Accounting Question
misbra
chapter 2
Cost Estimation and Cost–Volume– Profit Relationships
Learning Objectives
After studying Chapter 2, you will be able to:
• Understand the significance of cost behavior to decision making and control.
• Identify the interacting elements of cost-volume-profit analysis.
• Explain the break-even formula and its underlying assumptions.
• Calculate the effect on profits of changes in selling prices, variable costs, or fixed costs.
• Calculate operating leverage, determine its effects on changes in profit, and under- stand how margin of safety relates to operating leverage.
• Find break-even points and volumes that attain desired profit levels when multiple products are sold in combination.
• Obtain cost functions by account analysis, the engineering approach, the scatter- graph approach, and the high-low method.
• Estimate and examine cost functions using regression and correlation analyses.
kati/imagebroker/imagebroker.net/SuperStock
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CHAPTER 2Chapter Outline
Chapter Outline
2.1 Significance of Cost Behavior to Decision Making and Control Decision Making Planning and Control Trends in Fixed Costs
2.2 Cost–Volume–Profit (CVP) Analysis Basics of CVP Analysis A Desired Pretax Profit A Desired Aftertax Profit
2.3 Graphical Analysis The Break-Even Chart Curvature of Revenue and Cost Lines The Profit-Volume Graph
2.4 Analysis of Changes in CVP Variables Sales Volume Variable Costs Price Policy Fixed Costs Ethical Considerations When Changing CVP Variables
2.5 Measures of Relationship Between Operating Levels and Break-Even Points Operating Leverage Margin of Safety
2.6 The Sales Mix
2.7 Cost Estimation Account Analysis Engineering Approach Scattergraph and Visual Fit High-Low Method Ethical Considerations in Estimating Costs Other Issues for Cost Estimation
2.8 Regression and Correlation Analyses Linear Regression Quality of the Regression
2.9 Control Limits
2.10 Multiple Regression
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CHAPTER 2Section 2.1 Significance of Cost Behavior to Decision Making and Control
“Can You Lose a Little on Each One, But Make It Up on Volume?”
David Scott, owner of Baker Cruise Lines, began his cruise business fifteen years ago. During that time, he has enjoyed a number of upswings and weathered several downturns in the economy. Generally, the business has had profitable years and has provided a high standard of living for David and his family.
But times are changing. Personnel costs continue to rise, particularly fringe benefits like medical insurance premiums. Other costs continue to rise as well. At the same time, because of increased competition from new competitors, prices of cruises have fallen dramatically. David has begun to slash operating costs, but he still faces shrink- ing profit margins.
Because of high profitability in the past, David never analyzed how his costs change in relation to changes in activity levels, nor did he analyze the relationships among revenues, costs, and passenger volume to see how they relate to profit levels. Now, David is wondering how far revenues can drop before he sees the red ink in losses. With that information, he hopes to identify what changes will keep operations profit- able. David sees many other businesses closing their doors and is fearful he will have to follow suit someday. He just doesn’t know what factors will influence his future costs and revenues.
Introduction Managers like David Scott of Baker Cruise Lines need to understand cost behavior and cost estimation to be in a better position to plan, make decisions, and control costs. As we dis- cussed in Chapter 1, cost behavior describes the relationship between costs and an activity as the level of activity increases or decreases. Determining cost behavior is important to management’s understanding of overhead costs, marketing costs, and general and admin- istrative expenses and for proper implementation of budgets and budgetary controls. With knowledge of cost behavior, managers can also estimate how costs are affected as future activity levels change, which can lead to better decisions. In addition, knowledge of cost behavior can assist managers in analyzing the interactions among revenues, costs, and volume for profit-planning purposes. These interactions are covered later in this chapter.
2.1 Significance of Cost Behavior to Decision Making and Control
To understand more fully the significance of a manager’s analysis of cost behavior, we look at three areas: decision making, planning and control, and trends in fixed costs. Decision Making
Cost behavior affects the decisions management makes. Variable costs are the incremen- tal or differential costs in most decisions. Fixed costs change only if the specific decision includes a change in the capacity requirement, such as more floor space.
Cost-based pricing requires a good understanding of cost behavior because fixed costs pose conceptual problems when converted to per unit amounts. Fixed costs per unit
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CHAPTER 2Section 2.1 Significance of Cost Behavior to Decision Making and Control
assume a given volume. If the volume turns out to be different from what was used in determining the cost-based price, the fixed cost component of the total cost yields a mis- leading price. Managers must know which costs are fixed, as well as anticipated volume, to make good pricing decisions.
Planning and Control
A company plans and controls variable costs differently than it plans and controls fixed costs. Variable costs are planned in terms of input/output relationships. For example, for each unit produced, the materials cost consists of a price per unit of materials times the number of units of materials; the labor cost consists of the labor rate times the number of labor hours. Once operations are underway, levels of activities may change. The input/out- put relationships identify changes in resources necessary to respond to the change in activ- ity. If activity levels increase, this signals that more resources (materials, labor, or variable overhead) are needed. If activity levels decrease, the resources are not needed, and proce- dures can be triggered to stop purchases and reassign or lay off workers. In cases where more materials or labor time are used than are necessary in the input/output relationship, inefficiencies and waste are in excess of the levels anticipated, and managers must investi- gate causes and eliminate or reduce the financial impact of the unfavorable variances.
Fixed costs, on the other hand, are planned on an annual basis, if not longer. Control of fixed costs is exercised at two points in time. The first time is when the decision is made to incur a fixed cost. Management evaluates the necessity of the cost and makes the deci- sion to move forward or reject the proposal. Once fixed costs are incurred, another point of control enters, that being the daily decision of how best to use the capacity provided by the cost. For example, a university makes a decision to build a new classroom and faculty office building. That decision is the first point of control. After construction, control is implemented in using the building to its maximum capacity. This will occur if classes are scheduled throughout the day and evening.
Another difference in the planning and control of variable and fixed costs is the level at which costs are controllable. Variable costs can be controlled at the lowest supervisory level. Fixed costs are often controllable only at higher managerial levels.
Trends in Fixed Costs
Organizations are finding that an increasing portion of their total costs are fixed costs. The following are a few of the more critical changes taking place.
Implementation of more automated equipment is replacing variable labor costs and a major share of the variable overhead costs. Thus, fixed costs are becoming a more signifi- cant part of total costs. Costs associated with the additional automated equipment such as depreciation, taxes, insurance, and fixed maintenance charges are substantially higher. Some industries, like steel and automobile, are becoming essentially fixed cost industries, with variable costs playing a less important role than was once the case.
Another factor that has helped to increase fixed costs significantly is the movement in some industries toward a guaranteed annual wage for production workers. Employees
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CHAPTER 2Section 2.2 Cost–Volume–Profit (CVP) Analysis
who were once hourly wage earners are now becoming salaried. With the use of more automated equipment, the workers of a company may not represent “touch labor,” that is, work directly on the product. Instead, the production worker may merely observe that the equipment is operating as it should and is properly supplied with materials or may moni- tor production by means of a television screen. The production line employee is handling more of the functions normally associated with indirect labor, and the cost is a fixed cost.
2.2 Cost–Volume–Profit (CVP) Analysis
The separation of fixed costs from variable costs contributes to an understanding of how revenues, costs, and volume interact to generate profits. With this understanding, managers can perform any number of analyses that fit into a broad category called cost– volume–profit (CVP) analysis or, more commonly, break-even analysis. Examples of such analyses include finding the:
1. number of unit sales required to break even; 2. dollars of sales needed to achieve a specified profit level; 3. effect on profits if selling prices and variable costs increase or decrease by a spe-
cific amount per unit; 4. increase in selling price needed to cover a projected fixed cost increase.
CVP analysis, as its name implies, examines the interaction of factors that influence the level of profits. Although the name gives the impression that only cost and volume deter- mine profits, several important factors exist that determine whether we have profits or losses and whether profits increase or decrease over time. The key factors appear in the basic CVP equation:
(Selling price)(Sales volume) 2 (Unit variable cost)(Sales volume) 2Total fixed cost 5 Pretax profit
The basic CVP equation is merely a condensed income statement, in equation form, where total variable costs (Unit variable cost x Sales volume) and total fixed costs are deducted from total sales revenues (Selling price x Sales volume) to arrive at pretax profit. This equation, as well as other variations that will be discussed later, appears as formula (1) in Figure 2.1.
