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Index number and indexes

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Autor:  fahadsaleem  20 November 2011
Tags:  Statistics
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Introduction


Index numbers are a statistician's way of expressing the difference between two measurements by designating one number as the "base", giving it the value 100 and then expressing the second number as a percentage of the first.
or

A measure of the value of a variable relative to its value at some base date or state (the base period). The index is often scaled so that its base value is 100. Such an index may be described as a base-weighted index


Mathematical representation of Index

Index number is basically a ratio of two quantity.For volume q and price p and the value V of the transactions in any commodity,

Q = V / P

Other formulas of index number are as follow and few of them are explained later



Let be the price per unit in period , be the quantity produced in period , and be the value of the units. Let be the estimated relative importance of a product. There are several types of indices defined, among them those listed in the following table.





index abbr. formula
Bowley index



Fisher index



geometric mean index



harmonic mean index



Laspeyres' index



marshall-Edgeworth index



mitchell index



Paasche's index



Walsh index





Examples


If the population of a town increased from 20,000 in 1988 to 21,000 in 1991, the population in 1991 was 105% of the population in 1988. Therefore, on a 1988 = 100 base, the population index for the town was 105 in 1991.

An "index", as the term is generally used when referring to statistics, is a series of index numbers expressing a series of numbers as percentages of a single number.





Uses Of Index Number


Indexes can be used to express comparisons between places, industries, etc. but the most common use is to express changes over a period of time, in which case the index is also a time series or "series". One point in time is designated the base period—it may be a year, month, or any other period—and given the value 100. The index numbers for the measurement (price, quantity, value, etc.) at all other points in time indicate the percentage change from the base period.

If the price, quantity or value has increased by 15% since the base period, the index is 115; if it has fallen 5%, the index is 95. It is important to note that indexes reflect percentage differences relative to the base year and not absolute levels. If the price index for one item is 110 and for another is 105, it means the price of the first has increased twice as much as the price of the second. It does not mean that the first item is more expensive than the second.

Each index number in a series reflects the percentage change from the base period. It is important not to confuse an index point change and a percentage change between two index numbers in a series.

Example: if the price index for butter was 130 one year and 143 the next year, the index point change would be:
143 – 130 = 13
but the percentage change for the index would be:
(143 – 130) x 100) ÷ 130 = 10%



Calculating Index Numbers

There are a number of different ways of calculating index numbers. Again, the easiest way of explaining what these are and how to work them out is to look at an example.

The small firm of Tastynibbles Ltd likes to give all its workers a Christmas party. The snacks are provided from Tastynibbles stock (a useful way of running down any extra stock before the Christmas/New year holiday), but they have to buy the drinks. They buy bottles of red and white wine (all at one price), 6-packs of beer also all at one price and litre bottles of soft drinks which are also priced the same as each other.
The first Christmas party was in 1990. It was a great success and so has been held every year since. We'll now look at how we can use index numbers to compare the cost for the base period of 1990 against the cost for a later period, taking the particular year of 2000 for our example.
The table below shows the details of the purchases for these two parties.


The Tastynibbles Christmas parties of 1990 and 2000

The 1990 party The 2000 party
Drink Unit price Quantity Unit price Quantity
po qo pn qn
wine £2.50 25 £3 30
beer £4.50 10 £6.00 8
soft drinks £0.60 10 £0.84 15



Now we'll use this data to show how to work out various index numbers.




The expenditure index

In a simple situation like my example, where we are comparing only a few purchases made on just two occasions, we can obtain the most exact information about the changing cost of the party by working out the expenditure index.
To do this, we first calculate the total cost of the party in 1990 and then the total cost of the party in 2000.
Also, to help us make a general rule for finding the index number, we'll make use of the capital Greek S which is called sigma and written
.
Mathematicians use to mean "the sum of everything like..."

Now, the party's cost in 1990
= poqo = (2.5 x 25) + (4.5 x 10) + (0.6 x 10) = 113.5.
The party's cost in 2000
= pnqn = (3 x 30) + (6 x 8) + (0.84 x 15) = 150.6.

(I have left out the £ signs here since the method is the same for all currencies and the index number is independent of the currency.)


