ACCA - P5 Study text - Emile Woolf INT - 2010
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Chapter 2: Accounting for decision-making
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Answer
Suppose that the annual usage of a telephone is x minutes.
Tariff 1 = 600 + 0.05x
Tariff 2 = 360 + 0.08x
Total costs are the same when:
600 + 0.05x = 360 + 0.08x
0.03x = 240,
x = 8,000 minutes.
Below annual usage of 8,000 minutes, Tariff 2 is cheaper.
3.4 CVP analysis and more than one product
CVP analysis can be used when an entity sells more than one product, but only if it
is assumed that products are sold in a fixed proportion or ratio. Without this
assumption of a fixed selling ratio, it is impossible to calculate a unique break-even
point or target sales level to achieve a certain amount of profit.
To use CVP analysis, we must use either:
a weighted average contribution per unit, based on the budgeted units of sale, or
a weighted average contribution/sales ratio, based on the budgeted sales
revenue (in $).
These weighted average amounts can be used to calculate a break-even point, and
the units or sales revenues for each product are then calculated by applying the
budgeted sales proportions.
Example
Entity MNP has the following budget for next year.
Product
Sales
Sales price
Variable cost
Contribution
per unit
Total
contribution
units $ $ $ $
A 5,000 7 5 2 10,000
B 5,000 14 6 8 40,000
C 20,000 4 1 3 60,000
D 5,000 10 4 6 30,000
35,000 140,000
Fixed costs will be $100,000.
What is the break-even point?
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Answer
Break-even point in units
Total contribution = $140,000
Weighted average contribution per unit = $140,000/35,000 units = $4 per unit.
Break-even point = $100,000/$4 per unit = 25,000 units.
Break-even point: sales of each unit:
Product
Sales
Contribution
per unit
Total
contribution
units $ $
A (1/7) 3,571.4 2 7,142.8
B (1/7) 3,571.4 8 28,571.2
C (4/7) 14,285.8 3 42,857.4
D (1/7) 3,571.4 6 21,428.4
25,000.0 99,999.8
This is the break-even position, allowing for a rounding difference of $0.02.
Break-even point in sales revenue
Budgeted sales = (5,000 × $7) + (5,000 × $14) + (20,000 × $4) + (5,000 × $10) =
$235,000.
Budgeted weighted average C/S ratio = $140,000/$235,000 = 140/235 = 0.59574468.
Break-even point = $100,000/0.59574468. = $167,857.
This break-even point is 71.42857% of budgeted sales (167,857/235,000).
Product
Break-even sales
Contribution/
Sales ratio
Total
contribution
$ $
A 25,000 2/7 7,142.9
B 50,000 8/14 28,571.4
C 57,143 3/4 42,857.3
D 35,714 6/10 21,428.4
167,857 100,000.0
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Paper P5
Advanced performance management
CHAPTER
3
Decision-making with risk
and uncertainty
Contents
1 Expected value and other decision rules
2 Assessing risk in decision-making
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Expected value and other decision rules
Decision-making and risk and uncertainty
Risk preference
Worst, most likely and best possible outcomes
Maximax, maximin and minimax regret decision rules
Expected values
1 Expected value and other decision rules
1.1 Decision-making and risk and uncertainty
Decision-making, particularly long-term decision-making, has to be taken under
conditions of risk and uncertainty.
Uncertainty occurs when there is insufficient information about what will
happen, or what will probably happen, in the future. It is therefore likely that
estimates of future values (future sales, future costs, and so on) will be
inaccurate.
Risk occurs when future events of the future outcome could be any of several
different possibilities. However, the probabilities of each possible outcome can
be assessed with reasonable accuracy.
Uncertainty in decision-making can be reduced by obtaining more reliable
information. Risk cannot be removed from a decision, because risk exists in the
situation itself. A decision-maker can try to analyse the risk, and must make a
decision on the basis of whether the risk is justified or acceptable.
The Study Guide for the P5 syllabus states that you should be able to ‘evaluate how
risk and uncertainty play and especially important role in long term strategic
planning and decision making that relies on forecasts of exogenous variables’.