Contemporary Practice 2.1
Separating Variable and Fixed Costs
A survey of 148 German Companies and 130 U.S. companies found that 46 percent of U.S. companies separate variable and fixed costs for each cost center, as compared to 36 percent of German companies (Krum- wiede & Suessmair, 2007).
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CHAPTER 2Section 2.2 Cost–Volume–Profit (CVP) Analysis
Figure 2.1: Formulas for CVP analysis
The excess of total sales revenue over total variable cost is called the contribution margin or, more precisely, variable contribution margin. From the basic CVP equation, we see that the contribution margin contributes to covering fixed costs as well as generating net income. The contribution margin, as well as the contribution margin ratio, often plays an important role in CVP analysis. The latter measure is the ratio of the contribution mar- gin to total sales revenue (or, equivalently, the ratio of the contribution margin per unit to the selling price).
Before proceeding, several fundamental assumptions are made to strengthen CVP analysis:
1. Relevant range – CVP analysis is limited to the company’s relevant range of activity;
2. Cost behavior identification – fixed and variable costs can be identified separately; 3. Linearity – the selling price and variable cost per unit are constant across all sales
levels within the company’s relevant range of activity;
1. Basic CVP equation:
2. CVP equation taxes:
3. Break-even point in units:
4. Break-even point in dollars:
5. Target pretax profit, in dollars:
6. Target pretax profit, in units
7. Target aftertax profit, in units:
8. Target aftertax profit, in dollars:
(p – v)x – FC = PP
[(p – v)x – FC](1 – t) = AP
FC (p – v)
FC CM%
FC + PP p – v
FC + PP CM%
FC + (1 – t) p – v
AP FC + PP p – v
=
FC +
(1 – t) CM%
AP FC + PP CM%
=
= Fixed cost dollars
= Selling price per unit
= Variable cost per unit
= Contribution margin ration (p – v) / p
= Aftertax profit
= Pretax profit
= Income tax rate
= Sales volume in units
FC
p
v
CM%
AP
PP
t
x
Where:
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CHAPTER 2Section 2.2 Cost–Volume–Profit (CVP) Analysis
4. Equality of production and sales – all units are produced and sold and inventory changes are ignored;
5. Activity measure – the primary cost driver is volume of units; 6. Constant sales mix – the sales of each product in a multiproduct firm is a constant
percentage.
Assumptions 1, 2, and 3 are straightforward. Assumption 4 is required because if sales and production are not equal, some amount of variable and fixed costs are treated as assets (inventories) rather than expenses. As long as inventories remain fairly stable between adja- cent time periods, this assumption does not seriously limit the applicability of CVP analysis. Regarding assumption 5, factors other than volume may drive costs, as we have discussed earlier in this chapter, and as we will discuss further in later chapters. Costs that vary with cost drivers other than volume can be added to the fixed-cost component. Assumption 6 is discussed in detail later in this chapter. In many cases, these assumptions are and must be violated in real-world situations, but the basic logic and analysis adds value. While our dis- cussion may appear to presume that CVP analysis is applicable only to companies that sell physical products, these techniques are just as applicable to service organizations.
Basics of CVP Analysis
CVP analysis is often called break-even analysis because of the significance of the break- even point, which is the volume where total revenue equals total costs. It indicates how many units of product must be sold or how much revenue is needed to at least cover all costs. All break-even analysis can be approached by using Formula 1 and by creating its derivations, as shown in Figure 2.1.
Each unit of product sold is expected to yield revenue in excess of its variable cost and thus contribute to the recovery of fixed costs and provide a profit. The point at which profit is zero indicates that the contribution margin is equal to the fixed costs. Sales vol- ume must increase beyond the break-even point for a company to realize a profit.
Let’s look at CVP relationships in the context of Felsen Electronics, a wholesale distributor of calculators. Assume that price and costs for its calculators are as follows:
Dollars per unit Percentage of selling price
Selling price $25 100%
Variable cost 15 60%
Contribution margin $10 40%
Total fixed cost $100,000
Each calculator sold contributes $10 to covering fixed costs and the creation of a profit. Hence, the company must sell 10,000 calculators to break even. The 10,000 calculators sold will result in a total contribution margin of $100,000, equaling total fixed cost.
The break-even point can be calculated by using the basic CVP equations that appear as Formulas 3 and 4 in Figure 2.1. For Felsen Electronics, the break-even point is determined as follows:
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CHAPTER 2Section 2.2 Cost–Volume–Profit (CVP) Analysis
Break-even point in units 5 Total fixed cost / Contribution margin per unit
5 $100,000 / $10 per unit
5 10,000 units
A break-even point measured in sales dollars can be computed by directly using Formula 4 of Figure 2.1, as follows:
Total fixed cost / Contribution margin ratio 5 Break-even point in sales dollars
$100,000 / 0.40 5 $250,000
A Desired Pretax Profit
In business, only breaking even is not satisfactory, but the break-even relationships do serve as a base for profit planning. If we have a target profit level, we can insert that num- ber into the basic CVP equation. This yields the following general formulas, which appear as Formulas 5 and 6 in Figure 2.1:
(Total fixed cost + Pretax profit) / Contribution margin ratio = Sales dollars required
(Total fixed cost + Pretax profit) / Contribution margin per unit = Unit sales required
Continuing with the Felsen Electronics illustration, suppose that Enoch Goodfriend, the president, had set a profit objective of $200,000 before taxes. The units and revenues required to attain this objective are determined as follows:
($100,000 1 $200,000) / $10 5 30,000 calculators
or
($100,000 1 $200,000) / 0.40 5 $750,000
A Desired Aftertax Profit
The profit objective may be stated as a net income after income taxes. Rather than chang- ing with volume of units, income taxes vary with profits after the break-even point. When income taxes are to be considered, the basic CVP equation is altered using Formulas 2, 7, and 8 in Figure 2.1. Formula 7 works as follows:
Target aftertax profit in units 5 (Fixed costs 1 (Aftertax profit / (1 2 Tax rate))) / Contribution margin per unit
Formula 8 yields the sales dollars needed to earn the desired aftertax profit.
Suppose Enoch Goodfriend had budgeted a $105,000 aftertax profit and that the income tax rate was 30 percent. We use the above general formulas to obtain the following volume and sales:
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CHAPTER 2Section 2.3 Graphical Analysis
($100,000 1 ($105,000 / (1 2 0.30))) / $10 5 25,000 calculators
or
($100,000 1 ($105,000 / (1 2 0.30))) / 0.40 5 $625,000
2.3 Graphical Analysis
Total sales dollars and total costs at different sales volumes can be estimated and plot-ted on a graph. The information shown on the graph can also be given in conven- tional reports, but it is often easier to grasp the fundamental facts when they are presented in graphic or pictorial form. Let’s look at two common forms of graphical analysis—the break-even chart and the profit-volume graph.
The Break-Even Chart
Dollars are shown on the vertical scale of the break-even chart, and the units of product to be sold are shown on the horizontal scale. The total costs are plotted for the various quantities to be sold and are connected by a line. This line is merely a combination of the fixed and variable cost diagrams from Chapter 1. Total sales at various levels are similarly entered on the chart.
The break-even point lies at the intersection of the total revenue and total cost lines. Losses are measured to the left of the break-even point; the amount of the loss at any point is equal to the dollar difference between the total cost line and the total revenue line. Profit is measured to the right of the break-even point, and, at any point, is equal to the dollar difference between the total revenue line and the total cost line. This dollar difference equals the contribution margin per unit multiplied by the volume in excess of the break- even point.
In Figure 2.2, a break-even chart has been prepared for Felsen Electronics using the follow- ing data associated with sales levels between 5,000 and 30,000 calculators.