Now, we work in a similar way to when we found the price relative for the cup of coffee.
The expenditure index = (party's cost in 2000)/(party's cost in 1990) x 100
= ( pnqn/ poqo) x 100 = (150.6/113.5) x 100 = 132.7 to 1 d.p.
Notice that we have taken account of the different quantities for wine, beer and soft drinks by multiplying the unit prices by the corresponding quantities. This process is called weighting.
We could have worked out what is called a simple aggregative index by just taking account of the unit prices as follows:
The simple aggregative index = ((3 + 6 + 0.84)/(2.5 + 4.5 +0.6)) x 100 = 129.5 to 1 d.p.
but it is not very useful for two reasons. Firstly, the quantities for different drinks differ so much from each other and, secondly, the unit prices are themselves for different quantities. We have single bottles of wine and soft drinks but 6-packs of beer. If we had calculated the index using the two prices for single cans of beer we would have got a different answer.
Expenditure is made up of two different elements, prices and quantities bought. We'll suppose first that we are particularly interested in price changes over time. In complicated situations, where we need to compare the prices of many items over many different time intervals (such as for the Retail Price Index), we work with the different prices, and use the quantities to weight them in different ways for different index numbers.
Here is how we would calculate two more index numbers using the Tastynibbles party example in each case.



The base weighted price index or Laspeyre's price index .
This index concentrates on measuring price changes from a base year. It is called a base weighted index because we use the quantities purchased in the base year (here 1990) to weight the unit prices in both years. Keeping the quantities constant in this way means that any change in the calculated expenditure is due solely to price changes.

The Laspeyre's price index is given by
( pnqo/ poqo) x 100.
In this particular case we have

pnqo = (3 x 25) + (6 x 10) + (0.84 x 10) = 143.4 and

poqo = (2.5 x 25) + (4.5 x 10) + (0.6 x 10) = 113.5

so Laspeyre's price index = (143.4/113.5) x 100 = 126.3 to 1 d.p.

Here's the table again so that you can check this.

The 1990 party The 2000 party
Drink Unit price Quantity Unit price Quantity
po qo pn qn
wine £2.50 25 £3 30
beer £4.50 10 £6.00 8
soft drinks £0.60 10 £0.84 15


In practice, the Laspeyre's price index is usually calculated using price relatives. For this method, we have to use the expenditures in the base year as weights. This sounds more complicated but the reason we do this is that it is easier to obtain data on expenditure than on actual quantities bought when we are dealing with a large complicated index. For example, cost of living weights are obtained by using sampling in the Survey of Household Expenditure. Indeed for some elements of the cost of living expenses, 'quantities' don't even make sense. You can't really talk about 'quantities' of public transport, for example.
I've shown the table again below, this time including the base year expenditures and the price relatives.
The 1990 party The 2000 party
Drink Unit price Quantity Expenditure Unit price Quantity Price relative
po qo po x qo pn qn (pn/po) x 100
wine £2.50 25 62.5 £3 30 120
beer £4.50 10 45 £6.00 8 133.3
soft drinks £0.60 10 6 £0.84 15 140


Here is the general rule for working out the base weighted or Laspeyre's price index using price relatives.


Notice that cancelling the po above and below on the top line and taking out the factor of 100 gives us
( pnqo/ poqo) x 100 as before.
Here's how the calculation now goes for the Tastynibbles example.
Substituting in the general rule, we have

giving the same answer as before.
The base weighted index has the advantage that we only have to work out the base year expenditures once. We can then use these in the calculation of the index in any subsequent period. However, this index can be misleading in telling us what is actually going on. For example, the fluctuations in fashion might have a considerable impact on an index. Suppose that skirts were considered as a separate item in a women's clothing manufacturer's index. The greatly increased relative popularity of trousers would dramatically affect the quantities sold and any index which used base year quantities from some time back would be misleading.The next index that we consider avoids this particular problem.

The end year weighted price index or Paasche's price index
This uses the end year quantities as weights. We'll now calculate this for the Tastynibbles parties. I've shown the table again below.



The 1990 party The 2000 party
Drink Unit price Quantity Unit price Quantity
po qo pn qn
wine £2.50 25 £3 30
beer £4.50 10 £6.00 8
soft drinks £0.60 10 £0.84 15



The end weighted or Paasche's price index is given by ( pnqn/ poqn) x 100.
In this particular case we have
pnqn = (3 x 30) + (6 x 8) + (0.84 x 15) = 150.6 and
poqn = (2.5 x 30) + (4.5 x 8) + (0.6 x 15) = 120
so Paasche's price index = (150.6/120) x 100 = 125.5.

