NZ Level 6 (NZC) Level 1 (NCEA)
Effective Interest Rates
Lesson

In More Moves with Compound Interest, we looked at how to use the compound interest formula for investments that accrue interest more than once a year, whether it be daily, weekly, monthly, quarterly or semiannually. If you remember, more interest is accrued on amounts that gather interest more than once a year. So does it really make sense just to divide the annual interest rate to express the interest rate of an amount that compounds more than once a year if they give different answers.

Well, not really. That is why we calculate the effective interest rate, the actual amount of interest that will be accrued in a year.

Remember, to calculate the value of an amount that compounds more than once the year, we can use the formula:

$A=P\left(1+\frac{r}{k}\right)^{nk}$A=P(1+rk)nk

However, if we just want to calculate the effective interest rate, we need to leave out the principal amount. We do this using the formula:

$\text{effective interest rate}=\left(1+\frac{r}{k}\right)^{nk}-1$effective interest rate=(1+rk)nk1

Then remember to convert this decimal answer to a percentage by multiplying by $100$100 and adding a $%$% symbol.

There is another way to work out the effective effect rate if you know the total amount of interest that a loan will incur. That is:

$\text{effective interest rate }=\frac{\text{total amount of interest}}{\text{loan amount }}\times100$effective interest rate =total amount of interestloan amount ×100

Being able to calculate the effective interest rate can come in handy when you're choosing bank accounts or home loans so you can work out how much interest you will actually earn (or have to pay in the case of loans).

#### Practice questions

##### Question 1

Neil invested $\$35003500 at $4.3%$4.3% p.a. compounded daily.

1. Find the amount of interest earned in a year.

Write your answer to the nearest cent.

2. Find the effective annual interest rate as a percentage.

Round your answer to two decimal places.

##### Question 2

An investment earns interest at a rate of $7.2%$7.2% compounding semiannually.

What is the effective rate correct to two decimal places?

### Outcomes

#### NA6-3

Apply everyday compounding rates

#### 91026

Apply numeric reasoning in solving problems