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The
importance of identifying and criticising the underlying assumptions of cost
volume profit analysis (CVP analysis) rests on the practical application of it:
anyone who has ever tried (or anyone who may wish) to apply CVP analysis in
reality, whilst trying to apply the substance of CVP theory will have found
severe difficulties. These notes will
help you solve those problems.
In
any discussion of CVP analysis, any lecturer, manual or accountant will be
frequently heard to say something along the lines of "Let's assume for a
moment that fixed costs remain fixed, even if output changes by a relatively
large amount ..." or "Of course, the selling price in this example is
constant over the whole range of output ...".
There
is little doubt that CVP analysis is useful in its proper context: there are
many decisions made which positively shout out that CVP analysis has been
employed: examples such as reduced price midday meals in restaurants compared
to evening meals in the same restaurant; reduced weekend rates in hotels. All such examples are based on CVP
reasoning; and there is little doubt that in the short term, at least, these
special deals attract clients who would otherwise not be attracted in, and thus
help to increase a business's contribution.
Nevertheless,
there are problems with CVP analysis when it comes to applying it. How many student accountants or young
accountants have gone to work following a riveting read of a chapter on CVP
analysis determined to calculate his firm's break even point only to find that
reality is much more complex than the theory might have them believe?! Such problems are centred around the
underlying assumptions on which CVP analysis is based. Nevertheless, it is frequently found that
students are quite happy to apply CVP analysis principles in theoretical
settings but may be unaware of these assumptions and how restrictive they
really are when it comes to when the examiner asks them to identify and criticise these underlying assumptions.
Let's
look at the assumptions one by one and analyse their limitations as we go:
When
we talk about fixed and variable costs, we usually assume that it is possible
to take a look at individual or total costs and split them into their fixed and
variable elements.
However,
if we look at any organisation of a reasonably large size we will quickly appreciate
that not only might there be several hundred costs comprising total cost but
also there are many forces acting on those costs (cost drivers in activity
based costing parlance). Consequently,
it can not be a simple matter of a few minutes' analysis and the fixed and
variable split has been fully explained.
Splitting
out fixed and variable costs can be a long, time consuming process; and
techniques such as the inspection of accounts method really are not
suitable if the analysis is to be realistic. At the very least, some kind of statistical or mathematical
analysis will have to be undertaken. I
have undertaken this kind of an exercise in a wide variety of companies and
industries; and it takes many man hours to research the organisation, set up and
work a spreadsheet, analyse the results and then present my findings.
This
is not to suggest that the splitting of fixed and variable costs is too
difficult for the average student or practitioner. Consider diagram one below (which we can assume for the sake of
the discussion is the regression line derived from an analysis of a business's
total costs) and suggest the level of fixed costs and hence calculate a
variable cost per unit:
The
graph is suggesting a regression equation of:
y
= a + bx = 1,000 + 3x
which,
in the present context, will be interpreted as: the fixed cost for the business
is £1,000; and the variable cost per unit is £3.
It
should be remembered that this graph refers to the whole business and, as we
have already agreed, a reasonably large business is complex: consequently,
although a statistical analysis can be carried out, its results will not always
be as simple to interpret as the assumptions on which CVP analysis, and the
example surrounding diagram one, would have us believe.
Imagine
the problems which must be faced by the analyst trying to cope with the kind of
cost portrayed in diagram five: no longer a straight line at all; and such cost
profiles are likely to be the normal: as opposed to straight lines, that
is.
More
than all of this, though: it is frequently the case that even the people
working in an organisation will have little or no idea
a)
of
their fixed/variable cost split; and
b)
b)
how to split their total costs into their fixed and variable components if
asked!
It
is these two aspects that often cause management accountants to assume
linearity and/or spend many hours analysing total costs.
Assessing the fixed and
variable cost split can be fraught with difficulties and can be time consuming.
Another
simplifying assumption which helps to keep the arithmetic of CVP analysis
simple but which does not help those of us who wish to apply the techniques.
