Savings annuities | Year 12

01 / LEARN

The idea, step by step

Follow each deposit to the common end date

For an ordinary savings annuity, deposit $R$ is made at the end of each period. The last deposit earns no interest before the balance is measured. Earlier deposits earn for progressively more periods, giving a geometric sum.

Draw the timeline before choosing a calculator setting. The formula below assumes a constant rate $i$ per deposit period and $n$ equal end-of-period deposits, with no fees.

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

Add initial savings or change deposit timing

An initial balance P earns interest for all n periods and is added separately: $P {(1 + i)}^{n}$ . If all n deposits occur at the beginnings of periods, each deposit earns one extra period, so multiply the deposit annuity component by 1+i.

A payment at time 0 must not be counted again as an end-of-period deposit. At zero interest, the annuity balance is simply nR; use this limit rather than dividing by zero.

A common trap

Giving the final end-of-period deposit a full period of interest.

Instead: The final deposit is included at its face value; its growth exponent is zero.

02 / WORKED EXAMPLES

See the thinking

Example 1

Build the model

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

The deposits earn for 4,3,2,1,0 years.
Balance is $200 ({(1.04)}^{5} - 1) / 0.04$ = AUD 1,083.26.

AUD 1,083.26

Example 2

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.

Initial savings grow to $1000 {(1.05)}^{4}$ .
Deposits grow to $150 ({(1.05)}^{4} - 1) / 0.05$ .
Add the two components: AUD 1,862.03.

AUD 1,862.03

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

AUD 1,500.00

At zero interest the balance is the sum of six deposits. $6 \times 250 = 1500$ .

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF 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 $80 ({(1.004)}^{12} - 1) / 0.004$ .
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 $s u m = 3000 {(1.05)}^{3}$ .
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

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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 ↗

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