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STUDENT ACCOUNTANT PAPER F5

ACCA 2010

THE USE OF QUANTITATIVE TECHNIQUES IN BUDGETING

RELEVANT TO ACCA QUALIFI CATION PAP ER F5

A budget can be defined as a quantified plan relating to a given period.

Its not surprising, therefore, that there are a number of quantitative

techniques available to help us prepare a budget. Most of these

techniques help address forecasting and planning issues such as:

If I make this quantity, what will costs be?

How much do we expect to sell in the first quarter of 2011?

How long will it take people to make these units?

This article looks at four quantitative techniques:

the high-low (or range) method

least squares linear regression

time series analysis

learning curves.

THE HIGH-LOW (OR RANGE) METHOD

The high-low method is a quick but crude technique. It is usually seen in

the context of separating fixed and variable costs.

EXAMPLE 1

Units produced Total factory

costs ($)

Quarter 1 1,000 35,000

Quarter 2 1,500 45,000

Quarter 3 2,000 50,000

Quarter 4 1,800 48,000

Just by looking at the figures in Example 1 you can see that the costs are

not purely variable otherwise they would double from Quarter 1 to Quarter

2 as output doubles. The assumption is that there are also fixed costs in


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ACCA 2010

each of the four quarters which can be removed by subtracting two pairs

of total costs.

The range method looks at the costs incurred at the lowest and highest

outputs and says that any increment in costs must be purely with the

result of additional variable costs. So:

Units Costs ($)

Highest output 2,000 50,000

Lowest output 1,000 35,000

Difference 1,000 15,000

If the extra 1,000 units are causing the extra $15,000 in costs then the

variable cost per unit is $15,000/1,000 = $15.

The other cost in the total costs must be fixed cost, and this can be

estimated using either of the highest or lowest data sets:

Lowest output:

Total costs = $35,000 of which 1,000 x $15 = $15,000 are variable. The

remaining $20,000 of the total costs must therefore be fixed.

Or

Highest output:

Total costs = $50,000 of which 2,000 x $15 = $30,000 are variable. The

remaining $20,000 of the total costs must therefore be fixed.

Armed with this information, we can estimate costs at any level of output.

For example:

Output = 1,200 units, costs = $20,000 + 1,200 x $15 = $38,000


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ACCA 2010

The high-low method is quick and easy but, as said earlier, crude. For

example, all the data falling between the highest and lowest values are

ignored.

You will see here that the predicted costs for output of 1,800 units is

$20,000 + 1,800 x $15 = $47,000, whereas the actual reading was

$48,000. No matter how sophisticated the forecasting tool, there will

always be anomalies between actual and forecast data: factors such as

efficiency, commodity prices, the weather, can all change unexpectedly

and cause forecasting anomalies.

LEAST SQUARES LINEAR REGRESSION

Linear regression is an objective way of fitting the best possible straight

line through any set of points.

Within that sentence lie two important warnings concerning the use of

linear regression:

It results in a straight line, even if a curved or kinked line might be

better.

It will work for any set of points, so, you could collate peoples ages

and apartment or house numbers and a linear regression would

give the best fit possible between where you live and your age (but

whether you are then expected to move every birthday is not

clear!).

So, just because you can perform a least squares regression, it gives you

no information whatsoever about how much you can rely on the predictive

power of the result. For that, you also have to calculate the coefficient of

correlation, or r.


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ACCA 2010

Month Volume

Costs

($)

1 1,000 8,500

2 1,200 9,600

3 1,800 14,000

4 900 7,000

5

2,000

16,000

6 400 5,000

You can see from Figure 1 that when plotted, the data does follow a

straight line relationship fairly well, and we could draw a fairly accurate

straight line to represent how cost depended on volume, but linear

regression takes the subjectivity out of that.

