Applications of Annuities
Master the applications of annuities for NSW Year 12 Mathematics Standard 2. An annuity is a series of equal payments with interest compounding, and a table of interest factors lets you find its future value — what regular superannuation or savings contributions grow to — or its present value, the single sum a retirement income or loan is worth now.
You will use the future-value and present-value tables to grow a superannuation balance, work out the contribution needed to reach a savings goal, value a retirement drawdown, and compare annuities as the amount, rate or duration changes — the decision-making skills at the heart of Standard 2 financial mathematics.
Theory
Applications of annuities put the future-value and present-value tables to work. This Year 12 Standard 2 (NSW) guide shows how to grow a superannuation or savings balance, find the deposit needed to reach a goal, value a retirement income or loan, and compare annuities when the amount, rate or duration changes.
An annuity is a sequence of equal payments made at regular intervals while interest compounds on the balance. It appears in two everyday forms: paying equal amounts in — superannuation or regular savings that grow to a future value — and a single sum paid to provide equal payments out, such as a loan you repay or a retirement income you draw down.
A table of interest factors does the compounding for you. The future-value factor is what an annuity of \(\$1\) grows to; the present-value factor is the single sum today that is worth \(\$1\) paid each period. You multiply the factor by your actual payment: \(FV = a\times F_{FV}\) or \(PV = a\times F_{PV}\).
In this Year 12 Standard 2 (NSW) topic you apply these to real decisions — how big a super balance regular contributions build, what yearly deposit reaches a savings goal, how much a retirement income costs today, and how changing the amount, rate or duration changes the result.
For a regular payment \(a\), read the factor for the rate per period and the number of periods \(n\), then:
To find the contribution needed for a target future value, rearrange:
Year by year, a super fund or loan can also be tracked with a recurrence rule (balance \(B\), rate \(r\), payment \(a\)):
How to solve an annuity application
- Choose FV or PV. Use the future value when money is growing to the future (savings, super); use the present value for the single sum a loan or an income stream is worth now.
- Rate and periods. Find the interest rate per period and the number of periods \(n\).
- Read the factor from the correct table at row \(n\) and the column for the rate.
- Multiply and interpret. Multiply the factor by the regular payment (or divide a target by the factor to get the payment). For a decision, repeat for each option and compare.
Future value of an annuity of \(\$1\) (contributions at the end of each period):
| n | 4% | 5% | 6% | 8% |
|---|---|---|---|---|
| 1 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
| 2 | 2.0400 | 2.0500 | 2.0600 | 2.0800 |
| 3 | 3.1216 | 3.1525 | 3.1836 | 3.2464 |
| 4 | 4.2465 | 4.3101 | 4.3746 | 4.5061 |
| 5 | 5.4163 | 5.5256 | 5.6371 | 5.8666 |
| 6 | 6.6330 | 6.8019 | 6.9753 | 7.3359 |
Present value of an annuity of \(\$1\) (payments at the end of each period):
| n | 4% | 5% | 6% | 8% |
|---|---|---|---|---|
| 1 | 0.9615 | 0.9524 | 0.9434 | 0.9259 |
| 2 | 1.8861 | 1.8594 | 1.8334 | 1.7833 |
| 3 | 2.7751 | 2.7232 | 2.6730 | 2.5771 |
| 4 | 3.6299 | 3.5460 | 3.4651 | 3.3121 |
| 5 | 4.4518 | 4.3295 | 4.2124 | 3.9927 |
| 6 | 5.2421 | 5.0757 | 4.9173 | 4.6229 |
Read the future-value factor at \(n=6\), \(8\%\), then multiply by the contribution.
| n | 4% | 5% | 6% | 8% |
|---|---|---|---|---|
| 1 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
| 2 | 2.0400 | 2.0500 | 2.0600 | 2.0800 |
| 3 | 3.1216 | 3.1525 | 3.1836 | 3.2464 |
| 4 | 4.2465 | 4.3101 | 4.3746 | 4.5061 |
| 5 | 5.4163 | 5.5256 | 5.6371 | 5.8666 |
| 6 | 6.6330 | 6.8019 | 6.9753 | 7.3359 |
| \(FV\) | \(=\) | \(a\times F_{FV}\) |
| \(F_{FV}(8\%,\,6)\) | \(=\) | \(7.3359\) |
| \(FV\) | \(=\) | \(3500\times 7.3359\) |
| \(\) | \(=\) | \(\$25{,}675.65\) |
Her super grows to \(\$25{,}675.65\).