We have now found out how to calculate two different price indexes to give us a measure of the fluctuations in price from a base year. But suppose the prices remain relatively stable and it is the quantities of items which are changing. In such circumstances, it may be more useful to calculate an index based on quantities, using prices as weights. The working out is then very similar to the calculations for the two price indexes.
The base weighted or Laspeyre's volume index is given by ( pnqo/ poqo) x 100 using the base period prices as weights.
The end weighted or Paasche's volume index is given by ( pnqn/ poqn) x 100 using the end period prices as weights.
Calculate these for yourself for the Tastynibbles party data. I have put in the table again here for your convenience.



The 1990 party The 2000 party
Drink Unit price Quantity Unit price Quantity
po qo pn qn
wine £2.50 25 £3 30
beer £4.50 10 £6.00 8
soft drinks £0.60 10 £0.84 15

You should find that Laspeyre's volume index is 105.7 to 1 d.p. and Paasche's volume index is 105.0 to 1 d.p.

One problem in the construction of any index number is choosing a suitable base period. We want a base where prices (or volumes) were not unnaturally high or low. An example of this would be if bad weather had caused an extreme shortage in a particular crop which then led to a very high price for it. Also, people are not happy with a base period which is too far in the past. Furthermore, tastes and availability can change a great deal over time so such an index could be seriously misleading. One way sometimes used to avoid these problems is to use a chain-based system where, in calculating successive index numbers, the base used is the previous period. A chain-based index number is particularly suited for period by period comparisons, but a fixed-base index number makes it easier to compare the movement of prices over time.


Index Numbers with Applications

The primary purposes of an index number are to provide a value useful for comparing magnitudes of aggregates of related variables to each other, and to measure the changes in these magnitudes over time. Consequently, many different index numbers have been developed for special use. There are a number of particularly well-known ones, some of which are announced on public media every day. Government agencies often report time series data in the form of index numbers. For example, the consumer price index is an important economic indicator. Therefore, it is useful to understand how index numbers are constructed and how to interpret them. These index numbers are developed usually starting with base 100 that indicates a change in magnitude relative to its value at a specified point in time.


Consumer Price index

A measure that examines the weighted average of prices of a basket of consumer goods and services, such as transportation, food and medical care. The CPI is calculated by taking price changes for each item in the predetermined basket of goods and averaging them; the goods are weighted according to their importance. Changes in CPI are used to assess price changes associated with the cost of living.

Sometimes referred to as "headline inflation".


Investopedia explains Consumer Price Index – CPI


The U.S. Bureau of Labor Statistics measures two kinds of CPI statistics: CPI for urban wage earners and clerical workers (CPI-W), and the chained CPI for all urban consumers (C-CPI-U). Of the two types of CPI, the C-CPI-U is a better representation of the general public, because it accounts for about 87% of the population.

CPI is one of the most frequently used statistics for identifying periods of inflation or deflation. This is because large rises in CPI during a short period of time typically denote periods of inflation and large drops in CPI during a short period of time usually mark periods of deflation

This diagram will help you in understanding





Quantity Index

We begin our discussion of quantity indexes by setting up some basic notation: Let be a vector of prices for n goods in period 0. Let be a vector of prices for the same n goods in period 1. Similarly, let and represent quantity vectors for the n goods in periods 0 and 1, respectively.
Total expenditure in the two periods is the sum (across all n goods) of the prices multiplied by the corresponding quantities: and . Thus, the ratio of total expenditures in the two periods equals Y 1/Y 0. If total expenditure is increasing from period 0 to period 1, then Y 1/Y 0 exceeds 1. If total expenditure is decreasing, then Y 1/Y 0 is less than 1. Total expenditure can increase from one period to another simply because prices are increasing. For example, suppose that the quantity vectors are identical in the two periods but the prices of all n goods increase from period 0 to period 1; then total expenditure will also increase.
Quantity indexes can be used to remove the effects of price changes in order to facilitate comparison of expenditures in different time periods. We will use the notation Q 01 to denote a quantity index between periods 0 and 1. If the quantity index exceeds 1, then it means that expenditure is increasing from period 0 to period 1 after the effects of price changes have been removed. Similarly, if it is less than 1, then it means that expenditure is decreasing after the effects of price changes have been removed. In the context of national income accounting, quantity indexes can best be thought of as measuring changes inreal or inflation-adjusted expenditure.

