# Risk and Uncertainty additional Qns

```B5-Risk and Uncertainty
Question 1 – CVP Analysis and Uncertainty
The accountant of Laburnum Ltd is preparing documents for a forthcoming meeting of the
budget committee. Currently, variable cost is 40% of selling price and total fixed costs are
&pound;40,000 per year.
The company uses an historical cost accounting system. There is concern that the level of
costs may rise during the ensuing year and the chairman of the budget committee has
expressed interest in a probabilistic approach to an investigation of the effect that this will
have on historic cost profits. The accountant is attempting to prepare the documents in a way
which will be most helpful to the committee members. He has obtained the following
estimates from his colleagues:
Pessimistic
Most likely
Optimistic
Average inflation rate over ensuing year
10%
5%
1%
Probability
0.4
0.5
0.1
1.0
Pessimistic
Most likely
Optimistic
Demand at current selling prices
&pound;50,000
&pound;75,000
&pound;100,000
Probability
0.3
0.6
0.1
1.0
The demand figures are given in terms of sales value at the current level of selling prices but
it is considered that the company could adjust its selling prices in line with the inflation rate
without affecting customer demand in real terms.
Some of the company’s fixed costs are contractually fixed and some are apportionments of
past costs; of the total fixed costs, an estimated 85% will remain constant irrespective of the
inflation rate.
You are required to analyse the foregoing information in a way which you consider will assist
management with its budgeting problem. Although you should assume that the directors of
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Laburnum Ltd are solely interested in the effect of inflation on historic cost profits, you
should comment on the validity of the accountant’s intended approach. As part of your
analysis you are required to calculate:
(a)
(b)
(c)
the probability of at least breaking even, and
the probability of achieving a profit of at least &pound;20,000.
(16 marks)
It can be argued that the use of point estimate probabilities (as above) is too unrealistic
because it constrains the demand and cost variables to relatively few values. Briefly
describe an alternative simulation approach which might meet this objection. (6 marks)
(Total 22 marks)
(ACCA Level 2 Management Accounting)
Question 2 - Maximax, Maximin and Expected Value
Shifters Haulage (SH) is considering changing some of the vans it uses to transport crates for
customers. The new vans come in three sizes; small, medium and large. SH is unsure about
which type to buy. The capacity is 100 crates for the small van, 150 for the medium van and
200 for the large van.
Demand for crates varies and can be either 120 or 190 crates per period, with the probability
of the higher demand figure being 0&middot;6.
The sale price per crate is \$10 and the variable cost \$4 per crate for all van sizes subject to the
fact that if the capacity of the van is greater than the demand for crates in a period then the
variable cost will be lower by 10% to allow for the fact that the vans will be partly empty
when transporting crates.
SH is concerned that if the demand for crates exceeds the capacity of the vans then customers
will have to be turned away. SH estimates that in this case goodwill of \$100 would be charged
against profits per period to allow for lost future sales regardless of the number of customers
that are turned away.
Depreciation charged would be \$200 per period for the small, \$300 for the medium and \$400
for the large van.
SH has in the past been very aggressive in its decision-making, pressing ahead with rapid
growth strategies. However, its managers have recently grown more cautious as the business
has become more competitive.
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Required:
(a)
(b)
(c)
(d)
Explain the principles behind the maximax, maximin and expected value criteria that
are sometimes used to make decisions in uncertain situations.
(4 marks)
Prepare a profits table showing the SIX possible profit figures per period.
(9 marks)
Using your profit table from (b) above discuss which type of van SH should buy taking
into consideration the possible risk attitudes of the managers.
(6 marks)
Describe THREE methods other than those mentioned in (a) above, which businesses
can use to analyse and assess the risk that exists in its decision-making.
