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hscscience Maths Adv · Y12
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Module 7 · L7 of 20 ~45 min ⚡ +95 XP available

Introducing Annuities, Future Value

Most people don't invest once and wait, they contribute regularly. Monthly into super, fortnightly into savings, annually into an education fund. An annuity is the mathematical model for this reality: regular payments plus compound interest, growing together into a future sum. You'll derive the formula from geometric sequences and see why it powers every retirement calculator on Earth.

Today's hook, Your employer deposits $\$9,200$ into your super every year for 40 years. That's $\$368,000$ in total contributions. But with compounding, the actual balance at retirement is closer to $1.84 million. Where does the extra $1.47 million come from?
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Get oriented

Recall what you already know, meet the key ideas and settle the terms.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question
01
Recall, your gut answer first
+5 XP warm-up

You deposit $500 at the end of every month into an account paying 0.5% per month. After 12 months, will your balance be: A) Exactly $6,000?   B) Slightly more than $6,000?   or C) Significantly more?

Make a prediction and explain your reasoning, no formula yet.

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02
What is an annuity?
+5 XP to read

An annuity is a series of equal payments made at regular intervals, with each payment earning compound interest. The two most common types in the HSC:

Ordinary annuity
Payments at the end of each period. This is the standard case, the formula in this lesson applies here.
Annuity due
Payments at the start of each period. Covered in Lesson 9. Result: multiply the ordinary FV by $(1+r)$.
Real examples
Superannuation, mortgage repayments, car loan instalments, regular savings deposits, all annuities.
Australian Superannuation. Employers contribute 11.5% of your salary to super. On an $\$80,000$ salary that's $\$9,200$/year. At 7% p.a. over 40 years, this grows to approximately $1.84 million, not from a lump sum, but from the quiet mathematics of regular contributions compounding decade after decade.
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What you'll master
Know

Key facts

  • The future value annuity formula
  • The difference between ordinary annuity and annuity due
  • That $r$ and $n$ must match the contribution frequency
Understand

Concepts

  • How the annuity formula is derived from the GP sum
  • Why regular contributions create accelerating growth
  • The difference between total contributions and future value
Can do

Skills

  • Calculate FV for any ordinary annuity
  • Convert annual rates and years to match monthly/quarterly contributions
  • Transpose the formula to find $a$, $r$, or $n$
  • Model superannuation and savings scenarios
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Key terms
AnnuityA series of equal payments made at regular intervals, each earning compound interest.
Future Value (FV)The total accumulated value of all payments plus interest at the end of $n$ periods.
Ordinary annuityPayments are made at the end of each period.
Contribution ($a$)The regular payment amount per period.
Rate per period ($r$)The interest rate expressed as a decimal, matched to contribution frequency.
Geometric seriesA sum of terms with a constant ratio, the mathematical backbone of the annuity formula.

Derive the future value formula

Build the annuity future value formula from the payments it sums.

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Deriving the future value formula
core concept

Consider $a$ deposited at the end of each period for $n$ periods at rate $r$ per period.

The last payment earns no interest. The second-last earns one period. The first earns $(n-1)$ periods of interest:

$$FV = a + a(1+r) + a(1+r)^2 + \dots + a(1+r)^{n-1}$$

This is a geometric series with:

  • First term: $a$
  • Common ratio: $(1+r)$
  • Number of terms: $n$

Using the GP sum formula $S_n = \dfrac{a(r^n - 1)}{r - 1}$ with ratio $(1+r)$:

Substitute into the sum formula and simplify, the denominator becomes simply $r$:

$$FV = a \times \dfrac{(1+r)^n - 1}{r}$$
Critical point. This formula assumes payments at the end of each period (ordinary annuity). If payments are at the start (annuity due), multiply the result by $(1+r)$.
Annuity Contribution Timeline a a a a a FV Period 1 Period 2 Period 3 Period 4 Period n End-of-period contributions compound forward to FV

Each contribution earns a different number of periods of interest, the first earns the most, the last earns none.

FV annuity formula: $FV = a \times \dfrac{(1+r)^n - 1}{r}$; Derived from GP sum with first term $a$, ratio $(1+r)$, $n$ terms

Pause, copy the FV annuity formula $FV = a \times \dfrac{(1+r)^n - 1}{r}$, derived from the GP sum with first term $a$ and ratio $(1+r)$, into your book.

Did you get this? True or false: in the future value annuity formula, the annuity payments form a geometric series with common ratio $(1+r)$.

Work the examples

Follow three worked future value problems end to end.

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · BASIC FV CALCULATION

You deposit $300 at the end of each month into an account earning 4.8% p.a. compounded monthly. What is the balance after 5 years?

1
$r = \dfrac{0.048}{12} = 0.004$, $\quad n = 5 \times 12 = 60$
Convert annual rate to monthly; convert years to months, frequency must match.
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PROBLEM 2 · FORTNIGHTLY SUPER

Jax contributes $450 fortnightly to his super fund returning 6.4% p.a. compounded fortnightly. He works for 25 years. Find (a) his balance at retirement and (b) how much is contributions vs interest.

1
$r = \dfrac{0.064}{26} = 0.002462$, $\quad n = 25 \times 26 = 650$
26 fortnights per year.
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PROBLEM 3 · QUARTERLY SAVINGS

Find the future value of $200 deposited at the end of each quarter for 8 years at 5.2% p.a. compounded quarterly.

