ICAI Notes- Index numbers | Quantitative Aptitude for CA Foundation PDF Download

 Page 1


INDEX NUMBERS
18
CHAPTER
Index Numbers
Issues Involved
Index Numbers
Types of
Index Numbers
Construction of
Index Numbers
Chain Index
Numbers
Value Index
Numbers
Price Index
Numbers
Quantity Index
Numbers
Splicing of Index
Numbers
Deflating Index
Numbers
Circular
Test
Factor
Reversal Text
Times
Reversal Text
Unit
Test
Often we encounter news of price rise, GDP growth, production growth, etc. It is important for
students of Chartered Accountancy to learn techniques of measuring growth/rise or decline of
various economic and business data and how to report them objectively.
After reading the chapter, students will be able to understand:
? Purpose of constructing index number and its important applications in understanding
rise or decline of production, prices, etc.
? Different methods of computing index number.
Usefulness of
Index Numbers
Tests of
Adequacy
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
Page 2


INDEX NUMBERS
18
CHAPTER
Index Numbers
Issues Involved
Index Numbers
Types of
Index Numbers
Construction of
Index Numbers
Chain Index
Numbers
Value Index
Numbers
Price Index
Numbers
Quantity Index
Numbers
Splicing of Index
Numbers
Deflating Index
Numbers
Circular
Test
Factor
Reversal Text
Times
Reversal Text
Unit
Test
Often we encounter news of price rise, GDP growth, production growth, etc. It is important for
students of Chartered Accountancy to learn techniques of measuring growth/rise or decline of
various economic and business data and how to report them objectively.
After reading the chapter, students will be able to understand:
? Purpose of constructing index number and its important applications in understanding
rise or decline of production, prices, etc.
? Different methods of computing index number.
Usefulness of
Index Numbers
Tests of
Adequacy
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
18.2
STATISTICS
Index numbers are convenient devices for measuring relative changes of differences from time
to time or from place to place. Just as the arithmetic mean is used to represent a set of values, an
index number is used to represent a set of values over two or more different periods or localities.
The basic device used in all methods of index number construction is to average the relative
change in either quantities or prices since relatives are comparable and can be added even though
the data from which they were derived cannot themselves be added. For example, if wheat
production has gone up to 110% of the previous year’s  producton and cotton production has
gone up to 105%, it is possible to average the two percentages as they have gone up by 107.5%.
This assumes that both have equal weight; but if wheat production is twice as important as
cotton, percentage should be weighted 2 and 1. The average relatives obtained through this process
are called the index numbers.
Definition: An index number is a ratio of two or more time periods are involved, one of which is
the base time period. The value at the base time period serves as the standard point of comparison.
Example: NSE, BSE, WPI, CPI etc.
An index time series is a list of index numbers for two or more periods of time, where each index
number employs the same base year.
Relatives are derived because absolute numbers measured in some appropriate unit, are often of
little importance and meaningless in themselves. If the meaning of a relative figure remains
ambiguous, it is necessary to know the absolute as well as the relative number.
Our discussion of index numbers is confined to various types of index numbers, their uses, the
mathematical tests and the principles involved in the construction of index numbers.
Index numbers are studied here because some techniques for making forecasts or inferences
about the figures are applied in terms of index number. In regression analysis, either the
independent or dependent variable or both may be in the form of index numbers. They are less
unwieldy than large numbers and are readily understandable.
These are of two broad types: simple and composite. The simple index is computed for one
variable whereas the composite is calculated from two or more variables. Most index numbers
are composite in nature.
Following are some of the important criteria/problems which have to be faced in the construction
of index Numbers.
Selection of data: It is important to understand the purpose for which the index is used. If it is used
for purposes of knowing the cost of living, there is no need of including the prices of capital goods
which do not directly influence the living.
Index numbers are often constructed from the sample. It is necessary to ensure that it is
representative. Random sampling, and if need be, a stratified random sampling can ensure this.
