P1 Management Accounting

Module: 14

Moving Average

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1. Moving averages In the previous chapter we saw how to forecast future results using both a graphical method, drawing a line of best fit, and mathematically, using regression analysis. There is one more method of time series analysis that you need to know and that's moving averages.

The moving average is a technique to reduce irregularities and smooth out the dispersion caused by variations. These components are difficult to identify, and the moving average technique makes the long-term trend stand out clearly.

Let's take the example we used in the previous chapter and work out a moving average for those same figures to see how this method works. Here are the results from Gavin's company for the past 4 years. His goal is to forecast the results for 20X4.

Time £m 20X0Q1 24.6 20X0Q2 38.4 20X0Q3 36.9 20X0Q4 48.0 20X1Q1 32.3 20X1Q2 44.8 20X1Q3 42.0 20X1Q4 60.3 20X2Q1 39.8 20X2Q3 54.9 20X2Q4 72.8 20X3Q1 56.9 20X3Q2 59.1 20X3Q3 59.9 20X3Q4 72.0

Three point moving average

Let's get straight into the calculation to demonstrate how moving averages work - they're much easier to show how they work rather than explain!

Let's start by taking the first 3 quarters of figures (those from 20X0) and calculate an average:

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24.6 + 38.4 + 36.9

3

= 33.3

Next, we move everything on one period, so now we take the last 3 quarters of 20X0:

38.4 + 36.9 + 48.0

3

= 41.1

And so, on until we get the 3 point moving averages as follows:

Time £m Moving Av. 20X0Q1 24.6

20X0Q2 38.4 33.3 20X0Q3 36.9 41.1 20X0Q4 48.0 39.1 20X1Q1 32.3 41.7 20X1Q2 44.8 39.7 20X1Q3 42.0 49.0 20X1Q4 60.3 47.4 20X2Q1 39.8 49.2 20X2Q2 47.6 47.4 20X2Q3 54.9 58.4 20X2Q4 72.8 61.5 20X3Q1 56.9 62.9 20X3Q2 59.1 58.6 20X3Q3 59.9 63.7 20X3Q4 72.0

Notice how we associate our first moving average with 20X0 Q2 (shown in bold), and the second with 20X0 Q3, those just being the midpoints of the three items we based our calculation on.

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Let's look at this data on a graph. The full line is the original data and the dotted line the moving average:

As you can see the moving average smooths out the distortions somewhat giving a straighter line, although it's far from perfect!

The reason is that we took an average of 3 quarters and sales in a business are more likely to vary every year, or 4 quarters. We'd hope that if we used a four-period average we would get a much smoother line. Let's see.

Four point moving average

Doing the calculation for 4 quarters is a little more complicated. Let's do our first 4 point moving average from the first 4 quarters.

24.6 + 38.4 + 36.9 + 48.0

4

= 37.0

Next move everything on one period, so now we take the last 3 quarters of 20X0 and the first quarter of 20X1:

38.4 + 36.9 + 48.0 + 32.3

4

= 38.9

Then we'll add these to our table as follows:

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Time £m Calculation 20X0Q1 24.6

20X0 Q2 20X0 Q3

38.4 36.9

37.0

20X0Q4

48.0

38.9

20X1Q1 32.3

Notice though that our calculations are not aligned to a particular quarter this time (unlike the 3-point average). That's a pain as when we do calculations later we do need them to be aligned.

The problem is easily solved. If we take the average of 37.0 and 38.9, we'll get a figure we can associate with 20X0 Q3 (which is midway between the two).

Doing this for all our data we get the following result:

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Time £m Calculation Moving Av. (Trend) 20X0Q1 24.6

20X0 Q2 20X0 Q3

38.4 36.9

37.0

37.9

20X0Q4

48.0

38.9 39.7

20X1Q1

32.3

40.5 41.1

20X1Q2

44.8

41.8 43.3

20X1Q3

42.0

44.9 45.8

20X1Q4

60.3

46.7 47.1

20X2Q1

39.8

47.4 49.0

20X2Q2

47.6

50.7 52.2

20X2Q3

54.9

53.8 55.9

20X2Q4

72.8

58.1 59.5

20X3Q1

56.9

60.9 61.6

20X3Q2

59.1

62.2 62.1

20X3Q3

59.9

62.0

20X3Q4 72.0

Let's look at the graph of this:

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The moving average (the dotted line) over 4 periods is much smoother than that for 3, and that's logical as sales are more likely to vary over the period of a year than over 3 quarters.

Seasonal variations

We can also calculate seasonal variations in much the same way as we did in the previous chapter.

Let's say we were using an additive model then the seasonal variation is the difference between the moving average and the actual figure.

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Time

£m

Moving Average Calculation (Trend)

Seasonal Variation

20X0Q1 24.6

20X0 Q2 20X0 Q3

38.4 36.9

37.0

37.9

-1.0

20X0Q4

48.0

38.9 39.7

8.3

20X1Q1

32.3

40.5 41.1

-8.8

20X1Q2

44.8

41.8 43.3

1.5

20X1Q3

42.0

44.9 45.8

-3.8

20X1Q4

60.3

46.7 47.1

13.2

20X2Q1

39.8

47.4 49.0

-9.2

20X2Q2

47.6

50.7 52.2

-4.6

20X2Q3

54.9

53.8 55.9

-1.0

20X2Q4

72.8

58.1 59.5

13.3

20X3Q1

56.9

60.9 61.6

-4.7

20X3Q2

59.1

62.2 62.1

20X3Q3

59.9

62.0

20X3Q4 72.0

As we did in the previous section we can work out an average seasonal variation. As an example, in Quarter 3, we have the seasonal variations of -1.0, -3.8, and -1.0, giving an average of -1.9. Doing a similar calculation, we get seasonal variations of: Quarter 1: -7.6 Quarter 2: -1.0

Quarter 3: -1.9 Quarter 4: 11.6

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Forecasting using moving averages

Finally, we can attempt to make a forecast.

