Effective annual rates | Year 11

01 / LEARN

Understand the idea.

Effective annual growth

A nominal annual rate states how the periodic rate is calculated; the effective annual rate measures actual growth over one complete year. If there are $m$ compounding periods, grow one unit by ${(1 + \frac{r}{m})}^{m}$ , then subtract its starting unit.

For positive interest with more than one compounding period, within-year interest also earns interest. The effective annual rate is therefore above the numerical nominal rate. The final subtraction of 1 separates the gain from the original principal.

Equivalent periodic rates

To find an equivalent monthly rate from an effective annual rate, require the twelve monthly factors to reproduce the annual factor: ${(1 + i)}^{12} = 1 + r_{eff}$ . Take the twelfth root, then subtract 1. Dividing an effective rate by 12 does not exactly reverse compounding.

Keep rates in decimal form while calculating and convert to percentages at the end. Effective annual rates compare growth over a common year, but fees and access conditions can still change which supplied account model has the higher final balance.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example

Effective annual rate for 6% nominal compounded monthly, to 3 decimal places?

Worked solution

1
Find the monthly rate from the nominal quote.
$i = \frac{0.06}{12} = 0.005$
A 6% nominal annual rate compounded monthly gives 0.5% per month.
2
Calculate the full-year growth factor.
$G = {(1.005)}^{12} \approx 1.061677812$
Twelve monthly multiplications produce one year's compound growth.
3
Remove the original principal and convert to a percentage.
$r_{eff} = (G - 1) \times 100 % \approx 6.168 %$
The part above 1 is the proportional gain, rounded to three decimal places as a percentage.

Answer

6.168%

EXAMPLE 02

Worked example

Equivalent monthly rate for 8% effective annually, to 4 decimal places as a percentage?

Worked solution

1
Set equal annual growth factors.
${(1 + i)}^{12} = 1.08$
The equivalent monthly rate must reproduce 8% growth over twelve months.
2
Take the twelfth root and subtract one.
$i = {(1.08)}^{1 / 12} - 1 \approx 0.006434030$
The root undoes twelve repeated factors, leaving one month's growth factor.
3
Convert the monthly rate to a percentage.
$i \approx 0.6434 % per month$
Multiply the decimal rate by 100 and round to four decimal places as requested.

Answer

0.6434%

03 / YOUR TURN

Try it yourself.

Check your understanding

Effective annual rate for 10% nominal compounded half-yearly?

Show answer & reasoning

10.25%

Half-yearly compounding gives $i = \frac{0.10}{2} = 0.05$ and two periods. The effective annual gain is ${(1.05)}^{2} - 1 = 0.1025$ , or $10.25 %$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Effective annual rate for 8% nominal compounded quarterly,3 decimal places?
2 marks
Effective annual rate for 1% per month,3 decimal places?
2 marks
Compare 7% effective annually with 6.9% nominal compounded monthly, ignoring fees.
2 marks
Quarterly rate equivalent to 12% effective annually,3 decimal places as a percentage?
2 marks
Why do nominal and effective annual rates agree when compounded annually?
2 marks
AUD 2000 becomes AUD 2120 in one year with no transactions or fees. Find effective annual rate.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Effective annual rate for 8% nominal compounded quarterly,3 decimal places?
2 marks
$({1.02}^{4} - 1) \times 100 %$ .
8.243%
Effective annual rate for 1% per month,3 decimal places?
2 marks
$({1.01}^{12} - 1) \times 100 %$ .
12.683%
Compare 7% effective annually with 6.9% nominal compounded monthly, ignoring fees.
2 marks
Its effective rate is 7.122%, above 7%.
The 6.9% nominal account grows faster.
Quarterly rate equivalent to 12% effective annually,3 decimal places as a percentage?
2 marks
$i = {1.12}^{\frac{1}{4}} - 1$ .
2.874%
Why do nominal and effective annual rates agree when compounded annually?
2 marks
Set $m = 1$ : ${(1 + r)}^{1} - 1 = r$ .
There is only one compounding period.
AUD 2000 becomes AUD 2120 in one year with no transactions or fees. Find effective annual rate.
2 marks
$\frac{(2120 - 2000)}{2000} = 0.06$ .
6%

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: investing and borrowing. Supplied educational models cover interest, investment comparisons, shares and credit; they are not recommendations of real products or statements of personal tax liability.

Official curriculum source ↗

In this lesson

Understand the idea.

Effective annual growth

Equivalent periodic rates

See the method in action.

Worked example

Find the monthly rate from the nominal quote.

Calculate the full-year growth factor.

Remove the original principal and convert to a percentage.

Worked example

Set equal annual growth factors.

Take the twelfth root and subtract one.

Convert the monthly rate to a percentage.

Try it yourself.

Check your understanding

Keep practising