Mathematics Standard • Year 12 • Module 7 • Lesson 6

Present Value of Annuities — Skill Drill

Build fluency with PV = M[1 − (1+r)^(−n)]/r and its rearrangement M = PV × r / [1 − (1+r)^(−n)] — the engine behind every mortgage repayment.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Write the present-value formula for an ordinary annuity.

PV = M × ____________________________________________

Q1.2 Write the loan-repayment formula (rearrangement to make M the subject).

M = PV × ____________________________________________

Q1.3 Write the formula for the total interest paid on a loan, and explain in one short phrase what an amortisation table shows.

Total interest = ____________________________________________

Amortisation table shows: ____________________________________________

Stuck? Revisit lesson § Key Ideas — PV formula, Loan Repayments and Amortisation.

2. Worked example — car loan repayment

Problem. A $25,000 car loan is taken at 7.2% p.a. compounded monthly over 5 years. Find the monthly repayment and the total interest paid.

Step 1 — Identify PV, r per period, n.

PV = $25,000 r = 0.072/12 = 0.006 per month n = 5 × 12 = 60 months

Step 2 — Apply M = PV × r / [1 − (1+r)^(−n)].

M = 25,000 × 0.006 / [1 − (1.006)^(−60)] = 150 / [1 − 0.69767] = 150 / 0.30233 = $496.20/month

Step 3 — Total repaid and total interest.

Total = 496.20 × 60 = $29,772.00. Interest = $29,772 − $25,000 = $4,772.00.

Conclusion. Monthly repayment ≈ $496.20; total interest ≈ $4,772.00.

3. Faded example — 30-year mortgage

A $400,000 mortgage is taken at 4.8% p.a. compounded monthly over 30 years. Find the monthly repayment and the total interest paid. Fill in the blanks. 4 marks

Step 1 — Identify PV, r per period, n:

PV = $ __________ r = 0.048/ ________ = ________ per month n = 30 × ________ = ________ months

Step 2 — Apply M = PV × r / [1 − (1+r)^(−n)]:

M = ________ × ________ / [1 − (1 + ________ )^(− ________ )]

M = ________ / [1 − ________ ] = ________ / ________ = $ ____________ /month

Step 3 — Total repaid and interest:

Total = $ ________ × ________ = $ ____________

Interest = $ ________ − $ ________ = $ ____________

Conclusion. Monthly = $ __________ Total interest = $ __________.

Stuck? Revisit lesson § Worked Example — $400,000 at 4.8% over 30 years.

4. Graduated practice — PV of annuities and loan repayments

Show one line of substitution. Round dollar amounts to 2 decimal places.

Foundation — identify M, r, n (4 questions)

Q	Problem	M, r, n
4.1 1	$20,000 loan at 6% p.a. monthly over 4 years.
4.2 1	$300,000 mortgage at 5.4% p.a. monthly over 25 years.
4.3 1	$180,000 mortgage at 4.8% p.a. monthly over 20 years.
4.4 1	$350,000 mortgage at 5% p.a. monthly over 25 years.

Standard — PV / repayment calculations (6 questions)

4.5 Find the PV of $300/month for 4 years at 5.4% p.a. compounded monthly. 2 marks

4.6 Find the monthly repayment on a $20,000 loan at 6% p.a. compounded monthly over 4 years. 2 marks

4.7 Find the total interest on a $300,000 mortgage at 5.4% p.a. monthly over 25 years. 3 marks

4.8 Find the monthly repayment on a $180,000 mortgage at 4.8% p.a. monthly over 20 years. 2 marks

4.9 Find the monthly repayment on a $350,000 mortgage at 5% p.a. monthly over 25 years. 2 marks

4.10 For Q4.9, calculate (i) total repaid over 25 years and (ii) total interest paid. 2 marks

Extension — amortisation snapshot (2 questions)

4.11 For a $15,000 personal loan at 8% p.a. compounded monthly over 3 years (monthly repayment $470.05), build the first 3 months of the amortisation table — for each month show opening balance, interest, principal and closing balance. 3 marks

4.12 In one sentence, explain why the interest component of each repayment decreases over the life of a reducing-balance loan, while the principal component increases. 2 marks

Stuck on 4.11? Each month: interest = balance × 0.08/12; principal = $470.05 − interest; closing balance = opening − principal.