Month Volume Costs ($)

x Y xy x2 y2

1 1,000 8,500 8,500,000 1,000,000 72,250,000

2 1,200 9,600 11,520,000 1,440,000 92,160,000

3 1,800 14,000 25,200,000 3,240,000 196,000,000

4 900 7,000 6,300,000 810,000 49,000,000

5 2,000 16,000 32,000,000 4,000,000 256,000,000

6 400 5,000 2,000,000 160,000 25,000,000

n = 6

x =

7,300

y =

60,100

xy =

85,520,000

x2 =

10,650,000

y2 =

690,410,000

b = 6 x 85,520,000 7,300 x 60,100 = 74,390,000 = 7

6 x 10,650,000 7,3002 10,610,000


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ACCA 2010

Figure 2

a = 60,100 - 7 x 7,300 = 1,500

6 6

This is interpreted as:

Total costs (y) = 1,500 + 7 x Volume (x)

Where $1,500 = fixed cost and $7 = variable cost per unit

The regression line is shown in Figure 2 (in red), superimposed on the

original Figure 2 points.

So, if we were asked to predict costs if for output of 1,500 units, we would

predict:

Total costs = 1,500 + 7 x 1,500 = $12,000

Even though we can see from Figure 2 that the line is a good fit, it is still

important that the coefficient of correlation is calculated.

r = 74,390,000

(6 x 10,650,000 7,300 x 7,300)(6 x 6,410,000 - 60,100 x

60,100)

= 0.99


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ACCA 2010

If r = 1, there is perfect positive correlation (all points fit on the

regression line) and as one variable increases, so does the other.

If r = -1, there is perfect negative correlation (all points fit on the

regression line) and as one variable increases, the other decreases.

If r = 0, there is no correlation.

More usefully, r2 is the coefficient of determination, and this can be

interpreted as the proportion of variation in one variable that is explainedby variation in the other. Here, r2 is 0.98, meaning that there is very good

association between volume and costs. In other words, the formula

derived should be good at predicting costs from volume.

But take heed of these warnings:

1 A linear regression based on very few points, even with a good

coefficient of correlation, is not necessarily reliable. In the extreme

case, if you had only two readings, anyone could draw a straight line

through them but it would prove nothing.

2 Good correlation does not prove cause and effect. In Example 2, few

would argue that producing more units would not cause more costs.

But what if we were to do a linear regression on advertising and sales

volume and found a strong correlation coefficient? That does not prove

that more advertising causes more sales. For example, we might have

seen the economy recovering so decide to advertise more, but the

extra sales could have happened anyway as the economy improved.

3 It is dangerous to extrapolate. What we mean by this is to go outside

the range of data used in the analysis. In Example 2, we forecast

costs of $12,000 for output of 1,500 units, which is an example of

interpolation because we have examined and used data on either sideof 1,500 units. If, however, we were asked to predict costs for an


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ACCA 2010

output volume of 3,000 units, then we have no experimental data

which we can use. Applying the formula would imply:

Total costs = 1,500 + 7 x 3,000 = 22,500

But how do we know that costs keep behaving in the same way at this

higher level of output? Fixed costs could step up, variable costs per

unit could increase, and employees may have to be paid overtime

rates.

4. You must remove other known effects before attempting to uncover

the association between two variables. If, for example, instead of our

data relating to months 1 6 it related to years 1 6, then other

effects are likely to interfere. The easiest to correct would be inflation

effects. Say that inflation runs at the rate of 5% for each year, then,

before the data is used, we should inflation-adjust all amounts of

money before attempting the regression. The data would then be:

Year Volume

Costs

($)

Inflationadjustment

Inflationadjusted

costs ($)

1 1,000 85,000 x (1.05)5 108,484

2 1,200 96,000 x (1.05)4 116,689

3 1,800 140,000 x (1.05)3 162,068

4 900 70,000 x (1.05)2 77,175

5 2,000 160,000 x 1.05 168,000

Now

6 400 50,000

Current $ 50,000

The regression should be carried out on the data in the Volume and

Inflation-adjusted costs columns.

Other effects should also be adjusted for, if possible, such as supply

difficulties affecting one periods costs. Remember, however, that over


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ACCA 2010

an extended period, the technologies used and products made can

change and these changes will also have confounding effects on thecost volume relationship.

5. Always be aware that a linear relationship might not be appropriate.

The relationship might be better described by a curve or a kinked line.

TIME SERIES ANALYSIS

Linear regression assumes that the relationship between two variables is

strictly linear explained by a straight line. Time series analysis is more

adaptable and recognises that the following effects could be present:

1 A trend: this is an underlying smooth increase or decrease of an

amount as time passes.

