Compound Interest & Future Value
Learn compound interest for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-02). With compound interest, each year's interest is added to the balance and then earns interest itself, so your money grows by a larger amount every year — interest earns interest.
You will use the compound interest formula to get an investment's future value from its present value, build the balance year by year, and work out the interest earned — skills for planning savings, term deposits and other investments over several years.
Theory
With compound interest, each year's interest is added to the balance and then earns interest itself, so the balance is multiplied by \((1+r)\) every year: \(FV=PV(1+r)^n\). This NSW Year 12 Mathematics Standard 1 guide shows how to find the future value of an investment, build the balance year by year, and work out the interest earned.
Compound interest is interest that is added to the balance each year, so that next year it earns interest too — interest earns interest. Each year the balance is multiplied by the growth factor \((1+r)\), where \(r\) is the rate per period (here per year) as a decimal.
The present value \(PV\) is the amount invested now. The future value \(FV\) is the balance at the end. After \(n\) years the balance has been multiplied by \((1+r)^n\), so \(FV=PV(1+r)^n\).
The interest earned is \(FV-PV\). Because each year's interest is bigger than the last, the graph of the balance is an upward-bending curve.
The future value of a present value \(PV\) at rate \(r\) per period (usually per year) for \(n\) periods is:
The interest earned is the increase in the balance:
How to find a future value
- Write \(PV\), the growth factor \(1+r\) (with \(r\) as a decimal) and the number of years \(n\).
- Calculate \((1+r)^n\) first, then multiply by \(PV\) to get \(FV\), rounding to the nearest cent at the end.
- Subtract \(PV\) if the question asks for the interest earned: \(I=FV-PV\).
Substitute \(PV=2400\), \(r=0.06\), \(n=3\).
| \(FV\) | \(=\) | \(PV(1+r)^n\) |
| \(\) | \(=\) | \(2400(1.06)^3\) |
| \(\) | \(=\) | \(2400\times1.191016\) |
| \(\) | \(=\) | \(2858.44\) |
The future value is \(\$2858.44\).
Multiply the balance by \(1.08\) each year.
| \(\text{Year 1: } 1500\times1.08\) | \(=\) | \(1620.00\) |
| \(\text{Year 2: } 1620\times1.08\) | \(=\) | \(1749.60\) |
| \(\text{Year 3: } 1749.60\times1.08\) | \(=\) | \(1889.57\) |
Check: \(1500(1.08)^3=1889.57\). The values are \(\$1620.00\), \(\$1749.60\) and \(\$1889.57\).
Find the future value, then subtract the amount invested.
| \(FV\) | \(=\) | \(7000(1.035)^5\) |
| \(\) | \(=\) | \(7000\times1.187686\) |
| \(\) | \(=\) | \(8313.80\) |
| \(I\) | \(=\) | \(FV-PV\) |
| \(\) | \(=\) | \(8313.80-7000\) |
| \(\) | \(=\) | \(1313.80\) |
Hannah earns \(\$1313.80\) interest.
Use \(FV=PV(1+r)^n\) with \(r=0.045\) and \(n=10\).
| \(FV\) | \(=\) | \(3000(1.045)^{10}\) |
| \(\) | \(=\) | \(3000\times1.552969\) |
| \(\) | \(=\) | \(4658.91\) |
The investment will be worth \(\$4658.91\).
Common pitfalls
Frequently asked questions
What is compound interest?
Compound interest is interest that is added to the balance each year, so it earns interest itself the following year. The balance is multiplied by 1 plus r every year, so it grows faster and faster.
What is the compound interest formula?
The future value is FV equals PV times (1 plus r) to the power n, where PV is the present value (the amount invested now), r is the rate per period (usually per year) as a decimal and n is the number of periods (years).
How do you calculate the interest earned with compound interest?
Find the future value with FV equals PV(1 plus r) to the power n, then subtract the amount invested: interest equals FV minus PV. For example, $7000 at 3.5% for 5 years grows to $8313.80, so the interest is $1313.80.
What does 1 + r mean in the compound interest formula?
1 plus r is the growth factor, the number the balance is multiplied by each year. For 8% per annum it is 1.08, which keeps the original 100% and adds 8%.
Why does a compound interest graph curve upwards?
Each year's interest is worked out on a bigger balance than the year before, so the yearly increase keeps growing. A growing increase makes the graph bend upwards instead of being a straight line.