Finance: nominal and effective interest rates

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This lesson was written by 
Free High School Science Texts Project
Objectives 
Students will find depreciation rate.
Students will calculate effective interest rates.
Procedure 

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.

TIP: 

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 after12months=P×(1+i)n

= R1000×(1+1%)12

= R1126,83

 

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 i12 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+i12)12

(1+i) =(1+i12)12divide both sides by P

i= (1+i12)12−1subtract1from both sides

 

For the example, this means that the effective annual rate for a monthly rate i12=1% is:

i=(1+i12)1

=2−1(1+1%)12−1

=0,12683

=12,683%

 

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

Closing balance after1year=P×(1+i)n

= R1000×(1+12,683%)1

= R1126,83

 

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!

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+i4)4

 

where i4 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+iT)T

 

where iT 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 annumnumber 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 annumnumber of periods per year12%12months1%per month

 

The same principle applies to other frequencies of payment.

EXERCISE 1: Nominal Interest Rate

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.

[ SHOW SOLUTION ]

EXERCISE 2: Nominal Interest Rate

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 amount have accumulated to in 3 years' time?

[ SHOW SOLUTION ]

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

iT         the interest rate paid T times per annum, i.e. iT=Nominal interest rateT

 

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+iT)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 Item 20 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 3 years. 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 he receive at the end of the 4 years?
  6. Paris opens accounts at a number of clothing stores and spends freely. She gets herself 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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