1. OMGT1117 - LECTURE 5: Index Numbers
Reading
Property Data Analysis – A Primer, Ch. 5

2. Objectives
To define as well as describe how various index numbers are calculated
To describe the properties of index numbers
To describe the pros and cons of various index numbers
To describe the most important adjustments that must be made to index number series
To highlight real-world index numbers that are monitored as well as routinely used by project managers and property specialists.
List the major considerations that should be brought to bear when devising an index number

3. Introduction to Index Numbers – Some Basic Definitions & Notation
Index Numbers may be used to measure the overall % change of a single variable or a group of related variables over time.
The Simple Index or Relative:
This is the name given to an index number that may be used to measure the overall percentage change in a single variable.
Examples:
A Single Price variable: might be the price of a particular tile
A Single quantity variable: might be the physical quantity of tiles of a given style that has been procured.
A Single Value Variable: might be the dollar expenditure on the procurement of tiles of a particular style.

4. Introduction to Index Numbers – Some Basic Definitions & Notation Continued
Composite Index Numbers:
This is the name given to index numbers that may be used to measure the overall
percentage change in a group of related variables.
Examples:
A group of related price variables: might be the prices of different styled tiles
A group of related quantity variables: might be the physical quantities of different tiles that have been procured
A group of related value variables: might be the dollar expenditures on the procurement of different tiles

5. Introduction to Index Numbers – Some Basic Definitions & Notation Continued
pn: price in period n
qn: quantity in period n
Vn (=pnqn) dollar value in period n
Note : n is normally set to zero for a base or reference period. Thus, p0 would represent price in the base period. Similarly q0 and v0 would respectively represent quantity and dollar value in the reference period.

6. The Simple Index or Relative Revisited
Let X denote p, q or v as the case may be. Then:
indicates the size of Xn relative to Xm For the example portrayed in the adjacent diagram:
In other words, X is 100% higher in period n than it is in period m. If m = 0. Then one could say that X is 100% higher than in the base period or reference period. m n
Xm = 10
Xn = 20
Time period X

7. Properties of Simple Index Numbers
1 Identity Property:
This states that the relative for any period n to the same period is 100.
Note: From the above it follows that the relative for the base period will always equal 100
2 Time Reversal Property:
This states that:
Suppose Xm = 10 and Xn =20 as in the
adjacent diagram then: m n
Xm = 10
Xn = 20
Time period X

8. Properties of Simple Index Numbers Continued
3 The Circular Property:
This states that:
As an example suppose that Xl = 2, Xm = 10 and Xn = 20, then:

9. Properties of Simple Index Numbers Continued
4 “Independence of Measurement Units” Property:
An index enjoys this property if its computed value is unaffected by the units in which variable X is measured:
eg.
price of a terracotta tile in 2001 is $0.80 (or 80¢)
price of a terracotta tile in 2006 is $1.00 (or 100¢)
Then:

10. Properties of Simple Index Numbers Continued
5 The Factor Reversal Property:
This property is satisfied if the value relative for period n with respect to period m equals the product of the analogous relatives for price and quantity:
That is:
Example:
Year p q
v = pq
2001
$10/unit
50 units
$500
2006
$20/unit
40 units
$800

11. x2006 y2006 x2001 z2006 y2001 z2001 Composite Indices
Sometimes there is a need to measure the overall change in a group of related variables not just one variable.
The complexity of the task is depicted in the following diagram where x, y and z are three related variables. For example, room tariffs collected over time for standard (z), double (y) and deluxe (x) rooms in a hotel.
x2001
y2001
z2001
x2006
y2006
z2006
2006
2001
The challenge is how to measure a cluster of related variables that has moved over time.
x, y and z

12. Various Composite Indices Continued
Simple Aggregative Price and Quantity Indices.
In what follows use will be made of the following data set where 2005 is the base year (m = 0) and 2006 (= n = 1). The regimen may be thought of as the basket of goods whose composite index the analyst wishes to determine.
Simple Aggregative Price Index
Simple Aggregative Quantity Index

13. Various Composite Indices Continued
Major Shortcomings of the Simple Aggregative Indices.
The relative importance of items in the regimen is ignored
The numerical value of the index will be affected by the units in which prices and quantities are quoted
These shortcomings may be overcome by Weighted Aggregative Indices of which there are several versions.

14. Various Composite Indices Continued
Laspeyres Weighted Aggregative Indices.
Regimen
Price
Quantity p0 pn q0 qn
Item
Unit
2005
2006 A kg $1
$1.5 5 6 B $2
$2.4 4 3
Laspeyres Weighted Aggregative Price Index (uses base year quantity weights).
Laspeyres Weighted Aggregative Quantity Index (uses base year price weights).

