Mathematics Advanced • Year 12 • Module 7 • Lesson 8

Present Value of an Annuity

Build procedural fluency in discounting future payment streams back to a present-value lump sum.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Complete the present value formula for an ordinary annuity:

PV = a × _____________________________________

Q1.2 Write the link between PV and FV (for the same annuity):

PV = FV / __________________

Q1.3 Decide PV or FV. "How much can I borrow today if my repayments are $a per month?" → ______. "How much will my superannuation grow to?" → ______.

Stuck? Revisit lesson § When to Use PV vs FV.

2. Worked example — pension PV at $2,000/month for 15 years

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

Problem. A pension pays $2,000 at the end of each month for 15 years. The discount rate is 4.8% p.a. compounded monthly. Find the present value.

Step 1 — Match the rate and time to monthly.

r = 0.048 / 12 = 0.004 ; n = 15 × 12 = 180.

Reason: PV formula needs r per period and n in periods.

Step 2 — Substitute into PV.

PV = 2,000 × [1 − (1.004)⁻¹⁸⁰] / 0.004

Reason: each future payment is discounted by (1+r)^(−k).

Step 3 — Evaluate the factor.

(1.004)⁻¹⁸⁰ = 0.4876. Factor = (1 − 0.4876) / 0.004 = 128.10.

Step 4 — Multiply by the payment.

PV = 2,000 × 128.10 = $256,200.

Conclusion. A pension of $2,000/month for 15 years at 4.8% p.a. is worth $256,200 today.

3. Faded example — fill in the missing steps

Find the present value of $800 paid at the end of each quarter for 6 years at 5.2% p.a. compounded quarterly. 4 marks

Step 1 — Convert to quarters.

r = 0.052 / ____ = ____________ ; n = 6 × ____ = ____________

Step 2 — Substitute.

PV = 800 × [1 − (1 + ____)^(− ____)] / ____

Step 3 — Evaluate.

(1.013)^(− ____) = ____________

Factor = (1 − ____________) / 0.013 = ____________

Step 4 — Multiply by the payment.

PV = 800 × ____________ = $____________

Conclusion. The present value is $____________.

Stuck? Revisit lesson § Try It Now ($800/quarter, 6 yrs).

4. Graduated practice

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

Foundation — direct PV evaluation (4 questions)

Q	Setup	PV
4.1 1	$1,000/year for 10 yrs at 5% p.a.
4.2 1	$500/year for 8 yrs at 4% p.a.
4.3 1	$200/quarter for 5 yrs at 4.8% p.a. compounded quarterly
4.4 1	$300/month for 2 yrs at 6% p.a. compounded monthly

Standard — typical HSC difficulty (6 questions)

Show formula line, factor, and final PV.

4.5 $500/month for 3 years at 6% p.a. compounded monthly. (Lesson example.) 2 marks

4.6 $50,000/year for 20 years at 5% p.a. (the lottery scenario). 2 marks

4.7 $400/year for 30 years at 4% p.a. State the FV/PV ratio (you do not need to compute both — see Activity 1). 2 marks

4.8 $80,000/year for 25 years at 6% p.a. State the fair lump-sum equivalent. 2 marks

4.9 $600/month for 5 years at 6% p.a. compounded monthly. (Max car-loan payment in the lesson Q8.) 2 marks

4.10 $2,500/month for 15 years at 4.8% p.a. compounded monthly. (Lesson Q10.) 2 marks

Extension — transpose and reason (2 questions)

4.11 A retiree has PV = $100,000 to fund 10 years of monthly withdrawals at 4.8% p.a. compounded monthly. Find the sustainable monthly withdrawal a, to the nearest cent. 3 marks

4.12 Show by direct algebra that PV(a, r, n) = FV(a, r, n) × (1 + r)⁻ⁿ. Use this to derive PV from FV for $500/year, n = 10, r = 6%. 3 marks

Stuck on 4.12? Multiply both bracketed factors by (1+r)⁻ⁿ and tidy.

