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.
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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The balance of an annuity behaves exactly like a loan balance: interest is added, then the payment is withdrawn.
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.
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.
A complete answer models the annuity with , uses 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.
2024 TASC General Mathematics4 marksQuestions about annuities. a) Annuity in Advance means the regular payments happen before the lump sum of money is delivered or achieved. Explain how a savings account might fall into this category. b) Annuity in Arrears means the lump sum of money 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.
2024 TASC General Mathematics3 marksA recent retiree, aged 65, has accumulated 4800 from this fund. How long will the superannuation 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 calculator's finance solver or the annuity formula with present value P = 600000, monthly rate i = 0.09/12 = 0.0075, withdrawal A = 4800, solving for n.
Each month the fund earns interest and then $4800 is withdrawn, so the balance falls slowly. Solving gives n = 371 months (approximately). Converting: 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.