Mathematics Advanced • Year 12 • Module 7 • Lesson 9
Annuities Due and Payment Timing
Practise HSC-style writing on annuity-due adjustments — including a derivation and a multi-step financial comparison.
1. Short-answer questions
1.1 Aria pays $500 at the start of each year into a savings account paying 6% p.a. Find the future value after 12 years. 2 marks Band 3
1.2 An ordinary annuity of $750/month for 8 years at 5.4% p.a. compounded monthly has a present value of $58,749. Without re-evaluating the bracketed factor, find the present value if the payments are instead made at the start of each month. 2 marks Band 3-4
1.3 A car-lease customer is choosing between paying $480/month at the end of each month or $470/month at the start of each month, for 4 years at 6.6% p.a. compounded monthly.
(a) Compute the PV of each option.
(b) Which option is cheaper for the customer, and by how many dollars in today's terms? 4 marks Band 4
2. Extended response
2.1 Let an ordinary annuity have payments $a at the end of each period, rate r per period, and n periods.
(a) By writing FV_ord = a + a(1+r) + … + a(1+r)ⁿ⁻¹ and FV_due = a(1+r) + a(1+r)² + … + a(1+r)ⁿ, prove that FV_due = FV_ord × (1 + r).
(b) Hence prove that the same multiplicative rule PV_due = PV_ord × (1 + r) holds, by using the identity PV = FV × (1+r)⁻ⁿ.
(c) A super fund customer makes $1,200 contributions at the end of each month for 30 years at 6% p.a. compounded monthly. Show that switching to start-of-month contributions increases the final balance by exactly 0.5%, and convert this to a dollar amount. 8 marks Band 5-6
Explicit marking criteria
Part (a) — 3 marks
• 1 mark — writes both FV_ord and FV_due sums explicitly.
• 1 mark — factors (1 + r) out of FV_due to identify FV_ord.
• 1 mark — concludes FV_due = (1 + r) × FV_ord.
Part (b) — 2 marks
• 1 mark — applies PV = FV × (1+r)⁻ⁿ to both annuities.
• 1 mark — concludes PV_due = (1 + r) × PV_ord.
Part (c) — 3 marks
• 1 mark — correct FV_ord with r = 0.005, n = 360.
• 1 mark — FV_due = FV_ord × 1.005, demonstrating the 0.5% gap.
• 1 mark — explicit dollar gap (≈ FV_ord × 0.005).
Your response:
Stuck on (c)? FV_ord = 1,200 × [(1.005)³⁶⁰ − 1]/0.005 ≈ $1,205,418; dollar gap ≈ 0.005 × 1,205,418 ≈ $6,027.How did this worksheet feel?
What I'll revisit before next class:
1.1 — FV due, $500/yr, 12 yrs, 6% (2 marks)
Sample response. FV_ord = 500 × [(1.06)¹² − 1]/0.06 = 500 × 16.870 = $8,434.97. FV_due = 8,434.97 × 1.06 = $8,941.07.
Marking notes. 1 mark — correct FV_ord; 1 mark — (1 + r) adjustment and correct final answer.
1.2 — PV due from given PV ord (2 marks)
Sample response. r = 0.054/12 = 0.0045. PV_due = 58,749 × 1.0045 = $59,013.37.
Marking notes. 1 mark — correct r per period; 1 mark — correct due-PV via the (1 + r) multiplier without recomputing the factor.
1.3 — Car-lease comparison (4 marks)
(a) Sample response. r = 0.066/12 = 0.0055; n = 48. Factor = [1 − (1.0055)⁻⁴⁸]/0.0055 = 42.0792.
End-of-month: PV = 480 × 42.0792 = $20,198.02.
Start-of-month: PV = 470 × 42.0792 × 1.0055 = 470 × 42.3107 = $19,886.04.
(b) Sample response. The start-of-month plan is cheaper for the customer by 20,198.02 − 19,886.04 = $311.98 in today's dollars.
Marking notes. (a) 1 mark — correct factor; 1 mark — both PVs. (b) 1 mark — correct dollar gap; 1 mark — chooses start-of-month plan.
2.1 — Extended response (8 marks): sample Band-6 response with annotations
Sample Band-6 response.
Part (a). Ordinary annuity (payments at the end of each period):
FV_ord = a + a(1 + r) + a(1 + r)² + … + a(1 + r)ⁿ⁻¹.
Annuity due (payments at the start of each period — every contribution earns one extra period of interest):
FV_due = a(1 + r) + a(1 + r)² + a(1 + r)³ + … + a(1 + r)ⁿ. [1 mark — both sums written.]
Factor (1 + r) out of every term in FV_due:
FV_due = (1 + r) × [a + a(1 + r) + … + a(1 + r)ⁿ⁻¹] = (1 + r) × FV_ord. [1 mark — factoring; 1 mark — conclusion.]
Part (b). Apply PV = FV × (1 + r)⁻ⁿ to each:
PV_ord = FV_ord × (1 + r)⁻ⁿ ; PV_due = FV_due × (1 + r)⁻ⁿ = (1 + r) × FV_ord × (1 + r)⁻ⁿ = (1 + r) × PV_ord. [1 mark — substitution; 1 mark — final result.]
Part (c). r = 0.06 / 12 = 0.005; n = 30 × 12 = 360. [1 mark — conversion.]
FV_ord = 1,200 × [(1.005)³⁶⁰ − 1] / 0.005 = 1,200 × 1,004.515 ≈ $1,205,417.65.
FV_due = FV_ord × 1.005 = 1,205,417.65 × 1.005 ≈ $1,211,444.74. [1 mark — both FVs.]
The percentage increase is FV_due / FV_ord − 1 = 1.005 − 1 = 0.5%, exactly the monthly rate. In dollar terms the gap is 1,211,444.74 − 1,205,417.65 = ≈ $6,027. [1 mark — interprets 0.5% gap and gives dollar amount.]
Total: 8/8.
Band descriptors for marker.
Band 3: Writes both sums but does not factor correctly; (c) attempted without applying the conversion. ≈ 2-3 marks.
Band 4: Part (a) complete; part (b) attempted but argument not closed; (c) numerically correct for FV_ord only. ≈ 4-5 marks.
Band 5: Both proofs complete; (c) FV_ord and FV_due correct but does not explicitly identify the 0.5% gap. ≈ 6-7 marks.
Band 6: All three parts complete; explicit factoring step; (c) links the percentage gap to the monthly rate algebraically. 8/8.