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So we know how to convert a monthly interest rate into an effective annual interest. Similarly, we can convert a quarterly interest, or a semi-annual interest rate or an interest rate of any frequency for that matter into an effective annual interest rate.
For a quarterly interest rate of say 3% per quarter, the interest will be paid four times per year (every three months). We can calculate the effective annual interest rate by solving for :
where is the quarterly interest rate.
So , and so . This is the effective annual interest rate.
In general, for interest paid at a frequency of times per annum, the follow equation holds:
where is the interest rate paid times per annum.
Market convention however, is not to state the interest rate as say 1% per month, but rather to express this amount as an annual amount which in this example would be paid monthly. This annual amount is called the nominal amount.
The market convention is to quote a nominal interest rate of “12% per annum paid monthly" instead of saying (an effective) 1% per month. We know from a previous example, that a nominal interest rate of 12% per annum paid monthly, equates to an effective annual interest rate of 12,68%, and the difference is due to the effects of interest-on-interest.
So if you are given an interest rate expressed as an annual rate but paid more frequently than annual, we first need to calculate the actual interest paid per period in order to calculate the effective annual interest rate.
For example, the monthly interest rate on 12% interest per annum paid monthly, is:
The same principle applies to other frequencies of payment.
Consider a savings account which pays a nominal interest at 8% per annum, paid quarterly. Calculate (a) the interest amount that is paid each quarter, and (b) the effective annual interest rate.
We are given that a savings account has a nominal interest rate of 8% paid quarterly. We are required to find:
We know that:
and
where is 4 because there are 4 payments each year.
The effective annual interest rate ( ) is calculated as:
The quarterly interest rate is 2% and the effective annual interest rate is 8,24%, for a nominal interest rate of 8% paid quarterly.
On their saving accounts, Echo Bank offers an interest rate of 18% nominal, paid monthly. If you save R100 in such an account now, how much would the amounthave accumulated to in 3 years' time?
Interest rate is 18% nominal paid monthly. There are 12 months in a year. We are working with ayearly time period, so . The amount we have saved is R100, so . We need the accumulated value, .
We know that
for converting from nominal interest rate to effective interest rate, we have
and for calculating accumulated value, we have
There are 12 month in a year, so
and then, we have
The accumulated value is R . (Remember to round off to the the nearest cent.)
As an easy reference, here are the key formulae that we derived and used during this chapter. While memorising them is nice (there are not many), it is the application that is useful. Financial experts are not paid a salary in order to recite formulae, they are paid a salary to use the right methods to solve financial problems.
| Principal (the amount of money at the starting point of the calculation) | |
| interest rate, normally the effective rate per annum | |
| period for which the investment is made | |
| the interest rate paid times per annum, i.e. |
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