APPENDIXACOST–VOLUME–PROFITANALYSISWITHUNCERTAINTY
Cost–volume–profit (CVP) analysis is often used to assess what future prospects
might be under various alternatives. Given its flexibility, CVP analysis is convenient
for conducting such assessments. Consider the following example.
Novelties Ltd. produces and sells products directed toward the teen market. A
new product has come onto the market that the company is anxious to produce and
sell. Enough capacity exists in the company’s plant to produce 15,000 units each
month. Variable costs to manufacture and sell one unit would be $1.60, and fixed
costs would total $16,000 per month.
The management of Novelties wants to assess the implications of various alternative sales volumes, selling prices, and variable expenses. Sales volumes would be 13,500
units or 15,000 units. Selling prices would be $3.50 or $4.00. Variable expenses were
estimated as being $1.28 or $1.60, depending on a series of outcomes. First, consider the
eight (2 sales volumes × 2 selling prices × 2 variable expenses) possible outcomes:
Alternative
1
2
3
4
5
6
7
8
Variable
Expenses
$1.28
1.28
1.28
1.28
1.60
1.60
1.60
1.60
Selling
Price
Sales
Volume
Fixed
Expenses
$3.50
3.50
4.00
4.00
3.50
3.50
4.00
4.00
13,500
15,000
13,500
15,000
13,500
15,000
13,500
15,000
$16,000
16,000
16,000
16,000
16,000
16,000
16,000
16,000
LEARNINGOBJECTIVE
10
Conduct a cost–volume–profit
analysis with uncertainty.
Operating
Income
$ 13,970
17,300
20,720
24,800
9,650
12,500
16,400
20,000
Given the alternative variable expenses and selling prices, the preceding table can
be represented in the form of a tree, commonly termed a decision tree, as shown in
Exhibit 4A–1.
Management would obviously prefer alternative 4, since it generates the highest operating income of $24,800. This operating income can be achieved if variable
EXHIBIT4A–1 A Decision Tree
Variable
Expenses
Selling
Price
$3.5
$4.0
$
8
1.2
0
0
$1
.60
0
$3.5
$4.0
0
Sales
Volume
13,500
15,000
13,500
15,000
13,500
15,000
13,500
15,000
Fixed
Expenses
Operating
Income
Alternative
Number
$16,000
$13,970
1
$16,000
$17,300
2
$16,000
$20,720
3
$16,000
$24,800
4
$16,000
$ 9,650
5
$16,000
$12,500
6
$16,000
$16,400
7
$16,000
$20,000
8
4A-2
Appendix 4A
Cost–Volume–Profit Analysis with Uncertainty
expenses are $1.28, selling price is $4.00, and sales volume is 15,000 units.
Unfortunately, managers do not have complete control over these outcomes.
Assume that the best a manager can do is to assess the chance of each alternative occurring. These chances are commonly called subjective probabilities and represent what the manager believes will occur. Each of the probabilities can also be
placed on the decision tree, as shown in Exhibit 4A–2.
Close inspection of Exhibit 4A–2 reveals several important aspects of the analysis.