2 Seasonal variations: cycles of variation repeating in less than a year,

such as spring, summer, autumn and winter, or sales for each day ofthe week.

3 Cyclical variations: cycles of variation repeating in more than a year,

typically, the long-term trade cycle.

4 Random effects: non-repetitive and non-predictable variations.

Time series analysis investigates the first two of these.

EXAMPLE 3

Look at this data: FIGURE 3

Year Qtr

Time

series

Sales

$000

2006 1 1 989.0

2 2 990.0

Sales

Time


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ACCA 2010

3 3 994.0

4 4 1,015.0

2007 1 5 1,030.0

2 6 1,042.5

3 7 1,036.0

4 8 1,056.5

2008 1 9 1,071.0

2 10 1,083.5

3 11 1,079.5

4 12 1,099.5

2009 1 13 1,115.52 14 1,127.5

3 15 1,123.5

4 16 1,135.0

2010 1 17 1,140.0

You can see from Figure 3 that there is some sort of trend (the line

increases overall) and there are seasonal variations with a dip occurring at

times 7, 11, and 15, corresponding to the third quarter of each year.

Quarter 2 tends to look high each year. So, if we are going to try to

forecast sales for the third quarter of 2010, we would first try to project

the trend, then superimpose the seasonal effect on to the trend in order

to decrease it appropriately.

Performing a time series analysis is rather tedious and it is likely, in any

exam question, that much of the work will have been done for you leaving

you to interpret and apply the results. However, for the purposes of

explanation, we will carry out the full process on this data.


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ACCA 2010

Year Qtr

Time

series

Raw

data

4-part

moving

average

Centred

moving

average

Seasonal

variation

(additive)

Seasonal

variation

(multiplicative)

2006 1 1 989.0

2 2 990.0

997.0

3 3 994.0 1,002.1 -8.1 0.9919

1,007.3

4 4 1,015.0 1,013.8 1.2 1.0012

1,020.4

2007 1 5 1,030.0 1,025.6 4.4 1.0043

1,030.9

2 6 1,042.5 1,036.1 6.4 1.0062

1,041.3

3 7 1,036.0 1,046.4 -10.4 0.9901

1,051.5

4 8 1,056.5 1,056.6 -0.1 0.9999

1,061.8

2008 1 9 1,071.0 1,067.2 3.8 1.0036

1,072.6

2 10 1,083.5 1,078.0 5.5 1.0051

1,083.4

3 11 1,079.5 1,088.9 -9.4 0.9913

1,094.5

4 12 1,099.5 1,100.0 -0.5 0.9995

1,105.5

2009 1 13 1,115.5 1,111.0 4.5 1.0041

1,116.5

2 14 1,127.5 1,120.9 6.6 1.0059

1,125.4

3 15 1,123.5

1,131.5

4 16 1,135.0

2010 1 17 1,140.0

-8.1 = 994 1002.1

0.9919 = 994/1002.1


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ACCA 2010

The first four columns are as before.

Column five is called 4-part moving average 4-part because we

believe the data repeats over four seasons. If we thought it repeatedover, say, five days in the week, we would create the 5-part moving

average.

The moving average is the average of the four components in the cycle.

So, in 2006, it is:

997 = (989 + 990 + 994 + 1015)/4

Then, moving down one season:

1,007.3 = (990 + 994 + 1,015 + 1,030)/4

Progressing down the data, the 4-part moving average contains one

element from each season. This is really where we can isolate the trend

because the high season and low season components tend to cancel out.

The trouble with 4-part moving averages (or any even periodicity) is that

the moving average is not really opposite any season. To get a figure

which is centred on a season, adjacent moving averages are themselves

summed. This is not necessary if we start with, say, five seasons in the

repetitive cycle.

Therefore:

1,002.1 = (997.0 + 1,007.3)/2

1,013.8 = (1007.3 + 1,020.4)/2

This data represents the trend line, and if plotted on a graph it would look

like Figure 4:


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ACCA 2010

This method of predicting future amounts is more sophisticated than

linear regression, but neither method, of course, guarantees an accurateanswer. However, they are at least based on historical evidence and this

must surely be better than pure guesswork.

