Perpetuities: QCE General Mathematics Unit 4 Loans, investments and annuities 2

Syllabus dot point

“Use the perpetuity formula A = d/i and solve practical problems involving perpetuities, including determining the total amount of the perpetuity, the periodic payment and the interest rate per compounding period”

QCEGeneral MathematicsUnit 4: Investing and networking13 min readUpdated 23 Sep 2026

Quick answer

A perpetuity pays the interest each period and never touches the principal, so it pays forever. Use $A = \frac{d}{i}$ for the amount needed, $d = Ai$ for the payment, and $i = \frac{d}{A}$ for the rate, with $i$ the rate per compounding period matching the payment period. An ordinary annuity from the same amount pays more per period but runs out.

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What this dot point is asking
The answer
Practice questions

What this dot point is asking

A perpetuity is an investment that pays a regular amount forever. It can do this because each payment is exactly the interest earned in that period, so the amount invested is never touched. The QCAA General Mathematics 2025 syllabus includes perpetuities in Topic 2 (Loans, investments and annuities 2): you must use the perpetuity formula $A = \frac{d}{i}$ and solve practical problems for the total amount of the perpetuity, the periodic payment and the interest rate per compounding period.

Perpetuities fund scholarships, prizes, charitable grants and some retirement plans. In the external assessment they usually appear as short simple familiar questions, or as part of a comparison with an ordinary annuity.

The answer

Why the balance never changes

Think of a perpetuity as an ordinary annuity in which the payment is set at exactly the right size. The balance follows the ordinary annuity recurrence:

A_{n+1} = rA_n - d, \qquad r = 1 + i.

Each period, interest of $iA_n$ is added and the payment $d$ is taken out. If $d = iA_n$ , these cancel:

A_{n+1} = (1 + i)A_n - iA_n = A_n.

The balance returns to the same value every period, so it can go on forever. For $2 500 000 at $1.3\%$ per quarter, the interest each quarter is $0.013 \times 2\,500\,000 = 32\,500$ , so a quarterly payment of $32 500 keeps the balance at $2 500 000:

A_1 = 1.013 \times 2\,500\,000 - 32\,500 = 2\,532\,500 - 32\,500 = 2\,500\,000.

The perpetuity formula

Since the payment equals the interest on the whole amount, $d = Ai$ , which rearranges to the formula in the QCAA formula book:

A = \frac{d}{i},

where $A$ is the total amount of the perpetuity (the amount that must be invested), $d$ is the periodic payment and $i$ is the interest rate per compounding period. The other two rearrangements follow:

d = Ai, \qquad i = \frac{d}{A}.

Key fact

Perpetuity: $A = \dfrac{d}{i}$ , so $d = Ai$ and $i = \dfrac{d}{A}$ . The payment equals the interest each period, so the balance never changes and the payments continue forever. $i$ must be the rate for the same period as the payment: monthly payments need the monthly rate.

Matching the rate to the payment period

The payment period and the compounding period must match. If payments are monthly and interest is quoted as 4.8% p.a. compounding monthly, use $i = \frac{0.048}{12} = 0.004$ . Using the annual rate by mistake gives a payment twelve times too large (and an amount twelve times too small).

Payment and compounding	Annual rate	$i$	Payment from $400 000
Annual	4.8%	0.048	$19 200 per year
Quarterly	4.8%	0.012	$4800 per quarter
Monthly	4.8%	0.004	$1600 per month

(Monthly and quarterly both give $19 200 over a year, because the interest is paid out as soon as it is earned rather than compounding.)

Perpetuity or ordinary annuity?

Compare the payment with the interest earned in the first period:

payment $=$ interest: perpetuity; the balance stays constant forever;
payment $>$ interest: ordinary annuity; the balance falls and eventually reaches zero;
payment $<$ interest: the balance grows; this is an investment still accumulating.

A perpetuity always pays less per period than an annuity from the same amount, because the annuity also spends the principal. From $600 000 at $0.35\%$ per month, the perpetuity pays $2100 per month forever, while an ordinary annuity lasting 25 years pays $3233.65 per month and then stops.

