Annuities providing an income | Year 12

01 / LEARN

The idea, step by step

Discount every withdrawal to the starting date

For an income annuity, a starting amount $P$ finances equal withdrawals $R$ at the end of each period. A withdrawal far in the future needs less money now because the allocated amount can earn interest first. Add the discounted withdrawals to obtain the present value.

The ordinary-annuity formula assumes constant $i$ , $n$ end withdrawals, and no balance after the final withdrawal. Use the same period for the withdrawal and the rate. If withdrawals begin immediately, the present-value requirement is multiplied by 1+ $i$ .

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

A remaining balance is not the sum of future withdrawals

After k withdrawals, the balance is $P {(1 + i)}^{k} - R ({(1 + i)}^{k} - 1) / i$ . It can also be found by discounting the remaining planned withdrawals to time k, provided the original plan exactly exhausts the account.

For a fixed term, solve $R = P / a_{n}$ . If $i = 0,$ divide the capital by the number of withdrawals. A payment above the interest earned can still be valid for a finite annuity; it gradually uses the capital.

A common trap

Using a future-value savings formula to find the starting retirement balance.

Instead: Income payments are discounted to the start; use a present-value annuity factor.

02 / WORKED EXAMPLES

See the thinking

Example 1

Build the model

What starting balance funds ten end-of-year withdrawals of AUD 5,000 at 4% annually, with no final balance?

Discount each of the ten withdrawals to time 0.
$P = 5000 (1 - {1.04}^{- 10}) / 0.04$ .

AUD 40,554.48

Example 2

Use the model to make a decision

An account contains AUD 60,000 and earns 5% annually. Find the equal year-end withdrawal that exhausts it after 12 withdrawals.

Solve $60000 = R (1 - {1.05}^{- 12}) / 0.05$ .
$R = 60000 / a_{12}$ .

AUD 6,769.52

Check your understanding

AUD 12,000 must provide 24 equal month-end withdrawals at zero interest. Find each withdrawal.

Show answer

AUD 500.00

At zero interest, total withdrawals equal starting capital. $R = 12000 / 24 = 500$ .

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Find the capital required for 8 end-of-year withdrawals of AUD 3,000 at 3%.
3 marks
AUD 12,000 must provide 24 equal month-end withdrawals at zero interest. Find each withdrawal.
3 marks
An account starts with AUD 40,000, earns 4% annually, and pays AUD 4,000 at each year end. Find the balance after 5 withdrawals.
3 marks
Compare the starting capital for five AUD 2,000 withdrawals at 5% when they occur at year ends and at year beginnings.
3 marks
A retirement model has AUD 100,000 earning 4% annually and pays AUD 6,000 at year end. Find the balance after the first year and explain why it falls.
3 marks
An exactly funded annuity has 6 withdrawals of AUD 2,500 still to come, starting next year, at 4%. Find its present balance.
3 marks

Read the worked solutions

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

Find the capital required for 8 end-of-year withdrawals of AUD 3,000 at 3%.
3 marks
Use $P = R a_{n}$ .
$P = 3000 (1 - {1.03}^{- 8}) / 0.03$ .
AUD 21,059.08
AUD 12,000 must provide 24 equal month-end withdrawals at zero interest. Find each withdrawal.
3 marks
At zero interest, total withdrawals equal starting capital.
$R = 12000 / 24 = 500$ .
AUD 500.00
An account starts with AUD 40,000, earns 4% annually, and pays AUD 4,000 at each year end. Find the balance after 5 withdrawals.
3 marks
Initial money grows for 5 years.
Subtract the accumulated value of the five withdrawals.
$B_{5} = 40000 {(1.04)}^{5} - 4000 ({(1.04)}^{5} - 1) / 0.04$ .
AUD 27,000.83
Compare the starting capital for five AUD 2,000 withdrawals at 5% when they occur at year ends and at year beginnings.
3 marks
Year-beginning withdrawals happen one period earlier.
Their present value is 1.05 times the ordinary-annuity present value.
End: AUD 8,658.95; beginning: AUD 9,091.90.
A retirement model has AUD 100,000 earning 4% annually and pays AUD 6,000 at year end. Find the balance after the first year and explain why it falls.
3 marks
Interest is $100000 \times 0.04 = 4000$ .
After withdrawal the balance is $100000 + 4000 - 6000 = 98000$ .
The withdrawal exceeds that year’s interest by AUD 2,000.
AUD 98,000.00
An exactly funded annuity has 6 withdrawals of AUD 2,500 still to come, starting next year, at 4%. Find its present balance.
3 marks
At the current date, the remaining payments form a new ordinary annuity.
Balance is $2500 (1 - {1.04}^{- 6}) / 0.04$ .
AUD 13,105.34

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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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