Contemporary Practice 2.2
Break-Even Attendance for Hockey Team
“The Wheeling Nailers announced on Thursday the team again will play 10 of its home dates in Johnstown next season . . . Going over the lease, the ticket prices, our travel and our expenses we came out with 2,500
(attendance) to break even.” (Mastovich, 2011)
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CHAPTER 2Section 2.3 Graphical Analysis
Number of calculators produced and sold
5,000 10,000 15,000 20,000 25,000 30,000
Total revenue $125,000 $250,000 $375,000 $500,000 $625,000 $750,000
Cost:
Variable $ 75,000 $150,000 $225,000 $300,000 $375,000 $450,000
Fixed 100,000 100,000 100,000 100,000 100,000 100,000
Total cost $175,000 $250,000 $325,000 $400,000 $475,000 $550,000
Profit (loss) ($50,000) 0 $ 50,000 $100,000 $150,000 $200,000
Figure 2.2: Break-even chart
Curvature of Revenue and Cost Lines
In some cases, revenues and costs cannot be represented by straight lines. If more units are to be sold, management may have to reduce selling prices. Under these conditions, the revenue function is a curve instead of a straight line. Costs may also be nonlinear depend- ing on what changes take place as volume increases. The cost curve may rise slowly at the start, then more steeply as volume is expanded. This occurs if the variable cost per unit becomes higher as more units are manufactured. Also, fixed costs might change as vol- ume increases. For example, volume increases might cause a jump in supervision, equip- ment, and space costs. Therefore, it may be possible to have two break-even points, as shown in Figure 2.3.
Number of Calculators (in Thousands)
Variable cost
Fixed cost
5 10 15 20 25 30
100
200
300
400
500
600
700
$800
D o lla
rs (
in T
h o u sa
n d s)
Loss area
Break-even point
Profit area
Total revenue line
Total cost line
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CHAPTER 2Section 2.3 Graphical Analysis
Figure 2.3: Break-even chart with two break-even points
The Profit-Volume Graph A profit-volume graph, or P/V graph, is sometimes used in place of, or along with, a break-even chart. Data used in the earlier illustration of a break-even chart in Figure 2.2 have also been used in preparing the P/V graph shown in Figure 2.4. In general, profits and losses appear on the vertical scale; units of product, sales revenue, and/or percent- ages of capacity appear on the horizontal scale. A horizontal line is drawn on the graph to separate profits from losses. The profit or loss at each of various sales levels is plotted. These points are then connected to form a profit line. The slope of the profit line is the contribution margin per unit if the horizontal line is stated as units of product and is the contribution margin ratio if the horizontal line is stated as sales revenue.
Units of Product Sold
D o lla
rs
Break-even point
Break-even point
To tal
co st
lin e
To tal
re ve
nu e l
ine
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CHAPTER 2Section 2.4 Analysis of Changes in CVP Variables
Figure 2.4: Profit-volume graph
The break-even point is the point where the profit line intersects the horizontal line. Dol- lars of profit are measured on the vertical scale above the line, and dollars of loss are mea- sured below the line. The P/V graph may be preferred to the break-even chart because profit or loss at any point is shown specifically on the vertical scale. However, the P/V graph does not clearly show how cost varies with activity. Break-even charts and P/V graphs are often used together, thus obtaining the advantages of both.
2.4 Analysis of Changes in CVP Variables
Break-even charts and P/V graphs are convenient devices to show how profit is affected by changes in the factors that impact profit. For example, if unit selling price, unit variable cost, and total fixed cost remain constant, how many more units must be sold to realize a greater profit? Or, if the unit variable cost can be reduced, what additional profit can be expected at any given volume of sales? The effects of changes in sales volume, unit variable cost, unit selling price, and total fixed cost are discussed in the following paragraphs. In all these cases, the starting point for analysis is the CVP formulas in Figure 2.1.
Sales Volume
For some companies, substantial profits depend on high sales volume. For example, if each unit of product is sold at a relatively low contribution margin, high profits are a function of selling in large quantities. This is more significant when the fixed cost is high.
Number of Calculators (in Thousands)
Revenue
(Thousands of Dollars)
$125 $250 $375 $500 $625 $750
L o ss
e s
P
ro fit
s
(T h o u sa
n d s
o f D
o lla
rs )
Break-even point
Profit line
5 10 15 20 25 30
100
100
$200
200
0
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CHAPTER 2Section 2.4 Analysis of Changes in CVP Variables
For an illustration, consider a company that handles a product with a selling price of $1 per unit. Assume a variable cost of $0.70 per unit and a fixed cost of $180,000 per year. The contribution margin, therefore, is $0.30 per unit ($1 - $0.70). Before any profit is realized, the company must sell enough units for the total contribution margin to recover the fixed cost. Therefore, 600,000 units must be sold just to break even:
$180,000 4 $.30 5 600,000 units
For every unit sold in excess of 600,000, a $0.30 profit before tax is earned. In such a situa- tion, the company must be certain that it can sell substantially more than 600,000 units to earn a reasonable profit on its investment.
When products sell for relatively high contribution margins per unit, the fixed cost is recaptured with the sale of fewer units, and a profit can be made on a relatively low sales volume. Suppose that each unit of product sells for $1,000 and that the variable cost per unit is $900. The fixed cost for the year is $180,000. The contribution margin ratio is only 10 percent, but this is equal to $100 from each unit sold. The break-even point will be reached when 1,800 units are sold. The physical quantity handled is much lower than it was in the preceding example, but the same principle applies. More than 1,800 units must be sold if the company is to produce a profit.
A key relationship between changes in volume and pretax profit is the following:
Contribution margin per unit 2 Change in sales volume 5 Change in net income
This relationship presumes that the contribution margin per unit remains unchanged when the sales volume changes. It also presumes that fixed costs have not been changed. Suppose that Enoch Goodfriend of Felsen Electronics wishes to know how the sale of an additional 500 calculators would impact profits. The above relationship reveals that net income would increase by $5,000 ($10 contribution margin per unit x 500 units).
Variable Costs
The relationship between the selling price of a product and its variable cost is important in any line of business. Even small savings in variable cost can add significantly to profits. A reduction of a fraction of a dollar in the unit cost becomes a contribution to fixed cost and profit. If 50,000 units are sold in a year, a $0.10 decrease in the unit cost becomes a $5,000 increase in profit. Conversely, a $0.10 increase in unit cost decreases profit by $5,000.
Management is continually searching for opportunities to make even small cost savings. What appears trivial may turn out to be the difference between profit or loss for the year. In manufacturing, it may be possible to save on materials cost by using cheaper materi- als that are just as satisfactory. Using materials more effectively can also result in savings. Improving methods of production may decrease labor and overhead costs per unit.
A small savings in unit cost can give a company a competitive advantage. If prices must be reduced, the low-cost producer will usually suffer less. At any given price and fixed cost structure, the low-cost producer will become profitable faster as sales volume increases.
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CHAPTER 2Section 2.4 Analysis of Changes in CVP Variables
The following operating results of three companies show how profit is influenced by changes in the variable cost pattern. Each of the three companies sells 100,000 units of one product line at a price of $5 per unit. Each has an annual fixed cost of $150,000. Company A can manufacture and sell each unit at a variable cost of $2.50. Company B has found ways to save costs and can produce each unit for a variable cost of $2, while Company C has allowed its unit variable cost to rise to $3.
Company A Company B Company C
Number of units sold 100,000 100,000 100,000
Unit selling price $5.00 $5.00 $5.00
Unit variable cost 2.50 2.00 3.00
Unit contribution margin 2.50 3.00 2.00
Contribution margin ratio 50% 60% 40%
Total sales revenue $500,000 $500,000 $500,000
Total variable cost 250,000 200,000 300,000
Total contribution margin $250,000 $300,000 $200,000
Fixed cost 150,000 150,000 150,000
Income before income tax $100,000 $150,000 $ 50,000
A difference of $0.50 in unit variable cost between Company A and Company B or between Company A and Company C adds up to a $50,000 difference in profit when 100,000 units are sold. The low-cost producer has a $1 per unit profit advantage over the high-cost pro- ducer. If sales volume should fall to 60,000 units per company, Company B would have a profit of $30,000, Company A would break even, and Company C would suffer a loss of $30,000. The same results are shown in the break-even chart in Figure 2.5.
