7Appendix 6A: Cost-Volume-Profit Uncertainty with Uncertainty Appendix 7A: with Cost-Volume-Profit CVP analysis is often employed to assess what future prospects might be under various arrangements. Given the compactness of the analysis, the CVP formula is a convenient approach to conducting such assessments. Consider the following example: Novelties Ltd. produces and sells highly faddish 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 alternatives. As part of the investigation, management wants an analysis of the operating income if various alternative sales volumes, selling prices, and variable expenses occur. 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 3 2 3 2) possible outcomes: Alternatives
1 2 3 4 5 6 7 8
Variable Expenses
Selling Prices
Sales Volumes
Fixed Expenses
Operating Income
$1.28 1.28 1.28 1.28 1.60 1.60 1.60 1.60
$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
$13,970 17,300 20,720 24,800 9,650 12,500 16,400 20,000
Learning Objective 10
Construct a cost-volume-profit analysis with uncertainty.
By noticing the repetitions of 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 7A–1.
Exhibit 7A–1 A Decision Tree Variable Expense
Selling Prices
0 $3.5
$4.0 8 1.2
0
$
$1 .60
0 $3.5
$4.0
0
Sales Volumes 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
7A–2
Appendix 7A Cost-Volume-Profit with Uncertainty
As a manager, one would like alternative 4, with a profit of $24,800. If a manager can force the future components of a profit to be the following—variable expenses, $1.28; selling price, $4.00; and sales volume, 15,000 units—an operating income of $24,800 can be achieved. Unfortunately, managers do not have such a luxury. Assume that the best the manager can do is assess the chances of each alternative occurring. These chances are commonly called subjective probabilities and can represent what the manager believes will occur. Each of the possible chances can also be placed on the tree, as shown in Exhibit 7A–2. Close observation reveals several important and general results to the manager of Novelties. First, the chances for each uncertain factor are expressed in decimal form and sum to one. Second, the chances are multiplied on the tree in the same sequence as the CVP elements. Third, no chance was assigned to fixed expenses because they are known in every case. The manager notes that if the subjective probabilities are correct, there is only a 2% chance, or 2 chances in 100, of having a profit of $24,800. To ascertain what Novelties might expect future profits to be, the expected value (often termed a mean) is computed as follows: Alternatives
Profits
Chances
Products
1 $13,970 .38 2 17,300 .04 3 20,720 .16 4 24,800 .02 5 9,650 .25 6 12,500 .03 7 16,400 .11 8 20,000 .01
$5,308.60 692.00 3,315.20 496.00 2,412.50 375.00 1,804.00 200.00
1.00 Total expected value
$14,603.30
($13,970 3 .38) ($17,300 3 .04)
Exhibit 7A–2 A Decision Tree Variable Expense Chances
Selling Price Chances
.70 .30
.60
.40 .70 .30
Sales Volume Chances
Product Results
Alternative Number
.60 � .70 � .90 � .38
1
.60 � .70 � .10 � .04
2
.90 .10
.60 � .30 � .90 � .16
3
.60 � .30 � .10 � .02
4
.90 .10
.40 � .70 � .90 � .25
5
.40 � .70 � .10 � .03
6
.90 .10
.40 � .30 � .90 � .11
7
.40 � .30 � .10 � .01 Total 1.00
8
.90 .10
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Appendix 7A Cost-Volume-Profit with Uncertainty
7A–3
The expected value, $14,603.30, is a reasonable estimate of what the profit of Novelties might be for next period, given the data and the chances supplied to the analysis. The decision tree analysis is very powerful. A computer can facilitate the tedious calculations. However, it is important to note that the number of calculations increases dramatically with both the number of alternatives (e.g., $1.28 or $1.60 for variable expenses) considered (e.g., 3 3 3 3 3 5 27, 4 3 4 3 4 5 64) and the number of factors (e.g., selling prices, sales volumes, etc.) considered in the CVP formulation (e.g., 2 3 2 3 2 3 2 5 16, 2 3 2 3 2 3 2 3 2 5 32). The number of calculations will increase rapidly.