‘Exogenous’ means ‘external’. For a company or non-profit making entity, an
exogenous variable is a factor or a future condition that is outside the current ability
of management to control.
Both risk and uncertainty relate to exogenous variables. Risk and uncertainty can be
analysed and assessed, before a decision is eventually taken. However, strategic
decisions and other decisions inevitably have to be taken in situations where
management are not at all sure what the actual outcome will be.
This chapter looks at a number of decision-making rules or methods that provide a
logical basis for decision-making under conditions of risk or uncertainty. In
addition, it discusses sensitivity analysis as a method of analysing risk or
uncertainty, and helping management to plan strategies for the future.
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1.2 Risk preference
Risk preference describes the attitude of a decision-maker towards risk. Decision-
makers might be described as risk averse, risk-seeking or possibly risk neutral.
A risk-averse decision maker considers risk in making a decision, and will not
select a course of action that is more risky unless the expected return is higher
and so justifies the extra risk. It is not correct to state that a risk-averse decision-
maker seeks to avoid risk as much as possible. However, a risk-averse decision-
maker might expect a substantially higher return to make the extra risk worth
taking.
A risk-neutral decision maker ignores risk in making a decision. The decision of
a risk neutral decision-maker is to select the course of action with the highest
expected return, regardless of risk. The highest expected return could mean the
highest EV of NPV or the highest EV of profit.
A risk-seeking decision maker also considers risk in making a decision. A risk
seeker, unlike a risk-averse decision-maker, will take extra risks in the hope of
earning a higher return. Given two options with the same EV of NPV, a risk
seeker will prefer the option with the higher risk.
Example
A company must make a choice between three different projects A, B and C. These
three projects are mutually exclusive. The net present values of each project have
been estimated as follows:
Project A Project B Project C
NPV NPV NPV
Probability 0.5 + 800,000 + 2,000,000 + 100,000
Probability 0.5 - 500,000 - 1,700,000 + 50,000
Expected value of NPV + 150,000 + 150,000 + 75,000
Projects A and B have a higher expected NPV than project C, but are more risky.
Project C would have a positive NPV, whereas projects A and B have a 50% chance
of a negative NPV. Project B is more risky than project A, although it has the same
expected return (NPV) of + 150,000. With project B, the NPV could be as high as +
2,000,000, but could also be as low as – 1,700,000.
So which project should be undertaken?
A risk-averse decision-maker would choose either project A or project C. Project
C offers a lower return but lower risk, and project A has a higher expected
return but higher risk. A risk-averse decision-maker would not choose project B,
because project A offers the same expected return for less risk.
A risk-seeking decision-maker would choose project B rather than project A, in
the hope of making a return of + 2,000,000.
A risk-neutral decision-maker might choose either project A or project B because
they both offer the same expected return, which is higher than the expected
return from project C.
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1.3 Worst, most likely and best possible outcomes
The choice between two or more alternative courses of action might be based on the
worst, most likely or best expected outcomes from each course of action. A simple
example will be used to explain this point.
Example
A company must make a decision between three possible courses of action, 1, 2 and
3. The expected profit or loss from each course of action has been estimated as
follows:
Course of action
Worst possible
outcome
Most likely
outcome
Best possible
outcome
Course 1 - 1,000 + 4,000 + 5,000
Course 2 - 500 + 2,500 + 7,000
Course 3 - 3,000 + 1,000 + 9,000
The course of action chosen by the company will depend on the attitude to risk of its
decision-maker.
A decision might be taken to select the project where the possible loss is the
lowest. This would be course of action 2, where the worst outcome is – 500. This
is not as bad as the worst outcome for courses of action 1 and 3. A risk-averse
decision-maker might choose this course of action.
A decision might be taken on the basis of the most likely outcome. The preferred
course of action would then be the one where the expected return is highest. In
this example, the preferred course of action would be course 1, where the most
likely return is + 4,000. This is better than course 2 (+ 2,500) or course 3 (+ 1,000).
