Mathematics Advanced • Year 12 • Module 7 • Lesson 8
Present Value of an Annuity
Apply the PV annuity formula to lottery comparisons, loans, pensions and withdrawals decision-problems.
Problem 1 — Lottery: lump sum or annuity?
A lottery winner can choose:
Option A: $2,000,000 lump sum today.
Option B: $150,000 at the end of each year for 20 years.
Money invested elsewhere earns 5% p.a.
Set up: What are we solving for?
(i) Find the present value of Option B at the 5% discount rate. 2 marks
(ii) Which option is worth more today? By how much? 1 mark
(iii) Re-run the comparison at a discount rate of 3%. Which option wins now, and why does lowering r make the annuity more valuable? 3 marks
Stuck on (iii)? A smaller r discounts future payments less aggressively, raising their PV.Problem 2 — Home loan approval
A bank uses PV to decide how much it will lend. Hayden can afford repayments of $2,800 at the end of each month for 25 years. The current home-loan rate is 6.6% p.a. compounded monthly.
Set up: What are we solving for?
(i) Find the maximum loan principal (PV) Hayden can support. 2 marks
(ii) If the rate rises to 7.8% p.a. (compounded monthly), how does Hayden's borrowing capacity change? 2 marks
(iii) In one sentence, explain why a rate rise hits borrowing capacity harder than it raises the monthly repayment on an existing loan. 1 mark
Problem 3 — Funding 20 years of retirement withdrawals
Greta retires with a lump sum of $600,000. She wants this to fund equal end-of-month withdrawals for 20 years. The fund earns 5.4% p.a. compounded monthly during retirement.
Set up: What are we solving for?
(i) Find Greta's sustainable monthly withdrawal a (to the nearest dollar). 3 marks
(ii) Greta is told she could withdraw $4,200/month. Show that this would deplete her savings before 20 years and estimate how many years the lump sum would actually last. 2 marks
(iii) If the fund return drops permanently to 3.8% p.a., recalculate the sustainable withdrawal. Discuss the implication in one line. 2 marks
Problem 4 — Valuing a defined-benefit pension
Lou is offered the choice: keep a defined-benefit pension of $80,000/year for 25 years, or accept a lump-sum buyout. The appropriate discount rate is 6% p.a.
Set up: What are we solving for?
(i) Compute the fair lump-sum equivalent (PV) of Lou's pension. 2 marks
(ii) Lou is offered a $900,000 lump sum. By how much is the offer below the fair value? 1 mark
(iii) Suppose Lou expects to live only 18 years instead of 25. Recompute the PV at the same rate and decide if the $900,000 offer is now reasonable. 3 marks
Stuck on (iii)? Re-use PV formula with n = 18.Problem 5 — Investing and borrowing are mirror images
An investor deposits $a at the end of each period and receives FV at the end of n periods. A borrower receives PV today and repays $a at the end of each period for n periods. Both use the same r per period.
Set up: What are we solving for?
(i) Show algebraically that FV = PV × (1 + r)ⁿ for the same a, r, n. 3 marks
(ii) Use this identity to compute the FV of an annuity of $1,000/year for 15 years at 5% p.a., given that PV = 1,000 × [1 − (1.05)⁻¹⁵]/0.05 = $10,379.66. 2 marks
(iii) Explain in one sentence how this identity tells you what happens to PV if a, r and n are the same but the bank also requires you to make payments at the start of each period (annuity due — Lesson 9). 1 mark
How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Lottery
Set up. We are comparing a lump sum offered today with the PV of a 20-year payment stream.
(i) PV = 150,000 × [1 − (1.05)⁻²⁰]/0.05 = 150,000 × 12.4622 ≈ $1,869,330.
(ii) Option A ($2,000,000) is worth $130,670 more in today's dollars. Choose A.
(iii) At r = 3%: PV = 150,000 × [1 − (1.03)⁻²⁰]/0.03 = 150,000 × 14.8775 ≈ $2,231,623. Option B now wins by about $231,623. Lower r means each future payment is discounted by a smaller factor, so the annuity stream becomes more valuable today.
Problem 2 — Home loan approval
Set up. We are using PV to translate a monthly repayment ability into a maximum borrowable principal.
(i) r = 0.066/12 = 0.0055; n = 300. PV = 2,800 × [1 − (1.0055)⁻³⁰⁰]/0.0055 = 2,800 × 145.96 ≈ $408,696.
(ii) r = 0.0065; n = 300. PV = 2,800 × [1 − (1.0065)⁻³⁰⁰]/0.0065 = 2,800 × 131.94 ≈ $369,427. Borrowing capacity falls by about $39,269.
(iii) The PV formula multiplies the same monthly payment by a discount-factor that is highly sensitive to r over a long horizon, so a small rate rise compresses many years of future payments and shrinks the loan amount more than it would the per-month cost.
Problem 3 — Retirement withdrawals
Set up. We are running the PV formula in reverse — given PV, r and n, solve for the payment a.
(i) r = 0.0045; n = 240. Factor = [1 − (1.0045)⁻²⁴⁰]/0.0045 = 145.42. a = 600,000 / 145.42 ≈ $4,126/month.
(ii) At $4,200/month: 600,000 = 4,200 × [1 − (1.0045)⁻ⁿ]/0.0045 ⇒ [1 − (1.0045)⁻ⁿ]/0.0045 = 142.857 ⇒ (1.0045)⁻ⁿ = 1 − 0.6429 = 0.3571 ⇒ n = ln(0.3571) / (−ln(1.0045)) ≈ 229 months ≈ 19.1 years. The lump sum runs out about 11 months early.
(iii) r = 0.003167; n = 240. Factor ≈ 168.36. a = 600,000 / 168.36 ≈ $3,564/month — a noticeable cut, showing how sensitive retirement income is to fund performance.
Problem 4 — Pension valuation
Set up. We are computing the fair PV of a defined-benefit stream and comparing with a lump-sum offer.
(i) PV = 80,000 × [1 − (1.06)⁻²⁵]/0.06 = 80,000 × 12.7834 ≈ $1,022,672.
(ii) Offer is $122,672 below the fair value (Lou is losing money by accepting).
(iii) PV = 80,000 × [1 − (1.06)⁻¹⁸]/0.06 = 80,000 × 10.8276 ≈ $866,206. The $900,000 offer now exceeds the PV (assuming the shortened life expectancy). Lou should take the lump sum if confident in the 18-year horizon, but keep in mind life-expectancy estimates carry uncertainty.
Problem 5 — PV–FV symmetry
Set up. We are proving the algebraic mirror between PV and FV annuity formulas and using it to convert one to the other.
(i) PV × (1 + r)ⁿ = a · [1 − (1+r)⁻ⁿ]/r · (1+r)ⁿ = a · [(1+r)ⁿ − 1]/r = FV. ✓
(ii) FV = 10,379.66 × (1.05)¹⁵ = 10,379.66 × 2.0789 ≈ $21,578.56. (Direct: 1,000 × [(1.05)¹⁵ − 1]/0.05 = 1,000 × 21.5786 = $21,578.56 ✓.)
(iii) Pushing payments to the start of each period gives every payment one extra period of interest, so both PV and FV scale up by exactly (1 + r) — the same multiplier we use to convert ordinary into annuity-due (Lesson 9).