Time series analysis and seasonal variations

Time Series Analysis Definition

When preparing a cash budget, or the forecasts on which it is based. there are some ‘statistical techniques that may help to arrive at valid estimates.  Time series analysis concerned with numerical ways that the past can be used to forecast the future. The term trend analysis also used to describe the technique that we now examine. It is useful as a tool to help us to forecast future sales units, but It can also be used in other circumstances.

At its simplest, the idea is based on the assumption that data will continue to move in the same direction in the future as it has in the past.


For example, suppose the following are the number of pairs of shoes that are sold by a shoe manufacturer over the last few months:

Month 1 10,000
Month 2 11,000
Month 3 12,000
Month 4 13,000
Month 5 14,000
Month 6 15,000
Month 7 16,000

It does not require a great deal of arithmetic to calculate that if the trend continues at the previous rate then shoe sales could be forecast at 17,000 pairs in Month 8 and 18,000 pairs in Month 9. Of course, this is a very simple example, and life is rarely this straightforward. It is also worth wondering for how long this rate of increase is sustainable.

Before we move on, we will examine the techniques available to forecast simple data. They will become very useful to understand the whole concept.

When each monthly change is identical …

What you probably did was to see that in each month the sales were 1,000 pairs of shoes more than the previous month. Using the last known figure of 16,000 pairs of shoes in Month 7.  we can then add 1,000 to arrive at a forecast of 17.000 pairs in Month 8. and so on.

Calculating the average monthly change …

A slightly more complicated technique could have been used to arrive at the same answer. If we compare the number of sales in Month 7 with the number in Month 1. we can see that It has risen by 6,000 pairs. By dividing that figure by the number of times that the month changed in our data we can arrive at an eragc change per month. The number of times that the month changes Is 6, which is the same as the number of “spaces” between the months, or the total number of months minus 1. Shown as an equation this becomes:

Average Monthly Sales Change = (Sales in Final Month – Sales in First Month) / (Number of Months – 1)

= (16,000 – 10,000) / (7 – 1)

= +1,000 (Which is what we could expect)

The +1,000 would then be added to the sales data in Month 7 of 16,000 (the last actual data) to arrive at a forecast of 17,000 for Month 8.

This technique is useful when all the increases are not quite identical, yet we want to use the average increase to forecast the trend.

Time Series Analysis  Graph

The same result can be produced graphically. Using the same shoe shop example we can extend the graph based on the actual data to form a forecast line.

Time Series Analysis Graph

If in another situation the actual data does not produce exactly equal increases, the graph will produce the same answer as the average annual change provided the straight-line runs through the first and last data points.

Time series analysis using a formula

The data in the example could have been expressed in the following formula:

y = mx + c


y is the forecast amount
m is 1,000 (the amount by the data increases each month)
x is the number of months since the start month
c is 10,000 (which is the sales figure in the start month)

If we wanted a forecast for Month 8, we could calculate it as:

Forecast = (1,000 x  number of months since Month 1) + 10,000

y (the forecast) = (1,000 x 7) + 10,000

= 17,000, which is what we would expect.

This formula works because the formula is based on the equation of a straight line.

Time Series and Seasonal Variations

In the above section we saw how simple historical data can be used to create an estimate or forecast of its future movement. To do this we assumed that there were no cyclical influences called seasonal variations that would have an impact on the data. We are now going to examine how historical data that we believe is affected by regular cyclical variations can be used to generate a forecast.

We can often use moving averages to analyze our historical data into its two main components of:

the trend – the general direction that the data is moving, and
the seasonal variations — the predictable movements in the data that occur in regular cycles.

The technique of moving averages that we are going to demonstrate can be used to generate the following types of data for cash budgets in certain circumstances. In all these situations the technique will only provide valid data if the past movements of data provide a good basis for a forecast of the future.

  • forecasts of sales units
  • forecasts of prices
  • forecasts of production levels

how do moving averages work?

A moving average is a term used for a series of averages calculated from a series of data (eg sales per month, labour costs per month) so that

  • every average is based on the same number of pieces of data, (eg three pieces of data in a ‘three-point moving average’), and
  • each subsequent average moves along with that series of data by one piece of data so that compared to the previous average it uses one new piece of data and abandons one old piece of data.


For example, suppose a factory used a manufacturing process that operated on a three-week cycle for technological reasons. At the end of each three-week period, the production vessels are cleaned and the process starts again.
Meanwhile, as the operatives become more competent at controlling the process, the output is gradually rising. The production figures for the last few weeks are as follows:

Time series analysis and seasonal variations

As the three-week cycle will influence the output, we can calculate a three-point moving average, with workings as follows:

Seasonal variations example

Notice how we move along the list of data. In this simple example with nine pieces of data we can’t work out any more three-point averages since we have arrived at the end of the numbers after seven calculations.

