Mathematics Advanced • Year 12 • Module 7 • Lesson 8
Present Value of an Annuity
Practise HSC-style writing on PV annuities — including a multi-part scenario with derivation and comparison.
1. Short-answer questions
1.1 Find the present value of $600 paid at the end of each month for 5 years at 6% p.a. compounded monthly. State the maximum car loan this would support. 2 marks Band 3
1.2 A retiree wants to receive $3,500 at the end of each month for 20 years. The fund earns 4.8% p.a. compounded monthly. Find the lump sum required at the start of retirement. 3 marks Band 3-4
1.3 Consider a 30-year ordinary annuity of $1,200 per month at 6% p.a. compounded monthly.
(a) Find both PV and FV.
(b) Hence verify that FV / PV = (1 + r)ⁿ for this annuity and explain in one sentence why this identity is exact. 4 marks Band 4
2. Extended response
2.1 Vega wins a lottery and is offered two options at her current discount rate r = 5% p.a.:
Option A: $1,800,000 lump sum today.
Option B: $130,000 at the end of every year for 20 years.
(a) Show, by writing the PV as a discounted sum of n cash flows and identifying it as a GP, that the present value of Option B is given by PV = a × [1 − (1+r)⁻ⁿ] / r.
(b) Hence compute the PV of Option B at r = 5% and decide which option Vega should prefer at that rate.
(c) Vega's financial adviser warns that if interest rates fall to r = 3.5% p.a. (the discount rate she uses on alternative investments), her decision should reverse. Recompute PV at r = 3.5% to justify or reject the adviser's claim, and explain in one sentence why the PV of a long annuity is so sensitive to r. 8 marks Band 5-6
Explicit marking criteria
Part (a) — 3 marks
• 1 mark — writes PV = a/(1+r) + a/(1+r)² + … + a/(1+r)ⁿ.
• 1 mark — identifies as GP with first term a/(1+r), ratio 1/(1+r), n terms.
• 1 mark — uses GP sum formula and simplifies to a × [1 − (1+r)⁻ⁿ]/r.
Part (b) — 2 marks
• 1 mark — numerical PV at r = 5%.
• 1 mark — explicit choice and dollar comparison with Option A.
Part (c) — 3 marks
• 1 mark — recomputes PV at r = 3.5%.
• 1 mark — states whether the decision reverses (with dollar comparison).
• 1 mark — sentence explaining sensitivity (each future payment is discounted by (1+r)⁻ᵏ, so a small change in r is amplified over 20 years).
Your response:
Stuck on (a)? Use the GP formula S = a₁(1 − Rⁿ)/(1 − R) with a₁ = a/(1+r) and R = 1/(1+r).How did this worksheet feel?
What I'll revisit before next class:
1.1 — Car loan PV (2 marks)
Sample response. r = 0.005, n = 60. PV = 600 × [1 − (1.005)⁻⁶⁰]/0.005 = 600 × 51.7256 = $31,035.41. Hayden could borrow up to $31,035.41 at this repayment level.
Marking notes. 1 mark — correct r and n; 1 mark — correct PV and explicit interpretation as "maximum loan". Lesson Q8 gives $31,015.59 — accept any value within rounding ranges.
1.2 — Required retirement lump sum (3 marks)
Sample response. r = 0.004, n = 240. PV = 3,500 × [1 − (1.004)⁻²⁴⁰]/0.004 = 3,500 × 154.0933 = $539,326.45 (use of slightly different rounding may give $539,000–$540,000).
Marking notes. 1 mark — correct conversion of r and n. 1 mark — correct PV factor. 1 mark — correct PV value rounded sensibly. Common error: using the FV formula instead.
1.3 — PV, FV and the identity (4 marks)
(a) Sample response. r = 0.005; n = 360. PV = 1,200 × [1 − (1.005)⁻³⁶⁰]/0.005 = 1,200 × 166.7916 = $200,149.96. FV = 1,200 × [(1.005)³⁶⁰ − 1]/0.005 = 1,200 × 1,004.515 = $1,205,417.65.
(b) Sample response. FV / PV = 1,205,417.65 / 200,149.96 = 6.0226 = (1.005)³⁶⁰ ✓. Algebraically, FV / PV = [(1+r)ⁿ − 1]/[1 − (1+r)⁻ⁿ] = (1+r)ⁿ, because multiplying top and bottom by (1+r)ⁿ gives ((1+r)²ⁿ − (1+r)ⁿ)/((1+r)ⁿ − 1) = (1+r)ⁿ × ((1+r)ⁿ − 1)/((1+r)ⁿ − 1) = (1+r)ⁿ.
Marking notes. (a) 1 mark — correct PV; 1 mark — correct FV. (b) 1 mark — numerical check; 1 mark — algebraic justification.
2.1 — Extended response (8 marks): sample Band-6 response with annotations
Sample Band-6 response.
Part (a). Each future payment a is discounted back to today by (1 + r)⁻ᵏ where k is the period it is received:
PV = a/(1 + r) + a/(1 + r)² + … + a/(1 + r)ⁿ. [1 mark — sum written out.]
This is a geometric series with first term a₁ = a/(1+r), common ratio R = 1/(1+r), and n terms. [1 mark — GP identified.]
Applying S_n = a₁ × (1 − Rⁿ)/(1 − R):
PV = [a/(1 + r)] × [1 − (1+r)⁻ⁿ] / [1 − 1/(1+r)] = [a/(1+r)] × [1 − (1+r)⁻ⁿ] × (1+r)/r = a × [1 − (1+r)⁻ⁿ] / r. [1 mark — simplified to the boxed formula.]
Part (b). At r = 0.05, n = 20:
PV_B = 130,000 × [1 − (1.05)⁻²⁰]/0.05 = 130,000 × 12.4622 = $1,620,086. [1 mark — PV.]
Option A is worth $1,800,000 today, so it exceeds Option B by about $179,914. Vega should choose Option A. [1 mark — explicit comparison.]
Part (c). At r = 0.035, n = 20:
PV_B = 130,000 × [1 − (1.035)⁻²⁰]/0.035 = 130,000 × 14.2124 = $1,847,617. [1 mark — recomputation.]
Now PV_B exceeds Option A ($1,800,000) by about $47,617, so the adviser is correct — the decision reverses, and Vega should prefer Option B at 3.5%. [1 mark — decision reverses, with dollar evidence.]
The PV of a long annuity is highly sensitive to r because each of the 20 future payments is multiplied by (1 + r)⁻ᵏ; small changes in r compound over many periods, so a 1.5-percentage-point fall in r raises the PV by hundreds of thousands of dollars. [1 mark — sensitivity explanation.]
Total: 8/8.
Band descriptors for marker.
Band 3: Writes PV sum with a sign error or missing simplification; computes (b) at r = 5% without justifying choice. ≈ 2-3 marks.
Band 4: Sum and GP identification correct; derivation has algebra slip in simplifying 1 − 1/(1+r); (b) and (c) numerically correct. ≈ 4-5 marks.
Band 5: Full derivation; (b) and (c) numerically correct; sensitivity remark stated but not linked to the (1+r)⁻ᵏ form. ≈ 6-7 marks.
Band 6: Complete derivation; both comparisons made with explicit dollar evidence; sensitivity explanation references the discount factor directly. 8/8.