9.4 Nominal and effective interest rates | Finance growth and decay

9.4 Nominal and effective interest rates (EMBJM)

We have seen that although interest is quoted as a percentage per annum it can be compounded more than once a year. We therefore need a way of comparing interest rates. For example, is an annual interest rate of \(\text{8}\%\) compounded quarterly higher or lower than an interest rate of \(\text{8}\%\) p.a. compounded yearly?

Nominal and effective interest rates

Calculate the accumulated amount at the end of one year if \(\text{R}\,\text{1 000}\) is invested at \(\text{8}\%\) p.a. compound interest:

\begin{align*} A &= P(1 + i)^n \\ &= \ldots \ldots \end{align*}

Calculate the value of \(\text{R}\,\text{1 000}\) if it is invested for one year at \(\text{8}\%\) p.a. compounded:

Frequency	Calculation	Accumulated amount	Interest amount
half-yearly	\(A = \text{1 000} \left( 1 + \frac{\text{0,08}}{2} \right)^{1 \times 2}\)	\(\text{R}\,\text{1 081,60}\)	\(\text{R}\,\text{81,60}\)
quarterly
monthly
weekly
daily

Use your results from the table above to calculate the effective rate that the investment of \(\text{R}\,\text{1 000}\) earns in one year:

Frequency	Accumulated amount	Calculation	Effective interest rate
half-yearly	\(\text{R}\,\text{1 081,60}\)	\(\begin{aligned} \text{1 081,60} &= \text{1 000}(1 + i) \\ \frac{\text{1 081,60}}{\text{1 000}} &= 1 + i \\ \frac{\text{1 081,60}}{\text{1 000}} - 1 &= i \\ \therefore i &= \text{0,0816} \end{aligned}\)	\(i = \text{8,16}\%\)
quarterly
monthly
weekly
daily

If you wanted to borrow \(\text{R}\,\text{10 000}\) from the bank, would it be better to pay it back at an interest rate of \(\text{22}\%\) p.a. compounded quarterly or \(\text{22}\%\) compounded monthly? Show your calculations.

An interest rate compounded more than once a year is called the nominal interest rate. In the investigation above, we determined that the nominal interest rate of \(\text{8}\%\) p.a. compounded half-yearly is actually an effective rate of \(\text{8,16}\%\) p.a.

Given a nominal interest rate \(i^{(m)}\) compounded at a frequency of \(m\) times per year and the effective interest rate \(i\), the accumulated amount calculated using both interest rates will be equal so we can write:

\begin{align*} P(1 + i) &= P \left( 1 + \frac{i^{(m)}}{m} \right)^m \\ \therefore 1 + i &= \left( 1 + \frac{i^{(m)}}{m} \right)^m \end{align*}

Worked example 15: Nominal and effective interest rates

Interest on a credit card is quoted as \(\text{23}\%\) p.a. compounded monthly. What is the effective annual interest rate? Give your answer correct to two decimal places.

Write down the known variables

Interest is being added monthly, therefore:

\begin{align*} m &= 12 \\ i^{(12)} &= \text{0,23} \end{align*}

\[1 + i = \left( 1 + \frac{i^{(m)}}{m} \right)^m\]

Substitute values and solve for \(i\)

\begin{align*} 1 + i &= \left( 1 + \frac{\text{0,23}}{12} \right)^{12} \\ \therefore i &= 1 - \left( 1 + \frac{\text{0,23}}{12} \right)^{12} \\ &= \text{25,59}\% \end{align*}

Write the final answer

The effective interest rate is \(\text{25,59}\%\) per annum.

Worked example 16: Nominal and effective interest rates

Determine the nominal interest rate compounded quarterly if the effective interest rate is \(\text{9}\%\) per annum (correct to two decimal places).

Write down the known variables

Interest is being added quarterly, therefore:

\begin{align*} m &= 4 \\ i &= \text{0,09} \end{align*}

\[1 + i = \left( 1 + \frac{i^{(m)}}{m} \right)^m\]

Substitute values and solve for \(i^{(m)}\)

\begin{align*} 1 + \text{0,09} &= \left( 1 + \frac{i^{(4)}}{4} \right)^{4} \\ \sqrt[4]{\text{1,09}} &= 1 + \frac{i^{(4)}}{4} \\ \sqrt[4]{\text{1,09}} - 1 &= \frac{i^{(4)}}{4} \\ 4 \left( \sqrt[4]{\text{1,09}} - 1 \right)&= i^{(4)}\\ \therefore i^{(4)} &= \text{8,71}\% \end{align*}

Write the final answer

The nominal interest rate is \(\text{8,71}\%\) p.a. compounded quarterly.