Figure 2.5: Effects of variable cost changes
Units of Product (in Thousands)
D o lla
rs (
in T
h o u sa
n d
s)
20 40 60 80 100
100
200
300
400
$500
Fixed cost line
Company B, total cost line
Company A, total cost line
Company C, total cost line
Total revenue line
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CHAPTER 2Section 2.4 Analysis of Changes in CVP Variables
The fixed cost line for each company is drawn at $150,000, the amount of the fixed cost. When 40,000 units are sold, a difference of $20,000 occurs between each total cost line. The lines diverge as greater quantities are sold. At the 100,000-unit level, the difference is $50,000 between each total cost line. Company B can make a profit by selling any quantity in excess of 50,000 units, but Company C must sell 75,000 units to break even. With its present cost structure, Company C will have to sell in greater volume if it is to earn a profit equal to profits earned by Company A or Company B. Company C is the inefficient producer in the group and, as such, operates at a disadvantage. When there is enough business for everyone, Company C will earn a profit, but will most likely earn less than the others. When business conditions are poor, Company C will be more vul- nerable to losses.
Price Policy
One of the ways to improve profit is to get more sales volume; to stimulate sales volume, management may decide to reduce prices. Bear in mind, however, that if demand for the product is inelastic or if competitors also reduce prices, volume may not increase at all. Moreover, even if the price reduction results in an increase in sales volume, the increase may not be enough to overcome the handicap of selling at a lower price. This point is often overlooked by optimistic managers who believe that a small increase in volume can compensate for a slight decrease in price.
Price cuts, like increases in unit variable costs, decrease the contribution margin. On a unit basis, price decreases may appear to be insignificant, but when the unit differential is mul- tiplied by thousands of units, the total effect may be tremendous. Many more units may need to be sold to make up for the difference. Company A, for example, hopes to increase profit by stimulating sales volume; to do so, it plans to reduce the unit price by 10 percent. The following tabulation portrays its present and contemplated situations:
Present situation Contemplated situation
Selling price $5.00 $4.50
Variable cost 2.50 2.50
Contribution margin $2.50 $2.00
Contribution margin ratio 50% 44.4%
At present, one-half of each revenue dollar can be applied to fixed cost and profit. When revenues are twice the fixed cost, Company A will break even. Therefore, 60,000 units yielding revenues of $300,000 must be sold if fixed cost is $150,000. But when the price is reduced, less than half of each dollar can be applied to fixed cost and profit. To recover $150,000 in fixed cost, unit sales must be 75,000 ($150,000 divided by the $2 contribution margin per unit). Thus, to overcome the effect of a 10 percent price cut, unit sales must increase by 25 percent:
(75,000 2 60,000) 4 60,000 5 25% increase
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CHAPTER 2Section 2.4 Analysis of Changes in CVP Variables
Similarly, revenue must increase to $337,500 (75,000 units 3 $4.50 per unit). This repre- sents a 12.5 percent increase, as follows:
($337,500 2 $300,000) 4 $300,000 5 12.5% increase
Not only must total revenue be higher, but with a lower price, more units must be sold to obtain that revenue. A break-even chart showing these changes appears in Figure 2.6.
Figure 2.6: Effects of price reduction
The present pretax income of $100,000 can still be earned if 125,000 units are sold. This is obtained using formula (5) of Figure 2.1:
($150,000 1 $100,000) 4 $2 5 125,000 units
Fixed Costs
A change in fixed cost has no effect on the contribution margin. Increases in fixed cost are recovered when the contribution margin from additional units sold is equal to the increase in fixed cost. Presuming that the contribution margin per unit remains unchanged, the fol- lowing general relationship holds:
(Contribution margin per unit 3 Change in sales volume) 2 Change in fixed cost 5 Change in net income
Suppose Enoch Goodfriend estimates that, if he spends an additional $30,000 on advertis- ing, he should be able to sell an additional 2,500 calculators. The above equation reveals that profits would decrease by $5,000 or [($10 3 2,500) 2 $30,000].
Units of Product (in Thousands)
D o lla
rs (
in T
h o u sa
n d s)
Fixed cost line
Total revenue line, price $4.50
To ta
l r ev
en ue
li ne
, p ric
e $5
30 60 90 120 150
100
200
300
400
$500
Tot al
co st
line
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CHAPTER 2Section 2.4 Analysis of Changes in CVP Variables
Because fixed cost is not part of the contribution margin computation, the slope of the total cost line on a break-even chart is unaffected by changes in fixed cost. The new total cost line is drawn parallel to the original line, and the vertical distance between the two lines, at any point, is equal to the increase or the decrease in fixed cost.
The break-even chart in Figure 2.7 shows the results of an increase in fixed cost from $100,000 to $130,000 at Felsen Electronics. Under the new fixed cost structure, the total cost line shifts upward, and, at any point, the new line is $30,000 higher than it was origi- nally. To break even or maintain the same profit as before, Felsen Electronics must sell 3,000 more calculators.
Figure 2.7: Effects of an increase in fixed cost
Decreases in fixed cost would cause the total cost line to shift downward. The total con- tribution margin can decline by the amount of the decrease in fixed cost without affecting profit. The lower sales volume now needed to maintain the same profit can be calculated by dividing the unit contribution margin into the decrease in fixed cost.
Ethical Considerations When Changing CVP Variables
Managers are often under pressure to reduce costs—both variable and fixed costs. Many companies have downsized in recent years by eliminating jobs and even closing plants. Managers must be careful not to cut costs in an unethical manner. Changing a price can also involve ethical issues. For instance, immediately after a hurricane hit southern Florida several years ago, some sellers of building supplies were accused of “price gouging.”
Units of Product (in Thousands)
D o lla
rs (
in T
h o u sa
n d s)
Fixed cost line, $130,000 Fixed cost line, $100,000
Total cost line, fixed cost $130,000
100
200
300
400
$500
5 10 15 20 25 30
Tot al c
ost lin
e, f ixe
d c ost
$1 00,
000 Total revenue line
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CHAPTER 2Section 2.5 Measures of Relationship Between Operating Levels and Break-Even Points
2.5 Measures of Relationship Between Operating Levels and Break-Even Points
Companies want to know where they are with respect to the break-even point. If they are operating around the break-even point, management may be more conservative in its approach to implementing changes and mapping out new strategies. On the other hand, if they are operating well away from the break-even point, management will be more aggres- sive because the downside risk is not as great. Two measures that relate to this distance between a break-even point and the current or planned operating volume are operating leverage and margin of safety. These measures are the subject of the following sections.
Operating Leverage
Operating leverage measures the effect that a percentage change in sales revenue has on pretax profit. It is a principle by which management in a high fixed cost industry with a relatively high contribution margin ratio (low variable costs relative to sales revenue) can increase profits substantially with a small increase in sales volume. We typically call this measure the operating leverage factor or the degree of operating leverage, and it is computed as follows:
Contribution margin
Net income before taxes 5 Operating leverage factor
As profit moves closer to zero, the closer the company is to the break-even point. This will yield a high operating leverage factor. As sales volume increases, the contribution margin and pretax profit both increase; consequently, the operating leverage factor becomes pro- gressively smaller. Hence, the operating leverage factor is related to the distance between the break-even point and a current or expected sales volume. With an increase in sales volume, profits will increase by the percentage increase in sales volume multiplied by the operating leverage factor.
Suppose Felsen Electronics is currently selling 15,000 calculators. With its unit contribu- tion margin of $10 and fixed costs of $100,000, its operating leverage factor is 3, computed as follows:
115,0002 1102 115,0002 1102 2 100,000 5
150,000 50,000
5 3
At a sales volume of 15,000, if sales volume can be increased by an additional 10 percent, profit can be increased by 30 percent:
Percentage increase in sales volume
10 3
Operating leverage factor
3 5
Percentage increase in pretax profit
30%
A 10 percent increase in sales volume will increase sales from 15,000 units to 16,500 units. Operating leverage suggests that the pretax profit should be $65,000 ($50,000 3 1.3). Indeed, when we deduct the $100,000 fixed cost from the new contribution margin of $165,000 (16,500 3 $10), we obtain a pretax profit of $65,000.
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CHAPTER 2Section 2.6 The Sales Mix
A company with high fixed costs will have to sell in large volume to recover the fixed costs. However, if the company also has a high contribution margin ratio, it will move into higher profits very quickly after the break-even volume is attained. Hence, a fairly small percentage increase in sales volume (computed on a base that is already fairly large) will increase profits rapidly.
