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).