15. Various Composite Indices Continued
Paasche Weighted Aggregative Indices.
Regimen
Price
Quantity p0 pn q0 qn
Item
Unit
2005
2006 A kg $1
$1.5 5 6 B $2
$2.4 4 3
Paasche Weighted Aggregative Price Index (uses current year quantity weights).
Paasche Weighted Aggregative Quantity Index (uses current year price weights).

16. Various Composite Indices Continued
Pros and Cons of Laspeyres and Paasche Indices.
Paasche
Laspeyres
Pros
Weights are always current
The weights do not have to be updated every period which is economical
Comparisons may be made between the current period index and those of any previous period
Cons
The weights have to be updated every period which may be costly and time consuming
Comparisons can only be made between the current period index and that of the base year
The weights may get out of date fairly quickly due to changing consumer tastes and the rapid pace of technological change and innovation.

17. Various Composite Indices Continued
Relationship between Laspeyres and Paasche Indices.
If for all items in the regimen, prices have risen by the same percentage and/or quantities have risen by the same percentage then it may be shown that:
In practice the comparative values of the Laspeyres and Paasche indices are dependant on the patterns underlying demand and supply over time. If it is supply that is shifting along a stable downward sloping demand curve then it will be the case that:
Conversely if it is demand that is shifting along a stable upward sloping supply curve then the following result will hold:

18. Various Composite Indices Continued
Other Weighted Aggregative Indices Continued.
Fixed Weighted Aggregative Price Index (uses fixed or typical quantity weights)
Fixed Weighted Aggregative Quantity Index (uses fixed or typical price weights)

19. Various Composite Indices Continued
Other Weighted Aggregative Indices Continued.
The Weighted Marshall/Edgeworth Aggregative Price Index:
This is a special fixed weighted aggregative price index that arises when the typical quantity weight: qt = (q0+qn)/2.
The Weighted Marshall/Edgeworth Aggregative Quantity Index
This is a special fixed weighted aggregative quantity index that arises when the typical price weight: pt = (p0+pn)/2.

20. Various Composite Indices Continued
Other Weighted Aggregative Indices Continued.
Fisher’s Ideal Weighted Aggregative Price Index:
This is taken as the geometric mean of the Laspeyres and Paasche Price Indices.
Fisher’s Ideal Weighted Aggregative Quantity Index:
This is taken as the geometric mean of the Laspeyres and Paasche Quantity Indices.
Note: It may be shown that the Factor Reversal Property is enjoyed by these set of indices when expressed in proportionate rather than percentage format:

21. Various Composite Indices Continued
Properties of Weighted Aggregative Indices.
As indicated in the following table, weighted aggregative indices do not satisfy all the intuitively appealing properties that are enjoyed by the simple index numbers. However, as emphasized by Lombardo (2006, pp ) this does not mitigate against the preference of a weighted aggregative index over a simple aggregative index
Index
Identity Property
Time Reversal Property
Circular
Property
Factor Reversal Property
Laspeyres  
Paasche
Fixed
Ideal

22. Various Composite Indices Continued
”Composite of Relatives” Indices.
So far only composite aggregative indices have been considered. Attention now turns to a treatment of “Composite of Relatives” indices.
Simple Average of Relatives Indices
The Simple Average of Price Relatives Index
The Simple Average of Quantity Relatives Index
Note: In the above formulae m refers to the number of items in the regimen and for the current example m = 2

23. Various Composite Indices Continued
”Composite of Relatives” Indices Continued.
Shortcomings of Simple Average of Relatives Indices
Each relative receives the same weight (i.e. 1/m) irrespective of how important the associated item is in relation to the total expenditure on the entire basket of goods.
The shortcoming is overcome by weighted average of relatives indices of which there are as many versions as there are weighted aggregative indices.

24. Various Composite Indices Continued
”Composite of Relatives” Indices Continued.
The Laspeyres Weighted Average of Price Relatives Index
Weight

25. Various Composite Indices Continued
“Composite of Relatives” Indices Continued.
The Laspeyres Weighted Average of Price Relatives Index Continued
Note: Laspeyres weighted average of price relatives index equals the value of the Laspeyres weighted aggregative price index. The former however, allows one to determine which items contributed most (least) to the movement in the index
Laspeyres weighted aggregative price index (refer to computations on Slide 14)
Laspeyres Weighted Average of Price Relatives Index
weight

26. Various Composite Indices Continued
Laspeyres weighted aggregative quantity index (refer to computations on Slide 14)
”Composite of Relatives” Indices Continued.
The Laspeyres Weighted Average of Quantity Relatives Index
weight
Laspeyres weighted average of quantity relatives index
Note: the Laspeyres weighted … quantity relatives index equals its weighted quantity aggregative but unlike the latter may be used to gauge the impact that each item in the regimen has on the move in either version of the index