Margin of Safety
The margin of safety is the excess of actual (or expected) sales over sales at the break- even point. The excess may also be computed as a percentage of actual (or expected) sales. The margin of safety, expressed either in dollars or as a percentage, shows how much sales volume can be reduced without sustaining losses. The formulas for calculating margin of safety are:
Margin of safety in dollars 5 Actual (or expected) sales 2 Break-even sales
Margin of safety in percentage form 5 Margin of safety in dollars
Actual 1or expected2 sales
For our purposes, margin of safety is the percentage form. Therefore, unless otherwise specified, a reference to margin of safety will mean a percentage.
Recall that the break-even sales level for Felsen Electronics was $250,000. At an actual sales level of 15,000 calculators, its safety margin is one-third, calculated as follows:
3 115,0002 1$252 2 $250,000 4 115,0002 1$252 5
$125,000 $375,000
5 33% or 1/3
Note that one-third is the reciprocal of the operating leverage factor computed earlier for Felsen Electronics. The margin of safety will always be the reciprocal of the operating leverage factor.
2.6 The Sales Mix
When selling more than one product line, the relative proportion of each product line to the total sales is called the sales mix. With each product line having a different contribution margin, management will try to maximize the sales of the product lines with higher contribution margins. However, a sales mix results because limits on either sales or production of any given product line may exist.
When products have their own individual production facilities and fixed costs are specifi- cally identified with the product line, cost-volume-profit analysis is performed for each product line. However, in many cases, product lines share facilities, and the fixed costs relate to many products. For such a situation, cost-volume-profit analysis requires aver- aging of data by using the sales mix percentages as weights. Consequently, a break-even point can be computed for any assumed mix of sales, and a break-even chart or P/V graph can be constructed for any sales mix.
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CHAPTER 2Section 2.6 The Sales Mix
Let’s consider cost-volume-profit analysis with a sales mix. Suppose that Bijan’s Exotic Catering Service offers three basic dinners for events that it caters: duck, venison, and shark. Assume that the following budget is prepared for these three product lines. Fixed costs are budgeted at $500,000 for the period:
Contribution margin
Product lines Sales volume
(meals) Price per meal Unit variable cost Dollars Ratio
Duck 20,000 $50 $20 $30 60%
Venison 10,000 50 30 20 40%
Shark 10,000 50 40 10 20%
Total 40,000
The break-even point in units (i.e., meals) is computed using a weighted average contribu- tion margin as follows:
Product lines Sales mix
proportions Contribution
margin per meal Weighted
contribution margin
Duck 50% 3 $30 5 $15.00
Venison 25% 3 20 5 5.00
Shark 25% 3 10 5 2.50
Weighted contribution margin $22.50
Fixed cost Weighted contribution margin
5 $500,000
$22.50 5 22,222 total units
The detailed composition of this overall break-even point of 22,222 units (and contribu- tion margins at this level) is as follows:
Product lines Sales mix
proportions Total units
No. of meals
Margin per meal
Contribution margin
Duck 50% 3 22,222 5 11,111 3 $30 5 $333,330
Venison 25% 3 22,222 5 5,556 3 20 5 111,120
Shark 25% 3 22,222 5 5,556 3 10 5 55,560
Break-even contribution margin*
$500,010
* Approximately equal to fixed cost of $500,000. Difference is due to rounding.
To obtain the sales revenue at the break-even point directly, we calculate it as we did ear- lier in the chapter. Simply divide the weighted contribution margin ratio into the fixed
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CHAPTER 2Section 2.6 The Sales Mix
costs. Individual product line revenues will be total revenues multiplied by individual sales mix proportions. Continuing with our illustration, we have:
Product lines
Duck Venison Shark Total
Units (number of meals) 20,000 10,000 10,000 40,000
Revenues $1,000,000 $500,000 $500,000 $2,000,000
Variable cost 400 000 300,000 400,000 1,100,000
Contribution margin $600,000 $200,000 $100,000 $ 900,000
Less fixed cost 500,000
Budgeted net income before taxes $ 400,000
As shown in the following calculations, the weighted contribution margin ratio is 45 per- cent, and revenue at the break-even point is $1,111,111.
Total contribution margin
Total revenues 5 5
$900,000 $2,000,000
5 45%
Fixed cost Weighted contribution margin ratio
5 500,000
45% 5 $1,111,111
Another way to obtain the number of meals needed to break even is by using an equa- tions approach. Let “d” represent the number of duck meals, “v” represent the number of venison meals, and “s” represent the number of shark meals. The following equation is an extension of the basic CVP equation to this scenario with multiple products:
$30d 1 $20v 1 $10s 5 $500,000
Next, we need to write the equation in terms of one variable rather than three by using the information from the sales mix proportions. Since there are half as many venison and shark meals as duck meals, we can rewrite the equation as follows:
$30d 1 $20(.5d) 1 $10(.5d) 5 $500,000
Solving this equation, we obtain d 5 11,111, and therefore, v 5 s 5 .5(11,111) 5 5,556. These are the same numbers of meals we derived earlier.
If the actual sales mix changes from the budgeted sales mix, the break-even point and other factors of cost-volume-profit analysis may change. Suppose that Bijan’s Exotic Catering Service actually operated at the budgeted capacity with fixed costs of $500,000. The unit selling prices and variable costs were also in agreement with the budget. Yet, with the same volume of 40,000 meals and total revenue of $2,000,000, the pretax profit was considerably lower than anticipated. The difference was due to a changed sales mix. Assume the following actual results:
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CHAPTER 2Section 2.7 Cost Estimation
Product lines Sales volume
(meals) Unit contribution
margin Total contribution
margin Revenue
Duck 5,000 $30 $150,000 $250,000
Venison 20,000 20 400,000 1,000,000
Shark 15,000 10 150,000 750,000
Total 40,000 $700,000 $2,000,000
Less fixed cost (500,000)
Actual net income after taxes
$200,000
Instead of earning $400,000 before taxes, the company earned only $200,000. Sales of veni- son and shark, the less profitable products, were much better than expected. At the same time, sales of the best product line, duck, were less than expected. As a result, the total contribution margin was less than budgeted, so net income before taxes was also less than budgeted.
One way to encourage the sales force to sell more of the high contribution margin lines is to compute sales commissions on the contribution margin rather than on sales revenue. If sales commissions are based on sales revenue, a sales force may sell a high volume of less profitable product lines and still earn a satisfactory commission. But if sales com- missions are related to contribution margin, the sales force is encouraged to strive for greater sales of more profitable products and, in doing so, will help to improve total company profits.
2.7 Cost Estimation
Cost estimation is the process of determining a cost relationship with activity for an individual cost item or grouping of costs. We typically express this relationship as an equation that reflects the cost behavior within the relevant range. In Chapter 1, we referred to this equation as a cost function 2 a 1 b (X). The dependent variable (Y) of the equation is what we want to predict (i.e., costs, in our case). The independent variable (X) (i.e., the activity) is used to predict the dependent variable. The cost function can be written as follows:
Y 5 a 1 b (X)
This equation states that Y, the total cost, is equal to a value a plus a factor of variability applied to the activity level X. The value a represents the fixed cost. The factor b represents the change in Y in relation to the change in X, i.e., the variable cost per unit of activity.
Although a number of techniques exist for estimating a cost-to-activity relationship, we will discuss four techniques: (1) account analysis, (2) engineering approach, (3) scatter- graph and visual fit, and (4) high-low method. A fifth method, regression analysis, is dis- cussed in section 2.10 since it requires some knowledge of statistics.
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CHAPTER 2Section 2.7 Cost Estimation
Account Analysis
In account analysis, accountants estimate variable and fixed cost behaviors of a particu- lar cost by evaluating information from two sources. First, the accountant reviews and interprets managerial policies with respect to the cost. Second, the accountant inspects the historical activity of the cost. All cost accounts are classified as fixed or variable. If a cost shows semi-variable or semi-fixed cost behavior, the analyst either (1) makes a subjective estimate of the variable and fixed portions of the cost or (2) classifies the account accord- ing to the preponderant cost behavior. Unit variable costs are estimated by dividing total variable costs by the quantity of the cost driver.
