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The general formula

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 i :

P ( 1 + i ) = P ( 1 + i 4 ) 4

where i 4 is the quarterly interest rate.

So ( 1 + i ) = ( 1 , 03 ) 4 , and so i = 12 , 55 % . This is the effective annual interest rate.

In general, for interest paid at a frequency of T times per annum, the follow equation holds:

P ( 1 + i ) = P ( 1 + i T ) T

where i T is the interest rate paid T times per annum.

De-coding the terminology

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.

monthly interest rate = Nominal interest rate per annum number of periods per year

For example, the monthly interest rate on 12% interest per annum paid monthly, is:

monthly interest rate = Nominal interest rate per annum number of periods per year = 12 % 12 months = 1 % per month

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.

  1. We are given that a savings account has a nominal interest rate of 8% paid quarterly. We are required to find:

    • the quarterly interest rate, i 4
    • the effective annual interest rate, i
  2. We know that:

    quarterly interest rate = Nominal interest rate per annum number of quarters per year

    and

    P ( 1 + i ) = P ( 1 + i T ) T

    where T is 4 because there are 4 payments each year.

  3. quarterly interest rate = Nominal interest rate per annum number of periods per year = 8 % 4 quarters = 2 % per quarter
  4. The effective annual interest rate ( i ) is calculated as:

    ( 1 + i ) = ( 1 + i 4 ) 4 ( 1 + i ) = ( 1 + 2 % ) 4 i = ( 1 + 2 % ) 4 - 1 = 8 , 24 %
  5. 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?

  1. Interest rate is 18% nominal paid monthly. There are 12 months in a year. We are working with ayearly time period, so n = 3 . The amount we have saved is R100, so P = 100 . We need the accumulated value, A .

  2. We know that

    monthly interest rate = Nominal interest rate per annum number of periods per year

    for converting from nominal interest rate to effective interest rate, we have

    1 + i = ( 1 + i T ) T

    and for calculating accumulated value, we have

    A = P × ( 1 + i ) n
  3. There are 12 month in a year, so

    i 12 = Nominal annual interest rate 12 = 18 % 12 = 1 , 5 % per month

    and then, we have

    1 + i = ( 1 + i 12 ) 12 i = ( 1 + i 12 ) 12 - 1 = ( 1 + 1 , 5 % ) 12 - 1 = ( 1 , 015 ) 12 - 1 = 19 , 56 %
  4. A = P × ( 1 + i ) n = 100 × ( 1 + 19 , 56 % ) 3 = 100 × 1 , 7091 = 170 , 91
  5. The accumulated value is R 170 , 91 . (Remember to round off to the the nearest cent.)

Nominal and effect interest rates

  1. Calculate the effective rate equivalent to a nominal interest rate of 8,75% p.a. compounded monthly.
  2. Cebela is quoted a nominal interest rate of 9,15% per annum compounded every four months on her investment of R 85 000. Calculate the effective rate per annum.

Formulae sheet

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.

Definitions

P Principal (the amount of money at the starting point of the calculation)
i interest rate, normally the effective rate per annum
n period for which the investment is made
i T the interest rate paid T times per annum, i.e. i T = Nominal interest rate T

Equations

Simple increase : A = P ( 1 + i × n ) Compound increase : A = P ( 1 + i ) n Simple decrease : A = P ( 1 - i × n ) Compound decrease : A = P ( 1 - i ) n Effective annual interest rate ( i ) : ( 1 + i ) = ( 1 + i T ) T

End of chapter exercises

  1. Shrek buys a Mercedes worth R385 000 in 2007. What will the value of the Mercedes be at the end of 2013 if
    1. the car depreciates at 6% p.a. straight-line depreciation
    2. the car depreciates at 12% p.a. reducing-balance depreciation.
  2. Greg enters into a 5-year hire-purchase agreement to buy a computer for R8 900. The interest rate is quoted as 11% per annum based on simple interest. Calculate the required monthly payment for this contract.
  3. A computer is purchased for R16 000. It depreciates at 15% per annum.
    1. Determine the book value of the computer after 3 years if depreciation is calculated according to the straight-line method.
    2. Find the rate, according to the reducing-balance method, that would yield the same book value as in [link] after 3 years.
  4. Maggie invests R12 500,00 for 5 years at 12% per annum compounded monthly for the first 2 years and 14% per annum compounded semi-annually for the next 3years. How much will Maggie receive in total after 5 years?
  5. Tintin invests R120 000. He is quoted a nominal interest rate of 7,2% per annum compounded monthly.
    1. Calculate the effective rate per annum correct to THREE decimal places.
    2. Use the effective rate to calculate the value of Tintin's investment if he invested the money for 3 years.
    3. Suppose Tintin invests his money for a total period of 4 years, but after 18 months makes a withdrawal of R20 000, how much will hereceive at the end of the 4 years?
  6. Paris opens accounts at a number of clothing stores and spends freely. She gets heself into terrible debt and she cannot pay off her accounts. She owes Hilton Fashion world R5 000 and the shop agrees to let Paris pay the bill at a nominal interest rate of 24% compounded monthly.
    1. How much money will she owe Hilton Fashion World after two years ?
    2. What is the effective rate of interest that Hilton Fashion World is charging her ?

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
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LITNING
scanning tunneling microscope
Sahil
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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Damian Reply
absolutely yes
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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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