Problems PROBLEM 7A–1 CVP under Uncertainty (Appendix 7A) [LO10] The marketing manager for Forestem Inc. wants to decide which of two market strategies to adopt in marketing a new product. He has assessed three levels of potential buyers: a small, moderate, or large number. The problem is to decide which strategy to choose based on estimates of profits associated with the two strategies for each level of potential buyers. The payoffs in net profits and the probabilities for the three levels are:
Number of Potential Buyers
Small Moderate Large
Profits for Marketing Strategy ($000) Probability
A
B
0.3 $–50 $ 10 0.5 100 75 0.2 400 250
Required: 1. Construct the decision tree for this problem. 2. Calculate the expected profits for the two strategies. What decision should the marketing manager take? PROBLEM 7A–2 CVP under Uncertainty (Appendix 7A) [LO10] A firm producing stereo amplifiers can manufacture a subassembly or purchase it from another company. Anticipated profits for each alternative, make or buy, and for three levels of demand for the stereo amplifier, are given in the following table: Demand
High Medium Low
Probability of Demand
Profits ($000) Make
Buy
0.40 $ 50.0 $35.0 0.30 30.0 30.0 0.30 –10.0 5.0
Required: 1. Draw and label the decision tree for this problem. 2. Which action—make or buy—should the firm take to maximize profits? (CGA-Canada, adapted) PROBLEM 7A–3 CVP, Uncertainty, and Bidding (Appendix 7A) [LO10] The city has just announced plans to build a library and arts centre complex. To encourage development of creative design concepts, the city has indicated its intention to hold a design competition. The best entry will win the architectural contract which will generate a revenue of $200,000 before the design costs. A local firm of architects is considering submitting a proposal. They know that a well-thoughtout design would greatly enhance their chance of winning. However, such a design is costly. On the other hand, a less costly design proposal has a limited chance of winning. www.mcgrawhill.ca/olc/garrison
7A–4
Appendix 7A Cost-Volume-Profit with Uncertainty
The architectural firm has two proposals under consideration. Each proposal has the following cost and probabilities associated with it:
Proposal A Proposal B
Cost of Design Proposal
Probability of Winning Contract
$60,000 $20,000
.50 .30
Design costs are assumed to be incurred at the beginning of the current year. Income taxes are 40 percent. Required: Which proposal would you recommend the architectural firm submit to the design competition? Show all calculations. (SMAC, adapted)
Case CASE 7A–1 CVP under Uncertainty (Appendix 7A) [LO10] Brunswick Limited (BL) is a manufacturer of small household 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 wishes to begin manufacturing and marketing its newly developed cordless steam iron. In order to evaluate properly the performance of this new product, management has decided to create a new division for its production and distribution. Two of BL’s competitors have recently introduced their own brands of cordless steam 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 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, Mr. T.C. Edward, is not sure whether this pricing strategy would be appropriate for the new iron and Is considering other proposals as follows: a. Variable product cost plus a 200% markup. b. A price of $27 to undercut the competition. Mr. Edward has hired a market research firm to study the likely demand for BL’s cordless steam iron at the three proposed 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 7A–3. The research firm, however, made it clear that there were no guarantees that the market would respond according to the projections. Mr. 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 (see Exhibit 7A–4). Mr. Edward then called on Joan Help, the chief financial officer, to analyze the situation and recommend a five-year pricing strategy for the new cordless steam iron. As a first step, Joan assembled some relevant data which are presented in Exhibit 7A–5. Required: As Joan Help, comply with Mr. Edward’s request. Include in your analysis consideration of both quantitative and qualitative factors in determining a five-year pricing strategy for the new iron. (SMAC, adapted)
Exhibit 7A–3 Market Research Data for BL’s Cordless Steam Iron
Selling Price
Volume
Probability
$24.00
500,000 400,000 300,000
20% 50 30
27.00
400,000 350,000 250,000
25 45 30
31.50
300,000 250,000 200,000
30 50 20
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7A–5
Appendix 7A Cost-Volume-Profit with Uncertainty
Selling Price
Volume
Probability
$24.00
500,000 400,000 300,000
10% 50 40
27.00
400,000 350,000 250,000
20 40 40
31.50
300,000 250,000 200,000
40 50 10
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 steam iron. The plant has capacity available to produce 500,000 units per year. Inventory Levels Just-in-time inventory management will result in virtually no inventory being stored at any particular time.
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Exhibit 7A–4 President’s Probability Data for BL’s Cordless Steam Iron
Exhibit 7A–5 Other Relevant Data for BL’s Cordless Steam Iron