© The Institute of Chartered Accountants of India
Page 3


INDEX NUMBERS
18
CHAPTER
Index Numbers
Issues Involved
Index Numbers
Types of
Index Numbers
Construction of
Index Numbers
Chain Index
Numbers
Value Index
Numbers
Price Index
Numbers
Quantity Index
Numbers
Splicing of Index
Numbers
Deflating Index
Numbers
Circular
Test
Factor
Reversal Text
Times
Reversal Text
Unit
Test
Often we encounter news of price rise, GDP growth, production growth, etc. It is important for
students of Chartered Accountancy to learn techniques of measuring growth/rise or decline of
various economic and business data and how to report them objectively.
After reading the chapter, students will be able to understand:
? Purpose of constructing index number and its important applications in understanding
rise or decline of production, prices, etc.
? Different methods of computing index number.
Usefulness of
Index Numbers
Tests of
Adequacy
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
18.2
STATISTICS
Index numbers are convenient devices for measuring relative changes of differences from time
to time or from place to place. Just as the arithmetic mean is used to represent a set of values, an
index number is used to represent a set of values over two or more different periods or localities.
The basic device used in all methods of index number construction is to average the relative
change in either quantities or prices since relatives are comparable and can be added even though
the data from which they were derived cannot themselves be added. For example, if wheat
production has gone up to 110% of the previous year’s  producton and cotton production has
gone up to 105%, it is possible to average the two percentages as they have gone up by 107.5%.
This assumes that both have equal weight; but if wheat production is twice as important as
cotton, percentage should be weighted 2 and 1. The average relatives obtained through this process
are called the index numbers.
Definition: An index number is a ratio of two or more time periods are involved, one of which is
the base time period. The value at the base time period serves as the standard point of comparison.
Example: NSE, BSE, WPI, CPI etc.
An index time series is a list of index numbers for two or more periods of time, where each index
number employs the same base year.
Relatives are derived because absolute numbers measured in some appropriate unit, are often of
little importance and meaningless in themselves. If the meaning of a relative figure remains
ambiguous, it is necessary to know the absolute as well as the relative number.
Our discussion of index numbers is confined to various types of index numbers, their uses, the
mathematical tests and the principles involved in the construction of index numbers.
Index numbers are studied here because some techniques for making forecasts or inferences
about the figures are applied in terms of index number. In regression analysis, either the
independent or dependent variable or both may be in the form of index numbers. They are less
unwieldy than large numbers and are readily understandable.
These are of two broad types: simple and composite. The simple index is computed for one
variable whereas the composite is calculated from two or more variables. Most index numbers
are composite in nature.
Following are some of the important criteria/problems which have to be faced in the construction
of index Numbers.
Selection of data: It is important to understand the purpose for which the index is used. If it is used
for purposes of knowing the cost of living, there is no need of including the prices of capital goods
which do not directly influence the living.
Index numbers are often constructed from the sample. It is necessary to ensure that it is
representative. Random sampling, and if need be, a stratified random sampling can ensure this.
© The Institute of Chartered Accountants of India
18.3
INDEX NUMBERS
It is also necessary to ensure comparability of data. This can be ensured by consistency in the
method of selection of the units for compilation of index numbers.
However, difficulties arise in the selection of commodities because the relative importance of
commodities keep on changing with the advancement of the society. More so, if the period is
quite long, these changes are quite significant both in the basket of production and the uses made
by people.
Base Period: It should be carefully selected because it is a point of reference in comparing various
data describing individual behaviour. The period should be normal i.e., one of the relative stability,
not affected by extraordinary events like war, famine, etc. It should be relatively recent because
we are more concerned with the changes with reference to the present and not with the distant
past. There are three variants of the base fixed, chain, and the average.
Selection of Weights: It is necessary to point out that each variable involved in composite index
should have a reasonable influence on the index, i.e., due consideration should be given to the
relative importance of each variable which relates to the purpose for which the index is to be
used. For example, in the computation of cost of living index, sugar cannot be given the same
importance as the cereals.
Use of Averages: Since we have to arrive at a single index number summarising a large amount
of information, it is easy to realise that average plays an important role in computing index
numbers. The geometric mean is better in averaging relatives, but for most of the indices arithmetic
mean is used because of its simplicity.
Choice of Variables: Index numbers are constructed with regard to price or quantity or any
other measure. We have to decide about the unit. For example, in price index numbers it is
necessary to decide whether to have wholesale or the retail prices. The choice would depend on
the purpose. Further, it is necessary to decide about the period to which such prices will be
related. There may be an average of price for certain time-period or the end of the period. The
former is normally preferred.