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 I noted PV = $20,000, r = 0.005 per month, n = 48.

For 4.6 I used the rearranged formula M = PV × r / [1 − (1+r)^(−n)], not the PV formula.

For 4.10 my "total interest" = total repaid − PV (not just M × n).

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — PV formula

PV = M × [1 − (1 + r)^(−n)] / r.

Q1.2 — Loan repayment formula

M = PV × r / [1 − (1 + r)^(−n)].

Q1.3 — Total interest and amortisation

Total interest = (M × n) − PV. Amortisation table shows: opening balance, interest, principal and closing balance for each repayment period.

Q3 — Faded example ($400,000 mortgage at 4.8% over 30 years)

Step 1: PV = $400,000, r = 0.048/12 = 0.004, n = 30 × 12 = 360 months.
Step 2: M = 400,000 × 0.004 / [1 − (1.004)^(−360)] = 1,600 / [1 − 0.23763] = 1,600 / 0.76237 = $2,098.71/month.
Step 3: Total = $2,098.71 × 360 = $755,535.39. Interest = $755,535.39 − $400,000 = $355,535.39.
Conclusion: Monthly = $2,098.71; Total interest ≈ $355,535.

Q4.1 — Identify

PV = $20,000; r = 0.005; n = 48.

Q4.2 — Identify

PV = $300,000; r = 0.0045; n = 300.

Q4.3 — Identify

PV = $180,000; r = 0.004; n = 240.

Q4.4 — Identify

PV = $350,000; r = 0.05/12 = 0.004167; n = 300.

Q4.5 — PV of $300/month at 5.4% for 4 years

r = 0.0045, n = 48. PV = 300 × [1 − (1.0045)^(−48)] / 0.0045 = 300 × [1 − 0.80569] / 0.0045 = 300 × 43.18 = $12,954.85.

Q4.6 — $20,000 at 6% monthly for 4 years

r = 0.005, n = 48. M = 20,000 × 0.005 / [1 − (1.005)^(−48)] = 100 / [1 − 0.78710] = 100 / 0.21290 = $469.70/month.

Q4.7 — Total interest on $300,000 at 5.4% over 25 years

r = 0.0045, n = 300. M = 300,000 × 0.0045 / [1 − (1.0045)^(−300)] = 1,350 / [1 − 0.26129] = 1,350 / 0.73871 = $1,827.51/month.
Total = $1,827.51 × 300 = $548,253. Interest = $548,253 − $300,000 = $248,253.43.

Q4.8 — $180,000 at 4.8% over 20 years

r = 0.004, n = 240. M = 180,000 × 0.004 / [1 − (1.004)^(−240)] = 720 / [1 − 0.38350] = 720 / 0.61650 = $1,167.89/month.

Q4.9 — $350,000 at 5% over 25 years

r = 0.004167, n = 300. M = 350,000 × 0.004167 / [1 − (1.004167)^(−300)] = 1,458.33 / [1 − 0.28720] = 1,458.33 / 0.71280 = $2,046.06/month.

Q4.10 — Totals for Q4.9

(i) Total = $2,046.06 × 300 = $613,817.97.
(ii) Interest = $613,817.97 − $350,000 = $263,817.97.

Q4.11 — First 3 months amortisation, $15,000 at 8% monthly, M = $470.05

r per month = 0.08/12 = 0.006667.
Month 1: Open $15,000.00; Interest = 15,000 × 0.006667 = $100.00; Principal = 470.05 − 100.00 = $370.05; Close = 15,000 − 370.05 = $14,629.95.
Month 2: Open $14,629.95; Interest = $97.53; Principal = $372.52; Close = $14,257.43.
Month 3: Open $14,257.43; Interest = $95.05; Principal = $375.00; Close = $13,882.43.

Q4.12 — Why interest shrinks and principal grows

Because interest each period is charged on the reducing balance: as the balance falls, the interest portion of a fixed-dollar repayment falls, so more of each successive payment is left to pay down the principal.