# 6.03 Effective annual interest rate

Lesson

The table below compares the effect of changing the number of compounding periods when $\$1000$$1000 is invested for one year at a nominal rate of 5%5% p.a. The final amount is calculated using the compound interest formula A=P\times(1+\frac{r}{n})^{nt}A=P×(1+rn)nt Number of compounding periods 11 (compounded annually) 44 (compounded quarterly) 365365 (compounded daily) Calculation A=1000\times(1+\frac{0.05}{1})^1A=1000×(1+0.051)1 =1000\times(1.05)=1000×(1.05) A=1000\times(1+\frac{0.05}{4})^4A=1000×(1+0.054)4 =1000\times(1.05095)=1000×(1.05095) A=1000\times(1+\frac{0.05}{365})^{365}A=1000×(1+0.05365)365 =1000\times(1.05127)=1000×(1.05127) Final amount \1050$$1050 $\$1050.95$$1050.95 \1051.27$$1051.27
Amount of interest $\$50$$50 \50.95$$50.95 $\$51.27$$51.27 Effective annual interest rate \frac{50}{1000}=0.05=5%501000=0.05=5% \frac{50.95}{1000}=0.05095=5.095%50.951000=0.05095=5.095% \frac{51.27}{1000}=0.05127=5.127%51.271000=0.05127=5.127% From the table we can see that the amount of interest earned increases when the number of compounding periods increases. The effective annual interest rate has been calculated using the formula: \text{effective interest rate}=\frac{\text{Interest earned in one year}}{\text{Balance at start of year }}\times100%effective interest rate=Interest earned in one yearBalance at start of year ×100% The published rate of 5%5% per annum is called the nominal interest rate. Note: If we only compound once per year then this nominal interest rate is the same as the effective interest rate. Effective interest formula i_{effective}=(1+\frac{i}{n})^n-1ieffective=(1+in)n1 where i_{effective}ieffective is the effective interest rate per annum, expressed as a decimal ii is the nominal (or published) interest rate per annum, expressed as a decimal nn is the number of compounding periods per annum Being able to calculate the effective interest rate can come in handy when we are choosing and comparing investments or loans. They allow us to more easily work out how much interest the investment or loan will actually earn and quickly compare rates that have different compounding periods. When investing money, we want to have the highest possible effective interest rate. When borrowing money, we want to have the lowest possible effective interest rate. #### Worked examples ##### example 1 A bank advertises a nominal interest rate of 5.6%5.6% per annum, compounded quarterly. Calculate the effective interest rate. Think: Using the effective interest rate formula, we are solving for i_{effective}ieffective where i=0.056i=0.056 and n=4n=4. Do:  i_{effective}ieffective​ == \left(1+\frac{i}{n}\right)^n-1(1+in​)n−1 == (1+\frac{0.056}{4})^4-1(1+0.0564​)4−1 \approx≈ 1.057187-11.057187−1 Evaluate using a calculator == 0.05720.0572 to 33 significant figures Therefore, the effective interest rate is 5.72%5.72% p.a. to 22 decimal places. Another option is to use the interest rate conversion facility in the financial application of your calculator. See below for instructions on how to do this. Remember: If you use the app you must make sure to write down the numbers you put in your calculator as your working. ##### example 2 The effective rate for an investment account which compounds monthly is 6.4%6.4% p.a.. Calculate the nominal rate for this account correct to two decimal places. Think: Using the effective interest rate formula, we want to solve for ii, where n=12n=12 and i_{effective}=0.064ieffective=0.064. Do:  i_{effective}ieffective​ == (1+\frac{i}{n})^n-1(1+in​)n−1 0.0640.064 == (1+\frac{i}{12})^{12}-1(1+i12​)12−1 Using the solve facility of our calculator. ii == 0.06220.0622 to 33 significant figures Therefore, the nominal rate of interest is 6.22%6.22% p.a. to two decimal places. Alternatively, use interest rate conversion facility in the financial application of your calculator. ### Using technology to find effective rates Select the brand of calculator you use below to work through an example of using a calculator to find effective rates. Casio Classpad How to use the CASIO Classpad to find effective interest rates for compound interest investments or loans. Consider the following loan options: Option A: 5.70% p.a. compounded quarterly Option B: 5.65% p.a. compounded monthly 1. What is the effective rate of option A correct to two decimal places? 2. What is the effective rate of option B correct to two decimal places? 3. Which option offers a better rate? TI Nspire How to use the TI Nspire to find effective interest rates for compound interest investments or loans. Consider the following loan options: Option A: 5.70% p.a. compounded quarterly Option B: 5.65% p.a. compounded monthly 1. What is the effective rate of option A correct to two decimal places? 2. What is the effective rate of option B correct to two decimal places? 3. Which option offers a better rate? #### Practice questions ##### Question 1 James invested \3000$$3000 at $4.6%$4.6% p.a. compounded daily.

1. Find the amount of interest earned in a year. You may assume that there are $365$365 days in a year (ignoring leap years).

2. Find the effective annual interest rate as a percentage 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

#### ACMGM095

calculate the effective annual rate of interest and use the results to compare investment returns and cost of loans when interest is paid or charged daily, monthly, quarterly or six-monthly