(6 marks)
(25 marks)
(ACCA F5 Performance Management December 2008 Q2)
Question 3 – Payoff Tables, Maximin, Maximax and Expected Values
Cement Co is a company specialising in the manufacture of cement, a product used in the
building industry. The company has found that when weather conditions are good, the demand
for cement increases since more building work is able to take place. Last year, the weather
was so good, and the demand for cement was so great, that Cement Co was unable to meet
demand. Cement Co is now trying to work out the level of cement production for the coming
year in order to maximise profits. The company doesn’t want to miss out on the opportunity to
earn large profits by running out of cement again. However, it doesn’t want to be left with
large quantities of the product unsold at the end of the year, since it deteriorates quickly and
then has to be disposed of. The company has received the following estimates about the
probable weather conditions and corresponding demand levels for the coming year:
Weather
Good
Average
Poor
Probability
25%
45%
30%
Demand
350,000 bags
280,000 bags
200,000 bags
Each bag of cement sells for \$9 and costs \$4 to make. If cement is unsold at the end of the
year, it has to be disposed of at a cost of \$0&middot;50 per bag.
Cement Co has decided to produce at one of the three levels of production to match forecast
demand. It now has to decide which level of cement production to select.
Required:
(a)
(b)
Construct a pay off table to show all the possible profit outcomes.
(8 marks)
Decide the level of cement production the company should choose, based on the
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following decision rules:
(c)
(i) Maximin
(1 mark)
(ii) Maximax
(1 mark)
(iii) Expected value
(4 marks)
You must justify your decision under each rule, showing all necessary calculations.
Describe the ‘maximin’ and ‘expected value’ decision rules, explaining when they might
be used and the attitudes of the decision makers who might use them.
(6 marks)
(20 marks)
(ACCA F5 Performance Management June 2011 Q1)
Question 4 – Pricing and purchase contract decisions based on uncertain demand and
calculation of maximum price to pay for perfect information
Z Ltd is considering various product pricing and material purchasing options with regard to a
new product it has in development. Estimates of demand and costs are as follows:
If selling price per unit is
Forecasts
Optimistic
Most likely
Pessimistic
Variable manufacturing costs
(excluding materials) per unit
General fixed costs
Probability
0.3
0.5
0.2
&pound;15 per unit
Sales volume
(000 units)
36
28
18
&pound;20 per unit
Sales volume
(000 units)
28
23
13
&pound;3
&pound;25,000
&pound;40,000
&pound;3
&pound;96,000
&pound;40,000
Each unit requires 3kg of material and because of storage problems any unused material must
be sold at &pound;1 per kg. The sole suppliers of the material offer three purchase options, which
must be decided at the outset, as follows:
(i) any quantity at &pound;3 per kg, or
(ii) a price of &pound;2.75 per kg for a minimum quantity of 50 000 kg, or
(iii) a price of &pound;2.50 per kg for a minimum quantity of 70 000 kg.
You are required, assuming that the company is risk neutral, to
(a)
prepare calculations to show what pricing and purchasing decisions the company should
make, clearly indicating the recommended decisions;
(15 marks)
(b)
calculate the maximum price you would pay for perfect information as to whether the
demand would be optimistic or most likely pessimistic.
(5 marks)
150
(Total 20 marks)
Question 5 – Selling price decision based on expected values and value of additional
information
Warren Ltd is to produce a new product in a short-term venture which will utilize some
obsolete materials and expected spare capacity. The new product will be advertised in quarter
I with production and sales taking price in quarter II. No further production or sales are
anticipated.
Sales volumes are uncertain but will, to some extent, be a function of sales price. The possible
sales volumes and the advertising costs associated with each potential sales price are as
follows:
The resources used in the production of each unit of the product are:
Production labour:
2 hours
1 hour
Materials: X
1 unit
Y
2 units
The normal cost per hour of labour is:
&pound;2
&pound;3
However, before considering the effects of the current venture, there is expected to be 4,000
hours of idle time for each grade of labour in quarter II. Idle time is paid at the normal rates.
Material X is in stock at a book value of &pound;8 per unit, but is widely used within the firm and
any usage for the purposes of this venture will require replacing. Replacement cost is &pound;9 per
unit.
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Material Y is obsolete stock. There are 16,000 units in stock at a book value of &pound;3.50 per unit
and any stock not used will have to be disposed of at a cost, to Warren, of &pound;2 per unit. Further
quantities of Y can be purchased for &pound;4 per unit.
&pound;2 per direct labour hour worked
&pound;3 per direct labour hour worked
Total fixed overheads will not alter as a result of the current venture.
