# 1.4 Nominal and effective interest rates

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## Nominal and effective interest rates

So far we have discussed annual interest rates, where the interest is quoted as a per annum amount. Although it has not been explicitly stated, we have assumed that when the interest is quoted as a per annum amount it means that the interest is paid once a year.

Interest however, may be paid more than just once a year, for example we could receive interest on a monthly basis, i.e. 12 times per year. So how do we compare a monthly interest rate, say, to an annual interest rate? This brings us to the concept of the effective annual interest rate.

One way to compare different rates and methods of interest payments would be to compare the Closing Balances under the different options, for a given Opening Balance. Another, more widely used, way is to calculate and compare the “effective annual interest rate" on each option. This way, regardless of the differences in how frequently the interest is paid, we can compare apples-with-apples.

For example, a savings account with an opening balance of R1 000 offers a compound interest rate of 1% per month which is paid at the end of every month. We can calculate the accumulated balance at the end of the year using the formulae from the previous section. But be careful as our interest rate has been given as a monthly rate, so we need to use the same units (months) for our time period of measurement.

Remember, the trick to using the formulae is to define the time period, and use the interest rate relevant to the time period.

So we can calculate the amount that would be accumulated by the end of 1-year as follows:

$\begin{array}{ccc}\hfill \mathrm{Closing balance after}\phantom{\rule{3pt}{0ex}}12\phantom{\rule{3pt}{0ex}}\mathrm{months}& =& P×{\left(1+i\right)}^{n}\hfill \\ & =& \mathrm{R}1\phantom{\rule{3.33333pt}{0ex}}000×{\left(1+1%\right)}^{12}\hfill \\ & =& \mathrm{R}1\phantom{\rule{3.33333pt}{0ex}}126,83\hfill \end{array}$

Note that because we are using a monthly time period, we have used $n$ = 12 months to calculate the balance at the end of one year.

The effective annual interest rate is an annual interest rate which represents the equivalent per annum interest rate assuming compounding.

It is the annual interest rate in our Compound Interest equation that equates to the same accumulated balance after one year. So we need to solve for the effective annual interest rate so that the accumulated balance is equal to our calculated amount of R1 126,83.

We use ${i}_{12}$ to denote the monthly interest rate. We have introduced this notation here to distinguish between the annual interest rate, $i$ . Specifically, we need to solve for $i$ in the following equation:

$\begin{array}{ccc}\hfill P×{\left(1+i\right)}^{1}& =& P×{\left(1+{i}_{12}\right)}^{12}\hfill \\ \hfill \left(1+i\right)& =& {\left(1+{i}_{12}\right)}^{12}\phantom{\rule{2em}{0ex}}\mathrm{divide both sides by P}\hfill \\ \hfill i& =& {\left(1+{i}_{12}\right)}^{12}-1\phantom{\rule{2em}{0ex}}\mathrm{subtract}\phantom{\rule{3pt}{0ex}}1\phantom{\rule{3pt}{0ex}}\mathrm{from both sides}\hfill \end{array}$

For the example, this means that the effective annual rate for a monthly rate ${i}_{12}=1%$ is:

$\begin{array}{ccc}\hfill i& =& {\left(1+{i}_{12}\right)}^{12}-1\hfill \\ & =& {\left(1+1%\right)}^{12}-1\hfill \\ & =& 0,12683\hfill \\ & =& 12,683%\hfill \end{array}$

If we recalculate the closing balance using this annual rate we get:

$\begin{array}{ccc}\hfill \mathrm{Closing balance after}\phantom{\rule{3pt}{0ex}}1\phantom{\rule{3pt}{0ex}}\mathrm{year}& =& P×{\left(1+i\right)}^{n}\hfill \\ & =& \mathrm{R}1\phantom{\rule{3.33333pt}{0ex}}000×{\left(1+12,683%\right)}^{1}\hfill \\ & =& \mathrm{R}1\phantom{\rule{3.33333pt}{0ex}}126,83\hfill \end{array}$

which is the same as the answer obtained for 12 months.

Note that this is greater than simply multiplying the monthly rate by 12 ( $12×1%=12%$ ) due to the effects of compounding. The difference is due to interest on interest. We have seen this before, but it is an important point!

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