CIMA - C3 Fundamentals Of Business Mathematics

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150 5: Index numbers ⏐ Part B Summarising and analysing data

1 Basic terminology

1.1 Price indices and quantity indices

An index is a measure, over time, of the average changes in the values (price or quantity) of a group of items.

An index comprises a series of index numbers and may be a price index or a quantity index.

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

• A quantity index (also called a volume index) measures the change in the non-monetary values of a group

of items over time.

It is possible to prepare an index for a single item, but such an index would probably be unnecessary. An index is a

most useful measure of comparison when there is a group of items.

1.2 Index points

The term 'points' refers to the difference between the index values in two years.

1.3 Example: Index points

For example, suppose that the index of food prices in 20X1 – 20X6 was as follows.

20X1 180

20X2 200

20X3 230

20X4 250

20X5 300

20X6 336

The index has risen 156 points between 20X1 and 20X6 (336 – 180). This is an increase of:

⎟

⎠

⎞

⎜

⎝

⎛

180

156

× 100 = 86.7%.

Similarly, the index rose 36 points between 20X5 and 20X6 (336 – 300), a rise of 12%.

1.4 The base period, or base year

Index numbers normally take the value for a base date as 100. The base period is usually the starting point of the

series, though this is not always the case.

2 Index relatives

2.1 Assessment formulae

An index relative (sometimes just called a relative) is the name given to an index number which measures the

change in a single distinct commodity.

Key term

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Part B Summarising and analysing data ⏐ 5: Index numbers 151

• A price relative is calculated as 100 ×

• A quantity relative is calculated as 100 ×

2.2 Example: Single-item indices

(a) Price index number

If the price of a cup of coffee was 40c in 20X0, 50c in 20X1 and 76c in 20X2, then using 20X0 as a base

year the

price index numbers for 20X1 and 20X2 would be as follows.

20X1 price index = 100 ×

= 125

20X2 price index = 100 ×

= 190

(b)

Quantity index number

If the number of cups of coffee sold in 20X0 was 500,000, in 20X1 700,000 and in 20X2 600,000, then

using 20X0 as a base year, the

quantity index numbers for 20X1 and 20X2 would be as follows.

20X1 quantity index = 100 ×

000,500

000,700

= 140

20X2 quantity index = 100 ×

000,500

000,600

= 120

Question

Price index

The price of a kilogram of raw material was $80 in year 1 and $120 in year 2. Using year 1 as a base year, the price

index number for year 2 is

A 67

B 140

C 150

D 167

Answe

C Year 2 price index = 120/80

× 100 = 150

Option C is therefore correct.

If you selected option A, you have confused the numerator with the denominator and calculated 80/120

instead of 120/80.

If you selected option B, you simply took the price difference of $40 ($120 – $80 = $40) and added this to

100.

If you selected option D, you added 80/120

× 100% = 67% to 100 which equals 167 which is incorrect.

Assessment

formula

152 5: Index numbers ⏐ Part B Summarising and analysing data

Question

Quantity index

A company used 20,000 litres of a raw material in year 1. In year 5 the usage of the same raw material amounted

to 25,000 litres. Using year 1 as a base year, the quantity index number for year 5 is

A 105

B 120

C 125

D 180

Answe

C Year 5 quantity index =

000,20

000,25

× 100 = 125

If you selected option A, you took the difference in litres (25,000 – 20,000 = 5,000) and interpreted this as a

five point increase, ie 100 + 5 = 105.

If you selected option D you calculated

000,25

000,20

× 100 = 80 and added this to 100 which is not the correct

method

to use.

3 Time series of index relatives

Index relatives can be calculated using the fixed base method or the chain base method.

(a) The fixed base method. A base year is selected (index 100), and all subsequent changes are

measured against this base. Such an approach should only be used if

the basic nature of the

commodity is unchanged over time

Fixed base index =

Value in any given year

Value in base year

× 100

(b) The chain base method. Changes are calculated with respect to the value of the commodity in the

period immediately before. This approach can be used for any set of commodity values but must be

used if

the basic nature of the commodity is changing over time.

Chain base index =

This year's value

Last year's value

× 100

3.1 Example: Fixed base method

The price of a commodity was $2.70 in 20X0, $3.11 in 20X1, $3.42 in 20X2 and $3.83 in 20X3. Construct a fixed

base index

for the years 20X0 to 20X3 using 20X0 as the base year.

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Part B Summarising and analysing data ⏐ 5: Index numbers 153

Solution

Fixed base index

20X0 100

20X1 115

(3.11/2.70

× 100)

20X2 127

(3.42/2.70

× 100)

20X3 142

(3.83/2.70

× 100)

3.2 Example: Chain base method

Using the information in Paragraph 3.1 construct a chain base index for the years 20X0 to 20X3 using 20X0 as the

base year.

