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".