What is the annual effective interest rate of the annual nominal interest rate is 12?

Nominal and effective interest rates

1. 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*}

2. 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

3. 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

4. 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.

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: