1.04)¹⁰ ≈ $44,407. Real = 4 − 2 = 2% p.a. PP = 30,000 × (1.02)¹⁰ ≈ $36,570. The 6% investment has the higher real return (4% vs 2%) and the higher real FV.
Q4.7 — 100% shares vs diversified
(a) Difference in average = 8 − 6.5 = 1.5 pp higher for shares. (b) Worst-year difference = 30 − 12 = 18 pp larger loss for shares. The diversified portfolio is more suited to a 1-year horizon because the smaller worst-case loss matters far more than the small average-return gap over a short period.
Q4.8 — Real FV
Real = 5 − 2.8 = 2.2%. PP = 25,000 × (1.022)⁶ ≈ 25,000 × 1.1392 ≈ $28,480 in today's dollars.
Q4.9 — Mixed portfolio
(a) TD: 140,000 × 0.045 = $6,300. Shares: 60,000 × 0.08 = $4,800. Total = $11,100. (b) Combined rate = 11,100 / 200,000 = 5.55% p.a. (= 0.7 × 4.5 + 0.3 × 8 — the weighted average).
Q4.10 — Below inflation
Purchasing power shrinks. The savings account earns 2.5% but prices are rising at 3%, so each dollar buys less each year — the real return is −0.5% p.a.
Q4.11 — Option A vs Option B
(a) Option A real PP = 20,000 × (1.02)⁸ ≈ $23,434 in today's dollars (real = 4.5 − 2.5 = 2%). (b) Option B best case: 20,000 × (1.075)⁸ ≈ 20,000 × 1.7835 ≈ $35,670. (c) Option B with year-1 loss: 20,000 × 0.80 × (1.075)⁷ ≈ 16,000 × 1.6590 ≈ $26,544. Even with a 20% loss in year 1, Option B nominally beats Option A's real PP, but with substantial volatility.
Q4.12 — Mortgage vs shares
(a) After-tax share return = 7% × (1 − 0.30) = 4.9% p.a. (b) Mortgage gives 5.4% guaranteed, after tax-free — better than 4.9% after-tax expected. (c) For a risk-averse investor the mortgage repayment wins: a guaranteed 5.4% beats an uncertain 4.9% expected.