Mathematics Standard • Year 12 • Module 7 • Lesson 6
Present Value of Annuities — Past-Paper Style
Practise HSC-style short answers and one structured extended response on the present value of annuities — mortgages, amortisation and PV comparisons.
1. Short-answer questions
1.1 Calculate the monthly repayment on a $200,000 mortgage at 5.4% per annum compounded monthly over 25 years. 3 marks Band 3
1.2 A pension pays $2,500 at the end of each month for 12 years. The discount rate is 6% p.a. compounded monthly.
(a) Calculate the present value of the pension.
(b) State in one sentence what this present value represents in plain language. 3 marks Band 3-4
1.3 A $25,000 personal loan is taken at 7.2% p.a. compounded monthly over 4 years.
(a) Calculate the monthly repayment.
(b) For Month 1, calculate the interest, principal and closing balance.
(c) Briefly explain why the principal repaid in Month 1 is small relative to the total monthly repayment. 4 marks Band 4
2. Extended response
2.1 Aiyana and Marcus are buying a home and need a $500,000 mortgage. The lender offers 4.5% p.a. compounded monthly. They are considering two options.
Option 30: 30-year term, monthly repayments.
Option 20: 20-year term, monthly repayments.
(a) Calculate the monthly repayment under Option 30.
(b) Calculate the monthly repayment under Option 20.
(c) Calculate the total interest paid under each option.
(d) Calculate the additional monthly repayment Option 20 requires compared to Option 30.
(e) Recommend an option for Aiyana and Marcus, naming the option and the interest saved if they take it, in a one-sentence conclusion that also mentions the trade-off (higher monthly repayment). 7 marks Band 5-6
Explicit marking criteria
Part (a) — 1 mark
• 1 mark — correct monthly repayment for Option 30 using M = PV × r / [1 − (1+r)^(−n)].
Part (b) — 1 mark
• 1 mark — correct monthly repayment for Option 20 with n = 240.
Part (c) — 2 marks
• 1 mark — correct total interest for Option 30 using total − PV.
• 1 mark — correct total interest for Option 20.
Part (d) — 1 mark
• 1 mark — correct difference in monthly repayments.
Part (e) — 2 marks
• 1 mark — recommendation sentence naming the chosen option and the dollar interest saved.
• 1 mark — explicit mention of the trade-off: the higher monthly repayment required by the shorter-term option.
Your response:
Stuck on (e)? Once the totals are calculated, the recommendation sentence must (i) name the chosen option, (ii) state the dollar interest saved, and (iii) note the higher monthly repayment trade-off.How did this worksheet feel?
What I'll revisit before next class:
1.1 — Monthly repayment on $200,000 at 5.4% over 25 years (3 marks)
Sample response. r = 0.0045, n = 300. M = 200,000 × 0.0045 / [1 − (1.0045)^(−300)] = 900 / [1 − 0.26129] = 900 / 0.73871 = $1,218.34/month.
Marking notes. 1 mark — correct r and n. 1 mark — correct substitution into the M-formula. 1 mark — correct final value to the nearest cent.
1.2 — PV of $2,500/month pension for 12 years at 6% monthly (3 marks)
(a) Sample response. r = 0.005, n = 144. PV = 2,500 × [1 − (1.005)^(−144)] / 0.005 = 2,500 × [1 − 0.48788] / 0.005 = 2,500 × 102.42 = $256,057.95.
(b) Sample response. The present value $256,057.95 is the lump-sum amount today that, invested at 6% p.a. compounded monthly, would exactly fund $2,500 per month for 12 years.
Marking notes. (a) 1 mark — correct r and n. 1 mark — correct PV. (b) 1 mark — explanation connects PV to "lump-sum equivalent to fund the stream".
1.3 — $25,000 loan at 7.2% over 4 years (4 marks)
(a) Sample response. r = 0.006, n = 48. M = 25,000 × 0.006 / [1 − (1.006)^(−48)] = 150 / [1 − 0.75103] = 150 / 0.24897 = $602.49/month.
(b) Sample response. Month 1: Interest = 25,000 × 0.006 = $150.00. Principal = 602.49 − 150.00 = $452.49. Closing balance = $25,000 − $452.49 = $24,547.51.
(c) Sample response. In Month 1 the interest charged on the full $25,000 balance is $150 — about 25% of the total $602.49 repayment — so only the remaining $452.49 goes towards reducing the principal; this fraction shifts towards principal in later months as the balance falls.
Marking notes. (a) 1 mark — correct M. (b) 1 mark — interest. 1 mark — principal and closing balance. (c) 1 mark — explanation links the small principal-share to interest being charged on the full original balance.
2.1 — $500,000 mortgage at 4.5%, 30 vs 20 year (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Option 30 monthly repayment.
r = 0.045/12 = 0.00375. n = 360. M = 500,000 × 0.00375 / [1 − (1.00375)^(−360)] = 1,875.00 / [1 − 0.25988] = 1,875.00 / 0.74012 = $2,533.43/month. [1 mark.]
(b) Option 20 monthly repayment.
n = 240. M = 500,000 × 0.00375 / [1 − (1.00375)^(−240)] = 1,875.00 / [1 − 0.40747] = 1,875.00 / 0.59253 = $3,163.92/month. [1 mark.]
(c) Total interest each option.
Option 30: total = 2,533.43 × 360 = $912,034.80; interest = $912,034.80 − $500,000 = $412,034.80. [1 mark.]
Option 20: total = 3,163.92 × 240 = $759,340.80; interest = $759,340.80 − $500,000 = $259,340.80. [1 mark.]
(d) Additional monthly repayment for Option 20 vs Option 30.
Extra = $3,163.92 − $2,533.43 = $630.49/month. [1 mark.]
(e) Recommendation and trade-off.
Recommendation: if cash flow allows, Aiyana and Marcus should take Option 20 (20-year term), saving $152,694 in interest over the life of the loan — though this requires an extra $630.49 per month in repayments. [1 mark — names Option 20 and the interest saved; 1 mark — explicitly mentions the higher monthly repayment trade-off.]
Total: 7/7.
Band descriptors for marker.
Band 3: Option 30 monthly correct, Option 20 monthly correct, but total interest not calculated for either; no recommendation. ≈ 2-3 marks.
Band 4: Both monthlies and both interest totals correct, but no trade-off mention in the recommendation. ≈ 5 marks.
Band 5: Full numerical solution including interest comparison; recommendation names Option 20 and the saving, but omits the higher-monthly trade-off. ≈ 6 marks.
Band 6: Complete: both monthlies, both totals, dollar difference, AND recommendation that names Option 20, the interest saved, and the trade-off of higher monthly repayments. 7/7.