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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 ?

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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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