CIMA - C3 Fundamentals Of Business Mathematics
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Topic list Syllabus references
1 The components of time series E, (i), (iii), (1)
2 Finding the trend E, (ii), (2)
3 Finding the seasonal variations E, (v), (3)
4 Forecasting E, (iv), (vi), (vii), (4), D, (iv)
5 The limitations of forecasting models E, (vii), (4)
Forecasting
Introduction
In some situations, there are no independent variables from which to forecast a dependent
variable. In this chapter we will be looking at a technique called time series analysis. With
this forecasting method we look at past data about the variable which we want to forecast
(such as sales levels) to see if there are any patterns. We then assume that these patterns
will continue into the future. We are then able to forecast what we believe will be the value of
a variable at some particular point of time in the future.
We will also use linear regression analysis (from Chapter 10) to make forecasts.
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1 The components of time series
A time series is a series of figures or values recorded over time. Any pattern found in the data is then assumed to
continue into the future and an extrapolative forecast is produced.
1.1 Examples of time series
• Output at a factory each day for the last month
• Monthly sales over the last two years
• Total annual costs for the last ten years
• The Retail Prices Index each month for the last ten years
• The number of people employed by a company each year for the last 20 years
There are four components of a time series: trend, seasonal variations, cyclical variations and random variations.
1.2 The trend
The trend is the underlying long-term movement over time in the values of the data recorded.
1.3 Example: Preparing time series graphs and identifying trends
Output per Number of
labour hour Cost per unit employees
Units $
20X4 30 1.00 100
20X5 24 1.08 103
20X6 26 1.20 96
20X7 22 1.15 102
20X8 21 1.18 103
20X9 17 1.25 98
(A) (B) (C)
(a) In time series (A) there is a downward trend in the output per labour hour. Output per labour hour did not
fall every year, because it went up between 20X5 and 20X6, but the long-term movement is clearly a
downward one.
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(b) In time series (B) there is an upward trend in the cost per unit. Although unit costs went down in 20X7
from a higher level in 20X6, the basic movement over time is one of rising costs.
(c) In time series (C) there is no clear movement up or down, and the number of employees remained fairly
constant around 100. The trend is therefore a static, or level one.
1.4 Seasonal variations
Seasonal variations are short-term fluctuations in recorded values, due to different circumstances which affect
results at different times of the year, on different days of the week or at different times of day etc.
1.5 Examples of seasonal variations
(a) Sales of ice cream will be higher in summer than in winter, and sales of overcoats will be higher in
autumn than in spring.
(b) Shops might expect higher sales shortly before Christmas, or in their winter and summer sales.
Key term
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(c) Sales might be higher on Friday and Saturday than on Monday.
(d) The telephone network may be heavily used at certain times of the day (such as mid-morning and
mid-afternoon) and much less used at other times (such as in the middle of the night).
1.6 Example: The trend and seasonal variations
The number of customers served by a company of travel agents over the past four years is shown in the following
historigram (time series graph).
In this example, there would appear to be large seasonal fluctuations in demand, but there is also a basic upward
trend.
1.7 Cyclical variations
Cyclical variations are medium-term changes in results caused by circumstances which repeat in cycles.
In business, cyclical variations are commonly associated with economic cycles, successive booms and slumps in
the economy. Economic cycles may last a few years. Cyclical variations are longer term than seasonal variations.
Though you should be aware of the cyclical component, you will not be expected to carry out any calculation
connected with isolating it. The mathematical models which we will use, therefore exclude any reference to C.
1.8 Summarising the components
The components of a time series combine to produce a variable in one of two ways.
Additive model: Series = Trend + Seasonal + Random
Y = T + S + R
Multiplicative model: Series = Trend × Seasonal × Random
Y = T × S × R
Assessment
formulae
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2 Finding the trend
2.1 Methods of finding the trend
The main problem we are concerned with in time series analysis is how to identify the trend and seasonal
variations.
Main methods of finding a trend
(a) A line of best fit (the trend line) can be drawn by eye on a graph.
(b) Linear regression analysis can be used. (We covered this in Chapter 10)
(c) A technique known as moving averages can be used.
2.2 Finding the trend by moving averages
One method of finding the trend is by the use of moving averages.
• A moving average is an average of the results of a fixed number of periods
• The moving averages method attempts to remove seasonal variations from actual data by a process of
averaging
2.3 Example: Moving averages of an odd number of results
Year Sales
Units
20X0 390
20X1 380
20X2 460
20X3 450
20X4 470
20X5 440
20X6 500
Required
Take a moving average of the annual sales over a period of three years.
Solution
(a) Average sales in the three year period 20X0 – 20X2 were
⎟
⎠
⎞
⎜
⎝
⎛
++
3
460380390
=
3
230,1
= 410
This average relates to the middle year of the period, 20X1.
(b) Similarly, average sales in the three year period 20X1 – 20X3 were
⎟
⎠
⎞
⎜
⎝
⎛
++
3
450460380
=
3
290,1
= 430
This average relates to the middle year of the period, 20X2.
