The table below compares the effect of changing the number of compounding periods when \$1000 is invested for one year at a nominal rate of 5\% p.a. The final amount is calculated using the compound interest formula A=P\times \left(1+\dfrac{r}{n} \right)^{nt}.

No. periods	Calculation	Final amount	Interest	Effective annual interest rate
1	1000 \times \left(1+\frac{0.05}{1} \right)^{1}	\$1050	\$50	\dfrac{50}{1000} = 5\%
4	1000 \times \left(1+\frac{0.05}{4} \right)^{4}	\$1050.95	\$50.95	\dfrac{50.95}{1000} =5.095\%
365	1000 \times \left(1+\frac{0.05}{365} \right)^{365}	\$1051.27	\$51.27	\dfrac{51.27}{1000} = 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}=\dfrac{\text{Interest earned in one year}}{\text{Balance at start of year }} \times 100\%

The published rate of 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.

The effective interest formula: i_{\text{effective}}=\left(1+\dfrac{i}{n} \right)^n-1 where i_{\text{effective}} is the effective interest rate per annum, expressed as a decimal, i is the nominal (or published) interest rate per annum, expressed as a decimal, and n 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.

Examples

Example 1

James invested \$3000 at 4.6\% p.a. compounded daily.

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

Worked Solution

Create a strategy

Use the compound interest formula: A=P×\left(1+ \dfrac{r}{n} \right)^{nt}, then subtract the principal to find the interest.

Apply the idea

We are given that P=3000, r=0.046, n=365 and t=1 since we only want to find the interest for 1 year.

\displaystyle A	\displaystyle =	\displaystyle P\left(1+ \dfrac{r}{n} \right)^{nt}	Use the formula
	\displaystyle =	\displaystyle 3000 \times \left(1+ \dfrac{0.046}{365} \right)^{365 \times 1}	Substitute the values
	\displaystyle =	\displaystyle \$3141.21	Evaluate

To find the interest, we can use the formula I=A-P.

\displaystyle I	\displaystyle =	\displaystyle 3141.21-3000	Subtract the principal from the final amount
	\displaystyle =	\displaystyle \$141.21	Evaluate

Find the effective annual interest rate as a percentage to two decimal places.

Worked Solution

Create a strategy

Use the effective interest formula: i_{\text{effective}}=\left(1+\dfrac{i}{n} \right)^n-1.

Apply the idea

We are given i=4.6\%=0.046, and n=365.

\displaystyle i_{\text{effective}}	\displaystyle =	\displaystyle \left(1+\frac{0.046}{365} \right)^{365}-1	Substitute the values
	\displaystyle \approx	\displaystyle 0.0471	Evaluate
	\displaystyle =	\displaystyle 4.71\%	Write as a percentage

Idea summary

The effective interest formula:

\displaystyle i_{\text{effective}}=\left(1+\dfrac{i}{n} \right)^n-1

\bm{i}

is the effective interest rate per annum

\bm{i}

is the published interest rate per annum

\bm{n}

is the number of compounding periods per annum

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

6.03 Effective annual interest rate

Ideas

Effective annual interest rate

Examples

Example 1

Outcomes

ACMGM095

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