Mathematics Standard • Year 12 • Module 7 • Lesson 4

Annuities — Skill Drill

Build fluency identifying an annuity, applying FV = M[(1+r)^n − 1]/r and PV = M[1 − (1+r)^(−n)]/r, and distinguishing ordinary annuities from annuities due.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Define an annuity in one sentence.

An annuity is ____________________________________________________________.

Q1.2 State the difference between an ordinary annuity and an annuity due in one short sentence.

Ordinary: ______________________________________________________

Due: ______________________________________________________

Q1.3 Write the future-value and present-value formulas for an ordinary annuity.

FV = M × ____________________________________________

PV = M × ____________________________________________

Stuck? Revisit lesson § Key Ideas — Annuities, Ordinary vs Due, FV and PV formulas.

2. Worked example — future value of regular deposits

Problem. $200 is deposited at the end of each month into an account paying 4.8% p.a. compounded monthly for 3 years. Find the future value and the total interest earned.

Step 1 — Identify M, r per period, n.

M = $200/month r = 0.048/12 = 0.004 per month n = 3 × 12 = 36 months

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

FV = 200 × [(1.004)^36 − 1] / 0.004 = 200 × [1.15473 − 1] / 0.004 = 200 × 38.68 = $7,736.41

Step 3 — Total interest = FV − total deposited.

Deposited = 200 × 36 = $7,200. Interest = $7,736.41 − $7,200 = $536.41.

Conclusion. FV ≈ $7,736.41; interest earned ≈ $536.41.

3. Faded example — present value of a stream of payments

What lump sum today would generate $500/month for 5 years at 6% p.a. compounded monthly? Fill in the blanks. 4 marks

Step 1 — Identify M, r per period, n:

M = $ ________ r = 0.06/ ________ = ________ per month n = 5 × ________ = ________ months

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

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

PV = ________ × [1 − ________ ] / ________ = ________ × ________ = $ ____________

Conclusion. A lump sum of $ ____________ today is equivalent to $500 per month for 5 years at this rate.

Stuck? Revisit lesson § Present Value — discounting future payments to today's dollars.

4. Graduated practice — Annuities

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

Foundation — identify and convert (4 questions)

Q	Problem	Answer
4.1 1	Identify the variables M, r per period, n for: $100/month at 6% p.a. compounded monthly for 4 years.
4.2 1	Identify the variables M, r per period, n for: $250/quarter at 4.8% p.a. compounded quarterly for 6 years.
4.3 1	Identify the variables M, r per period, n for: $50/week at 5.2% p.a. compounded weekly for 2 years (52 weeks/year).
4.4 1	Classify the following as ordinary or annuity due: rent paid on the 1st of every month.

Standard — calculate FV or PV (6 questions)

4.5 $150 deposited monthly at 3.6% p.a. compounded monthly for 5 years. Find FV. 2 marks

4.6 $800 deposited quarterly at 6% p.a. compounded quarterly for 10 years. Find FV. 2 marks

4.7 Find the present value of $600 per month for 3 years at 5.4% p.a. compounded monthly. 2 marks

4.8 Find the present value of $1,000 paid semi-annually for 8 years at 4.8% p.a. compounded semi-annually. 2 marks

4.9 $400 deposited monthly at 7.2% p.a. compounded monthly for 4 years. Find FV. 2 marks

4.10 For Q4.9, calculate the total interest earned. 2 marks

Extension — interpret and convert (2 questions)

4.11 $300 is paid at the start of every month (annuity due) into an account earning 4.8% p.a. compounded monthly for 2 years. (i) Find the FV of the equivalent ordinary annuity. (ii) Adjust to FV_due using FV_due = FV_ord × (1 + r). 3 marks

4.12 Mia is offered $10,000 today or $500 per month for 2 years at 6% p.a. compounded monthly. Calculate the PV of the monthly stream and state which is more valuable today. 3 marks

Stuck on 4.12? Compute PV of $500/month for 24 months, then compare to $10,000.