27. Various Composite Indices Continued
Paasche weighted aggregative price index (refer to computations on Slide 15)
”Composite of Relatives” Indices Continued.
The Paasche Weighted Average of Price Relatives Index
weight
Paasche weighted average of price relatives relative
Note: the Paasche weighted … price relatives index equals its weighted price aggregative but unlike the latter may be used to gauge the impact that each item in the regimen has on the move in either version of the index

28. Various Composite Indices Continued
Paasche weighted aggregative quantity index (refer to computations on Slide 15)
Various Composite Indices Continued
”Composite of Relatives” Indices Continued.
The Paasche Weighted Average of Quantity Relatives Index
weight
Paasche weighted average of quantity relatives Index
Note: the Paasche weighted … quantity relatives index equals its weighted quantity aggregative but unlike the latter may be used to gauge the impact that each item in the regimen has on the move in either version of the index

29. Various Composite Indices Continued
”Composite of Relatives” Indices Continued.
Fixed Weighted ……. Relatives Indices
As with the Laspeyres and Paasche weighted …. relatives indices, much the same tasks may be achieved with fixed weighted …. relatives indices. That is they can be used to determine how much each item in the regimen contributed to the move in its associated fixed weighted aggregative index

30. Important Adjustments to Index Number Series
The 3 most important adjustments that must be made to indices are as follows:
A change in base or reference period.
original series:
new series:
The new series is obtained by dividing the original series by 125 and multiplying by 100
Splicing or melding together index series (based on different regimens) say for the CPI into one continuous series.
Series 1: X Y
Series 2: Z
Spliced Series: or
where: X=125(144/100); Y = 125(200/100) and Z = 100(100/125)

31. Important Adjustments to Index Number Series Continued
The last of the 3 most important adjustments that must be made to indices is:
to adjust the series for the quality of the goods in the regimen as it may change through time. Whilst this is covered for completeness in Lombardo ( p ), students are not expected to study this section for assessment purposes.

32. Important Real World Index Numbers
The Consumer Price Index (CPI):
Discussed at length in Lombardo (2006, pp ), this index may be used to gauge the quarterly and/or annual price change of a fixed basket of goods and services purchased by Australian Metropolitan Households.
The CPI may be viewed as a particular variant of the Laspeyres price index.
The CPI may be used in escalator clauses for say rent adjustments using a formula like:

33. Important Real World Index Numbers Continued
The Consumer Price Index (CPI) continued:
The CPI may be used to deflate money or nominal wages into real wages. The latter represents the real purchasing power of wages over household goods and services. The formula for deflating (or converting) Money wages (i.e. wages measured in current dollars) into real wages (i.e. wages in dollars of a reference year) is given by:

34. Important Real World Index Numbers Continued
The Chain Volume Measure of Gross Domestic Output (GDP):
This replaces an earlier concept known as Real GDP which was obtained by converting GDP measured in current prices (often called Money or Nominal GDP) into GDP measured in constant reference or base year prices. The purpose of the adjustment (known as deflating Money GDP) was to remove the distortion that a rising price-level would otherwise wreak on GDP in its capacity to be used as a reliable measure of economic activity
Where: GDPd is the implicit price deflator taken as the ratio of GDP at current prices to GDP at base year prices. As such GDPd took the form of a Paasche Index

35. Important Real World Index Numbers Continued
The Chain Volume Measure of Gross Domestic Output (GDP) Continued:
In contrast to the old method of obtaining Real GDP the Chain Volume Measure of GDP is effectively arrived at by deflating Money GDP by a variant of a Laspeyres Price Index. For a deeper discussion refer to Lombardo (2006, pp )
Share Market Indices and Property Performance Indices:
Professionals in the area of Property Finance must be able monitor the performance of different asset classes by reference to various published indices like the Dow Jones industrial Average. The S&P/ASX All ordinaries, the S&P/ASX 100, S&P/ASX 200 etc. For a reasonably detailed discussion of these as well as various specific Property Performance Indices for direct property investments, refer to Lombardo (2006, pp ).

36. Important Real World Index Numbers Continued
Residential Property Price Indices:
There are 6 residential property price indices currently used in Australia. These are described in Lombardo (2006, pp ). In principle these may be used to gauge the % change in price levels of residential dwellings as well as deflate house price data that may be used in valuation related studies which apply a technique known as regression analysis (studied at an elementary level in Topics 11 and 12).
Construction Cost Indices:
The ABS produces various Laspeyres weighted aggregative price indices of selected materials or groups of materials in the construction of houses. For a discussion of these indices read Lombardo (2006, p. 221)

37. Important Considerations in the Use and Construction of Indices
Data Validity
Choice of Reference Period
Nature of Sampling
Suitability of Weighting System
Typicality of Index
Mathematical Properties of Index