Effective interest rates | Year 12

01 / LEARN

The idea, step by step

Nominal is a quotation; effective is the annual growth

A nominal annual rate j compounded m times per year means a periodic rate of $j / m$ . Because interest earns interest during the year, the actual one-year growth is generally larger than j. The effective annual rate is the one-year multiplier minus one.

Use rates as decimals. A nominal rate is incomplete without its compounding frequency. A larger number of compounding periods does not automatically win when the quoted rates are different.

i_{e f f} = {(1 + j / m)}^{m} - 1

Compare equivalent periods

Compare products using the same one-year effective rate before considering fees or other conditions. To obtain a nominal rate equivalent to an effective annual rate e, solve $j = m ({(1 + e)}^{1 / m} - 1)$ .

For several years, the growth factor is ${(1 + e)}^{t}$ . Do not multiply an effective annual rate by the number of years unless the intended model is simple interest. Round a displayed rate only after finding the required equivalent value.

A common trap

Dividing an effective annual rate by 12 and calling the result an exactly equivalent monthly rate.

Instead: For equivalence use ${(1 + e)}^{1 / 12} - 1$ ; dividing by 12 applies to a quoted nominal rate.

02 / WORKED EXAMPLES

See the thinking

Example 1

Build the model

Find the effective annual rate for a nominal 6% annual rate compounded monthly.

Monthly rate = $0.06 / 12 = 0.005$ .
Effective rate = ${(1.005)}^{12} - 1$ = 0.061678.

6.1678%

Example 2

Use the model to make a decision

Account A quotes 5.9% compounded monthly; B quotes 6% compounded quarterly. Compare effective annual rates.

A uses 12 periods; B uses 4.
Calculate each annual multiplier, then subtract 1.
The higher effective rate is B; no conclusion about fees has yet been made.

A: 6.0622%; B: 6.1364%. B has the higher growth rate before fees.

Check your understanding

A balance grows by a factor of 1.072 in one year. State its effective annual interest rate.

Show answer

7.2%

Effective rate is annual multiplier minus 1. $1.072 - 1 = 0.072 = 7.2$ %.

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Convert 4.8% nominal per year, compounded quarterly, to an effective annual rate.
3 marks
A balance grows by a factor of 1.072 in one year. State its effective annual interest rate.
3 marks
Find the nominal annual rate, compounded monthly, equivalent to 6% effective per year.
3 marks
What monthly effective rate is equivalent to an 8% effective annual rate?
3 marks
AUD 4,000 is invested for 3 years at an effective annual rate of 5.2%. Find its balance.
3 marks
A nominal rate is 0% with monthly compounding. Find its effective annual rate and explain the result.
3 marks

Read the worked solutions

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

Convert 4.8% nominal per year, compounded quarterly, to an effective annual rate.
3 marks
Quarterly rate is 0.012.
Effective rate is ${(1.012)}^{4} - 1$ .
4.8871%
A balance grows by a factor of 1.072 in one year. State its effective annual interest rate.
3 marks
Effective rate is annual multiplier minus 1.
$1.072 - 1 = 0.072 = 7.2$ %.
7.2%
Find the nominal annual rate, compounded monthly, equivalent to 6% effective per year.
3 marks
Monthly rate is ${1.06}^{1 / 12} - 1$ .
Multiply that periodic rate by 12 to quote the nominal annual rate.
5.8411%
What monthly effective rate is equivalent to an 8% effective annual rate?
3 marks
Let i be the monthly rate, so ${(1 + i)}^{12} = 1.08$ .
$i = {1.08}^{1 / 12} - 1$ .
0.6434%
AUD 4,000 is invested for 3 years at an effective annual rate of 5.2%. Find its balance.
3 marks
Use the effective annual multiplier for each of three years.
Balance is $4000 {(1.052)}^{3}$ .
AUD 4,657.01
A nominal rate is 0% with monthly compounding. Find its effective annual rate and explain the result.
3 marks
${(1 + 0 / 12)}^{12} - 1 = 0$ .
Every monthly multiplier is 1.
0%; changing the frequency cannot create interest from a zero periodic rate.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 General Mathematics financial models: investments and annuities, superannuation, income streams and borrowing. Original practice uses supplied hypothetical rates, taxes and fees; statutory or commercial conditions are not inferred.

Official curriculum source ↗

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