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.)