As an example, suppose for cost control purposes, the managing partner of Azran & Asso- ciates, a stock brokerage firm, wishes to estimate the brokers’ automobile expenses as a function of miles driven. Costs for the past quarter, during which 58,000 miles were driven, are classified as follows:
Item Cost Classification
Fuel $3,200 Variable
Depreciation 11,200 Fixed
Insurance 3,900 Fixed
Maintenance 1,800 Variable
Parking 2,100 Fixed
The fixed costs total $17,200, while the variable cost per mile is 8.62 cents [($3,200 1 $1,800) 4 58,000]. Hence, the cost function would be expressed as:
Y 5 $17,200 1 $.0862 (X)
The managing partner believes that costs should be stable for the coming quarter for which the brokers’ auto travel budget would be 65,000 miles. Accordingly, the auto expenses should total $22,803 [$17,200 1 ($.0862 3 65,000)]. The managing partner intends to inves- tigate any significant deviation from this total cost.
Account analysis is fairly accurate for determining cost behavior in many cases. Vendor invoices, for instance, show that direct materials have a variable cost behavior; leasing costs are fixed. A telephone bill is a semi-variable cost. One portion is fixed for the mini- mum monthly charge, and the remainder may be variable with usage.
Account analysis has limited data requirements and is simple to implement. The judgment necessary to make the method work comes from experienced managers and accountants who are familiar with the operations and management policies. Because operating results are required for only one period, this method is particularly useful for new products or situations involving rapid changes in products or technologies.
The two primary disadvantages of this method are its lack of a range of observations and its subjectivity. Using judgment generates two potential issues: (1) different analysts may
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CHAPTER 2Section 2.7 Cost Estimation
develop different cost estimates from the same data, and (2) the results of analysis may have significant financial consequences for the analyst, therefore the analyst will likely show self-serving estimates. Another potential weakness in the method is that data used in the analysis may reflect unusual circumstances or inefficient operations, as is likely to occur with new products. These factors then become incorporated in the subsequent cost estimates. This method is also dependent on the quality of the detailed chart of accounts and transaction coding.
Engineering Approach
The engineering approach uses analysis and direct observation of processes to identify the relationship between inputs and outputs and then quantifies an expected cost behav- ior. A basic issue in manufacturing a product is determining the amount of direct materi- als, direct labor, and overhead required to run a given process. For a service, the question relates primarily to the labor and overhead costs.
One method of applying this approach to a product is to make a list of all materials, labor tasks, and overhead activities necessary for the manufacturing process. Engineering spec- ifications and vendor information can be used to quantify the units of the various mate- rials and subassemblies. Time and motion studies can help estimate the amount of time required for the tasks to be performed. Other analyses are used to assess the overhead relationships to the process. Once quantities and time are determined, those amounts are priced at appropriate materials prices, labor rates, and overhead rates.
A major advantage of the engineering approach is that it details each step required to perform a task. This permits transfer of information to similar tasks in different situations. It also allows an organization to review productivity and identify strengths and weak- nesses in the process. Another advantage is that it does not require historical accounting data. It is, therefore, a useful approach in estimating costs of new products and services. The major disadvantage is the expensive nature of the approach. For example, time and motion studies require in-depth examinations of tasks and close observations of indi- viduals performing each task. An additional disadvantage is that estimates made by the engineering approach are often based on near-optimal working conditions. Since actual working conditions are generally less than optimal, cost estimates using the engineering approach will generally understate the actual costs.
Scattergraph and Visual Fit
An approach that yields rough approximations to fixed and variable costs is called scattergraph and visual fit. With the advent of personal computers, the mechanical nature of this approach loses its appeal. When the analyst has data, the computer can graph the observations and estimate cost behavior quickly. However, we present the details below because, in many cases, it can be used in a preliminary analysis and can easily be applied.
The first step in applying the approach is to graph each observation, with cost on the verti- cal axis and activity or cost driver on the horizontal axis. The second step is to fit visually
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CHAPTER 2Section 2.7 Cost Estimation
and judgmentally a line to the data. Care should be taken so that the distances of the observations above the line are approximately equal to the distances of the observations below the line. Figure 2.8 shows a graph of maintenance cost and hours of operation for ten weeks of activity. The data for this graph are as follows:
Hours (X) Maintenance cost (Y)
50 $120
30 110
10 60
50 150
40 100
30 80
20 70
60 150
40 110
20 50
Figure 2.8 A visually fitted cost function
The third step is to estimate the cost behavior from the plotted line. The variable cost per hour is indicated by the slope of the line, and the fixed cost is measured where the line begins at zero hours of activity. In the maintenance cost example, the fixed cost is at the Y-intercept and is $30. The variable costs can be calculated by subtracting fixed costs from total costs at some point along the line. Let’s select 40 hours of operation. The cost indi- cated by the line at 40 hours is approximately $110. Compute variable costs as follows:
Hours
M a in
te n a n ce
C o st
2010 30 40 50 60
100
200
300
400
$500
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CHAPTER 2Section 2.7 Cost Estimation
Total costs at 40 hours of operation $110
Less fixed costs 30
Variable costs $80
$80 variable costs 4 40 hours 5 $2 per hour of operation
This analysis yields the following cost function:
Y 5 $30 1 $2 (X)
When used by itself, the scattergraph and visual fit approach is limited by the judgment of the person drawing the line through the data. Even reasonable people will disagree on the slope and intercept for a given graph. However, lines that are drawn visually will tend to be fairly consistent near the center of the data. Therefore, a visual fit may be a useful way to obtain rough approximations near the center of the data. Care should be taken with estimates away from the center. The further from the central area, the larger are the errors that may occur in estimates of fixed and variable costs.
High-Low Method
Another method for obtaining rough approximations to fixed and variable costs is the high-low method. The first step is to list the observed costs for various levels of activ- ity from the highest level in the range to the lowest. This method chooses observations associated with the highest and the lowest activity levels, not the highest and lowest costs. The second step is to divide the difference in activity between the highest and the lowest levels into the difference in cost for the corresponding activity levels in order to arrive at the variable cost rate. As an example, suppose a manager for Edlin’s Pest Control Service wishes to estimate supplies costs as input for bidding on jobs. Its costs of supplies for sev- eral recent jobs, along with the hours of activity, are as follows:
Hours of activity Supplies cost
High 95 $397
90 377
87 365
82 345
78 329
75 317
66 281
58 239
Low 50 217
The difference in hours is 45 (95 2 50), and the difference in cost is $180 ($397 2 $217). The variable supplies cost per hour is computed below:
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CHAPTER 2Section 2.7 Cost Estimation
Cost at highest activity 2 Cost at lowest activity
Highest activity 2 Lowest activity 5
$180 45
5 $4 Variable cost per hour
The fixed cost is estimated by using the total cost at either the highest or lowest level and subtracting the estimated total variable cost for that level:
Total fixed cost 5 Total cost at highest activity 2 (Variable cost per unit 3 Highest activity)
or
Total fixed cost 5 Total cost at lowest activity 2 (Variable cost per unit 3 Lowest activity)
If the variable cost is calculated correctly, the fixed cost will be the same at both the high and low points. For the above illustration, the calculation of total fixed cost is as follows:
Total fixed cost 5 $397 2 ($4 Variable cost per hour 3 95 hours)
5 $397 2 $380
5 $17
or
Total fixed cost 5 $217 2 ($4 Variable cost per hour 3 50 hours)
5 $217 2 $200
5 $17
The cost function that results from the high-low method is:
Y 5 $17 1 $4 (X)
If Edlin’s manager forecasts that a particular job would require 75 hours, the bid would include an estimate of $317 for supplies cost.
Occasionally, either the highest or lowest activity or the cost associated with one of those points is obviously an outlier to the remaining data. When this happens, use the next highest or lowest observation that appears to align better with the data. Sometimes only two data points are available, so in effect, these are used as the high and low points.
The high-low method is simple and can be used in a multiplicity of situations. Its primary disadvantage is that two points from all of the observations will only produce reliable estimates of fixed and variable cost behavior if the extreme points are representative of the points in between. Otherwise, distorted results may occur.
If enough quality data are available, statistical techniques can provide more objective cost estimates than we have discussed thus far. These techniques are covered in section 2.10.
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CHAPTER 2Section 2.8 Regression and Correlation Analyses
Ethical Considerations in Estimating Costs
All cost estimation methods involve some degree of subjectivity. With account analysis, managers judgmentally classify costs. In the engineering method, the manager must often subjectively adjust data obtained from near-optimal working conditions to normal work- ing conditions. The scattergraph approach involves a subjective fitting of a line to data points. With the high-low method, one must decide whether the high and low points are representative data points.
