Future Value of an Annuity (FV Tables)
Master the future value of an annuity using a table of interest factors for NSW Year 12 Mathematics Standard 2. An annuity is a series of equal contributions earning compound interest; you read a future-value factor from the table and multiply by the contribution, since \(FV = C\times F\).
You will find the future value, the interest earned (future value minus the contributions), and the contribution needed to reach a savings or superannuation goal, and learn to handle non-annual compounding such as quarterly and half-yearly annuities — core skills for Standard 2 financial mathematics.
Theory
Future value of an annuity is part of the annuities topic in Year 12 Standard 2 (NSW). You read a future-value interest factor from a table and multiply by the contribution, since \(FV = C\times F\). This guide shows how to find the future value, the interest earned, the contribution needed for a target amount, and how to handle non-annual compounding for superannuation and savings.
An annuity is a series of equal contributions made at regular intervals, where each contribution earns compound interest until the end of the term. Its future value is the total amount the annuity is worth at that point.
Rather than adding up each deposit separately, you read a future-value interest factor from a table. The table is built for a \(\$1\) contribution, so the factor \(F\) is what one dollar per period grows to; the actual future value is \(FV = C\times F\), where \(C\) is the real contribution.
In this Year 12 Standard 2 (NSW) topic you use the table to find the future value, the interest earned, or the contribution needed for a target amount, and you handle non-annual compounding such as quarterly or half-yearly annuities.
For a contribution \(C\) each period and a future-value factor \(F\) read from the table (row = number of periods \(n\), column = rate per period):
The factor itself is the future value of an annuity of \(\$1\):
The interest earned is the future value less every contribution paid in:
To reach a target future value, rearrange for the contribution:
How to find the future value of an annuity from a table
- Match the period and rate. Work out the number of periods \(n\) and the interest rate per period. For non-annual compounding, divide the p.a. rate by the number of periods per year and use \(n = m\times t\).
- Read the factor. Find the row for \(n\) periods and read across to the rate column — that entry is the factor \(F\).
- Multiply. The future value is \(FV = C\times F\), where \(C\) is the contribution each period.
- Answer the question. For interest earned, subtract the contributions \(C\times n\); for a target future value, divide instead: \(C = FV / F\).
For example, an annuity of \(\$2000\) per year for \(3\) years at \(8\%\) reads the factor \(3.2464\), so \(FV = 2000\times 3.2464 = \$6492.80\).
Future value of an annuity of $1
| Period | 2% | 4% | 6% | 8% | 10% |
|---|---|---|---|---|---|
| 1 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
| 2 | 2.0200 | 2.0400 | 2.0600 | 2.0800 | 2.1000 |
| 3 | 3.0604 | 3.1216 | 3.1836 | 3.2464 | 3.3100 |
| 4 | 4.1216 | 4.2465 | 4.3746 | 4.5061 | 4.6410 |
| 5 | 5.2040 | 5.4163 | 5.6371 | 5.8666 | 6.1051 |
| 6 | 6.3081 | 6.6330 | 6.9753 | 7.3359 | 7.7156 |
Read the factor for \(5\) periods in the \(6\%\) column, then multiply by the contribution.
Future value of an annuity of $1
| Period | 2% | 4% | 6% | 8% | 10% |
|---|---|---|---|---|---|
| 1 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
| 2 | 2.0200 | 2.0400 | 2.0600 | 2.0800 | 2.1000 |
| 3 | 3.0604 | 3.1216 | 3.1836 | 3.2464 | 3.3100 |
| 4 | 4.1216 | 4.2465 | 4.3746 | 4.5061 | 4.6410 |
| 5 | 5.2040 | 5.4163 | 5.6371 | 5.8666 | 6.1051 |
| 6 | 6.3081 | 6.6330 | 6.9753 | 7.3359 | 7.7156 |
| \(FV\) | \(=\) | \(C \times F\) |
| \(\text{factor } (n=5,\ 6\%)\) | \(=\) | \(5.6371\) |
| \(FV\) | \(=\) | \(4000 \times 5.6371\) |
| \(\) | \(=\) | \(22{,}548.40\) |
The future value is \(\$22{,}548.40\).
Find the future value, then subtract the total contributions paid in.
| \(\text{factor } (n=4,\ 8\%)\) | \(=\) | \(4.5061\) |
| \(FV\) | \(=\) | \(1500 \times 4.5061 = 6759.15\) |
| \(\text{contributions}\) | \(=\) | \(4 \times 1500 = 6000\) |
| \(\text{interest}\) | \(=\) | \(6759.15 - 6000 = 759.15\) |
The interest earned is \(\$759.15\).
Since \(FV = C\times F\), rearrange for \(C\) and divide by the factor.
| \(\text{factor } (n=6,\ 4\%)\) | \(=\) | \(6.6330\) |
| \(C\) | \(=\) | \(\dfrac{FV}{F} = \dfrac{30000}{6.6330}\) |
| \(\) | \(=\) | \(4522.84\) |
Ben must contribute \(\$4522.84\) each year.
Convert to a per-quarter rate and count the periods, then read that factor.
| \(\text{rate per quarter}\) | \(=\) | \(\dfrac{24\%}{4} = 6\%\) |
| \(\text{periods}\) | \(=\) | \(4 \times 1 = 4\) |
| \(\text{factor } (n=4,\ 6\%)\) | \(=\) | \(4.3746\) |
| \(FV\) | \(=\) | \(600 \times 4.3746 = 2624.76\) |
The future value is \(\$2624.76\).
Common pitfalls
Frequently asked questions
What is the future value of an annuity?
It is the total amount a series of equal, regular contributions grows to by the end of the term, once each contribution has earned compound interest. You read a future-value factor from a table and multiply it by the contribution: future value equals contribution times factor.
How do you use a future-value annuity table?
Find the row for the number of periods and read across to the column for the interest rate per period. That entry is the factor, which is the future value of an annuity of one dollar. Multiply it by your actual contribution to get the future value of the annuity.
How do you find the interest earned on an annuity?
First find the future value from the table, then subtract the total of all the contributions you paid in. If you contribute C each period for n periods, the contributions total C times n, so the interest earned equals the future value minus C times n.
How do you find the contribution needed to reach a savings goal?
Rearrange the formula future value equals contribution times factor. Read the factor for the number of periods and rate, then divide your target future value by that factor. The result is the contribution you must make each period to reach the goal.
How do you use the table when interest is not compounded annually?
Convert the annual rate to a rate per period by dividing by the number of periods each year, and count the total number of periods. For example, 8 percent per year compounded half-yearly is 4 percent per period, and 2 years is 4 periods. Then read that row and column as normal.
What is the difference between the future value and the contributions?
The contributions are just the money you put in, added up. The future value is larger because the money earns interest while it is invested. The difference between the two is the interest earned, which is why the fund value curve rises above the straight contributions line.