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Year 12 Maths Advanced (2027) Financial mathematics

Annuities & Future Value

20 practice questions 0 video lessons Theory + worked examples
NSW · Year 12 Mathematics Advanced · Financial mathematics

Annuities are series of equal payments; their future value is the total an investment grows to, found by summing the compounded contributions as a geometric series.

Part of the NSW Year 12 Mathematics Advanced course, in the Financial mathematics focus area of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.

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Theory

An annuity is built from regular equal contributions earning compound interest; its future value is the total plus interest. This Year 12 Mathematics Advanced topic (MAV-12-08).

Each contribution grows by compound interest for the time it is invested; the future value is the sum of these grown contributions (a geometric series). A single amount grows by \(P(1+r)^{n}\) over \(n\) periods.

Contributions growEach contribution grows by compound interest for the time it is invested. xy yr3 yr2 yr1
Each contribution grows for a different length of time.
\[\text{single amount: }P(1+r)^{n};\quad \text{FV}=\text{sum of grown contributions}\]
a single amount grows by P times (1 plus r) to the n; future value sums the grown contributions

Method

  1. Grow each contribution to the end date.
  2. Add them (a geometric series) for the future value.
  3. Total contributed is payment times number of payments.
Example 1 — One amount
\(\$1000\) for \(3\) years at \(10\%\). Find its value.
Solution
\(A\)\(=\)\(1000(1.10)^{3}=\$1331\)
Example 2 — Two payments
\(\$1000\) at the end of year 1 and year 2 at \(10\%\). Value at end of year 2?
Solution
\(\ \)\(=\)\(1000(1.10)+1000=\$2100\)
Example 3 — Future value
\(\$1000\) at the end of each year for \(3\) years at \(8\%\). Find the future value.
Solution
\(\ \)\(=\)\(1000(1.08^{2}+1.08+1)\)
\(=\)\(\$3246.40\)
Example 4 — Total contributed
\(\$150\) monthly for \(12\) months. How much is contributed (before interest)?
Solution
\(\ \)\(=\)\(150\times12=\$1800\)

Common pitfalls

Each contribution grows for a different time.
The future value is a geometric-series sum.
Total contributed ignores interest.

Frequently asked questions

What is an annuity?

An investment built from regular equal contributions that each earn compound interest.

What is the future value of an annuity?

The total of all the contributions plus the interest they have earned, found by summing a geometric series.

How does a single investment grow?

By P times (1 plus r) to the power n over n periods of compound interest.

How do you find the total contributed?

Multiply the regular payment by the number of payments, ignoring interest.

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