The
common view of fixed costs is given in diagram two:
However,
the major error contained in such charts is that it ignores (or merely assumes
away) the importance of the relevant
range.
The
relevant range is the range of levels of activity over which the business has
direct experience. That is, it has
probably produced at or over that range of outputs; or it has studied such
levels of output carefully. Hence, no
business will know with certainty what its fixed costs will be outside its
relevant range; and there is no guarantee that fixed costs will remain fixed if
the business produces at a level of output outside its relevant range: whether
through expansion or contraction.
Diagram three illustrates a more realistic scenario: where a fixed cost
can change as a result of a change in output level to a level outside the
relevant range. The relevant range in
diagram three is represented as 401 units to 800 units.
The
reasons why fixed costs will change in such a way include, for a reduction in
output: managers and supervisors being
laid off as no longer required at reduced levels of output; machinery sold;
buildings sold or not rented any more.
A similar analysis applies to an increase in output and fixed costs.
Fixed
costs behaving in this step cost fashion is another cause for concern over
glibly trying to apply CVP analysis. We may not, in fact, know how our fixed
costs will behave outside our relevant range unless and until we carry out
detailed cost analysis of extra relevant range scenarios.
A
nice neat assumption which might be true in some circumstances. It is possible for a cost to be truly
variable and behave in a perfectly linear way: think of examples such as making
one standard design of wooden tables and chairs. However, it is still useful to explore here the more likely
exceptions to that behaviour.
Diagram
four demonstrates how a perfectly variable cost behaves:
In
reality, of course, a whole host of forces can act upon a cost which is deemed
to be variable. For example, once a
business grows beyond a certain size it can then enjoy the benefits of greater
volume: such benefits are known as economies of scale and include being awarded
trade discounts, being offered cash discounts now that it can obtain credit;
and quantity discounts because it can now buy in greater bulk. Consequently,
even though the quantity of components in a product remains standard and fixed,
their cost per unit can fall as a result of these economies of scale.
These
changes to the basic assumption of linearity mean that when diagram four shows
a perfectly straight line, reality could be more like diagram five where we can
easily be dealing with a situation where variable costs are essentially
variable but which are not perfectly variable.
In the case of diagram five, we see a true curve; and any analysis of an
estimation of a precise relationship between variable cost and output will
yield a solution but not a linear one.
Again, since any reasonably large business will have many such costs,
isolating the variability of all such costs can be a major task.
There
are many variations on the possible shapes which a variable cost curve might
assume. For example, it might be the
case that at higher levels of output a variable cost curve starts to slope
upwards again, having initially behaved like the curve in diagram five: such a
situation would hold when diseconomies of scale or increasing import tariffs
were being imposed.
A
very similar series of arguments holds for selling prices as held for variable
costs. There is no reason why any
business needs to sell to all of its customers at the same price for all
products. We could easily demonstrate
that different prices are offered for different levels of purchasing: for
example, discounts for bulk buying. The
hypothesis of supply and demand also dictates that the higher the price the
fewer will be sold; and the lower the price the more will be sold. Diagram six combines the basic assumed sales
curve and a more realistic sales curve based on the arguments just put forward:
Again,
when we consider the realistic side of total sales a true curve emerges; and
again, this means that any analysis of sales immediately becomes more difficult
than the basic assumptions of CVP analysis would have us believe.
As
with the variable cost curve, there are potentially many shapes which the sales
curve could take on: diagram six gives only one variation from the usually
assumed straight line.
This,
to some extent, is the worst of all of the assumptions from the point of view
of a realistic application of CVP analysis.
It is the worst because it denies there being such things as labour
efficiency and changes to labour efficiency: the learning effect is ignored, or
assumed away, by this assumption, of course.
Along
with all of the discussion so far, there are many reasons why a total cost or a
cost per unit might change; and changes in the level of output is only
one. Consider your own environment: why
might any one of the costs with which you are associated change?