First, the probabilities for the uncertain factors are expressed in decimal form and sum
to one. For example, the probabilities for variable expenses being $1.28 and $1.60 are
respectively 0.60 and 0.40, which sum to 1.0. Second, the probabilities are multiplied
on the decision tree in the same sequence as the CVP elements, so that the chance of
each alternative occurring can be computed. For example, the probability of alternative
1 (variable expense $1.28, selling price $3.50, and sales volume 13,500) is 0.38 (0.60 ×
0.70 × 0.90). Third, no probability is assigned to fixed expenses because they are
known with certainty in every case to be $16,000. Note from Exhibit 4A–2 that if the
subjective probabilities are correct, there is only a 2% chance of realizing a profit of
$24,800. To estimate expected future profits, the expected value is computed as follows:
Alternative
(a)
Profits
(b)
Probability
(a) × (b)
Expected Value
1
2
3
4
5
6
7
8
$13,970
17,300
20,720
24,800
9,650
12,500
16,400
20,000
0.38
0.04
0.16
0.02
0.25
0.03
0.11
0.01
1.00
$ 5,308.60
692.00
3,315.20
496.00
2,412.50
375.00
1,804.00
200.00
Total expected value
$14,603.30
EXHIBIT4A–2 A Decision Tree with Probabilities
Variable
Expense
Probability
Selling
Price
Probability
0.70
0.30
0
.6 0
0.4
0
0.70
0.30
Sales
Volume
Probability
0.90
0.10
0.90
0.10
0.90
0.10
0.90
0.10
Product
Result
Alternative
Number
0.60 3 0.70 3 0.90 5 0.38
1
0.60 3 0.70 3 0.10 5 0.04
2
0.60 3 0.30 3 0.90 5 0.16
3
0.60 3 0.30 3 0.10 5 0.02
4
0.40 3 0.70 3 0.90 5 0.25
5
0.40 3 0.70 3 0.10 5 0.03
6
0.40 3 0.30 3 0.90 5 0.11
7
0.40 3 0.30 3 0.10 5 0.01
Total 1.00
8
Appendix 4A
Cost–Volume–Profit Analysis with Uncertainty
The total expected value, $14,603.30, is a reasonable estimate of profit for the
next period, given the data and the probabilities provided above.
Decision tree analysis is very powerful, and computers can facilitate the various
calculations. However, it is important to note that the number of calculations
increases dramatically with both the number of alternatives considered for each factor (e.g., three selling prices versus two) and the number of factors (e.g., selling
prices, sales volumes) considered in the CVP formulation. For example, changing to
2 alternatives for fixed costs in the above example (say, $16,000 and $14,000) would
double the number of total alternatives to 16.
APPENDIXAPROBLEMSANDCASES
PROBLEM 4A–1 Cost–Volume–Profit under Uncertainty [LO10]
The advertising manager for Kastoff Kings wants to decide which of two advertising campaigns to adopt in marketing the services of his company, which specializes in the collection
of unwanted items from owners of commercial and residential properties. He has assessed
three levels of business activity for each campaign: small, moderate, and large. The problem is
to decide which advertising campaign to choose, based on estimated profits associated with
the two campaigns for each level of potential activity. The payoffs in operating profits and the
probabilities for the three levels are as follows:
Operating Profits for
Advertising Campaign
($000)
Level of
Business Activity
Probability
A
B
Small
Moderate
Large
0.3
0.6
0.1
$(100)
200
800
$ 20
150
900
Required:
1.
2.
Construct the decision tree for this problem.
Calculate the expected profits for the two advertising campaigns. Which campaign should
the advertising manager choose?
PROBLEM 4A–2 Cost–Volume–Profit under Uncertainty [LO10]
DL Electronics produces a line of 3D DVD players. DL can manufacture a subassembly used in
the players or purchase it from an external supplier. Anticipated operating profits for each
alternative, make or buy, and for three levels of demand for the 3D DVD players for the three
months ended June 30 are given in the following table:
Demand
Probability
of Demand
High
Medium
Low
0.30
0.40
0.30
Operating Profits ($000)
Make
Buy
$ 150.0
90.0
(30.0)
$105.0
90.0
25.0
Required:
1.
2.
Draw and label the decision tree for this problem.
Which action—make or buy—should the firm take to maximize profits?
(CGA–Canada, adapted)
PROBLEM 4A–3 Cost–Volume–Profit, Uncertainty, and Bidding [LO10]
Edmonton has decided to build a new stadium for use by its professional hockey team, the
Oilers, as well as for concerts and other events requiring a large venue. To solicit creative
4A-3
4A-4
Appendix 4A
Cost–Volume–Profit Analysis with Uncertainty
design concepts, city council has decided to have a competition. The firm submitting a design
that best fulfills the criteria developed by the city planning committee will earn revenues of
$1,000,000 before consideration of design costs.
