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Annuities & perpetuities
Same idea, four ways to study it. Tap a style and find the one that clicks for you.

What they are

An annuity is a pot of money that earns interest while you draw a fixed payment out of it each period. It runs down over a set number of periods, the same machine as a loan, just with the money flowing to you. A perpetuity is the special case where you only ever take out the interest, so the pot never shrinks and pays out forever.

The two key tools

annuity, each period: new = old × (1 + i) − withdrawal
perpetuity: PV = payment ÷ i

i = interest rate per period (as a decimal)  ·  PV = the lump sum you need up front.

A perpetuity works because the payment equals exactly one period's interest, so the balance is untouched.

Worked example, perpetuity

You want $2,000 a year, forever, from a fund paying 5% per annum. How much must you invest now?

  1. Use PV = payment ÷ i
  2. PV = 2000 ÷ 0.05
  3. PV = $40,000
  4. Check: 5% of $40,000 = $2,000, exactly the payout, so the fund never drops.

Worked example, annuity drawdown

$10,000 invested at 5% per year, withdrawing $3,000 at the end of each year.

End of yearCalculationBalance
Startopening$10,000.00
110000 × 1.05 − 3000$7,500.00
27500 × 1.05 − 3000$4,875.00
34875 × 1.05 − 3000$2,118.75

Because you withdraw more than the interest earned ($3,000 vs about $500), the pot runs down.

How a marker wants it laid out

The drawdown question, written the QCAA way. Each line earns its own tick.

i = 5% = 0.05,   withdrawal = $3000 ✓ states the rate and payment
An+1 = 1.05 An − 3000 ✓ writes the recurrence
A0 = 10 000
A1 = 1.05 × 10 000 − 3000 = 7500
A2 = 1.05 × 7500 − 3000 = 4875 ✓ shows the iteration
After 2 years the fund holds $4875. ✓ answer in context

A perpetuity is shorter: PV = payment ÷ i, then the answer in a sentence. Add a reasonableness line if the question makes a claim.

Watch out

• An annuity runs out after a set number of periods. A perpetuity pays forever. Read which one the question wants.
• For a perpetuity, divide by i as a decimal (5% → 0.05), not by 5.
• In the drawdown table, interest goes on first, then the withdrawal comes off.

Perpetuity, colour coded

PV = payment ÷ i
payment = what you take each period i = interest rate, as a decimal

$2,000 a year at 5% → 2000 ÷ 0.05 = $40,000 in the fund.

Annuity: one step per year (it runs down)

$10,000×1.05 −3000 → $7,500×1.05 −3000 → $4,875×1.05 −3000 → $2,118.75

Same machine as a loan, money flows to you.

Annuity drops, perpetuity holds

perpetuity (flat, forever) annuity (runs out) 0 time $

Take only the interest and the line stays flat (perpetuity). Take more, and it slopes down to zero (annuity).

Warm up first

Don't read yet, just have a go in your head:

Pays out forever without shrinking, what's it called?
A perpetuity. An annuity stops after a set number of periods.
$1,000 a year forever at 10%. Lump sum needed?
PV = 1000 ÷ 0.10 = $10,000.

Faded example: perpetuity, then a drawdown

Rung 1 · watch one done fully

$5,000 a year forever at 4%: PV = 5000 ÷ 0.04 = $125,000.

Rung 2 · you fill the gaps

$3,000 a year forever at 6%: PV = 3000 ÷ ? = ?

Check my gaps
3000 ÷ 0.06 = $50,000.
Rung 3 · all you

Annuity: $10,000 at 5%, withdraw $3,000 a year. Find the balance after 2 years, then check.

Check my answer
10000 × 1.05 − 3000 = $7,500, then 7500 × 1.05 − 3000 = $4,875.

Say it back

In one sentence, out loud: what makes a perpetuity last forever when an annuity runs out? If you can say it, you've got it.

⚡ Annuities & perpetuities, one look

PerpetuityPV = payment ÷ i  (forever)
Annuitynew = old × (1 + i) − withdrawal  (runs out)
Tell apart"forever" → perpetuity · "for N years" → annuity
Example$2,000/yr forever at 5% → 2000 ÷ 0.05 = $40,000
Trapdivide by 0.05, not 5 · interest before the withdrawal
On the Casio10000= then Ans×1.05−3000= =

The Ans key trick: type the starting fund, press =, then build the rule once and keep pressing = to step the drawdown table.