Quantitative Methods for

Business Decision Making

Index Number

CONTENT

What is Index NumberType of Index NumberSome DefinitionsUn-weighted Average of Price Relative IndexWeighted Average of Price Relative IndexUn-weighted Aggregate IndexWeighted Aggregate Index Problems related to Index Number

What is Index Number

Index number measures how much a variable changes over time

Simple Index (Base as 1980) : One variableYear # of Business started Index

1980 9,300 100

1985 6,500 79

1990 9,600 103

1995 10,100 109

Type of Index Number

Price Index: e.g. Consumer Price Index Quantity Index: e.g. example for Simple Index Value Index: Combines Price Index and

Quantity Index, e.g.

Year Value (Crore) Index

1980 18.4 100

1985 14.6 79

1990 26.2 142

1995 29.4 160

Type of Index Number (Cont)

Simple Vs Composite Composite Index: reflects more than one

changing variable, e.g. Consumer price index consists of individual prices of various goods and services

Some Notations

Base Year: Year from which comparisons are made (Subscript 0)

Current Year: Year under consideration (Subscript 1)

Simple & Weighted of Prices Relative Methods

Simple Prices Relative Methods Example 5 of PTU Arithmetic Mean P01 = Σ(P1i/P0i)* 100 / n

Geometric Mean P01 = (∏ (P1i/P0i)* 100)1/n

= Antilog [ΣlogPi/n]

Why Weighted method? (All items may not be equally important)

Weighted Prices Relative Methods Example 6 of PTU Arithmetic Mean P01 = Σ[(P1i/P0i)* 100* wi] / Σwi

Geometric Mean P01 = (∏ (P1i/P0i)* 100* wi)1/ Σwi

= Antilog ([Σwi*logPi]/ Σwi)

Simple & Weighted Aggregative Methods

Simple Aggregative Method Example 4 of PTU P01 = (ΣP1i)/(ΣP0i)*100

Weighted Aggregative Method Example 6 of PTU P01 = Σ(P1iwi) / Σ(P0iwi)*100

Unweighted Aggregates Index

Elements in Composite

Prices 1990

P0

Prices 1995

P1

Milk 1 liter 19.20 34.00

Egg 1 Dozen 8.10 10.00

Hamburger 1 Pound

14.90 20.00

Gasoline 1 Liter 14.90 20.00

TOTAL 52.20 75.70

Unweighted Aggregates Price Index

=52.20/52.20*100 =100

=75.70/52.20*100 =145

Simple (Un-weighted) Aggregates Index

Disadvantage: It does not attach greater importance (or weight) to the price change of a high use item (Family might be taking 100 litres of milk but only 25 pound of Hamburger)

Advantage: Easy to calculate

Weighted Aggregates Index

Element Q (Vol) P0 1990 Price

P1 1995 Price

P0*Q P1*Q

Milk 20,000 19.20 34.00 384.00 680.00

Eggs 3,500 8.10 10.00 28.40 35.00

Hamburger 11,000 14.90 20.00 163.90 220.00

Petrol 154,000 10.00 11.70 1540.00 1801.80

Calculator 0.002 150.00 110.00 0.30 0.20

TOTAL 2116.60 2737.00

INDEX 100 100*2737 / 2116.60 =129

Weighted Aggregates Index (Cont)

Laspeyres Method: Take quantity of base period (Q = Q0)

ΣPiQ0 / ΣP0Q0

Quantity required only for base period Comparison easy Does not consider change in consumption pattern

Paasche Method: Take quantity of current period (Q = Qi)

ΣPiQi / ΣP0Qi

Quantity required for each period Comparison difficult Change in consumption pattern accounted for

Weighted Aggregates Index (Cont)

Fixed Weight Aggregates Method: Take quantity of some fixed period (Q = Q2)

ΣPiQ2 / ΣP0Q2

Quantity required only for one period Comparison easy Does not consider change in consumption pattern Flexibility in deciding fixed period

Fisher Index = Geometric Mean of Laspeyres Index and Paasche Index

Dorbish and Bowley Index = Airthmetic Mean of Laspeyres Index and Paasche Index

Weighted Aggregates Index (Cont)

Marshall and Edgeworth Index Weights are taken as arithmetic mean of base and

current year quantity, namely (q0 + q1) / 2

Walsh Index Weights are taken as geometric mean of base and

current year quantity, namely (q0 * q1) 1/2

Problems related to Index Number

Difficulty in finding suitable data: objective is to find seasonal pattern in sale, but availability is of annual data

Incompatibility of indices: Basic changes have occurred over time. E.g. transportation cost index has increased, but so is quality

Inappropriate Weighting factors: For CPI (Consumer Price Index), weighting factors are changing

Selection of improper base: Base year should not be a very good / very bad year for the relevant aspect, e.g. choosing recessionary year as a base to represent profitability