Selection of Formula: The question of selection of an appropriate formula arises, since different
types of indices give different values when applied to the same data. We will see different types
of indices to be used for construction succeedingly.
Notations: It is customary to let P
n
(
1
), P
n
(
2
), P
n
(
3
) denote the prices during n
th
 period for the first,
second and third commodity. The corresponding price during a base period are denoted by P
o
(
1
),
P
o
(
2
), P
o
(
3
), etc. With these notations the price of commodity j during period n can be indicated by
P
n
(
j
). We can use the summation notation by summing over the superscripts j as follows:
k
? P
n
 ( j)    or    ? P
n
( j )
j = 1
We can omit the superscript altogether and write as ? P
n
 etc.
Relatives: One of the simplest examples of an index number is a price relative, which is the ratio
of the price of single commodity in a given period to its price in another period called the base
period or the reference period. It can be indicated as follows:
© The Institute of Chartered Accountants of India
Page 4


INDEX NUMBERS
18
CHAPTER
Index Numbers
Issues Involved
Index Numbers
Types of
Index Numbers
Construction of
Index Numbers
Chain Index
Numbers
Value Index
Numbers
Price Index
Numbers
Quantity Index
Numbers
Splicing of Index
Numbers
Deflating Index
Numbers
Circular
Test
Factor
Reversal Text
Times
Reversal Text
Unit
Test
Often we encounter news of price rise, GDP growth, production growth, etc. It is important for
students of Chartered Accountancy to learn techniques of measuring growth/rise or decline of
various economic and business data and how to report them objectively.
After reading the chapter, students will be able to understand:
? Purpose of constructing index number and its important applications in understanding
rise or decline of production, prices, etc.
? Different methods of computing index number.
Usefulness of
Index Numbers
Tests of
Adequacy
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
18.2
STATISTICS
Index numbers are convenient devices for measuring relative changes of differences from time
to time or from place to place. Just as the arithmetic mean is used to represent a set of values, an
index number is used to represent a set of values over two or more different periods or localities.
The basic device used in all methods of index number construction is to average the relative
change in either quantities or prices since relatives are comparable and can be added even though
the data from which they were derived cannot themselves be added. For example, if wheat
production has gone up to 110% of the previous year’s  producton and cotton production has
gone up to 105%, it is possible to average the two percentages as they have gone up by 107.5%.
This assumes that both have equal weight; but if wheat production is twice as important as
cotton, percentage should be weighted 2 and 1. The average relatives obtained through this process
are called the index numbers.
Definition: An index number is a ratio of two or more time periods are involved, one of which is
the base time period. The value at the base time period serves as the standard point of comparison.
Example: NSE, BSE, WPI, CPI etc.
An index time series is a list of index numbers for two or more periods of time, where each index
number employs the same base year.
Relatives are derived because absolute numbers measured in some appropriate unit, are often of
little importance and meaningless in themselves. If the meaning of a relative figure remains
ambiguous, it is necessary to know the absolute as well as the relative number.
Our discussion of index numbers is confined to various types of index numbers, their uses, the
mathematical tests and the principles involved in the construction of index numbers.
Index numbers are studied here because some techniques for making forecasts or inferences
about the figures are applied in terms of index number. In regression analysis, either the
independent or dependent variable or both may be in the form of index numbers. They are less
unwieldy than large numbers and are readily understandable.
These are of two broad types: simple and composite. The simple index is computed for one
variable whereas the composite is calculated from two or more variables. Most index numbers
are composite in nature.
Following are some of the important criteria/problems which have to be faced in the construction
of index Numbers.
Selection of data: It is important to understand the purpose for which the index is used. If it is used
for purposes of knowing the cost of living, there is no need of including the prices of capital goods
which do not directly influence the living.
Index numbers are often constructed from the sample. It is necessary to ensure that it is
representative. Random sampling, and if need be, a stratified random sampling can ensure this.
© The Institute of Chartered Accountants of India
18.3
INDEX NUMBERS
It is also necessary to ensure comparability of data. This can be ensured by consistency in the
method of selection of the units for compilation of index numbers.
