We've already learnt about simple interest, where interest is calculated only on the principal (that is, the initial amount) so the amount of interest remains constant or fixed. However, most of the time when banks and financial institutions calculate interest, they are using compound interest.
Compound interest is calculated at the end of each compounding period, which is typically a day, month, quarter, or year. At the end of each compounding period, the total amount (principal plus interest) from previous compounding periods is used to calculate the new quantity of interest.
Worked example
Example 1
Suppose $\$500$$500 is invested in a compound interest account with an interest rate of $10%$10% p.a. compounded annually (that is, with a compounding period of one year) for $3$3 years. Then after one year, the interest is calculated:
Interest$=500\times10%=500\times0.1=\$50$=500×10%=500×0.1=$50
This interest is then added to the account.
Balance after $1$1 year$=500+50=\$550$=500+50=$550.
After the second year interest is calculated again, but this time the interest rate is applied to the balance from the previous year:
| Interest | $=550\times0.1=\$55$=550×0.1=$55 |
| Balance after $2$2 years | $=550+55=\$605$=550+55=$605 |
Finally, after the third year we have:
| Interest | $=605\times0.1=\$60.50$=605×0.1=$60.50 |
| Balance after $3$3 years | $=605+60.5=\$665.50$=605+60.5=$665.50 |
Practice question
Question 1
$\$8000$$8000 is invested for $3$3 years at a rate of $3%$3% p.a. compounded annually.
Complete the table below, rounding to the nearest cent.
Number of periods Interest ($\$$$) Balance ($\$$$) After $0$0 years - $8000$8000 After $1$1 year $\editable{}$ $\editable{}$ After $2$2 years $\editable{}$ $\editable{}$ After $3$3 years $\editable{}$ $\editable{}$ Calculate the total interest accumulated over $3$3 years in dollars.
Round your answer to the nearest cent.
The compound interest formula
Notice that in the above example, at the end of each compounding period there is a two step process: calculate the interest and then add it to the account balance. We could treat this as a percentage increase and combine these two steps as follows:
Balance after $1$1 year $=500+500\times0.1=500\times(1+0.1)=550$=500+500×0.1=500×(1+0.1)=550
This suggests a rule:
New balance $=$= Previous balance $\times(1+0.1)$×(1+0.1)
In other words, we can find the balance at the end of each year by repeatedly multiplying by $(1+0.1)$(1+0.1)
| Balance after $1$1 years | $=500\times(1+0.1)$=500×(1+0.1) |
| Balance after $2$2 years | $=500\times(1+0.1)\times(1+0.1)=500\times(1+0.1)^2$=500×(1+0.1)×(1+0.1)=500×(1+0.1)2 |
| Balance after $3$3 years | $=500\times(1+0.1)\times(1+0.1)\times(1+0.1)=500\times(1+0.1)^3$=500×(1+0.1)×(1+0.1)×(1+0.1)=500×(1+0.1)3 |
This leads us to the compound interest formula.
$A=P\left(1+r\right)^t$A=P(1+r)t
where:
$A$A is the final amount of money (principal and interest together)
$P$P is the principal (the initial amount of money invested)
$r$r is the interest rate per year, expressed as a decimal or fraction
$t$t is the number of years
This formula gives us the total amount (ie. the principal and interest together). If we just want to know the value of the interest, we can work it out by subtracting the principal from the total amount of the investment. In symbols:
$I=A-P$I=A−P
Practice questions
Question 2
William's investment of $\$2000$$2000 earns interest at a rate of $6%$6% p.a, compounded annually over $4$4 years.
What is the future value of the investment to the nearest cent?
Question 3
Bob borrows $\$5000$$5000 at a rate of $5.2%$5.2% p.a. compounded annually. If he pays off the loan in a lump sum at the end of $6$6 years, how much interest does he pay?
Give your answer in dollars.
Round your answer to the nearest cent.
Question 4
Kathleen has just won $\$20000$$20000. When she retires in $21$21 years, she wants to have $\$52000$$52000 in her fund which earns $8%$8% interest per annum.
How much of her winnings, to the nearest cent, does she need to invest now to achieve this?