Mathematics Standard • Year 12 • Module 7 • Lesson 4
Annuities — Past-Paper Style
Practise HSC-style short answers and one structured extended response on annuities, FV and PV, with full marking criteria.
1. Short-answer questions
1.1 $250 is deposited at the end of each month into a savings account that pays 4.8% p.a. compounded monthly. Calculate the future value of the account after 4 years. 3 marks Band 3
1.2 A retirement fund pays $500 at the end of every month for 5 years. The account earns 6% p.a. compounded monthly.
(a) Calculate the present value of these payments.
(b) State in one sentence what this present value represents in plain language. 3 marks Band 3-4
1.3 Two workers start saving for retirement.
Worker P deposits $200 at the end of each month for 30 years at 6% p.a. compounded monthly.
Worker Q deposits $400 at the end of each month for 15 years at 6% p.a. compounded monthly.
Both stop depositing at the end of their saving period.
(a) Calculate the FV of Worker P's account at the end of 30 years.
(b) Calculate the FV of Worker Q's account at the end of 15 years.
(c) Explain in one sentence which worker contributes more and which ends with more, with the dollar gap. 4 marks Band 4
2. Extended response
2.1 Mei has won a $250,000 prize and is offered two payout options.
Option X (Lump sum): $250,000 today.
Option Y (Annuity): $1,800 per month at the end of each month for 15 years.
Mei can invest any lump sum at 5.4% p.a. compounded monthly, and the annuity's PV should also be calculated using this rate.
(a) State whether Option Y is an ordinary annuity or an annuity due, and justify in one short sentence.
(b) Calculate the present value of Option Y.
(c) Calculate the future value at the end of 15 years if Mei takes Option X and invests the $250,000 at 5.4% p.a. compounded monthly (single lump-sum compounding, A = P(1 + r/k)^(kn)).
(d) Calculate the future value at the end of 15 years if Mei takes Option Y and invests each $1,800 payment as it arrives at 5.4% p.a. compounded monthly (use the FV annuity formula).
(e) Recommend which option is better at the end of 15 years and state the dollar gap. 7 marks Band 5-6
Explicit marking criteria
Part (a) — 1 mark
• 1 mark — identifies Option Y as an ordinary annuity (end-of-period payments) with a one-line justification.
Part (b) — 1 mark
• 1 mark — correct PV using M = 1,800, r = 0.0045, n = 180.
Part (c) — 1 mark
• 1 mark — correct FV of $250,000 lump sum using A = P(1 + r/k)^(kn) with kn = 180.
Part (d) — 2 marks
• 1 mark — correct FV-annuity formula applied with M = 1,800, r = 0.0045, n = 180.
• 1 mark — correct numerical value.
Part (e) — 2 marks
• 1 mark — correct numerical comparison of the two 15-year FV figures.
• 1 mark — explicit recommendation sentence naming the better option and the dollar gap.
Your response:
Stuck on (e)? Subtract the two 15-year FV figures and write "Option ___ is better by $___ at the end of 15 years".How did this worksheet feel?
What I'll revisit before next class:
1.1 — FV of $250/month at 4.8% monthly for 4 years (3 marks)
Sample response. r = 0.048/12 = 0.004, n = 4 × 12 = 48. FV = 250 × [(1.004)^48 − 1] / 0.004 = 250 × [1.21039 − 1] / 0.004 = 250 × 52.60 = $13,149.66.
Marking notes. 1 mark — correct r and n. 1 mark — correct substitution into the FV formula. 1 mark — correct numerical answer (to the nearest cent).
1.2 — PV of $500/month for 5 years at 6% monthly (3 marks)
(a) Sample response. r = 0.005, n = 60. PV = 500 × [1 − (1.005)^(−60)] / 0.005 = 500 × [1 − 0.74137] / 0.005 = 500 × 51.726 = $25,862.78.
(b) Sample response. The present value $25,862.78 represents the lump sum you would need to deposit today, at 6% p.a. compounded monthly, in order to fund exactly $500 per month for 5 years.
Marking notes. (a) 1 mark — correct r and n. 1 mark — correct PV. (b) 1 mark — explanation must connect PV to the lump-sum equivalent / cost-of-the-stream concept.
1.3 — Two long-term savers (4 marks)
(a) Sample response. Worker P: r = 0.005, n = 360. FV = 200 × [(1.005)^360 − 1] / 0.005 = 200 × [6.0226 − 1] / 0.005 = 200 × 1004.5 = $200,902.62 (to nearest cent).
(b) Sample response. Worker Q: n = 180. FV = 400 × [(1.005)^180 − 1] / 0.005 = 400 × [2.4541 − 1] / 0.005 = 400 × 290.82 = $116,326.88 (to nearest cent).
(c) Sample response. Worker P contributes $200 × 360 = $72,000, and Worker Q contributes $400 × 180 = $72,000 — both contribute the same total. But Worker P ends with $200,902.62 vs Worker Q's $116,326.88, a gap of $84,575.74 in favour of Worker P, because P's contributions had longer to compound.
Marking notes. (a) 1 mark. (b) 1 mark. (c) 1 mark for noting equal contributions; 1 mark for naming the winning worker and the dollar gap.
2.1 — Lump sum vs annuity over 15 years (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Type of annuity.
Ordinary annuity — payments are at the end of each month. [1 mark — correctly identifies ordinary annuity with end-of-period reason.]
(b) PV of Option Y.
r = 0.054/12 = 0.0045, n = 180. PV = 1,800 × [1 − (1.0045)^(−180)] / 0.0045 = 1,800 × [1 − 0.44613] / 0.0045 = 1,800 × 123.08 = $221,549.18. [1 mark — correct PV.]
(c) Option X — FV of $250,000 lump sum at 5.4% monthly for 15 years.
A = 250,000 × (1.0045)^180 = 250,000 × 2.24158 = $560,394.06. [1 mark — correct compounded value.]
(d) Option Y — FV of monthly $1,800 deposits.
FV = 1,800 × [(1.0045)^180 − 1] / 0.0045 = 1,800 × [2.24158 − 1] / 0.0045 = 1,800 × 275.91 = $496,640.66. [1 mark — correct formula applied; 1 mark — correct value.]
(e) Comparison and recommendation.
Option X (lump sum invested): $560,394.06. Option Y (annuity invested as received): $496,640.66. Difference = $560,394.06 − $496,640.66 = $63,753.40. [1 mark — correct comparison.]
Recommendation: Mei should take Option X (the $250,000 lump sum) and invest it — she will be $63,753.40 better off at the end of 15 years compared to taking the monthly annuity, because the entire $250,000 is compounding from day one. [1 mark — explicit recommendation naming Option X and the dollar gap.]
Total: 7/7.
Band descriptors for marker.
Band 3: Identifies annuity type and calculates PV of Y correctly, but Options X and Y future values not attempted on the same 15-year basis. ≈ 2-3 marks.
Band 4: Both FV figures computed but with one wrong substitution (often using simple interest for X or omitting the annuity formula for Y); no recommendation. ≈ 4-5 marks.
Band 5: Full numerical solution with correct comparison, but recommendation is bare numbers without naming Option X and the dollar gap. ≈ 6 marks.
Band 6: Complete: annuity type, correct PV, both 15-year FV figures, comparison and recommendation sentence naming Option X and the $63,753 gap. 7/7.