Effective interest rates | Year 12

01 / LEARN

Understand the idea.

Nominal is a quotation; effective is the annual growth

A nominal annual rate $j$ compounded $m$ times per year quotes a periodic rate $\frac{j}{m}$ . It does not include the extra growth from interest being credited and then earning further interest within that year.

The annual multiplier is ${(1 + \frac{j}{m})}^{m}$ . Subtract $1$ to find the effective annual rate. This conversion places different compounding frequencies on the same one-year basis; a nominal quotation needs both its rate and its frequency.

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

Compare equivalent periods

Equivalent rates must produce the same growth over the same interval. If $e$ is an effective annual rate, solve ${(1 + i)}^{m} = 1 + e$ to obtain $i = {(1 + e)}^{1 / m} - 1$ . The matching nominal annual quotation is then $j = m i$ .

For a constant effective annual rate over $t$ years, use ${(1 + e)}^{t}$ . Compare unrounded effective rates before considering fees or other conditions. Greater compounding frequency alone does not identify the better growth rate when the nominal rates also differ.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Build the model

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

Worked solution

1
Convert the nominal quotation to a monthly rate.
$i = \frac{0.06}{12} = 0.005$
A nominal annual rate is divided among its twelve stated compounding periods.
2
Find the one-year growth factor.
${(1 + i)}^{12} = {(1.005)}^{12} \approx 1.0616778119$
One year contains twelve successive monthly multipliers.
3
Convert the annual increase to a percentage.
$\begin{aligned} e & = {(1.005)}^{12} - 1 \\ \approx 0.0616778119 = 6.1678 % \end{aligned}$
Subtracting one removes the original principal from the growth factor.

Answer

6.1678%

EXAMPLE 02

Use the model to make a decision

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

Worked solution

1
Calculate A using twelve monthly periods.
$\begin{aligned} e_{A} & = {(1 + \frac{0.059}{12})}^{12} - 1 \\ \approx 0.0606218977 = 6.0622 % \end{aligned}$
A full year must include all twelve monthly interest credits.
2
Calculate B using four quarterly periods.
$\begin{aligned} e_{B} & = {(1 + \frac{0.06}{4})}^{4} - 1 \\ = 0.061363550625 \approx 6.1364 % \end{aligned}$
The quarterly rate is 1.5%, compounded four times in a year.
3
Compare on the common annual basis.
$e_{B} > e_{A}$
B has the higher effective annual growth rate before fees; the compounding frequency alone would not establish this.

Answer

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

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

7.2%

The factor $1.072$ comprises the original $1$ plus the annual increase. Hence $e = 1.072 - 1 = 0.072 = 7.2 %$ .

Keep practising

6 worksheet questions · 18 marks

Open worksheet 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 ${1.06}^{\frac{1}{12}} - 1$ .
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 ${(1 + i)}^{12} = 1.08$ .
$i = {1.08}^{\frac{1}{12}} - 1$ .
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
${(1 + \frac{0}{12})}^{12} - 1 = 0$ .
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 ↗

In this lesson

Understand the idea.

Nominal is a quotation; effective is the annual growth

Compare equivalent periods

See the method in action.

Build the model

Convert the nominal quotation to a monthly rate.

Find the one-year growth factor.

Convert the annual increase to a percentage.

Use the model to make a decision

Calculate A using twelve monthly periods.

Calculate B using four quarterly periods.

Compare on the common annual basis.

Try it yourself.

Check your understanding

Keep practising