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Year 12 Maths Advanced (2027) Financial mathematics

Annuities & Future Value

20 practice questions 0 video lessons Theory + worked examples

Learn annuities and future value for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-08). An annuity is a plan of equal, regular payments that earn compound interest, such as a savings scheme or superannuation, and its future value is what it has grown to at the end.

You will learn to treat each contribution as growing separately, add them using the sum of a geometric series to find the future value, and apply it to superannuation and other regular savings plans — practical saving skills.

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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.

In this course each payment is made at the end of a period, and the future value is found straight after the last payment. So with \(n\) payments of \(M\), the first payment earns interest for \(n-1\) periods and the last payment earns none.

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
\[\text{FV}=M+M(1+r)+\dots+M(1+r)^{n-1}=\dfrac{M\big[(1+r)^{n}-1\big]}{r}\]
for n end-of-period payments of M, the future value is M plus M(1 plus r) up to M(1 plus r) to the n minus 1, which sums to M times ((1 plus r) to the n minus 1) over r

Method

  1. Grow each contribution to the end date (the last payment, made at the end, earns no interest).
  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.
Payments are at the end of each period: with \(n\) payments the first grows for \(n-1\) periods, not \(n\).
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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