CAT / FIA FMA Syllabus B. Data Analysis And Statistical Techniques - The purpose of index numbers - Notes ^{13 / 14}

What is an index number?

An index number is a technique for comparing, over time, changes in some feature of a group of items (e.g. price, quantity consumed, etc) by expressing the property each year as a percentage of some earlier year.

The year that is used as the initial year for comparison is known as the base year. The base year should also be fairly recent on a regular basis.

Simple Indices

A simple index is one that measures the changes in either price or quantity of a single item in comparison to the base year.

Therefore there are two types of simple indices

• A price index – this measures the change in the money value of a group of items over time.

• A quantity (volume) index – this measures the change in the non-monetary values of a group of items over time.

A simple aggregate price index would be calculated:

∑p0 = 2 + 2.50 + 3.50 = $8.0

∑pn = 2.25 + 2.65 + 3.00 = $7.90

This index ignores the amounts of each product which was consumed. To overcome these problems, we can use a weighting which is an indicator of the importance of the component

The formulae for calculating simple indices are:

Simple price index =

pn x 100
----  
p0

Simple quantity index=

qn x100
----
q0

Where

p0 is the price for the base period
pn is the price for the period under consideration
q0 is the quantity for the base period
qn is the quantity for the period under consideration

Illustration

2 years ago:

Price index 150
Cost $15,000

Today, the price index is 250. 

What would the cost be of the same items?

Solution

$15,000/150*250 = $25,000

Composite indices

Composite indices are used when we have more than one item

Weighted aggregate Indices

A weighted index involves multiplying each component value by its corresponding weight and adding these products to form an aggregate.  

This is done for both the base period and the period in question.  

The aggregate for that period is then divided by the base period aggregate.

Weighted aggregate index = 

∑wvn
------- 
 ∑wv0

Where:

V0 is the value of the commodity in the base period
Vn is the value of the commodity in the period in question

Price indices are usually weighted by quantities and quantity indices are usually weighted by prices.

Laspeyre, Paasche and Fisher indices

Laspeyre and Paasche indices are special cases of weighted aggregate indices.  

Laspeyre index is a multi-item index using weights at the base date. It is sometimes called base weighted index.  

Paasche index is a multi-item index using weights at the current date. Hence, the weights are changed every time period.

Fisher’s ideal index is found by taking the geometric mean of the Laspeyre index and the Paasche index.

Fisher’s ideal index = √(Laspeyre x Paasche)

Advantages of Indices

• Indices present changes in data or information over time in percentage term, i.e. more meaningful information.

• The use of indices makes comparison between items of data easier and more meaningful- it is relatively easy to make comparisons and draw conclusions from figures when you are starting from a base of 100.

• The ability to calculate separate price and quantity indices, allows management to identify the relative importance of changes in each of two variables. 

  A typical application of this technique is to allow management to identify price and quantity effects and their relative influence on changes in total revenue and total costs.

Disadvantages of Indices

• The Laspeyre and Paasche approaches give different results. 

  This suggests that there may be no single correct way of calculating an index, especially the more sophisticated index numbers. 

  The user of the information should bear in mind the basis on which the index is calculated.

• The overall result obtained from multi-item index numbers, such as Laspeyre and Paasche are averages - they may hide quite significant variations in changes involved in the component items.

• An index number, to be meaningful, should only be applied to the items which are included in the index calculation.

• Index numbers are relative values, not absolute figures and may not give the whole picture. 

  For example, Division A has achieved growth of 10% compared to last year while Division B has only achieved 5%. 

  At first glance it may appear that Division A is performing better than Division B. 

  The actual sales figures for the period are $27,500 for Division A and $262,500 for Division B. 

  The absolute increase in sales revenue compared to last year is $2,500 for Division A  ($2,200/$25,000 x 100% = 10% increase) but $12,500 for Division B ($12,500/$250,000 x 100%= 5 % increase)