Solution

Chain base index

20X0 100

20X1 115

(3.11/2.70

× 100)

20X2 110

(3.42/3.11

× 100)

20X3 112

(3.83/3.42

× 100)

The chain base relatives show the rate of change in prices from year to year, whereas the fixed base relatives show

changes relative to prices in the base year.

3.3 Splicing a single index

Splicing involves redefining the base year of an index in a particular year and then restating the index values in

previous years.

It is sometimes necessary to change the base of a time series (to rebase) of fixed base relatives, perhaps because

the

base time point is too far in the past. The following table shows a price index which rebased its base year to

2000.

Year Price index

(1975 = 100)

1998 320

1999 325

2000 331

(2000 = 100)

2001 104

2002 108

We need to change the base of the original index to 2000.

1998 100

331

320

= 97

1999 100

× 98

331

325

The index now becomes:

Important!

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154 5: Index numbers ⏐ Part B Summarising and analysing data

Year Price index

(2000 = 100)

1998 97

1999 98

2000 100

2001 104

2002 108

Question

Rebased index

In 2001 a price index based on 1990 = 100 had a value of 132. In 2001 it was rebased at 2001 = 100. The value of

the new index in 2006 was 111.

For a continuous estimate of price changes since 1990, how should the new index be expressed in terms of the old

index (to 2 decimal places)?

A 118.92

B 146.52

C 132.00

D 121.50

Answe

The correct answer is B.

100

111

× 132 = 146.52

3.4 Comparing sets of fixed base relatives

(a) You may be required to compare two sets of time series relatives. For example, an index of the

annual number of advertisements placed by an organisation in the press and the index of the

number of the organisation's product sold per annum might be compared. If the base years of the

two indices differ, however, comparison is extremely difficult (as the illustration below shows).

20W8 20W9 20X0 20X1 20X2 20X3 20X4

Number of advertisements

Placed (20X0 = 100) 90 96 100 115 128 140 160

Volumes of sales (20W0 = 100) 340 347 355 420 472 515 572

(b) From the figures above it is impossible to determine whether sales are increasing at a greater rate

than the number of advertisements placed, or vice versa. This difficulty can be overcome by

rebasing one set of relatives so that the base dates are the same. For example, we could rebase the

index of volume of sales to 20X0.

20W8 20W9 20X0 20X1 20X2 20X3 20X4

Number of advertisements

Placed (20X0 = 100) 90 96 100 115 128 140 160

Volumes of sales (20X0 = 100) 96 98* 100 118 133*

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Part B Summarising and analysing data ⏐ 5: Index numbers 155

355

347

× 100

355

472

× 100

increasing at a slightly faster rate, in general, than the number of advertisements placed.

4 Time series deflation

4.1 Real value of a commodity

The real value of a commodity can only be measured in terms of some 'indicator' such as the rate of inflation

(normally represented by the Retail Prices Index (RPI)).

For example the cost of a commodity may have been $10 in 20X0 and $11 in 20X1, representing an increase of

10%. However, if we are told the prices

in general (as measured by the RPI) increased by 12% between 20X0 and

20X1, we can argue that the

real cost of the commodity has decreased.

Time series deflation is a technique used to obtain a set of index relatives that measure the changes in the real

value of some commodity with respect to some given indicator.

4.2 Example: Deflation

Mack Johnson works for Pound of Flesh Co. Over the last five years he has received an annual salary increase of

$500. Despite his employer assuring him that $500 is a reasonable annual salary increase, Mack is unhappy

because, although he agrees $500 is a lot of money, he finds it difficult to maintain the standard of living he had

when he first joined the company.

Consider the figures below.

(a) (b) (c)

Year Wages RPI Real wages

$ $

1 12,000 250 12,000

2 12,500 260 12,019

3 13,000 275 11,818

4 13,500 295 11,441

5 14,000 315 11,111

(a) This column shows Mack's wages over the five-year period.

(b) This column shows the current RPI.

The wages have been deflated relative to the new base period (year 1). Economists call these deflated wage

figures

real wages. The real wages for years 2 and 4, for example, are calculated as follows.

Year 2: $12,500

260

250

= $12,019

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156 5: Index numbers ⏐ Part B Summarising and analysing data

Year 4: $13,500 ×

295

250

= $11,441

Conclusion

The real wages index shows that the real value of Mack's wages has fallen by 7.4% over the five-year period

12,000 -11,111

12,000

× 100%). In real terms he is now earning $11,111 compared to $12,000 in year 1. He is

probably justified, therefore, in being unhappy.

Deflated/inflated cash flow = Actual cash flow in given year

Index number for base year

Index number for given year

This is a very important technique and very popular with examiners.

5 Composite index numbers

Composite index numbers cover more than one item.

5.1 Example: Composite index numbers

Suppose that the cost of living index is calculated from only three commodities: bread, tea and caviar, and that the

prices for 20X1 and 20X2 were as follows.