Solving practical problems

Most perpetuity questions reduce to one of three calculations:

How much must be invested? $A = \frac{d}{i}$ .
What payment will the investment provide? $d = Ai$ .
What interest rate is needed? $i = \frac{d}{A}$ , then multiply by the number of periods per year for the nominal annual rate.

Other questions test understanding: "What is the balance after 10 years?" (the same as the start), "How much of the principal is used?" (none), or "When does the total paid first exceed the amount invested?" (after more than $\frac{1}{i}$ payments, since each payment is $iA$ ).

Worked examples: amount, payment, rate and comparison

Amount needed for a scholarship

A $5000 scholarship is to be paid at the end of every year forever, from a fund earning 4% p.a. compounding annually.

Identify: $d = 5000$ , $i = 0.04$ .
Formula: $A = \frac{5000}{0.04} = 125\,000$ .
Answer: $125 000.

Marker's note: check reasonableness: $4\%$ of $125 000 is $5000, exactly one year's payment.

Payment from a perpetuity with monthly compounding

$400 000 is invested at 4.8% p.a. compounding monthly, paid monthly.

Rate per period. $i = 0.004$ .

Payment. $d = 400\,000 \times 0.004 = 1600$ per month.

Marker's note: $1600 per month is $19 200 per year, which is $4.8\%$ of $400 000. The annual check catches a wrong $i$ .

Rate needed

A retiree with $800 000 wants $3000 per month forever.

Rate per month. $i = \frac{3000}{800\,000} = 0.00375$ .

Nominal annual rate. $0.00375 \times 12 = 0.045$ , so 4.5% p.a. compounding monthly.

Marker's note: state whether your answer is a rate per period or per annum.

Perpetuity versus annuity

$600 000 at 4.2% p.a. compounding monthly: compare a perpetuity with a 25-year ordinary annuity.

Perpetuity: $d = 600\,000 \times 0.0035 = 2100$ per month, forever.
Annuity: $d = \frac{600\,000 \times 0.0035}{1 - 1.0035^{-300}} \approx 3233.65$ per month for 25 years.
Interpretation: The annuity pays about $1133.65 more each month because it also returns the principal, but it stops after 25 years; the perpetuity keeps the $600 000 intact.

Marker's note: in complex familiar questions, the comparison is where the marks are. Say what each option gives up.

Common traps

Using the annual rate with monthly payments: $i$ must match the payment period.
Multiplying instead of dividing: The amount needed is $\frac{d}{i}$ , which is much bigger than the payment. If your answer is smaller than one payment, you multiplied.
Using a percentage instead of a decimal: $4\%$ is $i = 0.04$ , not 4.
Thinking the balance falls: In a perpetuity it never falls; only the interest is paid.
Forgetting to convert the rate back: If asked for an annual rate, multiply the rate per period by the number of periods per year.

Exam technique

Write the formula, then list $A$ , $d$ and $i$ with the unknown marked, before substituting. Always do a one-line reasonableness check: one period's interest on your amount should equal one payment. In comparison questions, calculate both options, then write a sentence on what each option provides and for how long.

Note

Imagine putting money in the bank and only ever spending the interest. The original money stays there, so you can keep taking the same amount of interest forever. That is a perpetuity, and it is how many scholarships and prizes are funded. To work out how much you need to put in, divide the yearly payment you want by the interest rate: $5000 a year at 4% needs $125 000, because 4% of $125 000 is $5000.

Practice questions

Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.

foundation2 marks

A school wants a scholarship of $5000 to be paid at the end of every year forever. The fund earns 4% p.a. compounding annually. How much must be invested?

Show worked solution →

Use $A = \frac{d}{i}$ with $d = 5000$ and $i = 0.04$ (annual payment, annual compounding). (1 mark)

A = \frac{5000}{0.04} = 125\,000.

