When you invest money, you usually earn interest. There are two main ways you can earn interest: simple and compound.
- In simple interest, the interest earned each time period is the same because it is always based on the principal amount invested. For example, if you invest $\$1000$$1000 at $2%$2% per annum simple interest, you will earn $2%$2% of $\$1000$$1000 each year. The interest of $\$20$$20 earned each year does not change over the term of the investment.
- In compound interest, the interest earned in each time period is added to the principal. So the interest earned in each time period increases because it is calculated on a growing balance. For example, if you invest $\$1000$$1000 at $2%$2% per annum compounded yearly, you will earn $\$20$$20 interest in the first year, which is added to the principal. In the second year, interest of $2%$2% is calculated on a new balance of $\$1020$$1020, not $\$1000$$1000.
This means compound interest usually leads to greater returns than simple interest because the principal is updated each period after interest is earned.
Let's compare the two with an example of earning $5%$5% interest on an initial principal of $\$1000$$1000.
| Year | Principal Used | Simple interest earned | New total |
|---|---|---|---|
| 1 | $\$1000$$1000 | $\$50$$50 | $\$1050$$1050 |
| 2 | $\$1000$$1000 | $\$50$$50 | $\$1100$$1100 |
| 3 | $\$1000$$1000 | $\$50$$50 | $\$1150$$1150 |
| Year | Principal Used | Compound interest earned | New total |
|---|---|---|---|
| 1 | $\$1000$$1000 | $\$50$$50 | $\$1050$$1050 |
| 2 | $\$1050$$1050 | $\$52.50$$52.50 | $\$1102.50$$1102.50 |
| 3 | $\$1102.50$$1102.50 | $\$55.13$$55.13 | $\$1157.63$$1157.63 |
We can see that after just $2$2 years, there is already a financial benefit in earning compound interest over simple interest.
Using repeated multiplication
In the previous example on compound interest, the new total after $1$1 year was $1000+1000\times0.05$1000+1000×0.05. Another way to calculate the new total after $1$1 year would be $1000\times1.05$1000×1.05.
Doing this repeatedly, we can find a pattern and a formula for calculating the future value of an investment using compound interest.
Remember, to increase by $r%$r%, multiply by $\left(1+\frac{r}{100}\right)$(1+r100).
So if the interest rate is $6%$6% p.a., we multiply by $\left(1+\frac{6}{100}\right)$(1+6100) or $1.06$1.06.
For example, if you invest $\$500$$500 for $3$3 years at a rate of $6%$6% per annum, the investment would grow as follows:
- After the first year, the investment would be worth $\$500\times1.06$$500×1.06.
- After the second year, the investment would be worth $\left(\$500\times1.06\right)\times1.06$($500×1.06)×1.06, which we could also write as $\$500\times1.06^2$$500×1.062.
- After the third year, the investment would be worth $\left(\left(\$500\times1.06\right)\times1.06\right)\times1.06$(($500×1.06)×1.06)×1.06, which we could also write as $\$500\times1.06^3$$500×1.063.
After just a couple of calculations, we can start to see a pattern. The size of the investment after $n$n years can be written in the form $\$500\times1.06^n$$500×1.06n. This pattern holds true for all forms of compound interest.
$FV=PV\left(1+r\right)^n$FV=PV(1+r)n
$n$n is the number of periods (can be years, months, weeks)
$FV$FV is the future value (final amount of our investment after $n$n periods)
$PV$PV is the present value (the initial principal amount)
$r$r is the interest rate for the period, expressed as a decimal
Remember: This formula gives us the total amount (the principal and interest together). If we just want to know the value of the interest ($I$I), we subtract the principal from the total amount of the investment. In other words:
$I=FV-PV$I=FV−PV
Worked examples
Question 1
Han invests $\$6000$$6000 in a savings account that pays interest at $2%$2% p.a. for $3$3 years.
A) What is the value of the investment after $3$3 years if the interest is compounded annually? Use repeated multiplication to answer the question.
Think: We need to apply the interest rate to the new amount each year.
Do: $r=2%$r=2% p.a. so we multiply by $1.02$1.02.
| Year 1: | $FV$FV | $=$= | $6000\times1.02$6000×1.02 |
| $=$= | $\$6120$$6120 | ||
| Year 2: | $FV$FV | $=$= | $6120\times1.02$6120×1.02 |
| $=$= | $\$6240.40$$6240.40 | ||
| Year 3: | $FV$FV | $=$= | $6240.40\times1.02$6240.40×1.02 |
| $=$= | $\$6367.25$$6367.25 |
After $3$3 years the value of the investment is $\$6367.25$$6367.25.
B) Calculate the value of the investment after $3$3 years if the interest earned is simple interest.
Think: The interest is the difference between the total amount of the investment and the principal (the initial amount invested).
Do: Using simple interest, $2%$2% of $\$6000$$6000 is $\$120$$120 each year.
| Value after $3$3 years | $=$= | $6000+3\times120$6000+3×120 |
| $=$= | $\$6360$$6360 |
C) Does compound or simple interest lead to greater financial gain?
Compound interest ($\$6367.25$$6367.25) leads to greater gain than simple interest ($\$6360$$6360).
Question 2
A $\$2090$$2090 investment earns interest at $4.2%$4.2% p.a. compounded annually over $17$17 years. Use the compound interest formula to calculate the value of this investment to the nearest cent.
Think: We need to substitute these values into the compound interest formula.
| $PV=\$2090$PV=$2090 |
| $r=4.2%$r=4.2% per year $=$= $0.042$0.042 as a decimal |
| $n=17$n=17 years |
| $FV$FV $=$= ? |
Do: Using the compound interest formula,
| $FV$FV | $=$= | $PV\left(1+r\right)^n$PV(1+r)n |
| $=$= | $2090\times\left(1+0.042\right)^{17}$2090×(1+0.042)17 | |
| $=$= | $4206.273$4206.273... | |
| $=$= | $\$4206.27$$4206.27 (2 d.p.) |
After $17$17 years the investment is worth $\$4206.27$$4206.27, to the nearest cent.
Question 3
Caitlin invests $\$80000$$80000 in a term deposit that earns interest at $5%$5% p.a. compounded annually over $3$3 years.
Use repeated multiplication to calculate the value of the investment after $3$3 years. Write your answer to the nearest cent.
What is the amount of interest earned?