In
the case of a manufacturer, costs might change because someone has improved the
way an operation is performed. A friend
of mine, John, has a good eye for helping people to work more efficiently. One
day he hoticed that an operative in a factory was working on making components
for a Poly Tunnel (greenhouse type thing!) and was working on a bench but
keeping his metal rods on the floor. John brought a stand around to where the
operative was working and put the metal rods on there … the operative then
completed his jobs in half the time it used to take! The consequences of this
relate to time, productivity, possibly better quality output and the cost per
unit will have improved. None of the
reason for this change in cost is due to the restrictive assumption of output
being the only determinant of cost.
The
reason for this assumption rests on the mathematics involved if more than one
product is assumed to be made. Although
it is not the purpose of this paper to go too deeply into such issues, we
should be aware that trying to model a multi product business in terms of CVP
analysis can become very frustrating indeed.
Consider diagram seven, which represents a ten product business: all
products have different characteristics, as we can see from the three products
included in the graph.
Within
this multiproduct business, there are six prices, all of which are subject to
varying levels of variability.
The
purpose of the graph is to demonstrate that simply by analysing the total sales
curve, and ignoring its constituent parts, is likely to lead to serious errors
of judgement or decision making: the total sales curve is almost a straight
line, but any one of the individual sales curves for any product can be
significantly different to a straight line; as is the case, especially, with
products three and ten.
Any
simplistic attempt at unravelling
this business is destined to fail. The
mathematical model even for this relatively simple ten product business could
run to several complete lines across an A4 page. Such a model is not too
unmanageable for most of us, but it is unwieldy and cannot be readily
simplified just for the sake of argument; and the same arguments would apply
equally well to the variable and fixed costs
(although they have been excluded from diagram seven).
The
sales mix argument is a straightforward one and it deals with the contribution
to sales ratio (the C/S ratio). If a
business makes two products: one with a C/S ratio of 80% and the other with a
C/S ratio of 70%, the average C/S ratio will not be 75% (which would be
the simple average of the two C/S
ratios). The average C/S ratio has to
be based on the weighted average of
the two; and the value of this weighted average varies as the sales mix
varies.
Consider
the weighted averages in each of the following cases for the business just
introduced:
|
Sales
mix (i) |
Product 1 |
Product 2 |
|
Sales
(units) |
100,000 |
200,000 |
|
Sales
(£) |
500,000 |
300,000 |
|
C/S
ratio (as given above) |
80% |
70% |
The weighted average C/S
ratio is:
Total Contribution = (£500,000 x 80%) +
(£300,000 x 70%)
Total Sales £500,000 + £300,000
= 76.25%
|
Sales
mix (ii) |
Product 1 |
Product 2 |
|
Sales
(units) |
300,000 |
350,000 |
|
Sales
(£) |
1,500,000 |
525,000 |
|
C/S
ratio |
80% |
70% |
The weighted average C/S
ratio is:
Total Contribution = (£1,500,000 x 80%) +
(£525,000 x 70%)
Total Sales £1,500,000 + £525,000
= 77.41%
By
changing the sales mix, in a situation where the values of the C/S ratio change
from product to product, the weighted average value of all C/S ratios also
changes; and unless this point is appreciated, the results of any CVP analysis
could easily be invalidated.
The
final assumption underlying CVP analysis is that there is no such thing as
uncertainty. Everything is known and
knowable to 100% certainty levels.
Prices are sure; variability of cost is certain; and there is nothing so
certain as the level of fixed cost!
It
should be clear that the only certainty about certainty is that it is certain
not to exist! Indeed, as has been said
and widely quoted many times, the only things certain in this world are death
and taxes: CVP analysis was not included on that list!
In
this discussion, we have worked through a wide ranging view on the assumptions
underlying cost volume profit analysis.
We have done so not so that we can all now dismiss CVP theory but so
that when CVP analysis is being considered, it can now be done so from a much
firmer basis: by pointing out the weaknesses of the assumptions on which CVP
analysis is based, the requirements for a more rigorous study can be developed.
Finally,
those of you studying for the later stages of your examinations now have a very
good assortment of views on which to base your answer to the question:
Identify and criticise
the underlying assumptions of CVP analysis.
© Duncan Williamson