A decision might be taken on the basis of the best possible return. The preferred
course of action would then be the one that offers the highest return. Here, the
preferred course of action would be course 3 (+ 9,000). This course of action
would be the choice of a risk-seeking decision-maker.
The main disadvantage of choosing between mutually-exclusive courses of action
on the basis of the worst, most likely or best possible outcome is that the choice
ignores the likelihood or probability of the worst, most likely or best outcomes
actually happening.
1.4 Maximax, maximin and minimax regret decision rules
Choosing between mutually-exclusive courses of action on the basis of worst, most
likely or best possible outcome can be stated as ‘decision rules’. The choice may be
based on a maximax, maximin or a minimax regret decision rule.
Maximax decision rule. The decision-maker will select the course of action with
the highest possible pay-off. The decision-maker seeks the highest return,
assuming that events turn out in the best way possible. The maximax decision
rule is a decision rule for the risk-seeker.
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Maximin decision rule. The decision-maker will select the course of action with
the highest expected return under the worst possible conditions. This decision
rule might be associated with a risk adverse decision maker.
Minimax regret decision rule. The decision-maker selects the course of action
with the lowest possible ‘regret’. ‘Regret’ is the opportunity cost of having made
the wrong decision, given the actual conditions that apply in the future.
Any of these decision rules may be used to make a decision, particularly when there
is no reliable information about the probability of the different possible outcomes.
Example
A business entity has to decide which of three projects to select for investment. The
three projects are mutually exclusive, and only one of them can be selected.
The expected profits from investing in each of the projects will depend on the state
of the market. The following estimates of net present value have been prepared:
State of market Diminishing Static Expanding
Probability 0.4 0.3 0.3
$000 $000 $000
Project 1 100 200 950
Project 2 0 650 600
Project 3 180 190 200
Which options would be selected?
(a) under the maximax decision rule
(b) under the minimax decision rule
(c) under the minimax regret decision rule?
Answer
In this example, there is an estimate of the probability of the future state of the
market, diminishing (0.40 probability), static (0.30) and expanding (0.30). Using any
of these three decision rules, however, the probabilities of each possible future
outcome are ignored.
(a) Maximax decision rule. Using this decision rule, the preferred project is the
one that offers the possibility of the largest possible NPV. In this example, the
most favourable outcome is an NPV of $950,000 for Project 1, $650,000 for
Project 2 and $200,000 for Project 3. Project 1 offers the highest potential
return; therefore Project 1 will be selected.
(b) Minimax decision rule. Using this decision rule, the preferred project is the
one that offers the best NPV if the worst possible outcome occurs. The least
favourable outcome for all three projects will occur if the future market
diminishes. The worst possible outcome for Project 1 is an NPV of $100,000,
the worst possible is $0 for Project 2 and the worst possible is $180,000 for
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Project 3. If the worst happens, the best result would be obtained by selecting
Project 3. Using the minimax decision rule, Project 3 would be selected.
(c) Minimax regret decision rule. To establish the decision under the minimax
regret decision rule, we must first prepare a pay-off table showing the regret
the decision-maker would have choosing Project 1, Project 2 or Project 3, and
the actual market conditions turn out diminishing, static or expanding.
The regret with each course of action is shown in the pay-off table below. The right-
hand column of the table shows the maximum regret.
State of market Diminishing Static Expanding Maximum regret
Regret Regret Regret
$000 $000 $000
Project 1 80 450 0 450
Project 2 180 0 350 350
Project 3 0 460 750 750
If you are not sure how regret is calculated, the following notes might be helpful:
If Project 1 is selected and the market condition turns out to be diminishing, the
regret will be not having chosen Project 3 3, because the NPV with Project 1
would be $100,000 and with Project 3 it would be $180,000. The regret is
therefore $180,000 – $100,000 = $80,000.
If Project 1 is selected and the market condition turns out to be static, the regret
will be not having chosen Project 2, because the NPV with Project 1 would be
$200,000 and with Project 2 it would be $650,000. The regret is therefore $650,000
– $200,000 = $450,000.
If Project 1 is selected and the market condition turns out to be expanding, there
is no regret because the best possible option has been chosen.
