Mathematics Standard • Year 12 • Module 7 • Lesson 6
Present Value of Annuities — Problem Set
Apply PV-of-annuity reasoning to Australian mortgages, lottery payouts, pension comparisons and amortisation snapshots.
Problem 1 — 20-year vs 15-year mortgage
Sam is buying a home and considering two repayment terms on the same $180,000 mortgage at 4.8% p.a. compounded monthly.
Option 20: 20-year term.
Option 15: 15-year term.
Set up: What are we solving for?
(i) Calculate the monthly repayment under Option 20. 2 marks
(ii) Calculate the monthly repayment under Option 15. 2 marks
(iii) Calculate the total interest paid under each option, and state the dollar interest saved by choosing the shorter term. 2 marks
Stuck? Revisit lesson § Worked Example — $300,000 mortgage repayments and interest.Problem 2 — Lottery: lump sum vs annual payments
A lottery winner is offered two options at a 5% p.a. compounded annually discount rate.
Option L (Lump sum): $450,000 today.
Option A (Annuity): $40,000 per year at the end of each year for 20 years.
Set up: What are we solving for?
(i) Calculate the present value of Option A using r = 0.05 and n = 20. 3 marks
(ii) Recommend the better option in today's dollars, with a one-sentence justification including the dollar gap. 2 marks
Stuck? Revisit lesson § Revisit Your Initial Thinking — the PV of a long annuity is much less than the simple sum of payments.Problem 3 — Amortisation snapshot
A $300,000 mortgage is taken at 5.4% p.a. compounded monthly over 25 years, with a monthly repayment of $1,827.54.
Set up: What are we solving for?
(i) Calculate the interest, principal and closing balance for Month 1. 2 marks
(ii) Calculate the same three figures for Month 2. 2 marks
(iii) After 12 months the borrower has paid $1,827.54 × 12 = $21,930.48 in total. Calculate the closing balance after month 12 (use the PV of remaining repayments). 2 marks
(iv) Hence calculate the amount of principal repaid in the first 12 months, and the amount of interest paid. 2 marks
Stuck on (iii)? Treat remaining 288 payments as a new PV-of-annuity calculation.Problem 4 — Rate rise on a fixed-term loan
Eva has a $250,000 mortgage at 4.5% p.a. compounded monthly over 30 years. After a refinancing, her rate increases to 5.5% p.a. compounded monthly (term unchanged).
Set up: What are we solving for?
(i) Calculate Eva's original monthly repayment at 4.5%. 2 marks
(ii) Calculate her new monthly repayment at 5.5% (same term). 2 marks
(iii) Calculate the increase in her monthly repayment in dollars, and the additional total interest she will pay over the 30 years. 2 marks
Stuck? Revisit lesson § Loan Repayments — small rate increases lead to large total-interest increases over long terms.Problem 5 — Pension-style PV
A retired worker is offered a pension that pays $3,500 per month for 15 years (ordinary annuity, end of month) starting next month. The discount rate is 6% p.a. compounded monthly.
Set up: What are we solving for?
(i) Calculate the present value of this pension. 3 marks
(ii) If the worker is instead offered $400,000 today as a lump-sum equivalent, state which option is mathematically more valuable and by how much. 2 marks
(iii) Calculate the total dollar amount the worker would receive over the 15-year pension (3,500 × 180) and explain in one sentence why this total is far larger than the PV. 2 marks
Stuck on (iii)? PV discounts each future payment to today, while the total simply sums the nominal payments without time-value adjustment.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Sam's mortgage term
Set up. Compute M for each term, then total interest for each, then the dollar saving.
(i) 20-year: r = 0.004, n = 240. M = 180,000 × 0.004 / [1 − (1.004)^(−240)] = 720 / 0.61650 = $1,167.89/month.
(ii) 15-year: r = 0.004, n = 180. M = 180,000 × 0.004 / [1 − (1.004)^(−180)] = 720 / 0.51236 = $1,405.36/month.
(iii) Total 20-yr = $1,167.89 × 240 = $280,294. Interest = $100,294. Total 15-yr = $1,405.36 × 180 = $252,965. Interest = $72,965. Interest saved by choosing the 15-year term = $27,329.
Problem 2 — Lottery lump sum vs annuity
Set up. Compute PV of $40,000 per year for 20 years at 5%, compare to $450,000.
(i) r = 0.05, n = 20. PV = 40,000 × [1 − (1.05)^(−20)] / 0.05 = 40,000 × [1 − 0.37689] / 0.05 = 40,000 × 12.462 = $498,488.41.
(ii) PV of annuity ($498,488) > lump sum ($450,000). The annuity is the better option in today's dollars, by $48,488.
Problem 3 — Amortisation snapshot ($300,000 at 5.4% over 25 years)
Set up. r = 0.0045 per month. Compute month-by-month for months 1 and 2, then a PV-of-remaining computation for month 12.
(i) Month 1: Open $300,000; Interest = 300,000 × 0.0045 = $1,350.00; Principal = 1,827.54 − 1,350.00 = $477.54; Close = $299,522.46.
(ii) Month 2: Open $299,522.46; Interest = $1,347.85; Principal = $479.69; Close = $299,042.77.
(iii) After 12 payments, remaining n = 288. Balance = 1,827.54 × [1 − (1.0045)^(−288)] / 0.0045 = 1,827.54 × 161.21 = $294,562 (to nearest dollar).
(iv) Principal repaid in year 1 = $300,000 − $294,562 = $5,438. Interest paid in year 1 = $21,930.48 − $5,438 = $16,492.48. (So 75% of year-1 payments are interest.)
Problem 4 — Eva's rate-rise
Set up. Compute M at 4.5% then at 5.5%, both with n = 360, then totals.
(i) r = 0.00375, n = 360. M = 250,000 × 0.00375 / [1 − (1.00375)^(−360)] = 937.50 / 0.74010 = $1,266.71/month.
(ii) r = 0.0045833, n = 360. M = 250,000 × 0.0045833 / [1 − (1.0045833)^(−360)] = 1,145.83 / 0.80662 = $1,419.47/month.
(iii) Increase per month = $1,419.47 − $1,266.71 = $152.76. Extra total interest = $152.76 × 360 = $54,993.95 over 30 years.
Problem 5 — Pension PV
Set up. Apply PV with M = 3,500, r = 0.005, n = 180, then compare to $400,000, then sum nominal payments.
(i) PV = 3,500 × [1 − (1.005)^(−180)] / 0.005 = 3,500 × [1 − 0.40748] / 0.005 = 3,500 × 118.504 = $414,762.71.
(ii) PV of pension ($414,763) > $400,000 lump sum. The pension is more valuable today, by $14,763.
(iii) Nominal total = $3,500 × 180 = $630,000. This is much larger than the PV ($414,763) because nominal totals add future dollars at face value, while PV discounts each future payment back to today's dollars (so a payment 15 years away is worth much less than $3,500 today).