We've now learned about simple interest, where interest is calculated at a fixed rate on just the principal amount, so the amount of interest earned in each time period remains constant. This type of interest is very rare to encounter in most situations, however. Instead, banks and other financial institutions that calculate interest do so using compound interest instead.
Compound interest as repeated multiplication
Compound interest occurs when we earn interest on the principal amount and the interest earned so far, rather than just on the principal amount. That is, each time period we calculate interest earned by using the total amount from the previous year. This means that the interest increases at an exponential rate, as we earn interest on interest that has already been earned. The process is similar to when we were calculating repeated percentage increases.
For example, if we were to invest $\$500$$500 for $3$3 years at a rate of $6%$6% p.a. using compound interest, then:
- After the first year, the value of the investment would be $500\times1.06$500×1.06.
- After the second year, the value of the investment would be $\left(500\times1.06\right)\times1.06$(500×1.06)×1.06, which simplifies to $500\times1.06^2$500×1.062
- After the third year, the investment would be $\left(500\times1.06^2\right)\times1.06$(500×1.062)×1.06, which we can simplify to $500\times1.06^3$500×1.063
Worked example
Example 1
Han's investment of $\$6000$$6000 earns interest at $2%$2% p.a, compounded annually over $3$3 years.
Answer the following questions by using repeated multiplication.
(a) What is the value of the investment after $3$3 years? Round your answer to the nearest cent.
Think: We should apply the interest rate to the new amount each year.
Do:
| Year 1: | $A_1$A1 | $=$= | $6000\times1.02$6000×1.02 |
| $=$= | $\$6120$$6120 | ||
| Year 2: | $A_2$A2 | $=$= | $6120\times1.02$6120×1.02 |
| $=$= | $\$6240.40$$6240.40 | ||
| Year 3: | $A_3$A3 | $=$= | $6240.40\times1.02$6240.40×1.02 |
| $=$= | $\$6367.25$$6367.25 (to the nearest cent) |
(b) What is the amount of interest earned?
Think: The interest is the difference between the total amount of the investment and the principal (the initial amount invested).
Do: $6367.25-6000=\$367.25$6367.25−6000=$367.25
Notice that when we calculate compound interest, we are calculating the total value of an investment or loan. To find the amount of interest earned, $I$I, we subtract the principal $P$P from the total value $A$A. That is, $I=P-A$I=P−A.
This is different to simple interest calculations, where we calculated the interest earned. To find the total value of the investment, we needed to add the interest and the principal amount. That is, $A=P+I$A=P+I. Notice that this is the same relation between $A$A, $I$I and $P$P as above, just rearranged.
Practice question
Question 1
Dave's investment of $\$6000$$6000 earns interest at $2%$2% p.a, compounded annually over $3$3 years.
Answer the following questions by repeated multiplication.
What is the value of the investment after $3$3 years?
Write your answer to the nearest cent.
What is the amount of interest earned?
Using a spreadsheet to model compound interest
We can use a spreadsheet to calculate compound interest for any amount and interest rate more efficiently.
For example:
Using the spreadsheet we can quickly see the interest earned and the total amount of an investment after any number of years. We can also adjust the initial amount and interest rate.
Your turn
- Open a spreadsheet program, such as Microsoft Excel or Google Sheets.
- Copy the cell contents and formulas below into your blank spreadsheet (make sure to copy the exact cells to ensure the formulas work).
Note that in the "Interest" and "Total Balance" columns, once you have entered the first three rows you can drag the corner of the cell down to copy the formula into the rows below.
- Use the spreadsheet to adjust the initial amount and rate and answer the following:
(a) What is the final amount when $\$3500$$3500 is invested for $6$6 years at a rate $4%$4% p.a.?
Do: Change the initial investment to $3500$3500 and the rate to $4$4. The final amount after $6$6 years will then be the value in the third column (the Total Balance column) of the End of year $6$6 row (that is, row $13$13). It should read $\$4428.62$$4428.62.
(b) What is the final amount when $\$9000$$9000 is invested for $10$10 years at a rate $2.1%$2.1% p.a?
Do: Change the initial investment to $9000$9000 and the rate to $2.1$2.1. The final amount after $10$10 years will then be the value in the third column (the Total Balance column) of the End of year $10$10 row (that is, row $17$17). It should read $\$11078.98$$11078.98.