1
$r = \dfrac{0.052}{4} = 0.013$, $\quad n = 8 \times 4 = 32$
Quarterly frequency: divide rate by 4, multiply years by 4.
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Quick check: $500 is deposited monthly for 2 years at 6% p.a. compounded monthly. Which values are correct for the formula?

Avoid the traps

Fix the timing and period-count slips that cost marks.

Common errors · the 3 traps that cost marks

Trap 01
Using the annual rate with monthly contributions
The rate $r$ in the formula must match contribution frequency. Monthly contributions need $r_\text{annual}/12$. Using the annual rate directly gives a wildly wrong answer, often 10× too large.
Trap 02
Confusing FV with total contributions
$n \times a$ gives only total contributions. The FV formula adds the interest every payment has earned. For $\$300$/month at 4.8% over 5 years, contributions are $\$18,000$ but FV is $20,287, the difference is compound interest.
Trap 03
Rounding $r$ early
If you round $r = 0.064/26$ to 3 decimal places as 0.002, you introduce compounding errors over 650 periods. Store the full value in your calculator memory and only round the final answer.

Write the two-step process to convert an annual interest rate and number of years into the correct $r$ and $n$ for quarterly contributions. Give an example with 5% p.a. over 3 years.

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Drill it and revisit

Run the drill, then return to your opening answer and name what has changed.

Quick-fire practice · 4 drills

1

Find $FV$ for $a = \$400$, $r = 5\%$ per period, $n = 12$ periods.

2

Find $FV$ for $a = \$1{,}000$, $r = 6\%$ per period, $n = 20$ periods.

3

$\$250$ per month for 4 years at 3.6% p.a. compounded monthly. Find $FV$.

4

$\$600$/month for 35 years at 7% p.a. compounded monthly. What is $FV$ and how much is interest?

Fill in the blanks. The future value of an ordinary annuity with contribution $a$, rate per period $r$, and $n$ periods is: $FV = a \times \dfrac{(\underline{\phantom{1+r}})^n - \underline{\phantom{1}}}{r}$. The formula is derived from a __________ series with common ratio $(1+r)$.

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Match each scenario to the correct $r$ and $n$ values.

Scenario A: Monthly contributions, 4% p.a., 3 years → $r = $ __, $n = $ __

Scenario B: Quarterly contributions, 6% p.a., 5 years → $r = $ __, $n = $ __

Scenario C: Fortnightly contributions, 5.2% p.a., 2 years → $r = $ __, $n = $ __

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Revisit your thinking

Earlier you predicted whether $\$500$/month at 0.5%/month for 12 months would give exactly $\$6,000$, slightly more, or significantly more. The answer is B, slightly more than $6,000:

$$FV = 500 \times \dfrac{(1.005)^{12} - 1}{0.005} = 500 \times 12.336 = \$6{,}168.03$$

The interest is only $\$168$ because the time is short and each contribution has limited time to compound. Over 10 years, interest would be $\$2,300$. Over 30 years, $14,000. Time is the critical variable in annuity growth.

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Practise annuity future value

Write full short-answer responses, then check them against the model answers.

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Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence. That tells the system what to drill next. Each retry pulls a fresh mix from the bank.

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Short answer
ApplyBand 43 marks

Q1. $250 is deposited at the end of each month for 5 years into an account earning 6% p.a. compounded monthly. Find the future value and the total interest earned. (3 marks)

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ApplyBand 43 marks

Q2. Ari wants a super balance of $\$600,000$ at retirement in 30 years. The fund earns 6% p.a. compounded monthly. What monthly contribution $a$ does she need? (3 marks)

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AnalyseBand 54 marks

Q3. (a) $800 per month is invested for 20 years at 4.8% p.a. compounded monthly. Find the FV and interest earned. (b) Explain why the interest earned exceeds the total contributions. (4 marks)

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Comprehensive answers (click to reveal)

Drill 1: $FV = 400 \times [(1.05)^{12}-1]/0.05 = \$6{,}398.51$   2: $FV = 1{,}000 \times [(1.06)^{20}-1]/0.06 = \$36{,}785.59$   3: $r=0.003$, $n=48$, $FV = \$12{,}726.86$   4: $r=0.005833$, $n=420$, $FV \approx \$1{,}082{,}537$; contributions $= \$252{,}000$; interest $= \$830{,}537$

Match: A: $r=0.0\overline{3}$, $n=36$ | B: $r=0.015$, $n=20$ | C: $r=0.002$, $n=52$

Q1 (3 marks): $r = 0.06/12 = 0.005$, $n = 60$ [1]. $FV = 250 \times [(1.005)^{60}-1]/0.005 = \$17{,}442.51$ [1]. Interest $= \$17{,}442.51 - \$15{,}000 = \$2{,}442.51$ [1].

Q2 (3 marks): $600{,}000 = a \times [(1.005)^{360}-1]/0.005$ [1]. $a = 600{,}000/1{,}004.52 = \$597.30$/month [2].

Q3 (4 marks): (a) $r=0.004$, $n=240$. $FV = 800 \times [(1.004)^{240}-1]/0.004 = \$396{,}287.29$ [2]. Interest $= \$396{,}287.29 - \$192{,}000 = \$204{,}287.29$ [1]. (b) Later payments have up to 20 years to compound, generating interest on interest, the exponential effect of compounding over many periods [1].