Feedback from advertising will enable the exact demand to be determined at the end of
quarter I and production in quarter II will be set to equal that demand. However, it is
necessary to decide now on the sales price in order that it can be incorporated into the
Required:
(a)
(b)
(c)
Calculate the expected money value of the venture at each sales price and on the basis
of this advise Warren of its best course of action.
(12 marks)
Briefly explain why the management of Warren might rationally reject the sales price
leading to the highest expected money value and prefer one of the other sales prices.
(4 marks)
It will be possible, for the sales price of &pound;40 per unit only, to ascertain which of the four
levels of demand will eventuate. If the indications are that the demand will be low then
the advertising campaign can be cancelled at a cost of &pound;10,000 but it would then not be
possible to continue the venture at another sales price. This accurate information
concerning demand will cost &pound;5,000 to obtain.
Indicate whether it is worthwhile obtaining the information and ascertain whether it
would alter the advice given in (a) above.
(4 marks)
(Total 20 marks)
(ACCA Level 2 Management Accounting)
Question 6 – Decision Tree
In the market for one of its product, MD and its two major competitors (CN and KL) together
account for 95% of total sales.
The quality of MD’s products is viewed by customers as being somewhat better than that of
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its competitors and therefore at similar prices it has an advantage.
During the past year, however, when MD raised its price to &pound;1.2 per litre, competitors kept
their prices at &pound;1.0 per litre and MD’s sales declined even though the total market grew in
volume.
MD is now considering whether to retain or reduce its price for the coming year. Its
expectations about its likely volume at various prices charged by itself and its competitors are
as follows:
Prices per litre
CN
(&pound;)
1.2
1.2
1.2
1.1
1.1
1.0
1.1
1.1
1.0
1.0
MD
(&pound;)
1.2
1.2
1.2
1.2
1.2
1.2
1.1
1.1
1.1
1.0
KL
(&pound;)
1.2
1.1
1.0
1.2
1.0
1.0
1.1
1.0
1.0
1.0
MD’s expected sales
Million litres
2.7
2.3
2.2
2.4
2.2
2.1
2.8
2.4
2.3
2.9
Experience has shown that CN tends to react to MD’s price level and KL tends to react to
CN’s price level. MD therefore assesses the following probabilities:
If MD’s price per litre is
(&pound;)
1.2
1.1
there is a probability of
0.2
0.4
0.4
1.0
0.3
0.7
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that CN’s price per litre
will be
(&pound;)
1.2
1.1
1.0
1.1
1.0
1.0
1.0
1.0
1.0
If CN’s price per litre is
there is a probability of
that KL’s price per litre
will be
(&pound;)
1.2
1.1
1.0
(&pound;)
1.2
0.1
0.6
0.3
1.0
1.1
0.3
0.7
1.0
1.1
1.0
1.0
1.0
1.0
Costs per litre of the product are as follows:
Direct wages
&pound;0.24
Direct materials
&pound;0.12
Departmental expenses:
Indirect wages, maintenance and supplies
Supervision and depreciation
General works expenses (allocated)
16 2/3% of direct wages
&pound;540,000 per annum
16 2/3% of prime cost
50% of manufacturing cost
You are required to state whether, on the basis of the data given above, it would be most
advantageous for MD to fix its price per litre for the coming year at &pound;1.2, &pound;1.1 or &pound;1.0.
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(20 marks)
Suggested Solutions
(a)
Notes
a Demand at current selling prices &times; (1 + inflation rate) contribution percentage. For
example, with &pound;50,000 demand at current prices, sales revenue will increase to &pound;55,000 if the
inflation rate is 10%. The contribution margin percentage remains constant at 60%, and
therefore contribution will be &pound;33,000.
b &pound;40,000 fixed costs + (inflation rate &times; 0.15 fixed costs)
Summary of probability distribution
Probability of loss = 0.30
Probability of at least breaking even = 0.70
Probability of at least a profit of &pound;20,000 = 0.10
Alternatively the entire probability distribution could be presented:
Probability of a loss of more than &pound;9,000 = 0.03
Probability of a loss of more than &pound;8,000 = 0.18
Probability of a loss of more than &pound;7,000 = 0.30
Probability of a profit of at least &pound;5,000 = 0.70
Probability of a profit of at least &pound;6,000 = 0.64
Probability of a profit of at least &pound;7,000 = 0.34
Probability of a profit of at least &pound;8,000 = 0.34
Probability of a profit of at least &pound;20,000 = 0.10
Probability of a profit of at least &pound;22,000 = 0.09
Probability of a profit of at least &pound;25,000 = 0.04
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It should be noted that it is inappropriate to assume that all costs and selling prices will alter
in line with each other. Also the existence of stocks will introduce a lag in the system.