The subjectivity inherent in cost estimation can lead to biased cost estimates. Indeed in certain instances, incentives exist for managers to bias cost estimates. In developing bud- gets or cost-based prices, managers may want to overestimate costs. In developing pro- posals for projects or programs, incentives may exist to underestimate costs. Managers must take care not to use subjectivity as an opportunity to act unethically.
Other Issues for Cost Estimation
An overriding concern with any cost estimation technique is that of extrapolating beyond the relevant range of activity. Cost behavior may change drastically once the activity falls below or rises above the relevant range. For instance, assume the relevant range of activity for Edlin’s Pest Control Service is 40 to 100 hours. We wish to predict the supplies cost for the coming time period during which 130 hours of activity are expected. Since this level is above the relevant range, the cost function we derived earlier may not be appropriate to use.
We have presumed that our cost data are suitably represented by a straight line (linear) and not by a curve. In some situations, the linear relationship may not be appropriate. Costs, for example, may not increase at a constant rate but instead may change at an increasing or a decreasing rate as the measure of activity increases. Hence, the cost data would be represented by a curve rather than a straight line. The shape of the line or curve can be revealed by plotting a sufficient amount of data for various volumes of activity.
2.8 Regression and Correlation Analyses
Regression and correlation analyses are statistical techniques that can be used to esti-mate and examine cost functions. Regression analysis fits a line to the cost and activ- ity data using the least squares method. Correlation analysis deals with the “goodness of fit” in the relationship between costs and activity as identified by the regression line. Both analyses are important in finding relationships and establishing the significance of those relationships.
Linear Regression
Linear regression is a statistical tool for describing the movement of one variable based on the movement of another variable. In determining cost behavior, it is important to know if the movement in costs is related to the movement in activity. The cost behavior
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CHAPTER 2Section 2.8 Regression and Correlation Analyses
is expressed as a line of regression. A line of regression can be fitted precisely to a large quantity of data by the least squares method. The high-low method is a rough approxi- mation computed from data taken only at the high and low points of the range, but the least squares method includes all data within the range. The line of regression is deter- mined so that the algebraic sum of the squared deviations from that line is at a minimum.
We will explain the application of linear regression through an illustration. Suppose that the buildings manager of the Atlanta Juvenile Community Center is asked to assist in budgeting for building-related costs. During the past year, electricity costs for various hours of monthly activity in a particular building have been recorded as follows:
Hours (X) Electricity cost (Y)
30 $500
50 650
20 300
10 300
60 900
50 750
40 650
60 700
30 450
10 350
40 600
20 450
The advent of the personal computer and spreadsheets has greatly simplified the applica- tion of linear regression. By entering data on a spreadsheet and using a few function com- mands, the results appear on the screen. The data for our example of electricity cost and hours of activity yield the following results with a computer spreadsheet analysis:
Regression output:
Constant 200
Std err of Y est 67.08
r2 0.886
No. of observations 12
Degrees of freedom 10
X coefficient(s) 10
The information necessary to obtain a cost function can be read from the output. The constant is the fixed cost of $200. The X coefficient is our variable cost per hour, $10. Hence, our cost function is:
Y 5 $200 1 $10X
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CHAPTER 2Section 2.8 Regression and Correlation Analyses
The number of observations is given as a check. The degrees of freedom are 10 and come from 12 2 2, where 2 is the number of parameters we are estimating (a and b). The remain- ing terms in the regression output will be discussed later.
The buildings manager can now use the derived cost function to budget the electricity cost. Suppose there is expected to be 35 hours of activity in the building during the next month. In that case, $550 will be budgeted for electricity cost [$200 1 ($10 3 35)].
Quality of the Regression
In measuring the relationship between the cost and activity, we are interested in more than just an equation for estimating cost. We also want to know the “goodness of fit” for the correlation of the regression line to the cost and activity data and the “reliability” of the cost estimates. This section discusses some of the measures available for assessing “goodness of fit” and reliability.
Goodness of Fit
The relationship between cost and activity is called correlation. At times costs may be randomly distributed and are not at all related to the cost driver used in defining the rela- tionship. This is illustrated in Figure 2.9.
Figure 2.9: No correlation
At the other extreme, the relationship may be so close that the data can almost be plotted on a line, as shown in Figure 2.10.
Hours
C o st
X
Y
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CHAPTER 2Section 2.8 Regression and Correlation Analyses
Figure 2.10: Positive correlation
Between these extremes, the degree of correlation may not be so evident. A high degree of correlation exists when the regression line explains most of the variation in the data. The above example of electricity costs has all data lying relatively close to the regression line, as shown on the graph in Figure 2.11.
Figure 2.11: Explanation of correlation
Hours
C o st
X
Y
Hours of Activity
E le
ct ri
ci ty
C o st
10 20 30 40 50 60 70
100
200
300
400
500
600
700
800
$900
Average cost
Line of regression
Unexplained variance
Variance explained by hours
of activity
Variance explained by hours of activity
Unexplained variance
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CHAPTER 2Section 2.8 Regression and Correlation Analyses
The average cost is computed in the conventional way by adding the costs and dividing by the number of data points. In this case, the average is $550. Any variance between the line of regression and the average can be explained by hours of activity. The unexplained variances are the variances between the actual costs and the line of regression. In this illustration, a large part of the variance from the average can be explained by hours of activity; only a small amount is unexplained. Hence, a good correlation exists between cost and hours.
The degree of correlation is measured by the correlation coefficient, most frequently des- ignated as r. A related measure, r2, is often used to assess the “goodness of fit” between the data and the regression line. The r2 figure can vary from 0 to 1. An r2 close to 0 would indicate that the regression line does not describe the data. That is, the regression line is nearly horizontal, and little of the variation in Y is explained by the variation in X. If the regression line is very descriptive of the data, the r2 will be close to 1. Such is the case for the above example of electricity cost estimation. Recall that the regression output showed an r2 of 0.886. Thus, 88.6 percent of the variation in electricity costs could be explained by hours of activity.
Reliability
Because a regression equation will not result in a perfect fit on the data observations, a measure of variability in the data is necessary with respect to the regression equation. The standard error of the estimate (Se) is a measure of the deviation between the actual observations of Y and the values predicted by the regression equation. In other words, Se gives an estimate of the amount by which the actual observation might differ from the estimate. For our example of electricity costs, the regression output showed an Se of 67.08, labeled as “Std Err of Y Est.”
Statistical data often form a pattern of distribution designated as a normal distribution. In a normal distribution, data can be plotted as a smooth, continuous, symmetrically bell- shaped curve with a single peak in the center of distribution. We will assume that the cost data in this section are normally distributed. A table of probabilities for a normal distribu- tion shows that approximately two-thirds of the data (more precisely, 68.27%) lie within plus and minus one standard deviation of the mean. In our example, then, approximately two-thirds of the cost observations should lie within plus and minus one standard error of the estimate of the line of regression, or between $67.08 above the line of regression and $67.08 below it. To understand how this works with our data, examine the plot in Figure 2.12 of differences between the actual cost and predicted cost.
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CHAPTER 2Section 2.8 Regression and Correlation Analyses
Figure 2.12: Graph of differences
An interrelationship exists between r2 and Se. For example, as the deviations between actual cost and predicted cost decrease, our measure for “goodness of fit” (r2) increases in amount and Se decreases in amount. That is, the higher the r
2, the lesser the deviation, the higher the correlation, and the closer actual observations snug the line of regression. The significance of this interrelationship is that the higher the r2 and the lower the standard error of the estimate, the more reliable is our estimate of b (the variable cost per hour).
Sources of Errors
The cost equation derived from a set of data has a certain degree of error due to imperfec- tions in the data, data collection, and other processing issues. These imperfections will appear in the difference between an actual cost and a cost predicted by the regression equation. Understanding the sources of errors is a step toward eliminating the impact of those errors on the results. The most common sources of errors fall into one of the follow- ing three categories: (1) major errors in the original data, (2) errors in keying data, and (3) inappropriate measures of activity.
Major errors in the original data are minimized through (1) reviewing the cutoff proce- dures that separate costs into periods, and (2) examining the data for procedural errors, such as classification of transactions into the wrong account. For errors in keying data and calculation, look for cost outliers (i.e., observations with large differences between the actual cost and the cost predicted by the regression line). If multiple cost drivers are avail- able in the data, try other cost drivers to locate one with a higher r2 value.
