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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:

Closing balance after 12 months = P × ( 1 + i ) n = R 1 000 × ( 1 + 1 % ) 12 = R 1 126 , 83

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:

P × ( 1 + i ) 1 = P × ( 1 + i 12 ) 12 ( 1 + i ) = ( 1 + i 12 ) 12 divide both sides by P i = ( 1 + i 12 ) 12 - 1 subtract 1 from both sides

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

i = ( 1 + i 12 ) 12 - 1 = ( 1 + 1 % ) 12 - 1 = 0 , 12683 = 12 , 683 %

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

Closing balance after 1 year = P × ( 1 + i ) n = R 1 000 × ( 1 + 12 , 683 % ) 1 = R 1 126 , 83

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!

Questions & Answers

what is the stm
Brian Reply
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scanning tunneling microscope
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what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Damian Reply
what king of growth are you checking .?
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Stoney Reply
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yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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Damian Reply
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Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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Akash Reply
it is a goid question and i want to know the answer as well
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
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Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Other chapter Q/A we can ask
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Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
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