(c)
A continuous probability distribution should be prepared for each variable which is subject to
uncertainty; for example, sales demand, costs and inflation rate. Probabilities would be
assigned to ranges of sales demand, costs and the inflation rate. For example, sales demand
might be presented with probabilities attached for the following ranges:
(a)
Maximax stands for maximising the maximum return an investor might expect. An investor
that subscribes to the maximax philosophy would generally select the strategy that could give
him the best possible return. He will ignore all other possible returns and only focus on the
biggest, hence this type of investor is often accused of being an optimist or a risk-taker.
Maximin stands for maximising the minimum return an investor might expect. This type of
investor will focus only on the potential minimum returns and seek to select the strategy that
will give the best worst case result. This type of investor could be said to be being cautious or
pessimistic in his outlook and a risk-avoider.
Expected value averages all possible returns in a weighted average calculation.
For example if an investor could expect \$100 with a 0&middot;3 probability and \$300 with a 0&middot;7
probability then on average the return would be:
(0&middot;3 x \$100) + (0&middot;7 x \$300) = \$240
This figure would then be used as a basis of the investment decision. The principle here is that
if this decision was repeated again and again, then the investor would get the EV as a return.
Its use is more questionable for use on one-off decisions.
(Note: you were not asked for a critique of this method.)
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(b)
Profit calculations
Capacity
Low demand (12)
High demand (190)
Sales
VC
Goodwill
Depreciation
Profit
Small Van
100
300
300
W1
1,000
(400)
(100)
(200)
300
W1
W2
W2
1,000
(400)
(100)
(200)
300
Medium Van
150
468
500
W3
1,200
(480)
48
(300)
468
W4
1,500
(600)
(100)
(300)
500
W2
W4
Large Van
200
368
816
W5
W6
W5
1,200
(480)
W6
1,900
(760)
48
(400)
368
76
(400)
816
(c)
Which type of van to buy?
This depends on the risk attitude of the investor. If they are optimistic about the future then
the maximax criteria would suggest that they choose the large van as this has the potentially
greatest profit.
If they are more pessimistic, then they would focus on the minimum expected returns and
choose the medium van as the worst possible result is \$468, which is better than the other
options. We are also told that the business managers are becoming more cautious and so a
maximin criterion may be preferred by them.
Expected values could be calculated thus:
\$
300
487
637
Small van
Medium van (\$468 &times; 0.4) + (\$500 &times; 0.6)
Large van (\$368 &times; 0.4) + (\$816 &times; 0.6)
Given SH is considering replacing a number of vans you could argue that an EV approach has
merit (not being a one-off decision – assuming individual booking sizes are independent of
each other).
The final decision lies with the managers, but, given what we know about their cautiousness,
a medium sized van would seem the logical choice. The small van could never be the correct
choice.
(d)
Methods of uncertainty reduction:
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Market research. This can be desk-based (secondary) or field-based (primary). Desk-based
is cheap but can lack focus. Field-based research is better in that you can target your
customers and your product area, but can be time consuming and expensive. The internet is
bringing down the cost and speeding up this type of research, email is being used to gather
information quickly on the promise of free gifts etc.
Simulation. Computer models can be built to simulate real life scenarios. The model will
predict what range of returns an investor could expect from a given decision without having
risked any actual cash. The models use random number tables to generate possible values for
the uncertainty the business is subject to. Again, computer technology is assisting in bringing
down the cost of such risk analysis.
Sensitivity analysis. This can be used to assess the range of values that would still give the
investor a positive return. The uncertainty may still be there, but the affect that it has on the
investor’s returns will be better understood. Sensitivity calculates the % change required in
individual values before a change of decision results. If only a (say) 2% change is required in
selling price before losses result an investor may think twice before proceeding. Risk is
therefore better understood.
