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.