Mathematics Advanced • Year 12 • Module 7 • Lesson 9

Annuities Due and Payment Timing

Build fluency in converting ordinary annuity calculations to annuity-due calculations using the (1 + r) adjustment.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Complete the rule that converts an ordinary annuity to an annuity due:

Annuity Due = Ordinary × ______________

Q1.2 An ordinary annuity has payments at the ____________ of each period. An annuity due has payments at the ____________ of each period.

Q1.3 The (1 + r) adjustment applies to (circle one): PV only / FV only / both PV and FV.

Stuck? Revisit lesson § Ordinary vs Annuity Due.

2. Worked example — gym membership $500/month for 2 years

Follow each line. Reasons appear in italics on the right.

Problem. A gym charges $500 at the start of each month for 2 years at 6% p.a. compounded monthly. Find the PV (the lump-sum value of the contract today).

Step 1 — Convert the rate and time to monthly.

r = 0.06 / 12 = 0.005 ; n = 24 months.

Reason: r and n must be in the same unit as the payment.

Step 2 — Compute the ordinary PV first.

PV_ord = 500 × [1 − (1.005)⁻²⁴] / 0.005 = 500 × 22.5806 = $11,290.30.

Reason: this assumes end-of-month payments.

Step 3 — Multiply by (1 + r) to shift payments to start-of-period.

PV_due = 11,290.30 × 1.005 = $11,346.75.

Reason: every payment is one period earlier, so each is discounted by one less period.

Conclusion. The gym effectively receives $56.45 more in PV terms than if you paid at the end of each month — and this is built into the gym's pricing.

3. Faded example — fill in the missing steps

An insurance policy requires premiums of $250 paid at the start of each quarter for 5 years at 8% p.a. compounded quarterly. Find the PV. 5 marks

Step 1 — Match rate and time to quarters.

r = 0.08 / ____ = ____________ ; n = 5 × ____ = ____________

Step 2 — Compute ordinary PV.

PV_ord = 250 × [1 − (1 + ____)^(− ____)] / ____ = 250 × ____________ = $____________

Step 3 — Apply the (1 + r) due adjustment.

PV_due = ____________ × (1 + ____) = $____________

Conclusion. The present value of the premium stream is $____________.

Stuck? Revisit lesson § Worked Example — Insurance policy.

4. Graduated practice

Show every line of working. Round to the nearest cent.

Foundation — apply the (1 + r) factor (4 questions)

Q	Ordinary value given	Due value (compute)
4.1 1	FV_ord = $10,000; r = 6%	FV_due =
4.2 1	PV_ord = $15,000; r = 4%	PV_due =
4.3 1	FV_ord = $50,000; r = 0.5% per month	FV_due =
4.4 1	PV_ord = $25,000; r = 1.2% per quarter	PV_due =

Standard — full annuity-due calculations (6 questions)

Compute the ordinary value, then multiply by (1 + r).

4.5 FV of $500 paid at the start of each year for 15 years at 6% p.a. 2 marks

4.6 FV of $1,000 paid at the start of each year for 30 years at 3% p.a. 2 marks

4.7 PV of $800 paid at the start of each year for 20 years at 5% p.a. 2 marks

4.8 Rent: $1,500/month at the start of each month for 1 year at 4.8% p.a. compounded monthly. Find PV. 2 marks

4.9 Insurance: $250 paid at the start of each quarter for 5 years at 8% p.a. compounded quarterly. Find PV. 2 marks

4.10 Tutoring: $150 at the start of each month for 12 months at 4.8% p.a. compounded monthly. Find PV. 2 marks

Extension — derive and reason (2 questions)

4.11 Show algebraically that FV_due = a × [(1 + r)ⁿ − 1] / r × (1 + r). Hence prove that the percentage gap FV_due/FV_ord − 1 equals exactly r, regardless of a and n. 3 marks

4.12 A landlord asks: "Pay $1,500 at the end of each month, or $1,470 at the start." At 4.8% p.a. compounded monthly, over 12 months: which is the better deal for the tenant? Show both PVs. 3 marks

Stuck on 4.12? Compute PV_ord at $1,500 and PV_due at $1,470, then compare.

5. Self-check the easy 3

Tick once you've checked your method works.

For 4.1 I multiplied by (1 + 0.06) = 1.06, not by (1 + 6) = 7.

For 4.5 I computed ordinary FV first, then multiplied by 1.06 — I did not memorise a separate due formula.

For 4.8 I kept n = 12 (the number of payments) — annuity due does not change n.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Conversion rule

Annuity Due = Ordinary × (1 + r).

Q1.2 — Timing

Ordinary: end of each period. Due: start of each period.

Q1.3 — Where the (1 + r) applies

Both PV and FV.

Q3 — Faded example: $250/quarter due for 5 yrs at 8%

r = 0.02; n = 20. PV_ord = 250 × [1 − (1.02)⁻²⁰]/0.02 = 250 × 16.3514 = $4,087.85. PV_due = 4,087.85 × 1.02 = $4,169.61. (Lesson gives $4,169.51 — within rounding.)

Q4.1–4.4 — Due-from-ordinary

4.1: FV_due = 10,000 × 1.06 = $10,600.
4.2: PV_due = 15,000 × 1.04 = $15,600.
4.3: FV_due = 50,000 × 1.005 = $50,250.
4.4: PV_due = 25,000 × 1.012 = $25,300.

Q4.5

FV_ord = 500 × [(1.06)¹⁵ − 1]/0.06 = 500 × 23.276 = $11,637.86. FV_due = 11,637.86 × 1.06 = $12,336.13.

Q4.6

FV_ord = 1,000 × [(1.03)³⁰ − 1]/0.03 = 1,000 × 47.5754 = $47,575.42. FV_due = 47,575.42 × 1.03 = $49,002.68.

Q4.7

PV_ord = 800 × [1 − (1.05)⁻²⁰]/0.05 = 800 × 12.4622 = $9,969.78. PV_due = 9,969.78 × 1.05 = $10,468.27. (Lesson Activity gives $9,966.63 → $10,464.96 — within rounding.)

Q4.8

r = 0.004, n = 12. PV_ord = 1,500 × [1 − (1.004)⁻¹²]/0.004 = 1,500 × 11.7424 = $17,613.52. PV_due = 17,613.52 × 1.004 = $17,683.97.

Q4.9

(See Worked Example.) PV_due = $4,169.61.

Q4.10

r = 0.004, n = 12. PV_ord = 150 × [1 − (1.004)⁻¹²]/0.004 = 150 × 11.7424 = $1,761.35. PV_due = 1,761.35 × 1.004 = $1,768.40.

Q4.11 — Why the gap equals r

FV_due / FV_ord = (1 + r), so FV_due/FV_ord − 1 = r. This holds for any a and n because the (1 + r) factor is the same multiplicative shift applied to every payment (one extra period of interest each).

Q4.12 — Landlord choice

End-of-month: PV_ord = 1,500 × [1 − (1.004)⁻¹²]/0.004 = 1,500 × 11.7424 = $17,613.52.
Start-of-month: PV_ord at $1,470 = 1,470 × 11.7424 = $17,261.25, then × 1.004 = $17,330.30.
The start-of-month $1,470 deal has lower PV for the tenant by $283.22 — the better choice (assuming the tenant is paying out).