However, difficulties arise in the selection of commodities because the relative importance of
commodities keep on changing with the advancement of the society. More so, if the period is
quite long, these changes are quite significant both in the basket of production and the uses made
by people.
Base Period: It should be carefully selected because it is a point of reference in comparing various
data describing individual behaviour. The period should be normal i.e., one of the relative stability,
not affected by extraordinary events like war, famine, etc. It should be relatively recent because
we are more concerned with the changes with reference to the present and not with the distant
past. There are three variants of the base fixed, chain, and the average.
Selection of Weights: It is necessary to point out that each variable involved in composite index
should have a reasonable influence on the index, i.e., due consideration should be given to the
relative importance of each variable which relates to the purpose for which the index is to be
used. For example, in the computation of cost of living index, sugar cannot be given the same
importance as the cereals.
Use of Averages: Since we have to arrive at a single index number summarising a large amount
of information, it is easy to realise that average plays an important role in computing index
numbers. The geometric mean is better in averaging relatives, but for most of the indices arithmetic
mean is used because of its simplicity.
Choice of Variables: Index numbers are constructed with regard to price or quantity or any
other measure. We have to decide about the unit. For example, in price index numbers it is
necessary to decide whether to have wholesale or the retail prices. The choice would depend on
the purpose. Further, it is necessary to decide about the period to which such prices will be
related. There may be an average of price for certain time-period or the end of the period. The
former is normally preferred.
Selection of Formula: The question of selection of an appropriate formula arises, since different
types of indices give different values when applied to the same data. We will see different types
of indices to be used for construction succeedingly.
Notations: It is customary to let P
n
(
1
), P
n
(
2
), P
n
(
3
) denote the prices during n
th
 period for the first,
second and third commodity. The corresponding price during a base period are denoted by P
o
(
1
),
P
o
(
2
), P
o
(
3
), etc. With these notations the price of commodity j during period n can be indicated by
P
n
(
j
). We can use the summation notation by summing over the superscripts j as follows:
k
? P
n
 ( j)    or    ? P
n
( j )
j = 1
We can omit the superscript altogether and write as ? P
n
 etc.
Relatives: One of the simplest examples of an index number is a price relative, which is the ratio
of the price of single commodity in a given period to its price in another period called the base
period or the reference period. It can be indicated as follows:
© The Institute of Chartered Accountants of India
18.4
STATISTICS
Price relative = 
o
n
P
P
It has to be expressed as a percentage, it is multiplied by 100
Price relative =
100
P
P
o
n
?
There can be other relatives such as of quantities, volume of consumption, exports, etc. The
relatives in that case will be:
Quantity relative = 
o
n
Q
Q
Similarly, there are value relatives:
Value relative = 
?
?
?
?
?
?
?
?
? ? ?
o
n
o
n
o o
nn
o
n
Q
Q
P
P
QP
QP
V
V
When successive prices or quantities are taken, the relatives are called the link relative,
1n
n
2
3
1
2
o
1
P
P
,
P
P
,
P
P
,
P
P
?
When the above relatives are in respect to a fixed base period these are also called the chain relatives
with respect to this base or the relatives chained to the fixed base. They are in the form of :
o
n
o
3
o
2
o
1
P
P
,
P
P
,
P
P
,
P
P
Methods: We can state the broad heads as follows:
In this method of computing a price index, we express the total of commodity prices in a given
year as a percentage of total commodity price in the base year. In symbols, we have
Simple aggregative price index =
100
P
P
o
n
?
?
?
where ?P
n
 is the sum of all commodity prices in the current year and ?P
o
 is the sum of all
commodity prices in the base year.
Methods
Simple Weighted
Aggregative Relative                                                                   Aggregative Relative
© The Institute of Chartered Accountants of India
Page 5


INDEX NUMBERS
18
CHAPTER
Index Numbers
Issues Involved
Index Numbers
Types of
Index Numbers
Construction of
Index Numbers
Chain Index
Numbers
Value Index
Numbers
Price Index
Numbers
Quantity Index
Numbers
Splicing of Index
Numbers
Deflating Index
Numbers
Circular
Test
Factor
Reversal Text
Times
Reversal Text
Unit
Test
Often we encounter news of price rise, GDP growth, production growth, etc. It is important for
students of Chartered Accountancy to learn techniques of measuring growth/rise or decline of
various economic and business data and how to report them objectively.
