Mathematics Advanced • Year 12 • Module 7 • Lesson 7
Introducing Annuities — Future Value
Apply the FV annuity formula to superannuation, savings, and "early-start vs catch-up" decision problems.
Problem 1 — Australian Super projection
Sienna earns $80,000 per year. Her employer contributes 11.5% of her salary into super each year. The fund averages 7% p.a. compounded annually.
Set up: What are we solving for?
(i) Compute the annual employer contribution a. 1 mark
(ii) Find the FV of Sienna's super after 40 years. 2 marks
(iii) Split the FV into total contributions and total interest. State both figures and the ratio interest : contributions. 2 marks
Stuck? Revisit lesson § Australian Superannuation real-world anchor.Problem 2 — Start early vs catch up
The government proposes a "super catch-up": instead of $600/month for 35 years, you can contribute $1,200/month for the final 15 years. Both options assume 7% p.a. compounded monthly.
Set up: What are we solving for?
(i) Find the FV of the early start ($600/month for 35 years). 2 marks
(ii) Find the FV of the catch-up plan ($1,200/month for 15 years). 2 marks
(iii) State which strategy wins, by how much, and explain in one sentence why time in the formula dominates contribution size. 2 marks
Problem 3 — Holiday saving target
Asha wants $20,000 in 4 years for a gap year. Her bank offers 4.8% p.a. compounded monthly on regular deposits made at the end of each month.
Set up: What are we solving for?
(i) Transpose the FV formula and find the required monthly contribution a (to the nearest dollar). 3 marks
(ii) Suppose Asha can only afford $350/month. Use trial-and-improvement (or a direct calculation) to find how many years she would need, to the nearest whole year. 2 marks
(iii) The rate suddenly drops to 3.6% p.a. (still monthly) for the original 4-year plan. How much does her required monthly contribution change? 2 marks
Problem 4 — "Doubling a doubles FV" — is it really true?
Use the FV formula to investigate the misconception tested by Activity 2 in the lesson.
Set up: What are we solving for?
(i) Find FV for a = $500, r = 5%, n = 10. 1 mark
(ii) Find FV for a = $1,000, r = 5%, n = 10. 1 mark
(iii) Show algebraically (using the FV formula) that doubling a doubles FV exactly, regardless of r and n. 2 marks
(iv) Now investigate doubling n: find FV for a = $500, r = 5%, n = 20, and explain in one line why doubling n does not double FV. 2 marks
Stuck on (iii)? a is a linear factor; r and n live inside the bracket.Problem 5 — Jax's fortnightly super
Jax contributes $450 fortnightly into a super fund returning 6.4% p.a. compounded fortnightly. He works for 25 years.
Set up: What are we solving for?
(i) Find the FV at retirement (use 26 fortnights per year). 2 marks
(ii) Decompose into contributions and interest. 1 mark
(iii) Adjust the FV for inflation averaging 2.5% p.a. over 25 years. State the real (today's-dollar) value of Jax's retirement balance. 2 marks
(iv) In one sentence, state the lesson Jax should take from comparing his interest earned against his real value. 1 mark
Stuck on (iii)? Divide your nominal FV by (1.025)²⁵.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Australian Super projection
Set up. We are valuing 40 equal annual contributions plus compound interest.
(i) a = 0.115 × 80,000 = $9,200 per year.
(ii) FV = 9,200 × [(1.07)⁴⁰ − 1]/0.07 = 9,200 × 199.635 = $1,836,640 (rounded; ≈ $1.84 million as in the lesson).
(iii) Contributions = 40 × 9,200 = $368,000. Interest = 1,836,640 − 368,000 = $1,468,640. Ratio ≈ 4.0 : 1 (interest is roughly four times contributions over a 40-year horizon).
Problem 2 — Start early vs catch up
Set up. We are comparing two annuities with the same r but very different (a, n) pairs.
(i) r = 0.07/12 = 0.005833; n = 35 × 12 = 420. FV = 600 × [(1.005833)⁴²⁰ − 1]/0.005833 ≈ 600 × 1,801.05 ≈ $1,080,632.
(ii) r = 0.005833; n = 180. FV = 1,200 × [(1.005833)¹⁸⁰ − 1]/0.005833 ≈ 1,200 × 316.96 ≈ $380,353.
(iii) Early-start wins by about $700,000. Because n appears inside an exponent, extra time compounds the growth factor; doubling the contribution only doubles a linear factor, which cannot beat 20 extra years of compounding.
Problem 3 — Holiday saving target
Set up. We are transposing the FV formula to solve for a, then re-using it with a different rate / horizon.
(i) r = 0.004; n = 48. 20,000 = a × [(1.004)⁴⁸ − 1]/0.004 = a × 52.871. a = 20,000 / 52.871 ≈ $378/month.
(ii) 20,000 = 350 × [(1.004)ⁿ − 1]/0.004 ⇒ [(1.004)ⁿ − 1]/0.004 = 57.143 ⇒ (1.004)ⁿ = 1.2286 ⇒ n = ln(1.2286)/ln(1.004) ≈ 51.4 months ≈ 4 years and 4 months (round to 5 years).
(iii) r = 0.003; n = 48. Factor = [(1.003)⁴⁸ − 1]/0.003 = 51.5365. a = 20,000 / 51.5365 ≈ $388/month — an extra ≈ $10/month.
Problem 4 — Scaling test
Set up. We are checking which input "linearly scales" the output FV.
(i) FV = 500 × [(1.05)¹⁰ − 1]/0.05 = 500 × 12.578 = $6,288.95.
(ii) FV = 1,000 × 12.578 = $12,577.89. Exactly double of (i) (to rounding).
(iii) FV(a) = a · [(1+r)ⁿ − 1]/r. FV(2a) = 2a · [(1+r)ⁿ − 1]/r = 2 · FV(a). The bracketed factor depends only on r and n, so it is invariant under doubling a.
(iv) FV = 500 × [(1.05)²⁰ − 1]/0.05 = 500 × 33.066 = $16,533.00. That is more than double of (i)'s $6,289 because (1+r)ⁿ sits in an exponent — doubling n compounds the growth factor, not the contribution.
Problem 5 — Jax's fortnightly super
Set up. We are applying the FV formula at a fortnightly frequency, then splitting and adjusting for inflation.
(i) r = 0.064/26 ≈ 0.002462; n = 650. FV = 450 × [(1.002462)⁶⁵⁰ − 1]/0.002462 ≈ 450 × 1,595.0 = $717,750 (approx).
(ii) Contributions = 450 × 650 = $292,500. Interest ≈ $425,250.
(iii) (1.025)²⁵ = 1.85394. Real value ≈ 717,750 / 1.85394 ≈ $387,148 in today's dollars.
(iv) Even though the nominal balance looks large, inflation roughly halves it in real purchasing power — so retirement planning must account for the real, not the nominal, FV.