Annuities providing an income | Year 12

01 / LEARN

Understand the idea.

Discount every withdrawal to the starting date

An income annuity uses an initial balance to fund equal future withdrawals. Discount each withdrawal back to the starting date: money assigned to a later withdrawal has longer to earn interest, so it requires less capital now.

For $n$ equal end-of-period withdrawals that leave zero capital, use $P = R \frac{1 - {(1 + i)}^{- n}}{i}$ . The rate must match the withdrawal interval. If the same number of withdrawals instead begins immediately, each occurs one period earlier, multiplying the required present value by $1 + i$ .

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

A remaining balance is not the sum of future withdrawals

To find a sustainable withdrawal for a fixed term, divide the available capital by the present-value factor. A finite income stream can withdraw more than each period’s interest because it deliberately spends some principal as well.

After $k$ withdrawals, the balance is $P {(1 + i)}^{k} - R \frac{{(1 + i)}^{k} - 1}{i}$ . It is also the present value at time $k$ of the remaining withdrawals when the exact original plan exhausts the account. At zero interest, use $R = \frac{P}{n}$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Build the model

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

Worked solution

1
Identify the cash flows and valuation date.
$R = 5000, i = 0.04, n = 10$
All ten withdrawals occur at year ends, while the required balance is measured at time zero.
2
Discount all ten withdrawals to the start.
$P = 5000 \frac{1 - {(1.04)}^{- 10}}{0.04}$
The ordinary present-value annuity factor includes the discount for every withdrawal.
3
Evaluate the initial capital.
$P \approx 40554.478897 \approx AUD 40554.48$
Interest earned during the ten years funds part of the AUD 50,000 withdrawn.

Answer

AUD 40,554.48

EXAMPLE 02

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.

Worked solution

1
Equate capital to the present value of withdrawals.
$60000 = R \frac{1 - {(1.05)}^{- 12}}{0.05}$
Zero capital after the twelfth withdrawal means all starting capital finances the scheduled income stream.
2
Divide by the twelve-year present-value factor.
$\begin{aligned} R & = \frac{60000 \times 0.05}{1 - {(1.05)}^{- 12}} \\ \approx 6769.524601 \end{aligned}$
The factor measures how much starting capital is required for one unit of annual income.
3
State the annual withdrawal to cents.
$R \approx AUD 6769.52$
The model uses the unrounded withdrawal for exact exhaustion; the displayed payment is rounded to cents.

Answer

AUD 6,769.52

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

AUD 500.00

No interest is earned, so all income must come directly from the starting capital. Solve $24 R = 12000$ : $R = 500$ , giving AUD 500.00 for each month-end withdrawal.

Keep practising

6 worksheet questions · 18 marks

Open worksheet 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 $P = R a_{n}$ .
$P = \frac{3000 (1 - {1.03}^{- 8})}{0.03}$ .
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.
$R = \frac{12000}{24} = 500$ .
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.
$B_{5} = 40000 {(1.04)}^{5} - \frac{4000 ({(1.04)}^{5} - 1)}{0.04}$ .
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 $\frac{2500 (1 - {1.04}^{- 6})}{0.04}$ .
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 ↗

In this lesson

Understand the idea.

Discount every withdrawal to the starting date

A remaining balance is not the sum of future withdrawals

See the method in action.

Build the model

Identify the cash flows and valuation date.

Discount all ten withdrawals to the start.

Evaluate the initial capital.

Use the model to make a decision

Equate capital to the present value of withdrawals.

Divide by the twelve-year present-value factor.

State the annual withdrawal to cents.

Try it yourself.

Check your understanding

Keep practising