After reading the chapter, students will be able to understand:
? Purpose of constructing index number and its important applications in understanding
rise or decline of production, prices, etc.
? Different methods of computing index number.
Usefulness of
Index Numbers
Tests of
Adequacy
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
18.2
STATISTICS
Index numbers are convenient devices for measuring relative changes of differences from time
to time or from place to place. Just as the arithmetic mean is used to represent a set of values, an
index number is used to represent a set of values over two or more different periods or localities.
The basic device used in all methods of index number construction is to average the relative
change in either quantities or prices since relatives are comparable and can be added even though
the data from which they were derived cannot themselves be added. For example, if wheat
production has gone up to 110% of the previous year’s  producton and cotton production has
gone up to 105%, it is possible to average the two percentages as they have gone up by 107.5%.
This assumes that both have equal weight; but if wheat production is twice as important as
cotton, percentage should be weighted 2 and 1. The average relatives obtained through this process
are called the index numbers.
Definition: An index number is a ratio of two or more time periods are involved, one of which is
the base time period. The value at the base time period serves as the standard point of comparison.
Example: NSE, BSE, WPI, CPI etc.
An index time series is a list of index numbers for two or more periods of time, where each index
number employs the same base year.
Relatives are derived because absolute numbers measured in some appropriate unit, are often of
little importance and meaningless in themselves. If the meaning of a relative figure remains
ambiguous, it is necessary to know the absolute as well as the relative number.
Our discussion of index numbers is confined to various types of index numbers, their uses, the
mathematical tests and the principles involved in the construction of index numbers.
Index numbers are studied here because some techniques for making forecasts or inferences
about the figures are applied in terms of index number. In regression analysis, either the
independent or dependent variable or both may be in the form of index numbers. They are less
unwieldy than large numbers and are readily understandable.
These are of two broad types: simple and composite. The simple index is computed for one
variable whereas the composite is calculated from two or more variables. Most index numbers
are composite in nature.
Following are some of the important criteria/problems which have to be faced in the construction
of index Numbers.
Selection of data: It is important to understand the purpose for which the index is used. If it is used
for purposes of knowing the cost of living, there is no need of including the prices of capital goods
which do not directly influence the living.
Index numbers are often constructed from the sample. It is necessary to ensure that it is
representative. Random sampling, and if need be, a stratified random sampling can ensure this.
© The Institute of Chartered Accountants of India
18.3
INDEX NUMBERS
It is also necessary to ensure comparability of data. This can be ensured by consistency in the
method of selection of the units for compilation of index numbers.
However, difficulties arise in the selection of commodities because the relative importance of
commodities keep on changing with the advancement of the society. More so, if the period is
quite long, these changes are quite significant both in the basket of production and the uses made
by people.
Base Period: It should be carefully selected because it is a point of reference in comparing various
data describing individual behaviour. The period should be normal i.e., one of the relative stability,
not affected by extraordinary events like war, famine, etc. It should be relatively recent because
we are more concerned with the changes with reference to the present and not with the distant
past. There are three variants of the base fixed, chain, and the average.
Selection of Weights: It is necessary to point out that each variable involved in composite index
should have a reasonable influence on the index, i.e., due consideration should be given to the
relative importance of each variable which relates to the purpose for which the index is to be
used. For example, in the computation of cost of living index, sugar cannot be given the same
importance as the cereals.
Use of Averages: Since we have to arrive at a single index number summarising a large amount
of information, it is easy to realise that average plays an important role in computing index
numbers. The geometric mean is better in averaging relatives, but for most of the indices arithmetic
mean is used because of its simplicity.
Choice of Variables: Index numbers are constructed with regard to price or quantity or any
other measure. We have to decide about the unit. For example, in price index numbers it is
necessary to decide whether to have wholesale or the retail prices. The choice would depend on
the purpose. Further, it is necessary to decide about the period to which such prices will be
related. There may be an average of price for certain time-period or the end of the period. The
former is normally preferred.
