Mathematics Standard • Year 12 • Module 7 • Lesson 5
Future Value of Annuities — Problem Set
Apply FV-of-annuity reasoning to Australian super, house-deposit and goal-saving scenarios — including "start-early" vs "save-more" comparisons.
Problem 1 — Emma's house deposit
Emma is saving for her first home and needs $60,000 in 10 years. Her high-interest savings account pays 4.8% p.a. compounded monthly.
Set up: What are we solving for?
(i) Calculate Emma's required monthly contribution to hit the $60,000 target. 2 marks
(ii) If Emma can only afford $350/month, calculate the FV she will have after 10 years. 2 marks
(iii) State the shortfall against her $60,000 target at year 10. 1 mark
Stuck on (i)? Rearrange the FV formula to M = FV × r / [(1+r)^n − 1].Problem 2 — Start early vs save more (superannuation)
Two friends both retire at age 65 and earn 7% p.a. compounded monthly on their super.
Anya: contributes $500/month from age 25 to age 35 (10 years), then stops contributing. The balance keeps compounding to age 65.
Ben: contributes $500/month from age 35 to age 65 (30 years) continuously.
Set up: What are we solving for?
(i) Calculate the FV of Anya's account at age 35 (when she stops contributing). 2 marks
(ii) Calculate the value of Anya's account at age 65, after the balance from (i) compounds for 30 years (use A = P(1 + r/k)^(kn)). 2 marks
(iii) Calculate the FV of Ben's account at age 65. 1 mark
(iv) State who has more at age 65, and how much each has contributed in total. 2 marks
Stuck? Revisit lesson § Annuity Due / Finding Payment — early contributions compound for longer.Problem 3 — Target balance for a child's university fund
A parent wants to have $80,000 ready for their child's university costs in 18 years. They will make equal monthly contributions to a fund earning 6% p.a. compounded monthly.
Set up: What are we solving for?
(i) Calculate the required monthly contribution. 2 marks
(ii) Calculate the total contributed by the parent over the 18 years. 1 mark
(iii) Calculate the total interest the fund will earn. 2 marks
Stuck? Revisit lesson § Finding the Payment — M = FV × r / [(1+r)^n − 1].Problem 4 — Ordinary vs annuity-due deposits
Liam deposits $300/month into a savings fund paying 5.4% p.a. compounded monthly for 12 years.
Set up: What are we solving for?
(i) Calculate the FV if deposits are made at the end of each month (ordinary annuity). 2 marks
(ii) Calculate the FV if deposits are made at the start of each month (annuity due) using FV_due = FV_ord × (1 + r). 2 marks
(iii) Calculate the dollar gain from switching to start-of-month deposits. 1 mark
Stuck? Revisit lesson § Annuity Due — each payment earns one extra period of interest.Problem 5 — Pay more vs save longer
Two scenarios for a person targeting a $250,000 super balance at age 60. Both earn 6% p.a. compounded monthly.
Scenario S1: Save $500/month for 25 years.
Scenario S2: Save $800/month for 15 years.
Set up: What are we solving for?
(i) Calculate the FV under S1. 2 marks
(ii) Calculate the FV under S2. 2 marks
(iii) State which scenario reaches the $250,000 target and by how much, and which falls short and by how much. 2 marks
Stuck? Revisit lesson § Activity 3 — Compare $200/month vs $400/month over different periods.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Emma's house deposit
Set up. Use the FV-payment rearrangement, then the direct FV formula, then subtract.
(i) r = 0.004, n = 120. M = 60,000 × 0.004 / [(1.004)^120 − 1] = 240 / [1.61499 − 1] = 240 / 0.61499 = $390.25/month.
(ii) At $350: FV = 350 × [(1.004)^120 − 1] / 0.004 = 350 × 153.748 = $53,811.91.
(iii) Shortfall = $60,000 − $53,811.91 = $6,188.09.
Problem 2 — Anya vs Ben (start early vs save more)
Set up. Compute Anya's annuity FV at 35, then compound to 65 as a lump sum. Compute Ben's annuity FV directly at 65.
(i) Anya at 35: r = 0.07/12 = 0.005833, n = 120. FV = 500 × [(1.005833)^120 − 1] / 0.005833 = 500 × [2.00966 − 1] / 0.005833 = 500 × 173.084 = $86,541.96.
(ii) Anya at 65: A = 86,541.96 × (1.005833)^360 = 86,541.96 × 8.11649 = $702,510 (to the nearest dollar).
(iii) Ben at 65: r = 0.005833, n = 360. FV = 500 × [(1.005833)^360 − 1] / 0.005833 = 500 × [8.11649 − 1] / 0.005833 = 500 × 1220.0 = $610,001 (to the nearest dollar).
(iv) Anya has more ($702,510 vs $610,001). Contributions: Anya $500 × 120 = $60,000; Ben $500 × 360 = $180,000. Anya contributed 1/3 as much yet ended with more, because her early money compounded for 30 extra years.
Problem 3 — University fund
Set up. Find the monthly M for $80,000 in 18 years at 6% monthly; then totals.
(i) r = 0.005, n = 216. M = 80,000 × 0.005 / [(1.005)^216 − 1] = 400 / [2.93283 − 1] = 400 / 1.93283 = $206.95/month.
(ii) Total contributed = $206.95 × 216 = $44,701.20.
(iii) Interest = $80,000 − $44,701.20 = $35,298.80.
Problem 4 — Ordinary vs annuity due for Liam
Set up. Compute FV_ord then convert to FV_due.
(i) r = 0.0045, n = 144. FV_ord = 300 × [(1.0045)^144 − 1] / 0.0045 = 300 × [1.90993 − 1] / 0.0045 = 300 × 202.21 = $60,664.55.
(ii) FV_due = $60,664.55 × 1.0045 = $60,937.54.
(iii) Gain = $60,937.54 − $60,664.55 = $272.99 by switching to start-of-month deposits.
Problem 5 — Pay more vs save longer
Set up. Apply FV-annuity to each scenario and compare to the $250,000 target.
(i) S1: r = 0.005, n = 300. FV = 500 × [(1.005)^300 − 1] / 0.005 = 500 × [4.4650 − 1] / 0.005 = 500 × 692.99 = $346,496.98.
(ii) S2: r = 0.005, n = 180. FV = 800 × [(1.005)^180 − 1] / 0.005 = 800 × [2.4541 − 1] / 0.005 = 800 × 290.82 = $232,653.75.
(iii) S1 reaches the target with $96,497 to spare; S2 falls short by $17,346. The longer saving period beats the higher monthly contribution at this rate.