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TASMathematics ApplicationsSyllabus dot point

How does an investment pay out a regular income, and when can it last forever?

Model annuities and perpetuities using recurrence relations and analyse the regular payments they support.

Annuities that pay a regular income from a lump sum, the recurrence model, and perpetuities that pay forever, in TCE Mathematics Applications.

Generated by Claude Opus 4.88 min answerUpdated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking

An annuity reverses the idea of a loan. Instead of borrowing and repaying, you invest a lump sum that earns interest and then draws down a fixed regular payment until it runs out. A perpetuity is the limiting case that pays out forever.

The annuity recurrence

The balance of an annuity behaves exactly like a loan balance: interest is added, then the payment is withdrawn.

Drawing down an annuity

A retiree invests $\$50\,000$ at 0.5 percent per month and withdraws $\$1000$ each month.

Step 1: Identify the parameters

The initial balance is $A_0 = 50000$ , the monthly interest rate is $r = 0.005$ , and the monthly withdrawal is $d = 1000$ . These values go directly into the annuity recurrence $A_{n+1} = A_n(1 + r) - d$ , so it is worth checking them carefully before calculating.

Step 2: Apply the recurrence for the first two months

$A_1 = 50000(1.005) - 1000 = 50250 - 1000 = 49250$ .
$A_2 = 49250(1.005) - 1000 = 49496.25 - 1000 = 48496.25$ .

Step 3: Interpret the pattern

In month 1 the fund earns $\$250$ in interest, so only $\$750$ of the $\$1000$ withdrawal comes from principal. Because the withdrawal exceeds the interest earned, the balance falls a little each month. As the balance shrinks, the interest earned also falls, which means more and more principal is consumed with each subsequent withdrawal and the fund drains at an accelerating rate.

Perpetuities

A perpetuity pays forever because each payment is set exactly equal to the interest earned. The principal is never touched, so the balance stays constant period after period.

Funding a perpetual scholarship

A university wants to pay a $\$3000$ scholarship every year forever from a fund earning 5 percent per annum. How much must be invested?

Step 1: State the perpetuity condition

For a fund to last forever, the annual payment must equal the interest earned on the principal. This means each year the interest replaces exactly what is withdrawn, so the balance never falls. The perpetuity formula $A = \frac{d}{r}$ comes directly from setting $d = A \times r$ and solving for the required lump sum $A$ .

Step 2: Substitute the known values

$d = 3000$ , $r = 0.05$ .

A = \frac{d}{r} = \frac{3000}{0.05} = \$60\,000.

Step 3: Verify the result makes sense

Investing $\$60\,000$ at 5 percent per annum earns exactly $0.05 \times 60000 = \$3000$ each year. That $\$3000$ is paid out as the scholarship, leaving the principal untouched at $\$60\,000$ . Because the principal never changes, the interest earned each year is always $\$3000$ , and the scholarship can continue indefinitely without any additional funding.

How long an annuity lasts

For an ordinary annuity where the withdrawal exceeds the interest, the balance falls each period and eventually reaches zero. You can iterate the recurrence to find the period in which the balance first hits or passes zero, which tells you how long the income stream survives.

Using the finance solver for annuities

For long drawdowns it is impractical to iterate by hand, so TASC expects the finance (TVM) solver. Enter the present value as the lump sum invested, the per-period interest rate, the regular payment as the withdrawal, and solve for the unknown, usually the number of periods $n$ or the payment that empties the fund in a set time. The sign convention matters: the lump sum invested is entered as a negative present value (money paid in) and the withdrawal as a positive payment (money received), or consistently the other way round. Getting the signs consistent is what makes the solver return a sensible answer rather than an error.

Reading the finance-solver inputs

A $\$200\,000$ annuity earns $6\%$ p.a. compounding monthly and pays $\$1500$ per month.

Step 1: Convert the annual rate to a monthly rate

The annual rate of $6\%$ must be divided by 12 to match the monthly payment frequency. Monthly rate $i = \frac{0.06}{12} = 0.005$ . Using the annual rate directly would overstate the interest earned each period and give an incorrect answer for $N$ .

Step 2: Enter all inputs into the finance solver

Set present value $PV = 200000$ (the lump sum invested), payment $PMT = -1500$ (entered as negative because money is leaving the fund each month), and future value $FV = 0$ (the fund is fully drawn down at the end). The monthly rate $i = 0.005$ is also entered. With all other fields known, the solver can find $N$ , the number of monthly payments.

Step 3: Read and interpret the solver output

The solver returns a value of $N$ near 248 months, so the annuity lasts about 20 years and 8 months. In the early months the fund earns interest of about $\$1000$ per month at the $0.5\%$ rate, which means only about $\$500$ of the $\$1500$ withdrawal comes from principal. Because so little principal is consumed at the start, the balance remains large for many years and the fund sustains payments for a considerable period before it is exhausted.

A complete answer models the annuity with $A_{n+1} = A_n(1 + r) - d$ , uses $A = d/r$ for a perpetuity, and states clearly whether the fund runs down, holds steady, or grows by comparing the payment with the interest earned.

Exam-style practice questions

Practice questions written in the style of TASC exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

TCE 20244 marksa) An annuity in advance means the regular payments happen before the lump sum is delivered or achieved. Explain how a savings account might fall into this category. b) An annuity in arrears means the lump sum is delivered before the regular payments happen. Explain how a housing loan might fall into this category.

Show worked answer →

a) (2 marks) In a savings annuity you make the regular deposits first, over many periods, and the lump sum (the accumulated balance) is achieved at the end. Because each payment is made at the start of its period and then earns interest before the goal is reached, the deposits come before the lump sum, so it is an annuity in advance.

b) (2 marks) With a housing loan the bank delivers the lump sum (the loan) at the start, and the borrower then makes regular repayments afterwards. The lump sum precedes the stream of payments, which is the defining feature of an annuity in arrears.

TCE 20243 marksA retiree aged 65 has accumulated

\

600\,000

in superannuation. The fund averages

9\%

p.a. compounding monthly. They need a monthly payment of

4800

. How long will the fund last? Give your answer in years and months.

Show worked answer →

(3 marks) This is an annuity in arrears being drawn down. Use the finance solver (or the annuity formula) with present value $P = 600000$ , monthly rate $i = \frac{0.09}{12} = 0.0075$ , withdrawal $A = 4800$ , solving for $n$ .

Each month the fund earns interest then $\$4800$ is withdrawn, so the balance falls slowly. Solving gives $n = 371$ months (approximately). Converting: $\frac{371}{12} = 30.9$ years, which is 30 years and 11 months. Markers expect care converting the decimal months ( $0.9$ of a year is 11 months, not 9) and showing the finance-solver inputs.