Selection of Formula: The question of selection of an appropriate formula arises, since different
types of indices give different values when applied to the same data. We will see different types
of indices to be used for construction succeedingly.
Notations: It is customary to let P
n
(
1
), P
n
(
2
), P
n
(
3
) denote the prices during n
th
 period for the first,
second and third commodity. The corresponding price during a base period are denoted by P
o
(
1
),
P
o
(
2
), P
o
(
3
), etc. With these notations the price of commodity j during period n can be indicated by
P
n
(
j
). We can use the summation notation by summing over the superscripts j as follows:
k
? P
n
 ( j)    or    ? P
n
( j )
j = 1
We can omit the superscript altogether and write as ? P
n
 etc.
Relatives: One of the simplest examples of an index number is a price relative, which is the ratio
of the price of single commodity in a given period to its price in another period called the base
period or the reference period. It can be indicated as follows:
© The Institute of Chartered Accountants of India
18.4
STATISTICS
Price relative = 
o
n
P
P
It has to be expressed as a percentage, it is multiplied by 100
Price relative =
100
P
P
o
n
?
There can be other relatives such as of quantities, volume of consumption, exports, etc. The
relatives in that case will be:
Quantity relative = 
o
n
Q
Q
Similarly, there are value relatives:
Value relative = 
?
?
?
?
?
?
?
?
? ? ?
o
n
o
n
o o
nn
o
n
Q
Q
P
P
QP
QP
V
V
When successive prices or quantities are taken, the relatives are called the link relative,
1n
n
2
3
1
2
o
1
P
P
,
P
P
,
P
P
,
P
P
?
When the above relatives are in respect to a fixed base period these are also called the chain relatives
with respect to this base or the relatives chained to the fixed base. They are in the form of :
o
n
o
3
o
2
o
1
P
P
,
P
P
,
P
P
,
P
P
Methods: We can state the broad heads as follows:
In this method of computing a price index, we express the total of commodity prices in a given
year as a percentage of total commodity price in the base year. In symbols, we have
Simple aggregative price index =
100
P
P
o
n
?
?
?
where ?P
n
 is the sum of all commodity prices in the current year and ?P
o
 is the sum of all
commodity prices in the base year.
Methods
Simple Weighted
Aggregative Relative                                                                   Aggregative Relative
© The Institute of Chartered Accountants of India
18.5
INDEX NUMBERS
 ILLUSTRATIONS:
Commodities 1998 1999 2000
Cheese (per 100 gms) 12.00 15.00 15.60
Egg (per piece) 3.00 3.60 3.30
Potato (per kg) 5.00 6.00 5.70
Aggregrate 20.00 24.60 24.60
Index 100 123 123
Simple Aggregative Index for 1999 over 1998 = 
123 100
00 . 20
60 . 24
P
P
o
n
? ? ?
?
?
and for 2000 over 1998 = 
123 100
00 . 20
60 . 24
100
P
P
o
n
? ? ? ?
?
?
The above method is easy to understand but it has a serious defect. It shows that the first
commodity exerts greater influence than the other two because the price of the first commodity
is higher than that of the other two. Further, if units are changed then the Index numbers will
also change. Students should independently calculate the Index number taking the price of eggs
per dozen i.e., ` 36, ` 43.20, ` 39.60 for the three years respectively. This is the major flaw in using
absolute quantities and not the relatives. Such price quotations become the concealed weights
which have no logical significance.
One way to rectify the drawbacks of a simple aggregative index is to construct a simple average
of relatives. Under it we invert the actual price for each variable into percentage of the base
period. These percentages are called relatives because they are relative to the value for the base
period. The index number is the average of all such relatives. One big advantage of price relatives
is that they are pure numbers. Price index number computed from relatives will remain the same
regardless of the units by which the prices are quoted. This method thus meets criterion of unit
test (discussed later). Also quantity index can be constructed for a group of variables that are
expressed in divergent units.
 ILLUSTRATIONS:
In the proceeding example we will calculate relatives as follows:
Commodities 1998 1999 2000
A 100.0 125.0 130.0
B 100.0 120.0 110.0
C 100.0 120.0 114.0
Aggregate 300.0 365.0 354.0
Index 100.0 121.67 118.0
© The Institute of Chartered Accountants of India