5. Self-check the easy 3

Tick once you've checked your method works.

For 4.1 I used the PV formula (not FV) and got an answer less than n × a = $10,000.

For 4.6 I noticed PV ($623k) is less than total payments ($1 million) because each payment is discounted.

For 4.11 I divided $100,000 by the PV factor — I did not divide by 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 — PV formula

PV = a × [1 − (1 + r)⁻ⁿ] / r.

Q1.2 — PV–FV link

PV = FV / (1 + r)ⁿ (or equivalently FV = PV × (1 + r)ⁿ).

Q1.3 — PV vs FV decision

Borrowing → PV (today's lump sum). Super → FV (future accumulation).

Q3 — Faded example: $800/quarter for 6 yrs at 5.2%

r = 0.052/4 = 0.013; n = 24. (1.013)⁻²⁴ = 0.7345. Factor = (1 − 0.7345)/0.013 = 20.426. PV = 800 × 20.426 ≈ $16,340.80. (Lesson gives $16,146.40 using 20.183 — both within rounding ranges; either answer accepted to within $200.)

Q4.1

PV = 1,000 × [1 − (1.05)⁻¹⁰]/0.05 = 1,000 × 7.7217 = $7,721.73.

Q4.2

PV = 500 × [1 − (1.04)⁻⁸]/0.04 = 500 × 6.7327 = $3,366.37.

Q4.3

r = 0.012, n = 20. PV = 200 × [1 − (1.012)⁻²⁰]/0.012 = 200 × 17.659 = $3,531.78.

Q4.4

r = 0.005, n = 24. PV = 300 × [1 − (1.005)⁻²⁴]/0.005 = 300 × 22.563 = $6,768.87.

Q4.5

r = 0.005, n = 36. PV = 500 × [1 − (1.005)⁻³⁶]/0.005 = 500 × 32.871 ≈ $16,435.50.

Q4.6

PV = 50,000 × [1 − (1.05)⁻²⁰]/0.05 = 50,000 × 12.4622 ≈ $623,110.52. Total payments = $1,000,000, so discounting reduces them to about 62% in today's dollars.

Q4.7

PV = 400 × [1 − (1.04)⁻³⁰]/0.04 = 400 × 17.292 = $6,916.81. (Activity 1: ratio FV/PV ≈ (1.04)³⁰ ≈ 3.24.)

Q4.8

PV = 80,000 × [1 − (1.06)⁻²⁵]/0.06 = 80,000 × 12.7834 ≈ $1,022,672. (Lesson Q9: $1,022,962 — within rounding.)

Q4.9

r = 0.005, n = 60. PV = 600 × [1 − (1.005)⁻⁶⁰]/0.005 = 600 × 51.726 ≈ $31,035.41. (Lesson Q8: $31,015.59 — within rounding.)

Q4.10

r = 0.004, n = 180. PV = 2,500 × [1 − (1.004)⁻¹⁸⁰]/0.004 = 2,500 × 128.10 ≈ $320,250.

Q4.11 — Sustainable monthly withdrawal

r = 0.004, n = 120. Factor = [1 − (1.004)⁻¹²⁰]/0.004 = 94.946. a = 100,000 / 94.946 ≈ $1,053.23 per month.

Q4.12 — PV via FV

FV = a × [(1+r)ⁿ − 1]/r. Multiply by (1+r)⁻ⁿ: FV × (1+r)⁻ⁿ = a × [(1+r)ⁿ − 1] × (1+r)⁻ⁿ / r = a × [1 − (1+r)⁻ⁿ] / r = PV. ✓
For $500/yr, 6%, n=10: FV = 500 × [(1.06)¹⁰ − 1]/0.06 = 500 × 13.181 = $6,590.40. PV = 6,590.40 / (1.06)¹⁰ = 6,590.40 / 1.7908 ≈ $3,680.04. Direct PV check: 500 × [1 − (1.06)⁻¹⁰]/0.06 = 500 × 7.3601 = $3,680.04. ✓