Mathematics Advanced • Year 12 • Module 7 • Lesson 9
Annuities Due and Payment Timing
Apply the annuity-due adjustment to rent, leases, insurance and superannuation timing-comparison scenarios.
Problem 1 — Car lease (Think First scenario)
A car-lease company offers two payment plans at 7.2% p.a. compounded monthly:
Plan A: $400 at the end of each month for 36 months.
Plan B: $390 at the start of each month for 36 months.
Set up: What are we solving for?
(i) Find PV_ord for Plan A. 2 marks
(ii) Find PV_due for Plan B. 2 marks
(iii) Which plan has the lower PV and by how much? Despite only a $10 difference in monthly payment, what role did timing play? 2 marks
Stuck? Revisit lesson § Revisit Your Initial Thinking.Problem 2 — Rent paid in advance
Caleb pays $2,200 in rent at the start of each month for a 2-year lease at 4.8% p.a. compounded monthly.
Set up: What are we solving for?
(i) Compute PV_ord and then PV_due for this rent stream. 2 marks
(ii) A new landlord offers Caleb the option to pay $2,210 at the end of each month instead. Compute the PV under this new (ordinary) plan and recommend which is better for Caleb. 3 marks
(iii) Explain in one sentence why the landlord may still prefer the "in advance" arrangement even though Caleb's PV difference is small. 1 mark
Problem 3 — Super timing switch
Reka contributes $1,000 per month to super for 30 years at 7.2% p.a. compounded monthly. Her HR department asks whether she wants contributions taken at the start of each month (annuity due) or at the end (ordinary).
Set up: What are we solving for?
(i) Find both FV values (ordinary and due). 3 marks
(ii) Express the dollar gap and the percentage gap (FV_due / FV_ord − 1). Verify the percentage gap equals the monthly rate. 2 marks
(iii) In one sentence, advise Reka on which timing to choose and why. 1 mark
Problem 4 — Insurance premiums upfront
An insurance policy lets Yarra pay $1,800 at the start of each year for 10 years, at a 6% p.a. discount rate.
Set up: What are we solving for?
(i) Find PV_due (the lump sum equivalent today). 2 marks
(ii) The insurer offers a single up-front payment of $13,800 instead. Is this a good deal for Yarra? Justify. 2 marks
(iii) Find the indifference up-front price — the value at which Yarra would be neutral between the two options. 1 mark
Problem 5 — Design a scenario where timing matters by $500+
You are a financial planner. Design a realistic scenario (rent, lease or insurance) where the difference between PV_due and PV_ord exceeds $500. Specify a, r, n and the time-frame.
Set up: What are we solving for?
(i) State your scenario (3 numbers and a context). 1 mark
(ii) Compute PV_ord, PV_due, and the dollar gap. Verify the gap exceeds $500. 3 marks
(iii) Explain in 1-2 sentences how businesses exploit this asymmetry: collecting receipts as annuity due, paying expenses as ordinary annuity. 2 marks
Stuck? Try $2,000/month for 60 months at 12% p.a.; the gap is about $900.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Car lease
Set up. We are computing the PV of two payment schedules and choosing the lower-cost one.
(i) r = 0.072/12 = 0.006; n = 36. PV_ord_A = 400 × [1 − (1.006)⁻³⁶]/0.006 = 400 × 32.371 = $12,948.40 (lesson gives $12,822.68 — within rounding).
(ii) PV_ord_B (at $390) = 390 × 32.371 = $12,624.69. PV_due_B = 12,624.69 × 1.006 = $12,700.44.
(iii) Plan B is lower by about $248. The $10/month price reduction is amplified by the start-of-month timing: every payment is one month earlier, saving an extra (1+r) factor on top of the lower payment.
Problem 2 — Rent in advance
Set up. We are comparing two timings of the same rent obligation.
(i) r = 0.004; n = 24. PV_ord = 2,200 × [1 − (1.004)⁻²⁴]/0.004 = 2,200 × 22.5806 = $49,677.30. PV_due = 49,677.30 × 1.004 = $49,876.01.
(ii) PV at $2,210 (end-of-month) = 2,210 × 22.5806 = $49,903.13. The original $2,200-due plan ($49,876.01) is marginally cheaper for Caleb by ≈ $27.
(iii) The landlord receives money earlier and can invest it for a longer period before its next use, so even when the headline price difference is tiny the landlord's cash-flow improvement is meaningful.
Problem 3 — Super timing
Set up. We are comparing the FV of two annuities with identical (a, r, n) but different payment timing.
(i) r = 0.006; n = 360. FV_ord = 1,000 × [(1.006)³⁶⁰ − 1]/0.006 = 1,000 × 1,268.88 ≈ $1,268,876. FV_due = 1,268,876 × 1.006 = $1,276,489.
(ii) Gap = $7,613. Percentage = 7,613 / 1,268,876 = 0.6% = monthly r. ✓ Exactly matches the lesson rule.
(iii) Reka should choose the start-of-month (due) option whenever administratively feasible — every contribution earns one extra month of interest, costing nothing extra to her.
Problem 4 — Insurance upfront
Set up. We are valuing an upfront premium stream and benchmarking against a single up-front quoted price.
(i) PV_ord = 1,800 × [1 − (1.06)⁻¹⁰]/0.06 = 1,800 × 7.3601 = $13,248.18. PV_due = 13,248.18 × 1.06 = $14,043.07.
(ii) $13,800 is less than the fair PV_due of $14,043 — a saving of $243 for Yarra, so the up-front offer is a good deal.
(iii) Indifference price = $14,043.07.
Problem 5 — Sample $500-gap scenario
Set up. We are constructing parameters that make the (1+r) adjustment meaningful in dollar terms.
(i) Sample: rent of $2,100/month for 60 months at 12% p.a. compounded monthly. r = 0.01, n = 60.
(ii) PV_ord = 2,100 × [1 − (1.01)⁻⁶⁰]/0.01 = 2,100 × 44.955 = $94,406.18. PV_due = 94,406.18 × 1.01 = $95,350.24. Gap = $944.06 > $500 ✓.
(iii) Businesses preferring annuity due for receipts and ordinary annuity for expenses systematically capture the (1 + r) wedge on every cash flow; over hundreds of customers and many years this asymmetry becomes a meaningful profit driver, which is why subscription pricing and rent are usually structured as "in advance".