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# 9.4 Nominal and effective interest rates

## 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

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.

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*}

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*}

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

## Nominal and effect interest rates

Textbook Exercise 9.6

Determine the effective annual interest rate if the nominal interest rate is:

$$\text{12}\%$$ 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*}

Consider the following:

• $$\text{16,8}\%$$ p.a. compounded annually.
• $$\text{16,4}\%$$ p.a. compounded monthly.
• $$\text{16,5}\%$$ p.a. compounded quarterly.

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*}

Determine which of the following would be the better agreement for paying back a student loan:

$$\text{9,1}\%$$ 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 $$\text{8 000} \left( 1 + \frac{\text{0,06}}{12} \right)^{60}$$ $$\text{R}\,\text{10 790,80}$$ $$\left( 1 + \frac{\text{0,06}}{12} \right)^{12} - 1 = \text{6,17}\%$$