Nominal and Effective Interest | IEB Grade 11 Maths

A nominal rate is what the bank quotes. The effective rate is what you actually earn in a year once compounding is included. Converting between them is the fair way to compare offers, and it is used throughout Grade 12 finance.

Converting between nominal and effective rates

$1 + i_{eff} = (1 + \frac{i ^{(m)}}{m})^{m}$

where $m = 12$ (monthly), $4$ (quarterly) or $2$ (half-yearly).

Nominal to effective. 9% p.a. compounded quarterly:

$i_{eff} = (1, 0225)^{4} - 1 \approx 9, 31%$

Effective to nominal. An effective rate of 10% as a nominal rate compounded monthly:

$(1 + \frac{i ^{(12)}}{12})^{12} = 1, 1 \Rightarrow i^{(12)} = 12 (1, 1^{\frac{1}{12}} - 1) \approx 9, 57%$

Comparing investment options

Convert each option to an effective rate, then compare.

Example. R10 000 for 2 years:

Option A: 7,8% p.a. compounded monthly gives an effective rate of about 8,08%
Option B: 8% p.a. compounded half-yearly gives an effective rate of 8,16%

Option B is better. After 2 years: A gives about R11 682,36 and B gives about R11 698,59.

Equivalent rates. Two rates are equivalent if they give the same effective rate. For example, 8% p.a. compounded monthly is equivalent to about 8,05% p.a. compounded quarterly.

Key terms

Nominal rate: The quoted annual interest rate with a stated compounding period.
Effective rate: The annual rate that gives the same growth with yearly compounding.
Equivalent rates: Different nominal rates that give the same effective rate.

Common mistakes

Comparing nominal rates directly when the compounding periods differ.
Forgetting to raise to the power $m$ .
Using $n$ in years with a monthly rate.
Rounding the effective rate before using it again.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

3 marks total

knowledge

Convert 9% p.a. compounded quarterly to an effective annual rate.

Worked solution

Hidden - try the question first

Solution

$1 + i_{eff} = (1 + \frac{0 , 09}{4})^{4}$
$= 1, 022 5^{4}$
$i_{eff} \approx 9, 31%$

Question 2

5 marks total

routine

Convert an effective rate of 10% p.a. to a nominal rate compounded monthly.

Worked solution

Hidden - try the question first

Solution

$1 + i_{eff} = (1 + \frac{0 , 09}{4})^{4}$
$1 + \frac{i}{12} = 1, 1^{\frac{1}{12}}$
$\frac{i}{12} = 1, 1^{\frac{1}{12}} - 1$
$i = 12 (1, 1^{\frac{1}{12}} - 1)$
$\approx 9, 57%$ p.a. compounded monthly

Question 3

6 marks total

complex

Lindiwe wants to invest R10 000 for 2 years.

Option A: 7,8% p.a. compounded monthly
Option B: 8% p.a. compounded half-yearly

(a) Calculate the effective rate of each option. (2)

(b) Which option is better? (1)

Worked solution

Hidden - try the question first

Solution

(a) A: $(1 + \frac{0 , 078}{12})^{12} - 1 \approx 8, 08%$
B: $(1, 04)^{2} - 1 = 8, 16%$
(b) Option B
(c) A: $10000 (1 + \frac{0 , 078}{12})^{24} \approx$ R11 682,36
B: $10000 (1, 04)^{4}$
$\approx$ R11 698,59

Question 4

8 marks total

problem-solving

R15 000 is invested for 3 years at 8% p.a. compounded monthly.

(a) Calculate the value of the investment after 3 years. (2)

(b) Calculate the nominal rate, compounded quarterly, that would give the same value. (4)

Worked solution

Hidden - try the question first

Solution

(a) $(1 + \frac{0 , 078}{12})^{12} - 1 \approx 8, 08%$
$\approx$ R19 053,56
(b) Equal growth each quarter: $(1 + \frac{i}{4}) = (1 + \frac{0 , 08}{12})^{3}$
$i = 4 [(1 + \frac{0 , 08}{12})^{3} - 1]$
$\approx 8, 05%$ p.a. compounded quarterly
(c) Interest is added less often with quarterly compounding
so a slightly higher nominal rate is needed for the same growth

FAQs

Why compare effective rates instead of nominal rates?

Effective rates include the effect of compounding, so they show the true yearly growth.

Which compounding periods are tested?

Monthly, quarterly and half-yearly compounding.

Is the effective rate always higher than the nominal rate?

Yes, when interest compounds more than once a year.

Nominal and effective interest rates

Converting between nominal and effective rates

Comparing investment options

Key terms

Common mistakes

Practice questions

FAQs

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