Nominal and effect interest rates

Textbook Exercise 9.6

\(\text{12}\%\) p.a. compounded quarterly.

\begin{align*} 1 + i &= \left( 1 + \frac{i^{(m)}}{m} \right)^{m} \\ i &= \left( 1 + \frac{\text{0,12}}{4} \right)^{4} - 1 \\ &= \left( \text{1,03} \right)^{4} - 1 \\ &= \text{0,125508} \ldots \\ \therefore i &\approx \text{12,6}\% \end{align*}

\(\text{14,5}\%\) p.a. compounded weekly.

\begin{align*} 1 + i &= \left( 1 + \frac{i^{(m)}}{m} \right)^{m} \\ i &= \left( 1 + \frac{\text{0,145}}{52} \right)^{52} - 1 \\ &= \text{0,155806} \ldots \\ \therefore i &\approx \text{15,6}\% \end{align*}

\(\text{20}\%\) p.a. compounded daily.

\begin{align*} 1 + i &= \left( 1 + \frac{i^{(m)}}{m} \right)^{m} \\ i &= \left( 1 + \frac{\text{0,2}}{\text{365}} \right)^{\text{365}} - 1 \\ &= \text{0,221335} \ldots \\ \therefore i &= \text{22,1}\% \end{align*}

Determine the effective annual interest rate of each of the nominal rates listed above.

\begin{align*} 1 + i &= \left( 1 + \frac{i^{(m)}}{m} \right)^{m} \\ i &= \left( 1 + \frac{\text{0,164}}{12} \right)^{12} - 1 \\ &= \text{0,176906} \ldots \\ \therefore i &= \text{17,7}\% \end{align*} \begin{align*} 1 + i &= \left( 1 + \frac{i^{(m)}}{m} \right)^{m} \\ i &= \left( 1 + \frac{\text{0,165}}{4} \right)^{4} - 1 \\ &= \text{0,175493}\ldots \\ \therefore i &= \text{17,5}\% \end{align*}

Which is the best interest rate for an investment?

\(\text{17,7}\%\)

Which is the best interest rate for a loan?

\(\text{16,8}\%\)

Calculate the effective annual interest rate equivalent to a nominal interest rate of \(\text{8,75}\%\) p.a. compounded monthly.

\begin{align*} 1 + i &= \left( 1 + \frac{i^{(m)}}{m} \right)^{m} \\ i &= \left( 1 + \frac{\text{0,0875}}{12} \right)^{12} - 1 \\ &= \text{0,091095} \ldots \\ \therefore i &= \text{9,1}\% \end{align*}

Cebela is quoted a nominal interest rate of \(\text{9,15}\%\) per annum compounded every four months on her investment of \(\text{R}\,\text{85 000}\). Calculate the effective rate per annum.

\begin{align*} 1 + i &= \left( 1 + \frac{i^{(m)}}{m} \right)^{m} \\ i &= \left( 1 + \frac{\text{0,0915}}{3} \right)^{3} - 1 \\ &= \text{0,094319} \ldots \\ \therefore i &= \text{9,4}\% \end{align*}

\(\text{9,1}\%\) p.a. compounded quarterly.

\begin{align*} 1 + i &= \left( 1 + \frac{i^{(m)}}{m} \right)^{m} \\ i &= \left( 1 + \frac{\text{0,091}}{4} \right)^{4} - 1 \\ &= \text{0,094152} \ldots \\ \therefore i &= \text{9,42}\% \end{align*}

\(\text{9}\%\) p.a. compounded monthly.

\begin{align*} 1 + i &= \left( 1 + \frac{i^{(m)}}{m} \right)^{m} \\ i &= \left( 1 + \frac{\text{0,09}}{12} \right)^{12} - 1 \\ &= \text{0,093806} \ldots \\ \therefore i &= \text{9,38}\% \end{align*}

\(\text{9,3}\%\) p.a. compounded half-yearly.

\begin{align*} 1 + i &= \left( 1 + \frac{i^{(m)}}{m} \right)^{m} \\ i &= \left( 1 + \frac{\text{0,093}}{2} \right)^{2} - 1 \\ &= \text{0,095162} \ldots \\ \therefore i &= \text{9,52}\% \end{align*}

Miranda invests \(\text{R}\,\text{8 000}\) for \(\text{5}\) years for her son's study fund. Determine how much money she will have at the end of the period and the effective annual interest rate if the nominal interest of \(\text{6}\%\) is compounded:

	Calculation	Accumulated amount	Effective annual interest rate
yearly
half-yearly
quarterly
monthly

	Calculation	Accumulated amount	Effective annual interest rate
yearly	\(\text{8 000} \left( 1 + \frac{\text{0,06}}{1} \right)^5\)	\(\text{R}\,\text{10 705,80}\)	\(\text{6}\%\)
half-yearly	\(\text{8 000} \left( 1 + \frac{\text{0,06}}{2} \right)^{10}\)	\(\text{R}\,\text{10 751,33}\)	\(\left( 1 + \frac{\text{0,06}}{2} \right)^2 - 1 = \text{6,09}\%\)
quarterly	\(\text{8 000} \left( 1 + \frac{\text{0,06}}{4} \right)^{20}\)	\(\text{R}\,\text{10 774,84}\)	\(\left( 1 + \frac{\text{0,06}}{4} \right)^4 - 1 = \text{6,14}\%\)
monthly	\(\text{8 000} \left( 1 + \frac{\text{0,06}}{12} \right)^{60}\)	\(\text{R}\,\text{10 790,80}\)	\(\left( 1 + \frac{\text{0,06}}{12} \right)^{12} - 1 = \text{6,17}\%\)

9.3 Timelines

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9.5 Summary