Compound Interest Introduction
We've already learnt about simple interest, where interest is calculated on the principal (ie. the initial) amount, so the amount of interest remains constant or fixed. However, most of the times that banks and financial institutions calculate interest, they are using compound interest.
Compound interest as repeated multiplication
Compound interest means we earn interest on the principal and interest, rather than just on the principal as with simple interest, which means that the interest increases exponentially. In other words, each year, we multiple the previous year's total by the interest rate. We use a similar process to when we were calculating percentage increases. For example, if I invested $\$500$$500 for $3$3 years at a rate of $6%$6% p.a.:
After the first year, the investment would be $500\times1.06$500×1.06
After the second year, the investment would be $500\times1.06\times1.06$500×1.06×1.06, which we could also write as $500\times1.06^2$500×1.062
After the third year, the investment would be $500\times1.06\times1.06\times1.06$500×1.06×1.06×1.06, which we could also write as $500\times1.06^3$500×1.063
The compound interest formula
Instead of using repeated multiplication, we calculate compound interest using the formula:
$A=P\left(1+r\right)^n$A=P(1+r)n
where:
$A$A is the final amount of money (principal and interest together)
$P$P is the principal (the initial amount of money invested)
$r$r is the interest rate expressed as a decimal
$n$n is the number of time periods
Remember: This formula gives us the total amount (ie. the principal and interest together). If we just want to know the value of the interest, we can work it out by subtracting the principal from the total amount of the investment. In other words:
$I=A-P$I=A−P
Why Does Money Grow Faster With Compound Interest Than Simple Interest?
Simple interest is interest earned on the principal invested amount only, whereas compound interest is interest earned on the principal amount plus interest on the interest already earned. So instead of the value of your investment increasing it a straight line as with simple interest, it will exponentially grow something like this:
For example, suppose you deposit $\$1000$$1000 into an online account for $2$2 years that pays $10$10% pa simple interest. The interest you would earn in the $2$2 years is $\$1000\times10%\times2=\$200$$1000×10%×2=$200.
But suppose that, instead of simple interest, you were paid interest compounded annually. In this case, the interest earned in the first year would be $\$1000\times10%\times1=\$100$$1000×10%×1=$100.
The new principal at the end of the first year would be $\$1000+\$100=\$1100$$1000+$100=$1100.
The interest earned in the second year would then be $\$1100\times10%\times1=\$110$$1100×10%×1=$110.
So the total compound interest earned over the two years would be $\$100+\$110=\$210$$100+$110=$210 which is $\$10$$10 more than what you would have earned with simple interest. Although a $\$10$$10 difference may not seem like much, think of how much the difference would have been if you had invested a million dollars instead of a thousand, or if you had invested for twenty years instead of two.
Examples
Question 1
Han's investment of $\$6000$$6000 earns interest at $2%$2% p.a, compounded annually over $3$3 years.
Answer the following questions by repeated multiplication.
A) What is the value of the investment after $3$3 years? Write your answer to the nearest cent.
Think: We need to apply the interest rate to the new amount each year.
| Year 1: | $A$A | $=$= | $6000\times1.02$6000×1.02 |
| $=$= | $\$6120$$6120 | ||
| Year 2: | $A$A | $=$= | $6120\times1.02$6120×1.02 |
| $=$= | $\$6240.40$$6240.40 | ||
| Year 3: | $A$A | $=$= | $6240.40\times1.02$6240.40×1.02 |
| $=$= | $\$6367.25$$6367.25 |
B)What is the amount of interest earned?
Think: The interest is the difference between the total amount of the investment and the principal (the initial amount invested).
Do:$6367.25-6000=\$367.25$6367.25−6000=$367.25
Question 2
A $\$2090$$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.
Do:
| $A$A | $=$= | $P\left(1+r\right)^n$P(1+r)n |
| $=$= | $2090\times\left(1+0.042\right)^{17}$2090×(1+0.042)17 | |
| $=$= | $4206.273$4206.273... | |
| $=$= | $\$4206.27$$4206.27 |
Question 3
Scott wants to have $\$1500$$1500 at the end of $5$5 years. If the bank offers $2.3%$2.3% p.a. compounded annually, how much should he invest now? Give your answer correct to the nearest cent.