Key terms
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(c) The average sales can also be found for the periods 20X2–20X4, 20X3–20X5 and 20X4–20X6, to give the
following.
Moving total of Moving average of
Year Sales 3 years' sales
3 years' sales (
÷
3)
20X0 390
20X1 380 1,230 410
20X2 460 1,290 430
20X3 450 1,380 460
20X4 470 1,360 453
20X5 440 1,410 470
20X6 500
Note the following points.
(i) The moving average series has five figures relating to the years from 20X1 to 20X5. The original
series had seven figures for the years from 20X0 to 20X6.
(ii) There is an upward trend in sales, which is more noticeable from the series of moving averages than
from the original series of actual sales each year.
2.4 Over what period should a moving average be taken?
The above example averaged over a three-year period. Over what period should a moving average be taken? The
answer to this question is that
the moving average which is most appropriate will depend on the circumstances
and the nature of the time series
. Note the following points.
(a) A moving average which takes an average of the results in many time periods will represent results
over a longer term than a moving average of two or three periods.
(b) On the other hand, with a moving average of results in many time periods, the last figure in the
series will be out of date by several periods. In our example, the most recent average related to
20X5. With a moving average of five years' results, the final figure in the series would relate to 20X4.
(c) When there is a known cycle over which seasonal variations occur, such as all the days in the week
or all the seasons in the year, the most suitable moving average would be one which covers one full
cycle.
Question
Three-month moving average
Using the following data, complete the following table in order to determine the three-month moving average for
the period January-June.
Month
No of new
houses finished
Moving total 3 months
new houses finished
Moving average of
3 months new
houses finished
January 500
February 450
March 700
April 900
May 1,250
June 1,000
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Answe
r
Month
No of new
houses finished
Moving total 3 months
new houses finished
Moving average of
3 months new
houses finished
January 500
February 450 1,650 550
March 700 2,050 683
April 900 2,850 950
May 1,250 3,150 1,050
June 1,000
2.5 Moving averages of an even number of results
When finding the moving average of an even number of results, a second moving average has to be calculated so
that trend values can relate to specific actual figures.
In the previous example, moving averages were taken of the results in an odd number of time periods, and the
average then related to the
mid-point of the overall period. If a moving average were taken of results in an even
number of time periods
, the basic technique would be the same, but the mid-point of the overall period would
not relate to a single period
. For example, suppose an average were taken of the following four results.
Spring 120
Summer 90
Autumn 180
average 115
Winter 70
The average would relate to the mid-point of the period, between summer and autumn. The trend line average
figures need to relate to a particular time period; otherwise, seasonal variations cannot be calculated. To overcome
this difficulty, we take a
moving average of the moving average. An example will illustrate this technique.
2.6 Example: Moving averages over an even number of periods
Calculate a moving average trend line of the following results.
Year Quarter Volume of sales
'000 units
20X5 1 600
2 840
3 420
4 720
20X6 1 640
2 860
3 420
4 740
20X7 1 670
2 900
3 430
4 760
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Solution
A moving average of four will be used, since the volume of sales would appear to depend on the season of the
year, and each year has four
quarterly results.
The moving average of four does not relate to any specific period of time; therefore a second moving average of
two will be calculated on the first moving average trend line.
Moving Moving Mid-point of
total of 4 average of 4 2 moving
Actual volume quarters' quarters' averages
Year Quarter of sales sales sales Trend line
'000 units '000 units '000 units '000 units
(A) (B)
(B
÷ 4)
(C)
20X5 1 600
2 840
2,580 645.0
3 420 650.00
2,620 655.0
4 720 657.50
2,640 660.0
20X6 1 640 660.00
2,640 660.0
2 860 662.50
2,660 665.0
3 420 668.75
2,690 672.5
4 740 677.50
2,730 682.5
20X7 1 670 683.75
2,740 685.0
2 900 687.50
2,760 690.0
3 430
4 760
By taking
a mid point (a moving average of two) of the original moving averages, we can relate the results to
specific quarters
(from the third quarter of 20X5 to the second quarter of 20X7).
The time series information and moving average trend can be shown on a graph.
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Time series graph and moving average trend
0
100
200
300
400
500
600
700
800
900
1000
Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4
20X5 20X6 20X7 Time
Volume
of sales
'000 units
Actual series
Trend
3 Finding the seasonal variations
Seasonal variations are the difference between actual and trend figures. An average of the seasonal variations for
each time period within the cycle must be determined and then adjusted so that the total of the seasonal variations
sums to zero. Seasonal variations can be estimated using the additive model (Y = T + S + R, with seasonal
variations = Y – T) or the multiplicative model (Y = T × S × R, with seasonal variations = Y ÷ T).
3.1 Finding the seasonal component using the additive model
Once a trend has been established, by whatever method, we can find the seasonal variations.
Step 1 The additive model for time series analysis is Y = T + S + R.
Step 2 If we deduct the trend from the additive model, we get Y – T = S + R.
Step 3 If we assume that R, the random, component of the time series is relatively small and therefore
negligible, then S = Y – T.
Therefore, the seasonal component, S = Y – T (the de-trended series).
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