Firstly, we can work out the average increase in the moving averages from the start (37.9) to the end (62.1). There are 12 points of moving averages data and so the average increase is:

62.1 - 37.9 12

= 2.0

Now we'll increase the trendline by 2.0 from the last one we had in our previous table of 62.1 (for Q2 20X3) and then adapt for the average seasonal variations.

Seasonal Period Trend variations Forecast 20X3Q2 62.1

20X3Q3 64.1

20X3Q4 66.1

20X4Q1 68.1 -7.6 60.5 20X4Q2 70.1 -1.0 69.1 20X4Q3 72.1 -1.9 70.2 20X4Q4 74.1 11.6 85.7

Comparing predictions

Inevitably different approaches to forecasting will have slightly different predictions. We can see how close Gavin's forecasts were using both methods in the following table.

Period Moving averages Regression analysis

Actually, they're not too far off, and remember that the regression analysis was done using the multiplicative method while the moving average was done as an additive, so that would make a difference too.

£m £m

20X4 Q1 60.5 57.8

20X4 Q2 69.1 72.7

20X4 Q3 70.2 72.0

20X4 Q4 85.7 93.1

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So, we've sorted out Gavin's predictions for him using both regression analysis and moving averages, and in the previous chapter also shown how Riya could simply use a graph for her estimates. Hopefully if your boss asks you the same question as they were asked you'd now be able to do a reasonably accurate forecast yourself!

Perhaps more importantly, if you get time series analysis come up in your exam you'll now be in a good position to answer the question!

2. High – low method Just to finish with our forecasting techniques, we'll do one more – hopefully a relatively easy finishing point for forecasting. - the high-low method. It's the least accurate method but is quick and easy. As with other forecasting techniques we've covered this is quick and easy way to help predict costs at different output levels using the highest and lowest (hence the name the high-low method!) levels of production.

The key to it is that we need to have some cost data for real life production levels. In particular it seeks to ascertain the fixed and variable costs by considering data from the highest and lowest levels of recorded activity.

Consider our bakery, let’s assume we know none of the fixed and variable costs and instead know only the total costs and activity levels for the following time periods:

As we can see, in the month of November we produced the highest amount and August the least. These are our ‘high’ and ‘low’ points. (Note: you must

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always take the highest and lowest activity level NOT the highest and lowest costs.)

The next step is to take these two figures and work out the difference between them:

From this figure we can now calculate the variable cost per unit. This is done by dividing the total costs by the activity level:

Variable cost per unit =

Total cost

Activity

= £3

So, the variable cost is £3 per unit.

We know that total costs = variable costs + fixed costs and we already have our variable and total costs; therefore, the final step is to rearrange this equation to give us our fixed costs.

To illustrate this, let’s break down November’s data:

Total cost = £25,500

Variable cost = 3,500 x £3 = £10,500

Total cost = Fixed cost + Variable cost

£25,500 = Fixed cost + £10,500

Fixed cost = £25,500 - £10,500

Fixed cost = £15,000

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So, there you have it, we have identified the fixed costs and the variable cost per unit, using these figures you will now be able to work out the costs for any level of activity between 1,500 and 3,500 using the following formula:

Total cost = £15,000 + (£3 x No of units produced)

Forecasting

We can now use this to make predictions of costs depending on the number of units. If we were making 4,000 units for example the total cost would be:

= £15,000 + (£3 x No of units produced)

= £15,000 + (£3 x 4,000)

= £27,000

3. Limitations of forecasting techniques As you've probably worked out, although the estimates we can find are useful, there are some definite limitations of the methods used.

Difficulty forecasting an underlying trend

The biggest problem with time series arises in forecasting underlying trend. Other patterns may be mistaken for linear trends. A logistic trend, for example (where the straight line tails off as time goes on), is easily mistaken for linear trend and extrapolated beyond the saturation plateau, and the signs of stagnation ignored. In Gavin's company, in the previous section, it could be that the market reached saturation at the end of 20X3 and sales started falling off after this, yet we have ignored in our calculation.

Ignoring cyclic and random variations

In our calculations we have also conveniently ignored cyclic and random factors for our convenience, but they are there and that could cause errors in our predictions. What if the world was about to enter a recession. Would Gavin's trend line hold up in those circumstances?

The past does not equal the future

It's always also worth remembering that the past does not equal the future, and everything can change, so for instance a new competitor could enter the market with a competing product and our forecasting model is suddenly completely wrong.

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Different methods have different estimates

As we've seen, different methods produce different estimates. Should we use multiplicative or addition models? Moving averages or regression analysis? No one method is correct, so which do we use? These questions are hard to answer and therein lies one of the difficulties with time series analysis.

Depends on having past data

What if the business is launching a brand-new product with no past data? The models we've looked at are then completely useless. Other information such as sales of similar products or results from market research will need to be used to make accurate forecasts.