New Zealand
Level 6 - NCEA Level 1

# Compound Interest - an Introduction

Lesson

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 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$$500 for 33 years at a rate of 6%6% p.a.: After the first year, the investment would be 500\times1.06500×1.06 After the second year, the investment would be 500\times1.06\times1.06500×1.06×1.06, which we could also write as 500\times1.06^2500×1.062 After the third year, the investment would be 500\times1.06\times1.06\times1.06500×1.06×1.06×1.06, which we could also write as 500\times1.06^3500×1.063 ## The compound interest formula Instead of using repeated multiplication, we calculate compound interest using the formula: A=P\left(1+r\right)^nA=P(1+r)n where: AA is the final amount of money (principal and interest together) PP is the principal (the initial amount of money invested) rr is the interest rate expressed as a decimal nn is the number of time periods Remember: This formula gives us the total amount (ie. the principal and interest together). If we just want to know the value of the interest, we can work it out by subtracting the principal from the total amount of the investment. In other words: I=A-PI=AP ## Why Does Money Grow Faster With Compound Interest Than Simple Interest? Simple interest is interest earned on the principal invested amount only, whereas compound interest is interest earned on the principal amount plus interest on the interest already earned. So instead of the value of your investment increasing it a straight line as with simple interest, it will exponentially grow something like this: For example, suppose you deposit \1000$$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.

### Outcomes

#### NA6-3

Apply everyday compounding rates

#### 91026

Apply numeric reasoning in solving problems