Calculation of worst and best case figures. An investor will often be interested in range. It
enables a better understanding of risk. An accountant could calculate the worst case scenario,
including poor demand and high costs whilst being sensible about it. He could also calculate
best case scenarios including good sales and minimum running costs. This analysis can often
reassure an investor. The production of a probability distribution to show an investor the range
of possible results is also useful to explain risks involved. A calculation of standard deviation
is also possible.
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(a)
Supply
Prob. *
350,000
280,000
200,000
Weather
\$000
\$000
\$000
Good
0.25
1,750 (1)
1,400
1,000
Average
0.45
1,085 (2)
1,400
1,000
Demand
Poor
0.3
325
640
1,000
* The probability column is only shown so as to help in part (b)(iii)’s calculations.
Profit per bag sold in coming year = \$9 – \$4 = \$5
Loss per bag disposed of = \$4 + \$0.50 = \$4.50
(1) 350,000 &times; \$5 = \$1,7500,000
(2) [280,000 x \$5] – [70,000 x \$(4&middot;50)] = \$1,085,000 etc
(b)(i)
Maximin – identify the worst outcome for each level of supply and choose the highest of
these worst outcomes.
Supply (no. of bags)
350,000
280,000
200,000
\$000
\$000
\$000
Worst
325
640
1,000
The highest of these is \$1,000,000 therefore choose to supply only 200,000 bags to meet poor
conditions.
(b)(ii)
Maximax – identify the best outcome for each level of supply and choose the highest of these
best outcomes.
Supply (no. of bags)
350,000
280,000
200,000
\$000
\$000
\$000
Best
1,750
1,400
1,000
The highest of these is \$1,750,000 therefore choose to supply 350,000 bags to meet good
conditions.
(b)(iii)
Expected value – use the probabilities provided in order to calculate the expected value of
each of the supply levels.
Good (0&middot;25 x \$1,750,000) + (0&middot;45 x \$1,085,000) + (0&middot;30 x \$325,000) = \$1,023,250
Average (0&middot;7 x \$1,400,000) + (0&middot;3 x \$640,000) = \$1,172,000
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Poor 1 x \$1,000,000 = \$1,000,000
The expected value of producing 280,000 bags when conditions are average is the highest at
\$1,172,000, therefore this supply level should be chosen.
(c)
Maximin and expected value decision rules
The ‘maximin’ decision rule looks at the worst possible outcome at each supply level and then
selects the highest one of these. It is used when the outcome cannot be assessed with any level
of certainty. The decision maker therefore chooses the outcome which is guaranteed to
minimise his losses. In the process, he loses out on the opportunity of making big profits. It is
often seen as the pessimistic approach to decision-making (assuming that the worst outcome
will occur) and is used by decision makers who are risk averse. It can be used for one-off or
repeated decisions.
The ‘expected value’ rule calculates the average return that will be made if a decision is
repeated again and again. It does this by weighting each of the possible outcomes with their
relative probability of occurring. It is the weighted arithmetic mean of the possible outcomes.
Since the expected value shows the long run average outcome of a decision which is repeated
time and time again, it is a useful decision rule for a risk neutral decision maker. This is
because a risk neutral person neither seeks risk or avoids it; they are happy to accept an
average outcome. The problem often is, however, that this rule is often used for decisions that
only occur once. In this situation, the actual outcome is unlikely to be close to the long run
average. For example, with Cement Co, the closest actual outcome to the expected value of
\$1,172,000 is the outcome of \$1,085,000. This is not too far away from the expected value but
many of the others are really different.
There are two possible selling prices and three possible direct material costs for each selling
price. The contributions per unit before deducting direct material costs are &pound;12 (&pound;15 – &pound;3) for
a &pound;15 selling price and &pound;17 for a &pound;20 selling price. The purchase costs per unit of output are
&pound;9 (3 kg &times;&pound;3), &pound;8.25 (3 kg &times; &pound;2.75) and &pound;7.50. Where the firm contracts to purchase a
minimum quantity any surplus materials are sold at &pound;1 per kg.
