Mathematics Standard • Year 12 • Module 7 • Lesson 4
Annuities — Problem Set
Apply FV and PV of annuity reasoning to Australian super, lease and pension scenarios — converting streams of payments into single dollar figures.
Problem 1 — Voluntary super contributions (FV)
Akira contributes $250 at the end of every month to his super account. The fund pays 5.4% p.a. compounded monthly. He plans to contribute for 20 years.
Set up: What are we solving for?
(i) Identify M, r per period and n. 1 mark
(ii) Calculate the future value of Akira's contributions after 20 years. 2 marks
(iii) Calculate the total contributed and hence the total interest earned. 2 marks
Stuck? Revisit lesson § Future Value — FV = M[(1+r)^n − 1]/r.Problem 2 — Lump-sum offer vs monthly pension (PV)
A retiring teacher is offered two options.
Option A: a lump sum payout of $180,000 today.
Option B: $1,400 per month for 15 years, with the rest of the fund earning 5% p.a. compounded monthly.
Set up: What are we solving for?
(i) Calculate the present value of Option B at the time of retirement. 3 marks
(ii) Recommend which option is mathematically more valuable today, and by how much. 2 marks
(iii) State one non-mathematical reason a retiree might still choose the smaller option in (ii). 1 mark
Stuck? Revisit lesson § Present Value — discount the monthly stream back to today's dollars.Problem 3 — Rent as an annuity due (commercial lease)
A small business pays $4,500 in rent on the 1st of every month for a 3-year commercial lease. If this rent could instead be invested in a fund earning 4.8% p.a. compounded monthly, the business owner wants to know the future value of the entire stream of rent payments at the end of the lease.
Set up: What are we solving for?
(i) Identify whether this is an ordinary annuity or an annuity due, and justify in one short sentence. 1 mark
(ii) Calculate the FV of the equivalent ordinary annuity (end-of-month payments). 2 marks
(iii) Adjust to the FV of the annuity due using FV_due = FV_ord × (1 + r). 2 marks
Stuck? Revisit lesson § Types of Annuities — annuity due earns one extra period of interest per payment.Problem 4 — Saving for a house deposit
Priya is saving for a house deposit. She deposits $1,200 at the end of every month into an account earning 5.4% p.a. compounded monthly for 6 years.
Set up: What are we solving for?
(i) Calculate the future value of Priya's deposit after 6 years. 2 marks
(ii) Calculate the total amount Priya deposited (no interest). 1 mark
(iii) Calculate the total interest she earned. 1 mark
(iv) If she needs $90,000 for the deposit, will her savings be enough? State the shortfall or surplus. 2 marks
Stuck? Revisit lesson § Worked Example — $300 fortnightly super illustration.Problem 5 — Lease vs cash purchase (PV)
A small business is considering leasing a new delivery van.
Lease option: $850 per month at the end of each month for 5 years.
Cash purchase: $45,000 today.
The business's borrowing rate (used as the discount rate) is 6% p.a. compounded monthly.
Set up: What are we solving for?
(i) Calculate the present value of the lease payments. 3 marks
(ii) Compare the lease PV with the cash purchase price and recommend which is cheaper in today's dollars. 2 marks
Stuck? Revisit lesson § Present Value — PV converts a stream into a lump sum for direct comparison.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Akira's super FV
Set up. Use FV = M[(1+r)^n − 1]/r with monthly compounding for 20 years.
(i) M = $250/month, r = 0.054/12 = 0.0045 per month, n = 20 × 12 = 240 months.
(ii) FV = 250 × [(1.0045)^240 − 1] / 0.0045 = 250 × [2.93932 − 1] / 0.0045 = 250 × 431.0 = $107,758.00 (to nearest dollar).
(iii) Total contributed = $250 × 240 = $60,000.00. Interest = $107,758 − $60,000 = $47,758.00.
Problem 2 — Lump sum vs pension
Set up. Find PV of $1,400/month for 15 years at 5% monthly, compare to $180,000, then comment.
(i) r = 0.05/12 = 0.004167, n = 180. PV = 1,400 × [1 − (1.004167)^(−180)] / 0.004167 = 1,400 × [1 − 0.4738] / 0.004167 = 1,400 × 126.29 = $176,809.00 (to nearest dollar).
(ii) Option A ($180,000) is worth $3,191 more in today's dollars than Option B (PV ≈ $176,809). The lump sum is mathematically the better option at a 5% discount rate.
(iii) A retiree might still choose Option B for the guaranteed cash flow, longevity-risk protection (if they live longer than 15 years they cannot outlive the pension), or to remove the temptation/risk of investing a large lump sum themselves.
Problem 3 — Commercial rent as annuity due
Set up. Identify the timing, compute FV_ord then convert to FV_due.
(i) Annuity due — rent is paid on the 1st of the month (at the start of each period).
(ii) r = 0.048/12 = 0.004, n = 36. FV_ord = 4,500 × [(1.004)^36 − 1] / 0.004 = 4,500 × [1.15473 − 1] / 0.004 = 4,500 × 38.68 = $174,068.18.
(iii) FV_due = $174,068.18 × 1.004 = $174,764.45 — an extra $696.27 because each rent payment compounds for one additional month.
Problem 4 — House deposit savings
Set up. Compute FV of $1,200/month at 5.4% monthly for 6 years, then deposited total, interest and gap to $90,000.
(i) r = 0.0045, n = 72. FV = 1,200 × [(1.0045)^72 − 1] / 0.0045 = 1,200 × [1.38226 − 1] / 0.0045 = 1,200 × 84.95 = $101,937.99.
(ii) Deposited = $1,200 × 72 = $86,400.00.
(iii) Interest = $101,937.99 − $86,400 = $15,537.99.
(iv) $101,937.99 > $90,000 — Priya has a surplus of $11,938.00 above her deposit target.
Problem 5 — Lease vs cash purchase
Set up. Compute PV of $850/month for 60 months at 6% monthly, then compare to $45,000.
(i) r = 0.005, n = 60. PV = 850 × [1 − (1.005)^(−60)] / 0.005 = 850 × [1 − 0.74137] / 0.005 = 850 × 51.726 = $43,966.66.
(ii) Lease PV ($43,967) < cash price ($45,000) — the lease is cheaper in today's dollars by $1,033.34, assuming the 6% discount rate is correct. (Real decisions also depend on tax treatment, residual value, and whether the business needs the cash for other purposes.)