$125 000 must be invested. (1 mark)

foundation2 marks

$400 000 is invested in a perpetuity earning 4.8% p.a. compounding monthly, paying out monthly. Find the monthly payment.

Show worked solution →

$i = \frac{0.048}{12} = 0.004$ per month. (1 mark)

$d = Ai = 400\,000 \times 0.004 = 1600$ . The perpetuity pays $1600 per month. (1 mark)

foundation1 mark

A $250 000 perpetuity pays $2500 every quarter. Find the interest rate per quarter.

Show worked solution →

$i = \frac{d}{A} = \frac{2500}{250\,000} = 0.01$ , which is $1\%$ per quarter (a nominal $4\%$ p.a. compounding quarterly).

core3 marks

A perpetuity is modelled by

A_{n+1} = 1.013A_n - 32\,500

A_0 = 2\,500\,000

, with

n

in quarters. (a) State the interest rate per quarter and per annum. (b) Show that the balance after one quarter is still $2 500 000. (c) Explain why the balance never changes.

Show worked solution →

(a) $r = 1.013 = 1 + i$ , so $i = 0.013$ : $1.3\%$ per quarter, $5.2\%$ p.a. compounding quarterly. (1 mark)

(b) $A_1 = 1.013 \times 2\,500\,000 - 32\,500 = 2\,532\,500 - 32\,500 = 2\,500\,000$ . (1 mark)

(c) Each quarter the interest added is $0.013 \times 2\,500\,000 = 32\,500$ , exactly the payment taken out. Interest in equals payment out, so the balance returns to $2 500 000 every quarter, forever. (1 mark)

core2 marks

A community group wants to pay a $1500 prize every six months forever. The fund earns 3.2% p.a. compounding six-monthly. How much must be invested?

Show worked solution →

$i = \frac{0.032}{2} = 0.016$ per six months. (1 mark)

$A = \frac{1500}{0.016} = 93\,750$ . $93 750 must be invested. (1 mark)

core2 marks

A retiree has $800 000 and wants $3000 per month forever from a perpetuity with monthly compounding. What nominal annual interest rate is needed?

Show worked solution →

$i = \frac{d}{A} = \frac{3000}{800\,000} = 0.00375$ per month. (1 mark)

Nominal annual rate $= 0.00375 \times 12 = 0.045$ , which is $4.5\%$ p.a. compounding monthly. (1 mark)

exam4 marks

A retiree invests $600 000 at 4.2% p.a. compounding monthly. (a) Find the monthly payment if the investment is a perpetuity. (b) Find the monthly payment if it is instead an ordinary annuity that pays out over exactly 25 years, using

A_{PV} = d\left(\frac{1 - (1 + i)^{-n}}{i}\right)

. (c) Explain the difference.

Show worked solution →

(a) $i = \frac{0.042}{12} = 0.0035$ ; $d = 600\,000 \times 0.0035 = 2100$ . $2100 per month. (1 mark)

(b) $n = 300$ . Rearrange: $d = \frac{A_{PV} \times i}{1 - (1 + i)^{-n}} = \frac{600\,000 \times 0.0035}{1 - 1.0035^{-300}} \approx 3233.65$ . $3233.65 per month. (2 marks)

(c) The perpetuity pays only the interest, so the $600 000 is never touched and payments continue forever. The annuity pays interest plus part of the principal each month, so it pays more (about $1133.65 more per month) but the balance falls to zero after 25 years. (1 mark)

exam2 marks

A perpetuity earns 0.4% per month and pays the interest monthly. After how many monthly payments does the total amount paid out first exceed the amount invested?

Show worked solution →

Each payment is $d = 0.004A$ , so after $n$ payments the total paid is $0.004An$ . (1 mark)

$0.004An > A$ gives $n > \frac{1}{0.004} = 250$ . The total first exceeds the amount invested after 251 payments (about 21 years). The amount invested is still in the fund, untouched. (1 mark)

Practise this

Flashcards General Mathematics deck

Sources & how we know this

General Mathematics General senior syllabus 2025 — QCAA