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161
Notes
a 170,000 kg minimum purchases at &pound;2.50 per kg less 16 000 kg [70 000 –(3 kg &times; 18,000) at
&pound;1 per kg].
b 50,000 kg minimum purchases at &pound;2.75 per kg less 11 000 kg [50 000 – (3 kg &times;13,000) at
&pound;1 per kg].
c 70,000 kg minimum purchases at &pound;2.50 per kg less 1000 kg [70 000 –(3 kg &times; 23,000) at &pound;1
per kg].
d 70,000 kg minimum purchases at &pound;2.50 per kg less 31 000 kg [70 000 – (3 kg &times; 13 000) at
&pound;1 per kg].
If the objective is to maximize expected profits then the &pound;20 selling price combined with
purchasing option (iii) is recommended. On the other hand, if the maximin criterion is
adopted then the &pound;15 selling price combined with purchasing option (ii) is recommended. An
alternative approach is to examine the probability distributions (final column of the statement)
and adopt a combination which best satisfies the decision-maker’s risk/return preferences.
(b)
If demand is predicted to be optimistic, the highest payoff of &pound;130,000 (&pound;20 selling price and
&pound;2.50 purchase price) for the most optimistic demand level would be chosen. If the most
likely demand is predicted, the highest payoff is &pound;81,000 (&pound;20 selling price and &pound;2.50
purchase price). If the pessimistic demand level is predicted, the highest payoff is &pound;2,500. The
expected value of profits assuming it is possible to obtain perfect information is:
\$
&pound;130,000 &times; 0.3
39,000
&pound;81,000 &times; 0.5
40,500
&pound;2,500 &times; 0.2
500
80,000
The highest expected profit without perfect information in (a) is &pound;67,700. Therefore the
maximum price payable for perfect information is &pound;12,300 (&pound;80,000 – &pound;67,700).
(a)
The relevant costs per unit are as follows:
162
a Labour costs are only relevant when idle time has been exhausted. This occurs at 2000 units
for grade 1 labour (2,000 units &times; 2 hours) and 4000 units for grade 2 labour (4,000 units &times; 1
hour). It is assumed that beyond these output levels incremental labour costs of &pound;2 per hour
for grade 1 and &pound;3 per hour for grade 2 will be incurred.
b Replacement cost of &pound;9 per unit.
c Each unit of Y used saves the company &pound;2 disposal costs. The product requires 2 units of Y,
thus saving &pound;4 disposal costs. When the stock of 16,000 units has been used (8,000 units
produced) additional supplies will be purchased at &pound;4 per unit.
d Variable overheads are assumed to vary with hours of input.
The relevant production costs for various output levels are as follows:
The outcomes and expected values for each selling price are presented in the following
schedule:
163
(b)
On the basis of the expected value decision rule, a selling price of &pound;25 should be selected.
Management might use criteria other than maximizing expected value. For example the
decision might be based on the minimization of risk. The above probability distributions
indicate that &pound;20 is the only selling price at which a loss will not arise. The final decision
should be based on an examination of each of the above probability distributions and
management’s attitude towards risk.
(c)
Assuming that management is proposing a selling price of &pound;40, if the information indicated
that demand would be zero or 3000 units then Warren should cancel the advertising at a cost
of &pound;10 000. This would give the following expected value:
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It is worthwhile obtaining the information, since the expected value increases from &pound;11 300 to
&pound;27 800. The &pound;40 selling price now yields the highest expected value, and this selling price
should be selected if decisions are based on maximizing expected values. Nevertheless,
management might select another selling price, since the &pound;40 selling price still has a 0.7
probability of making a loss.
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(a)
The variable cost per litre is as follows:
&pound;
0.12
0.24
0.04
0.40
Direct materials
Direct wages
Indirect wages, etc. (16 2/3% &times; &pound;0.24)
And the range of contributions are:
&pound;0.80 for a selling price of &pound;1.20
&pound;0.70 for a selling price of &pound;1.10
&pound;0.60 for a selling price of &pound;1.00
The decision tree indicating the possible outcomes presented in the above Figure shows that
the expected value of the contribution is maximized at a selling price of &pound;1.20. Fixed costs are
common and unavoidable to all alternatives, and are therefore not included in the analysis.
However, management might prefer the certain contribution of &pound;1.74 million at a selling price
of &pound;1.00. From columns 6 and 7 of the decision tree it can be seen that there is a 0.60
probability that contribution will be in excess of &pound;1.74 million when a selling price of &pound;1.20 is
implemented. The final decision depends on management’s attitude towards risk.
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