Management at Big Sky Design is considering submitting a proposal. They know that a
carefully developed design proposal will greatly increase the likelihood of winning, but such
a design will be expensive to develop. Conversely, they could submit a less creative design that
will be considerably cheaper to develop, but less likely to win.
Big Sky Design is considering two proposals. The proposal cost and probability of success
for each are as follows:
Proposal A
Proposal B
Cost of
Design Proposal
Probability of
Winning
$300,000
$100,000
0.60
0.20
Design costs are assumed to be incurred at the beginning of the current year. Income
taxes are 30%. If Big Sky Design does not win the competition, the costs of the design proposal can be deducted from the revenues generated by other projects, which will reduce
taxes payable.
Required:
Which proposal would you recommend Big Sky Design submit to the design competition?
Show all calculations.
(SMAC, adapted)
CASE 4A–4 Cost–Volume–Profit under Uncertainty [LO10]
Brunswick Limited (BL) manufactures small personal-care appliances. The company has only
one manufacturing facility, which services all of Canada. BL is well established and sells its
products directly to department stores.
BL wants to begin manufacturing and marketing its newly developed cordless curling
iron. In order to properly evaluate the performance of this new product, management has
decided to create a separate division for its production and distribution.
Two of BL’s competitors have recently introduced their own brands of cordless curling
irons at a price of $28 each. BL’s usual pricing strategy for new products is full absorption cost
plus a 100% markup. For the new curling iron, at a production and sales volume of 350,000
units per year, this strategy would imply a price of $31.50. BL’s president, T. C. Edward, is not
sure whether this pricing strategy would be appropriate for the new curling iron and is considering other proposals, as follows (see Exhibit 4A–3 for details):
a. Variable product cost plus a 200% markup.
b. A price of $27 to undercut the competition.
Edward has hired a market research firm to study the likely demand for BL’s cordless curling iron at the three possible prices. The research firm conducted an extensive market test,
resulting in projected annual sales volumes over the next five years at these prices. These sales
projections are summarized in Exhibit 4A–4. The research firm, however, made it clear that
there were no guarantees that the market would respond according to the projections.
EXHIBIT4A–3 President’s
Probability Data for BL’s
Cordless Curling Iron
Selling Price
Volume
Probability
$24.00
500,000
400,000
300,000
400,000
350,000
250,000
300,000
250,000
200,000
10%
50
40
20
40
40
40
50
10
27.00
31.50
Appendix 4A
Selling Price
Volume
Probability
$24.00
500,000
400,000
300,000
400,000
350,000
250,000
300,000
250,000
200,000
20%
50
30
25
45
30
30
50
20
27.00
31.50
4A-5
Cost–Volume–Profit Analysis with Uncertainty
Expected Costs Based on Annual Production of 350,000 Units
Total variable costs $2,800,000
Total fixed overhead $2,712,500
Plant and Equipment
No additional machinery or plant space will be required to produce the cordless
curling iron. The plant has capacity available to produce 500,000 units per year.
Inventory Levels
JIT inventory management will result in virtually no inventory being stored at
any particular time.
Edward was not happy with the probabilities that the market research firm assigned to the
various price/volume levels. He therefore used his own knowledge and past experience to assign
different probabilities. Edward also called on Joan Help, the chief financial officer, to analyze the
situation and recommend a five-year pricing strategy for the new cordless curling iron. The
revised probabilities and other data put together by Help are shown in Exhibits 4A–3 and 4A–5.
Required:
As Joan Help, comply with Edward’s request. Include in your analysis consideration of both
quantitative and qualitative factors in determining a five-year pricing strategy for the new
curling iron.
(SMAC, adapted)
EXHIBIT4A–4 Market
Research Data for BL’s
Cordless Curling Iron
EXHIBIT4A–5 Other
Relevant Data for BL’s
Cordless Curling Iron