Savings annuities | Year 12

01 / LEARN

Understand the idea.

Follow each deposit to the common end date

An ordinary savings annuity has equal deposits at the ends of equal periods. Measure the final balance immediately after the last deposit: that deposit earns no interest, the previous one earns for one period, and earlier deposits earn for longer.

The accumulated deposits form a geometric sum, giving $F V = R \frac{{(1 + i)}^{n} - 1}{i}$ for $i \neq 0$ . The rate $i$ must cover one deposit interval. This formula assumes a constant rate, equal deposits and no fees.

F V = R \frac{{(1 + i)}^{n} - 1}{i}

Add initial savings or change deposit timing

Treat money already present at time zero separately. It grows for all $n$ periods, so the total is $P {(1 + i)}^{n} + R \frac{{(1 + i)}^{n} - 1}{i}$ . Adding these components puts every cash flow at the same final date.

If all $n$ deposits instead occur at the beginnings of periods, each earns one extra period: multiply only the deposit component by $1 + i$ . Count each deposit once. At zero interest use $F V = P + n R$ , since no money earns growth.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Build the model

Deposit AUD 200 at each year end for 5 years at 4% annually. Find the balance immediately after the fifth deposit.

Worked solution

1
Identify how long the five deposits earn interest.
$4, 3, 2, 1, 0 years$
The final balance is measured immediately after the fifth year-end deposit.
2
Accumulate all deposits to the fifth year end.
$\begin{aligned} F V & = 200 {(1.04)}^{4} + 200 {(1.04)}^{3} \\ + 200 {(1.04)}^{2} + 200 (1.04) + 200 \end{aligned}$
Each deposit is multiplied for exactly the time it remains in the account.
3
Evaluate the geometric sum.
$\begin{aligned} F V & = 200 \frac{{(1.04)}^{5} - 1}{0.04} \\ = 1083.264512 \approx AUD 1083.26 \end{aligned}$
The ordinary-annuity factor sums the five accumulated deposits.

Answer

AUD 1,083.26

EXAMPLE 02

Use the model to make a decision

Start with AUD 1,000 and add AUD 150 at each year end for 4 years at 5%. Find the final balance.

Worked solution

1
Grow the initial savings for four years.
$1000 {(1.05)}^{4} = 1215.50625$
The starting AUD 1,000 is present for all four annual interest periods.
2
Accumulate the four year-end deposits.
$150 \frac{{(1.05)}^{4} - 1}{0.05} = 646.51875$
These deposits earn for three, two, one and zero years respectively.
3
Combine amounts at the same final date.
$\begin{aligned} F V & = 1215.50625 + 646.51875 \\ = 1862.025 \approx AUD 1862.03 \end{aligned}$
The total balance contains both the grown initial savings and the grown deposits.

Answer

AUD 1,862.03

03 / YOUR TURN

Try it yourself.

Check your understanding

Deposit AUD 250 at each year end for 6 years in an account paying no interest. Find the final balance.

Show answer & reasoning

AUD 1,500.00

There is no interest, so add the six deposits directly: $F V = 6 \times 250 = 1500$ . The final balance is AUD 1,500.00; the fraction formula need not be used at $i = 0$ .

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

Read the questions online

Deposit AUD 80 at each month end for 12 months. Interest is 0.4% monthly. Find the final balance.
3 marks
Deposit AUD 250 at each year end for 6 years in an account paying no interest. Find the final balance.
3 marks
Deposit AUD 300 at each year beginning for 5 years at 3%. Measure the balance at time 5. Find it.
3 marks
An ordinary annuity ends with AUD 6,200 after 10 deposits of AUD 500. How much interest has been earned?
3 marks
At 5% annually, compare three end-of-year deposits of AUD 1,000 with one AUD 3,000 deposit at time 0, both measured at time 3.
3 marks
An account has AUD 2,000 now. With 4% annual interest and AUD 400 deposits at each year end, write and calculate its balance after 2 years.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Deposit AUD 80 at each month end for 12 months. Interest is 0.4% monthly. Find the final balance.
3 marks
Use $i = 0.004$ and $n = 12$ .
Balance is $\frac{80 ({(1.004)}^{12} - 1)}{0.004}$ .
AUD 981.40
Deposit AUD 250 at each year end for 6 years in an account paying no interest. Find the final balance.
3 marks
At zero interest the balance is the sum of six deposits.
$6 \times 250 = 1500$ .
AUD 1,500.00
Deposit AUD 300 at each year beginning for 5 years at 3%. Measure the balance at time 5. Find it.
3 marks
An ordinary annuity gives the end-deposit balance.
Multiply by 1.03 because each beginning deposit earns one extra year.
AUD 1,640.52
An ordinary annuity ends with AUD 6,200 after 10 deposits of AUD 500. How much interest has been earned?
3 marks
Total deposited is $10 \times 500 = 5000$ .
Interest is $6200 - 5000 = 1200$ .
AUD 1,200.00
At 5% annually, compare three end-of-year deposits of AUD 1,000 with one AUD 3,000 deposit at time 0, both measured at time 3.
3 marks
Annuity is $1000 ({1.05}^{2} + 1.05 + 1)$ .
Lump sum $= 3000 {(1.05)}^{3}$ .
The lump sum grows more because all money is invested earlier.
Annuity: AUD 3,152.50; lump sum: AUD 3,472.88.
An account has AUD 2,000 now. With 4% annual interest and AUD 400 deposits at each year end, write and calculate its balance after 2 years.
3 marks
Year 1: $2000 \times 1.04 + 400 = 2480$ .
Year 2: $2480 \times 1.04 + 400 = 2979.20$ .
The recurrence agrees with initial balance plus annuity formula.
AUD 2,979.20

Checking saved progress…

Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 General Mathematics financial models: investments and annuities, superannuation, income streams and borrowing. Original practice uses supplied hypothetical rates, taxes and fees; statutory or commercial conditions are not inferred.

Official curriculum source ↗

In this lesson

Understand the idea.

Follow each deposit to the common end date

Add initial savings or change deposit timing

See the method in action.

Build the model

Identify how long the five deposits earn interest.

Accumulate all deposits to the fifth year end.

Evaluate the geometric sum.

Use the model to make a decision

Grow the initial savings for four years.

Accumulate the four year-end deposits.

Combine amounts at the same final date.

Try it yourself.

Check your understanding

Keep practising