20X1 20X2

Bread 20c a loaf 40c a loaf

Tea 25c a packet 30c a packet

Caviar 450c a jar 405c a jar

(a) A simple index could be calculated by adding the prices for single items in 20X2 and dividing by the

corresponding sum relating to 20X1 (if 20X1 is the base year). In general, if the sum of the prices in the

base year is

∑P

and the sum of the prices in the new year is ∑P

, the index is 100 ×

∑

. The index,

known as a simple aggregate price index, would therefore be calculated as follows.

20X1

20X2

Bread

0.20

0.40

Tea

0.25

0.30

Caviar

4.50

4.05

∑P

4.95

∑P

4.75

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Assessment

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Part B Summarising and analysing data ⏐ 5: Index numbers 157

Year

∑

Simple aggregate price index

20X1

95.4

= 1.00

100

20X2

95.4

75.4

= 0.96

(b) The simple aggregate price index has a number of disadvantages.

(i) It ignores the

amounts of bread, tea and caviar consumed (and hence the importance of each item).

(ii) It ignores the units to which the prices refer. If, for example, we had been given the price of a cup of

tea rather than a packet of tea, the index would have been different.

5.2 Average relatives indices

To overcome the problem of different units we consider the changes in prices as ratios rather than absolutes so

that all price movements, whatever their absolute values, are treated as equally important. Price changes are

considered as ratios rather than absolutes by using the

average price relatives index. Quantity changes are

considered as ratios by using the

average quantity relatives index.

• Average price relatives index = 100 ×

× ∑

⎛⎞

⎜⎟

⎝⎠

•

Average quantity relatives index = 100 ×

× ∑

⎛⎞

⎜⎟

⎝⎠

where n is the number of goods.

The price relative P

(so called because it gives the new price level of each item relative to the base year price)

for a particular commodity will have the same value whatever the unit for which the price is quoted.

5.3 Example: Average relatives indices

Using the information in the example in Paragraph 5.1, we can construct the average price relatives index as

follows.

Commodity

Bread

0.20

0.40

2.00

Tea

0.25

0.30

1.20

Caviar

4.50

4.05

0.90

4.10

Year

⎟

⎠

⎞

⎜

⎝

⎛

∑

Average price relatives index

20X1

× 3.00 = 1.00

100

20X2

× 4.10 = 1.37

137

158 5: Index numbers ⏐ Part B Summarising and analysing data

There has therefore been an average price increase of 37% between 20X1 and 20X2.

No account has been taken of the

relative importance of each item in this index. Bread is probably more important

than caviar. To overcome both the problem of quantities in different units and the need to attach importance to

each item, we can use

weighting which reflects the importance of each item. To decide the weightings of different

items in an index, it is necessary to obtain information, perhaps by market research, about the

relative importance

of each item. The next section of this chapter looks at weighted index numbers.

6 Weighted index numbers

6.1 Weighting

Weighting is used to reflect the importance of each item in the index.

There are two types of index which give different weights to different items.

• Weighted average of relatives indices

• Weighted aggregate indices

The weighted average of relatives index is the one that you need to be able to calculate in your assessment.

6.2 Weighted average of relatives indices

Weighted average of relatives indices are found by calculating indices and then applying weights.

• Weighted average of price relative index =

WP/P

∑

100

• Weighted average of quantity relative index =

WQ/Q

∑

× 100

where W = the weighting factor

6.3 Example: Weighted average of relatives indices

Use both the information in Paragraph 5.1 and the following details about quantities purchased by each household

in a week in 20X1 to determine a weighted average of price relatives index number for 20X2 using 20X1 as the

base year.

Quantity

Bread 12

Tea 5

Caviar 3

Assessment

formula

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Part B Summarising and analysing data ⏐ 5: Index numbers 159

Solution

Price relatives

()

P/P

Bread

= 2.00

Tea

= 1.20

Caviar

450

405

= 0.90

Weightings (W) Bread 12.00

Tea 5.00

Caviar

3.00

∑W =

20.00

Index Bread 2 × 12 = 24.00

Tea 1.2

× 5 = 6.00

Caviar 0.9

× 3 =

2.70

∑W ×

P/P =

32.70

Index number =

0.20

7.32

× 100 = 163.5

Question

Weighted average of relatives method

The average prices of three commodities and the number of units used annually by a company are given below.

Commodity

20X1

Price per unit (P

)

20X2

Price per unit(P

)

Quantity

$ $ Units

X 20 22 20

Y 40 48 2

Z 100 104 10

The price for 20X2 based on 20X1, calculated using the weighted average of relatives method is (to the nearest

whole number)

A 107 C 109

B 108 D 110

Answe

Commodity

Price

P/P

Weight

(W)

Relative weight

()

P/PW×

= 1.1

22.0

= 1.2

2.4

100

104

= 1.04

10.4

∑W =

34.8