LEARNI NG CURVES

The learning curve phenomenon was first quantified for business use in

the early 1900s in the aircraft construction industry. It is perhaps no

accident that this was where learning was investigated in a commercial

context because the construction of aircraft was then both highly manual

and highly complex, and it is exactly these complex, manual tasks which

give the greatest opportunity for learning.

The rule for quantifying learning is simple to quote, but very subtle in

operation:

The cumulative average time taken to complete a task decreases

by (or to) a given proportion every time the cumulative output

doubles.

Learning curves are easy to tabulate by doubling cumulative

output and applying the learning curve factor to the cumulative

average time. For example, if the first item (or batch of items)

takes 20 hours to make and the learning effect is 80% or (0.8),

then the table will be:


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ACCA 2010

Cumulative

output

(a)

Cumulative

averageunit time

(b)

Cumulative

time

(a x b)1 20.00 20.00

2 16.00 32.00

4 12.80 51.20

8 10.24 81.92

16 8.19 131.04

32 6.55 209.60

64 5.24 335.36

EXAMPLE 4

A typical question would be: 'A company has already made the

first four units, how long will it take to produce the next four?'

Questions such as these can only be solved by working with

cumulative outputs. The table tells us that the first eight items

will take 81.92 hours in total; the first four of those would have

taken 51.20 hours. Therefore, the time needed for the second

four will be:

81.92 51.20 = 30.72.

Knowledge of the learning curve effect and its impacts on

times/unit change is important for the following:

1 Costing. Variable labour and overheads are directly affected;

fixed overheads are usually absorbed on a time basis.

2 Pricing. You need to make sure that your price is

competitive and, at the same time, that you make a profit.

0.8

0.8

0.8

0.8

0.8

0.8


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3 Scheduling work. If the learning effect is not taken into

account. machines might become idle because not enough

work is planned.

It is instructive to draw a graph of cumulative output against

cumulative average unit time (Figure 5). This shows that:

learning is particularly important in the early stages of

production, shown by a rapid fall off in the cumulative

average unit times

learning is much less marked in later stages, showing very

little improvement.

FIGURE 5

In fact, it is usually assumed that eventually the learning effect

will cease and that the time per unit will reach a steady state.

Improvement stops because:

there is a limit to human dexterity

some processes cannot be speeded up any more (for

example, a chemical reaction)

Cumulative average unit time

Cumulative production


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ACCA 2010

new, inexperienced staff will replace practised ones from

time to time, slowing things down again.

To find the steady state time for production, we usually have tofree ourselves from the tabular approach, which is linked to

cumulative output doubling. Lets say, in the Example 4, that it

is believed that a steady state is reached from the 80th item

onwards. This doesnt fall on a cumulative doubling stage so we

cant solve it by this using the tablebut there is a formula

(provided below). Students are expected to have a scientific

calculator in the exam, and be able to calculate b.

We can test this formula by using it to calculate the table figure

for a cumulative output of 32.

a = 20 x = 32

log 0.8 = -0.0969 log 2 = 0.3010

So, b = 0.0969/0.3010 = -0.3219

Therefore, y, the cumulative time per unit when production has

reached 32, is:

y = 20 x 32 -0.3219 = 6.55 (as derived previously in the

table).

y = axb

where

y = average time per batcha = time for first batchx = cumulative number of batches producedb = log learning rate/log2


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But now back to the steady state problem. We need to find the

time for the 80th item. This can only be calculated by finding the

total time for 80 items and then subtracting the total time for the

first 79 of those items:

.

Therefore, the time taken to make the 80th item must be:

390.41 387.09 = 3.32

and so, if you were asked to forecast the total time that 100items would take:

The first 80 would take (from above) 390.41 hours

The next 20 would take 20 x 3.32 = 66.40 hours

Total for 100 items 456.81 hours

Ken Garrett is a freelance write and lecturer

Cumulative production = 80

y (the cumulative average time for 1st 80)

= 20 x 80 -0.3219 = 4.8801

Total time for first 80:

= 80 x 4.8801 = 390.41

Cumulative production = 79

y (the cumulative average time for 1st 79)

= 20 x 79 -0.3219 = 4.899

Total time for first 79:

= 80 x 4.899 = 387.09

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