5. Self-check the easy 3

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

For 4.1 I divided 0.06 by 12 to get r per period (0.005), and multiplied years by 12 for n.

For 4.5 I used n = 60 (months) not n = 5, and applied the FV formula (not PV).

For 4.7 I applied the PV formula with the negative exponent (1 + r)^(−n), not the FV formula.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Definition

An annuity is a sequence of equal payments made at equal time intervals.

Q1.2 — Ordinary vs annuity due

Ordinary: payments at the end of each period (e.g. mortgage). Due: payments at the beginning of each period (e.g. rent, insurance).

Q1.3 — Formulas

FV = M × [(1 + r)^n − 1] / r.
PV = M × [1 − (1 + r)^(−n)] / r.

Q3 — Faded example (PV of $500/month for 5 years at 6% monthly)

Step 1: M = $500, r = 0.06/12 = 0.005, n = 5 × 12 = 60 months.
Step 2: PV = 500 × [1 − (1.005)^(−60)] / 0.005 = 500 × [1 − 0.74137] / 0.005 = 500 × 51.726 = $25,862.78.
Conclusion: A lump sum of $25,862.78 today is equivalent.

Q4.1 — Identify M, r, n

M = $100, r = 0.06/12 = 0.005 per month, n = 4 × 12 = 48 months.

Q4.2 — Identify M, r, n

M = $250, r = 0.048/4 = 0.012 per quarter, n = 6 × 4 = 24 quarters.

Q4.3 — Identify M, r, n

M = $50, r = 0.052/52 = 0.001 per week, n = 2 × 52 = 104 weeks.

Q4.4 — Classify

Rent paid on the 1st of every month = annuity due (payment at the start of the period).

Q4.5 — FV of $150/month at 3.6% monthly for 5 years

r = 0.003, n = 60. FV = 150 × [(1.003)^60 − 1] / 0.003 = 150 × [1.19668 − 1] / 0.003 = 150 × 65.56 = $9,834.20.

Q4.6 — FV of $800/quarter at 6% quarterly for 10 years

r = 0.015, n = 40. FV = 800 × [(1.015)^40 − 1] / 0.015 = 800 × [1.81402 − 1] / 0.015 = 800 × 54.27 = $43,411.79.

Q4.7 — PV of $600/month at 5.4% monthly for 3 years

r = 0.0045, n = 36. PV = 600 × [1 − (1.0045)^(−36)] / 0.0045 = 600 × [1 − 0.85051] / 0.0045 = 600 × 33.22 = $19,932.50.

Q4.8 — PV of $1,000/half-year at 4.8% semi-annually for 8 years

r = 0.024, n = 16. PV = 1,000 × [1 − (1.024)^(−16)] / 0.024 = 1,000 × [1 − 0.68466] / 0.024 = 1,000 × 13.14 = $13,139.34.

Q4.9 — FV of $400/month at 7.2% monthly for 4 years

r = 0.006, n = 48. FV = 400 × [(1.006)^48 − 1] / 0.006 = 400 × [1.33247 − 1] / 0.006 = 400 × 55.41 = $22,164.74.

Q4.10 — Interest on Q4.9

Deposited = $400 × 48 = $19,200. Interest = $22,164.74 − $19,200 = $2,964.74.

Q4.11 — Annuity due adjustment, $300/month at 4.8% monthly for 2 years

(i) r = 0.004, n = 24. FV_ord = 300 × [(1.004)^24 − 1] / 0.004 = 300 × [1.10059 − 1] / 0.004 = 300 × 25.148 = $7,544.34.
(ii) FV_due = $7,544.34 × 1.004 = $7,574.52.

Q4.12 — $10,000 today vs $500/month for 24 months at 6% monthly

r = 0.005, n = 24. PV = 500 × [1 − (1.005)^(−24)] / 0.005 = 500 × [1 − 0.88719] / 0.005 = 500 × 22.563 = $11,281.34.
The monthly stream is more valuable today: PV $11,281.34 > $10,000 lump sum.