PRICE INDEXES, QUANTITY INDEXES, AND THE WEAK FACTOR REVERSAL TEST

Price and quantity indexes are closely related concepts. A price index, P 01, is a function of the price and quantity vectors in periods 0 and 1, which measures the change in the prices of the n goods between the two periods. If it is greater than 1, it means that prices increased from period 0 to 1. If it is less than 1, it means that prices decreased.
A price index and a quantity index satisfy the weak factor reversal test if the following equation holds: P 01Q 01 = Y 1/Y 0. Weak factor reversal can be used to formalize our interpretation of quantity indexes as removing the effects of price changes in order to facilitate comparison of total expenditure in two different time periods.
Assume, just for simplicity, that both total expenditure and prices are increasing between the two periods, so that Y 1/Y 0 > 1 and P 01 > 1. If the percentage increase in prices, implied by the price index, is exactly the same as the percentage increase in total expenditure, then P 01 = Y 1/Y 0 and, consequently, the quantity index will equal 1. If the percentage increase in prices exceeds the percentage increase in total expenditure, then P 01 > Y 1/Y0, implying that the quantity index will be less than 1. Conversely, if the percentage increase in total expenditure exceeds the percentage increase in prices, then Y 1/Y 0 > P 01, implying that the quantity index will exceed 1.

Other indexes also have different practical applications and they are calculated accordingly.To summarize it
statistic which assigns a single number to several individual statistics in order to quantify trends. The best-known index in the United States is the consumer price index, which gives a sort of "average" value for inflation based on price changes for a group of selected products. The Dow Jones and NASDAQ indexes for the New York and American Stock Exchanges, respectively, are also index numbers
Practical Implications Of Index in Stock Exchange

A stock index is generally a portfolio of stocks, bonds or other kinds of investments which are used to represent either segments of an exchange or the whole exchange. One of the most common ways to understand a stock index is to have a look at the composition of the stocks it represents. Generally, the set of rules require the stocks to satisfy certain criteria, such that[1]:
 All the investments in the index are subject to selection.
 Includes calculations and rules for weighting of the index components.
 Provides specific instructions for adjustments to maintain consistency.

Classification of Market Index


The benchmark indices of various exchanges not only represent the stocks, but the scenario of the market as a whole. Hence, they are used to depict the overall health of the economy as well. Understanding how the index works, is a good way to begin analysis on various stocks and their importance to the economy. A good way to analyze an index is to understand the composition of the stocks it represents.
A broad based index or composite index is the one which covers almost all stocks on the exchange (or a certain majority percentage of the market capitalization on the exchange). The main purpose of the broad based index is to act as a proxy for the performance of the economic conditions of the entire market, or reveal investor sentiment towards the market.
Based on the definition of a stock index, a global index could include indices which span across exchanges and possibly countries. The Dow Jones Wilshire 5000 Total Stock Market Index spans all publicly traded index in America, excluding foreign stocks and ADRs. The Euronext 100 includes stocks from all over the European Union and the Russell Investment group have something called the Global Index.

An example is as under


Stock Index Calculation

Market Capitalization Based Stock Index Calculations
Capitalization Weighted Indices

These indices are also called market capitalization weighted indices or the value-weighted indices. They involve the total market capitalization of the companies weighted by their effect on the index, so the larger stocks would make more of a difference to the index as compared to a smaller market cap company. This is also called the free float method. The basic formula for any index is (be it capitalization weighted or any other stock index)[2]:
 Index level= Σ(Price of stock* Number of shares)*Free float factor/ Index Divisor.


The Free float Adjustment factor represents the proportion of shares that is free floated as a percentage of issued shares and then its rounded up to the nearest multiple of 5% for calculation purposes. To find the free-float capitalization of a company, first find its market cap (number of outstanding shares x share price) then multiply its free-float factor. The free-float method, therefore, does not include restricted stocks, such as those held by company insiders.







References

Fisher, I. The Making of Index Numbers: A Study of Their Varieties, Tests and Reliability, 3rd ed. New York: Augustus M. Kelly, 1967.

Kenney, J. F. and Keeping, E. S. "Index Numbers." Ch. 5 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 64-74, 1962.

Mudgett, B. D. Index Numbers. New York: Wiley, 1951

www.bank.lv/images/img_lb/izdevumi/latvian/quarter/2001/q1/pics/q1-

http://www.investopedia.com/terms/c/consumerpriceindex.asp

http://www.encyclopedia.com/doc/1G2-3045302145.html

http://www.statcan.gc.ca/pub/11-533-x/2005001/using-utiliser/4072242-eng.htm

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