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).