The maximum regret with Project 1 is the highest of $180,000; $450,000 and $0.
This is $450,000.
The regret can be calculated in a similar way for both Projects 2 and 3, to complete
the matrix.
The ‘maximum regret’ is minimised by selecting Project 2. The biggest regret that
would then happen is $350,000 (and this would happen if the future market
expands). Project 2 would therefore be selected if the minimax regret rule is applied.
1.5 Expected values
Expected values can be used to analyse information and make a decision where the
risk can be assessed in terms of the probability of each possible future outcome.
Where probabilities are assigned to different outcomes, we can evaluate the worth
of a decision as the expected value or weighted average of these outcomes.
Expected value (EV) = weighted average of possible outcomes.
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The weighted average value is calculated by applying the probability of each
possible outcome to the value of the outcome.
EV = ∑px
Where
p = the probability of each outcome and
x = the value of each outcome
An EV is a measurement of weighted average value.
A decision might be based on selecting the course of action that offers the highest
EV of profit, or the lowest EV of cost. In other words, the ‘decision rule’ is to select
the course of action with the highest EV of profit or the lowest EV of cost.
The main advantage of using EVs to make a decision, compared to the decision
rules explained earlier, is that it takes into consideration the probability or
likelihood of each different possible outcome.
Example
A business entity has to decide which of three projects to select for investment. The
three projects are mutually exclusive, and only one of them can be selected.
The expected profits from investing in each of the projects will depend on the state
of the market. The following estimates of net present value have been prepared:
State of market Diminishing Static Expanding
Probability 0.4 0.3 0.3
$000 $000 $000
Project 1 100 200 900
Project 2 0 650 600
Project 3 180 190 200
(Note: This type of table is called a ‘pay-off table’ or a ‘pay-off matrix’. It shows all
the possible ‘pay-offs’ or results – NPV, profit and so on – from different possible
decisions or strategies).
Required
Identify which project would be selected if the decision is to choose the project with
the highest expected value of NPV.
Answer
State of market Probability Project 1 Project 2 Project 3
NPV
EV of
NPV
NPV
EV of
NPV
NPV
EV of
NPV
$000 $000 $000 $000 $000 $000
Diminishing 0.4 100 40 0 0 180 72
Static 0.3 200 60 650 195 190 57
Expanding 0.3 900 270 600 180 200 60
370 375 189
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Based on expected values, Project 2 should be selected, because it has the highest EV
(although not much higher than the EV for Project 1).
Advantages of using expected values
Using EVs to make a decision has some advantages.
An EV is a weighted average value, that is based on all the different possible
outcomes and the probability that each will occur.
It recognises the risk in decisions, based on the probabilities of different possible
results or outcomes.
It expresses risk in a single figure.
Disadvantages of using expected values
Using EVs to make a decision also has some disadvantages.
The probabilities of the different possible outcomes may be difficult to estimate.
The EV is unlikely to be an actual outcome that could occur. In the example
above, the EVs for projects 1, 2 and 3 (370, 375 and 189) are not expected to
occur. They are simply weighted average values.
Unless the same decision has to be made many times, the average will not be
achieved. It is therefore not a valid way of making a decision about the future
when the outcome will happen only once.
An EV is an average value. It gives no indication of the range or spread of
possible outcomes. It is therefore an inadequate measurement of risk.
Example
A business entity has a choice of three different projects. They all involve the same
amount of capital expenditure, and the entity can only afford one of the three
projects. The expected NPV of the three projects will depend on future market
conditions, which may be poor (probability 30%) moderate (probability 50%) or
good (probability 20%).
The following table sets out the NPV for each product for each possible market
condition. It also shows the EV of the NPV. (Workings are not shown, but make
surethatyoucancalculatetheEVs).
Project Poor (0.3) Moderate (0.5) Good (0.2)
NPV NPV NPV
EV of NPV
$m $m $m $m
1 (20) 15 60 + 13.5
2 1 10 20 + 9.1
3 (10) 22 38 + 15.6
UsingEVasabasisfordecision‐making,thedecisionshouldbetoselectProject3.