The future value is the contribution times the factor, so divide the target by the factor.
| \(F_{FV}(5\%,\,6)\) | \(=\) | \(6.8019\) |
| \(a\) | \(=\) | \(\dfrac{FV}{F_{FV}} = \dfrac{40000}{6.8019}\) |
| \(\) | \(=\) | \(\$5880.71\) |
He must deposit \(\$5880.71\) each year.
The lump sum is the present value of the withdrawals: payment times the PV factor.
| n | 4% | 5% | 6% | 8% |
|---|---|---|---|---|
| 1 | 0.9615 | 0.9524 | 0.9434 | 0.9259 |
| 2 | 1.8861 | 1.8594 | 1.8334 | 1.7833 |
| 3 | 2.7751 | 2.7232 | 2.6730 | 2.5771 |
| 4 | 3.6299 | 3.5460 | 3.4651 | 3.3121 |
| 5 | 4.4518 | 4.3295 | 4.2124 | 3.9927 |
| 6 | 5.2421 | 5.0757 | 4.9173 | 4.6229 |
| \(PV\) | \(=\) | \(a\times F_{PV}\) |
| \(F_{PV}(8\%,\,5)\) | \(=\) | \(3.9927\) |
| \(PV\) | \(=\) | \(8000\times 3.9927\) |
| \(\) | \(=\) | \(\$31{,}941.60\) |
She must invest \(\$31{,}941.60\) now.
Find each future value, then subtract.
| \(FV_A\) | \(=\) | \(2000\times 6.6330 = \$13{,}266.00\) |
| \(FV_B\) | \(=\) | \(2000\times 7.3359 = \$14{,}671.80\) |
| \(\text{difference}\) | \(=\) | \(14671.80 - 13266.00\) |
| \(\) | \(=\) | \(\$1405.80\) |
Fund B gives \(\$1405.80\) more — a higher rate gives a larger future-value factor.
Common pitfalls
Frequently asked questions
What is an annuity in Maths Standard 2?
An annuity is a series of equal payments made at regular intervals while interest compounds on the balance. Paying a fixed amount into superannuation each year, or repaying a loan with equal instalments, are both annuities. You use a table of interest factors to find the future value or present value.
How do you use a future-value annuity table?
Find the factor at the row for the number of periods and the column for the interest rate per period, then multiply it by the regular contribution. For example, $3500 a year for 6 years at 8% uses the factor 7.3359, so the future value is 3500 times 7.3359, which is $25,675.65.
How do you use a present-value annuity table?
The present value is the single lump sum today that is worth a stream of equal payments. Read the present-value factor for the rate and number of periods, then multiply by the payment. It tells you how much to invest now to provide a retirement income, or how much can be borrowed for a set repayment.
How do you find the regular payment needed to reach a savings goal?
Rearrange the future-value rule: the contribution equals the target future value divided by the future-value factor. For example, to reach $40,000 in 6 years at 5% you divide 40000 by the factor 6.8019, giving about $5880.71 a year.
What happens to the future value if the interest rate or the number of years increases?
Both make the future-value factor larger, so the future value grows. A higher rate earns more interest on every contribution, and a longer term means more contributions and more compounding. That is why comparing funds or terms comes down to comparing their factors.
What is the difference between the future value and present value of an annuity?
The future value is what a series of equal payments grows to by the end — used for savings and superannuation. The present value is the single sum today that is worth those payments — used for loans and retirement incomes. You choose the table that matches whether the money is growing to the future or being valued now.