Here we chose the number of pieces of data to average each time so that it corresponded with the number of points in a full cycle (in this case the three-week production cycle of the manufacturing business). By choosing three points that correspond with the number of weeks in the production cycle we always had one example of the output of every type of week in our average.

This means that any influence on the average by including the first week is cancelled out by also including data from a second week and a third week in the production cycle. We must therefore always be careful to work out moving averages so that exactly one complete cycle is included in every average.

working out the trend line

By using moving averages we have worked out what is called the trend line, and we can use this to help with our forecast. A trend line is essentially a line showing a trend that can be plotted on a graph. When determining a trend
line, each average relates to the data from its midpoint, as is shown in the layout of the figures we have just calculated:

Seasonal variations graph

This means that the first average that we calculated (31 tonnes) can be used as the trend point of week 2, with the second point (32 tonnes) forming the trend point of week 3 (see dotted lines). The result is that we:

  • know exactly where the trend line is for each period of time. and
  • have a basis from which we can calculate ‘seasonal variations’

Calculating seasonal variations

Even using our limited data in this example we can see how seasonal variations can be calculated. They are the difference between the actual data at a point and the trend at the same point. These seasonal variations are shown in the right-hand column of the table below. The data used are the features already calculated on the previous page. All the data are in tonnes.

Calculating seasonal variations

We can see in this example that the seasonal variations repeat as follows:

  • the first week in the production cycle always has an output of 10 tonnes less than the trend
  • the second week in the production cycle has an output of 15 tonnes more than the trend, and
  • the third week in the cycle regularly has a production output of 5 tonnes less than the trend

Note the way that plus and minus signs are used to denote the seasonal variations, and be careful to calculate them accurately:

  • a plus sign, in this case, means that the actual production figure is higher than the trend (see week 2)
  • a minus sign, in this case, means that the actual production figure is less than the trend (see week 3)

Using the same data we can now go on to forecast the production for future weeks by:

  • estimating where the trend will be by the chosen week, and
  • incorporating the seasonal variation based on the appropriate week in the cycle

The forecast production for weeks 10 — 12 is carried out as follows:

Seasonal Variations Forecast

Here we can calculate the forecast trend easily, since it is consistently increasing by 1 tonne of output each week.

Using forecast data in the cash budget

We can now see how this forecast data can be used in a cash budget. Using the data from the above example, we will assume:

  • all output is sold immediately it is produced
  • the selling price is $2000 per tonne
  • sales are made on four weeks credit

The following receipts would appear in the cash budget for weeks 12 — 16.

Using forecast data in cash budget

Note that due to lagging, the receipts in weeks 12 and 13 relate to the actual production in weeks 8 and 9. The forecast data for weeks 10 — 12 is used to calculate the receipts in weeks 14 — 16.

We will now use a Case Study to demonstrate how the same technique could be used to forecast price data for a cash budget.

Case Study (Forecasting Seasonal Prices)



A supermarket obtains its supplies of the knawberry (a popular soft fruit) from three
sources, depending on the time of year.

  • In January—April it is bought from UK growers who raise the plants in glasshouses so that they will crop earlier than plants grown outdoors,
  • In May-August it is obtained from UK farmers who grow the fruit outdoors.
  • In September—December it is bought from overseas growers whose climate allows production at this time of the year.

Prices paid by the supermarket to the suppliers have been as follows over the last three years.

Forecasting Seasonal Prices - Case Study

The monthly demand for the fruit (which will form the basis of the purchasing requirements) has already been calculated for the months of April—October of year 4, as follows:

Case Study-1

All growers are paid on two months credit.


  • Using a three-point moving average and seasonal variations, forecast the buying prices that will be payable in year 4.
  • Show an extract from the payments section of the cash budget for June — November year 4 relating to the knawberry.



The trend in the prices is rising by $1 each ‘season’. We can use this with the seasonal variations to forecast the prices in year 4.

Solution - Time Series Analysis and Seasonal Variations

We can now use the demand data, together with the forecast prices and credit period to calculate our payments extract from the cash budget.


Note that the amounts paid in June relate to April purchases (due to the credit period), and are therefore based on 1,600 kgs at $16. Similarly, the July — October payments are based on the UK outside prices and quantities in May to August. The November payment is for overseas goods bought in September at $25 per kg.

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