Nominal and Effective Interest Rates

Banks quote a nominal annual rate, but interest compounded more often than once a year earns more than that rate suggests. The effective rate shows what you really earn or pay in a year, and it is the fair way to compare offers. This conversion appears in most Paper 1 finance questions.

Converting between nominal and effective rates

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

where $i^{(m)}$ is the nominal annual rate compounded $m$ times a year: $m = 12$ (monthly), $m = 4$ (quarterly), $m = 2$ (half-yearly).

Nominal to effective. 12% p.a. compounded monthly:

$i_{eff} = (1 + \frac{0 , 12}{12})^{12} - 1 \approx 0, 1268 = 12, 68%$

Effective to nominal. An effective rate of 9% p.a. as a nominal rate compounded quarterly:

$(1 + \frac{i ^{(4)}}{4})^{4} = 1, 09 \Rightarrow i^{(4)} = 4 (1, 0 9^{\frac{1}{4}} - 1) \approx 8, 71%$

Comparing offers. Convert every offer to an effective rate. The higher effective rate is better for an investor and worse for a borrower.

Keep full calculator values until the end. Rounding the rate early changes the rand answer.

Key terms

Nominal rate: The quoted annual rate, compounded a stated number of times per year.
Effective rate: The rate that, compounded once a year, gives the same growth.
Compounding period: How often interest is added: monthly, quarterly or half-yearly.
p.a.: Per annum: per year.

Common mistakes

Dividing the rate by $m$ but forgetting to raise to the power $m$ .
Subtracting 1 at the wrong point when converting effective to nominal.
Comparing nominal rates with different compounding periods directly.
Rounding the rate before using it in a further calculation.

Practice questions

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

Question 1

3 marks total

knowledge

Convert an interest rate of 12% p.a. compounded monthly to an effective annual rate.

Worked solution

Hidden - try the question first

Solution

$1 + i_{eff} = (1 + \frac{0 , 12}{12})^{12}$
$i_{eff} = 1, 0 1^{12} - 1$
$\approx 12, 68%$

Question 2

5 marks total

routine

An investment offers an effective rate of 9% p.a. Calculate the equivalent nominal rate compounded quarterly.

Worked solution

Hidden - try the question first

Solution

$1 + i_{eff} = (1 + \frac{0 , 12}{12})^{12}$
$1 + \frac{i ^{(4)}}{4} = 1, 0 9^{\frac{1}{4}}$
$\frac{i ^{(4)}}{4} = 1, 0 9^{\frac{1}{4}} - 1$
$i^{(4)} = 4 (1, 0 9^{\frac{1}{4}} - 1)$
$\approx 8, 71%$ p.a. compounded quarterly

Question 3

6 marks total

complex

Sipho wants to invest money for a year. Bank A offers 8,5% p.a. compounded monthly. Bank B offers 8,7% p.a. compounded half-yearly.

Which bank gives the better return? Show all calculations.

Worked solution

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Solution

Bank A: $i_{eff} = (1 + \frac{0 , 085}{12})^{12} - 1$
$\approx 8, 84%$
Bank B: $i_{eff} = (1 + \frac{0 , 087}{2})^{2} - 1$
$\approx 8, 89%$
Compare effective rates: $8, 89% > 8, 84%$
Bank B gives the better return

Question 4

8 marks total

problem-solving

Naledi invests R20 000. For the first year the interest rate is 7% p.a. compounded monthly. After that, the rate changes to 8% p.a. compounded quarterly for the next two years.

(a) Calculate the value of the investment after 3 years. (4)

(b) Calculate the single effective annual rate that would give the same result over the 3 years. (4)

Worked solution

Hidden - try the question first

Solution

(a) After 1 year: $i_{eff} = (1 + \frac{0 , 085}{12})^{12} - 1$
Next 2 years: 8 quarters at $\frac{0 , 08}{4} = 0, 02$
$A = 20000 (1 + \frac{0 , 07}{12})^{12} (1, 02)^{8}$
$\approx$ R25 127,17
(b) $20000 (1 + i)^{3} = 25127, 17$
$(1 + i)^{3} = 1, 25636$
$i = 1, 2563 6^{\frac{1}{3}} - 1$
$\approx 7, 90%$ p.a. effective

FAQs

Why is the effective rate higher than the nominal rate?

Because interest earns interest during the year. The more often it compounds, the bigger the difference.

Which compounding periods are examined?

Monthly, quarterly and half-yearly compounding are the standard ones in IEB finance questions.

Should I compare nominal or effective rates?

Always compare effective rates. Nominal rates with different compounding periods cannot be compared directly.

Converting between nominal and effective rates

Key terms

Common mistakes

Practice questions

FAQs

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