Effective annual rates | Year 11

01 / LEARN

The idea, step by step

Effective annual growth

A nominal annual rate does not yet account for within-year compounding. The effective annual rate is the proportional increase during one full year: $r_{e f f} = {(1 + r / m)}^{m} - 1$ .

For 6% nominal compounded monthly, the monthly rate is 0.5%. Compounding twelve times gives slightly more than 6% growth. Subtract 1 from the growth factor before expressing the result as a percentage.

Equivalent periodic rates

An effective annual rate and a monthly rate describe the same growth if ${(1 + i)}^{12} = 1 + r_{e f f}$ . Therefore $i = {(1 + r_{e f f})}^{1 / 12} - 1$ . Dividing an effective annual rate by 12 is only an approximation.

Effective rates compare interest growth, not net outcomes. Different account fees or conditions still require a final-balance comparison.

A common trap

Equal numerical nominal and effective rates always produce equal growth.

Instead: A nominal rate with multiple compounding periods generally gives a larger effective annual rate.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example

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

i=0.06/12=0.005.
r_eff=1.005^12−1.

6.168%

Example 2

Worked example

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

(1+i)^12=1.08.
i=1.08^(1/12)−1.

0.6434%

Check your understanding

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

Show answer

10.25%

1.05^2−1=0.1025.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF 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)×100%.
8.243%
Effective annual rate for 1% per month,3 decimal places?
2 marks
(1.01^12−1)×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^(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
(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.

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