Hours of Activity
A ct
u a l C
o st
M in
u s
P re
d ic
te d C
o st
10 20 30 40 50 60 70 80
–100
–50
0
50
100
$150
+ $67.08
– $67.08
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CHAPTER 2Section 2.10 Multiple Regression
2.9 Control Limits
From the data given in our example, the fixed electricity cost is estimated at $200, and the variable electricity cost is estimated to vary at the rate of $10 per hour. For 30 hours of activity, the total cost is estimated at $500. This is a predicted cost, however, and it is unlikely that the actual cost will be precisely $500. Because some variation in cost can be expected, management should establish an acceptable range of tolerance. Costs that lie within the limits of variation can be accepted. Costs beyond the limits, however, should be identified and investigated.
In deciding upon an acceptable range of cost variability, management may employ the standard error of the estimate discussed earlier. Recall that a table of probabilities for a normal distribution shows that approximately two-thirds of the data lie within plus and minus one standard error of the estimate of the regression line. For the electricity cost illustration, the standard error of the estimate amounts to $67.08. At 40 hours of activity, for example, the cost is expected to lie between $532.92 and $667.08 about two-thirds of the time:
Upper limit (11 standard error)
Lower limit (21 standard error)
Regression prediction [$200 1 ($10 3 40)] $600.00 $600.00
Standard error of the estimate 167.08 2 67.08
$667.08 $532.92
{{End Table}}
If more tolerance is permitted for control purposes, the limits may be extended. For instance, a 95 percent probability (also known as a confidence interval) occurs for a range of costs of plus and minus 1.96 standard deviations. From the data given, the 95 percent confidence interval is for a cost range between $468.52 and $731.48 at 40 hours of activity [$600 plus and minus $131.48 (1.96 3 $67.08)]. Management must make a decision by balancing two alternatives:
1. A relatively narrow range of cost variation with a relatively low probability of a cost being within the zone.
2. A relatively wide range of cost variation with a relatively high probability of a cost being within the zone.
In other words, the wider the range, the fewer the costs that will be considered for inves- tigation and the higher the likelihood that waste and inefficiencies will go uncorrected.
2.10 Multiple Regression
In many situations, more than one factor will be related to cost behavior. Electricity costs in our example above may vary not only with changes in the hours of activity but also with the number of people using the building, temperature changes, or other cost drivers.
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CHAPTER 2Section 2.10 Multiple Regression
Or, telephone service costs may be a function of the basic monthly charge, in-state long distance calls, out-of-state long distance calls, and features such as call waiting or call for- warding. Insofar as possible, all factors that are related to cost behavior should be brought into the analysis. This will provide a more effective approach to predicting and control- ling costs. In simple regression only one factor is considered, but in multiple regression several factors are considered in combination. The basic form of the multiple regression model is:
Y 5 a 1 b1X1 1 b2X2 1 . . . . . 1 bmXm
The Xs represent different independent variables, and the as and bs are the coefficients. Any b is the average change in Y resulting from a one-unit change in the corresponding Xi. Output from a multiple regression analysis with three independent variables might look like the following:
Multiple regression output
Constant 995
Std err of Y est 146
R2 0.724
No. of observations 19
Degrees of freedom 15
X1 X2 X3
X coefficient(s) 0.735 0.121 0.885
Note that with multiple regression, the “goodness of fit” measure uses a capital R rather than the lower case r we saw with simple regression. Also, note that there are three X coef- ficients given for the three independent variables. These represent b1, b2, and b3.
Concerns in Using Multiple Regression
Sometimes a factor affecting the amount of cost is not (or only partially) quantitative in nature. For example, bank charges for various services may be different for senior citizens than for people under 65 years of age. A multiple regression model will have one inde- pendent variable that will have a value of 1 for senior citizens and 0 for other customers. These variables are called “dummy variables.”
Another concern in using a multiple regression model is the potential existence of a very high correlation between two or more independent variables. The variables move so closely together that the technique cannot tell them apart. We call this situation multicol- linearity. For example, direct labor hours and direct labor costs would be highly corre- lated. Multicollinearity is not an issue if we are interested only in predicting the total costs. However, when we need accurate coefficients, a definite problem exists. The coefficients (the bs) in the model are variable costs for that independent variable, and accurate coef- ficients can be used in pricing decisions and cost-volume-profit analyses.
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CHAPTER 2 Chapter Summary
Multicollinearity, when severe, will be indicated by one or more of the following symptoms:
1. A coefficient is negative when a positive one is expected. 2. A coefficient is insignificant which, in theory, should be highly significant. 3. An unreasonably high coefficient exists that does not make economic sense.
If one or more of the symptoms appear, we need to think through our theory supporting the equation. Remove one of the independent variables that is less critical to the setting. Adding two problem variables together may be a solution in some cases.
Chapter Summary
In planning profit, management considers sales volume, selling prices, variable costs, fixed costs, and the sales mix. When the contribution margin is equal to fixed costs, the company breaks even. A desired profit level can be attained when the contribution margin is equal to the fixed costs plus the desired profit before income taxes. Break-even charts or profit-volume graphs are visual representations of profits or losses that can be expected at different volume levels.
In making plans, management can review various alternatives to see how they will affect profit. For example, what will likely happen if the selling price is increased or if the vari- able cost is decreased? Often a relatively small change in variable cost per unit will have a relatively large effect on profit. Prices may be cut to increase sales volume, but this will not necessarily increase profit.
Recent developments such as increased automation tend to increase the importance of fixed costs in the total cost structure. With a relatively high contribution margin ratio and relatively high fixed costs, a small percentage increase in sales volume can be translated into a substantial increase in profits. This principle is known as operating leverage. The reciprocal of the operating leverage factor is the margin of safety.
When more than one product line is sold, the relative proportion of total sales for each product line is known as the sales mix. To maximize profit, management will try to maxi- mize the sales of product lines with higher contribution margin ratios. A break-even point and break-even units for each product line can be computed for any given sales mix.
Four methods for identifying variable and fixed cost behavior are account analysis, the engineering approach, the scattergraph and visual fit, and the high-low method. In account analysis, the analyst determines the cost behavior of a specific cost by reviewing and interpreting managerial policies with respect to the cost and by inspecting the histori- cal activity of the cost. The engineering approach uses analysis and direct observation of processes to identify the relationship between inputs and outputs and then quantifies an expected cost behavior. In the scattergraph and visual fit method, the analyst graphs each observation, with cost on the vertical axis and activity or cost driver on the horizontal axis. Then, the analyst visually and judgmentally fits a line to the data. The Y-intercept of the line is the estimate of fixed cost, and the slope of the line is the estimate of variable cost per unit. The high-low method is a simple method in which the rate of cost variability is determined from data taken only at the high and low points of a range of data.
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CHAPTER 2Problem for Review
Problem for Review
Rodbell Roofing Co. installs residential roofs throughout Alabama. In 2013, the company installed 145 roofs at an average price of $30,000 per roof. Variable costs were $18,000 per roof and total fixed costs were $1,200,000.
In 2014, fixed costs are expected to increase to $1,350,000, while variable costs should decline to $16,000 per roof due to a new source of materials and supplies. The company forecasts a 20 percent increase in number of installations for 2014; however, the average price charged to customers is not planned to change.
The 2015 forecast is for 20 more installations than in 2014 and fixed costs are ex- pected to jump by $240,000 more than in 2014. The average price and variable cost is not expected to change from 2014.
Questions:
1. For 2013, calculate the number of installations needed to break even. 2. For 2014, calculate the operating leverage factor. 3. What is expected to be the amount of profit change from 2014 to 2015?
Solution:
1. For 2014, the break-even point:
$1,200,000 (Fixed cost) 5 100 installations $12,000 (Unit contribution margin)
2. For 2014, the operating leverage factor:
Number of installations in 2014 5 1.2 3 145 5 174
Unit contribution margin in 2014 5 $30,000 2 $16,000 5 $14,000
Total contribution margin in 2014 5 174 3 $14,000 5 $2,436,000
Pretax profit in 2014 5 $2,436,000 2$1,350,000 5 $1,086,000
Operating leverage factor in 2014 5 $2,436,000 4 $1,086,000 5 2.24
3. Amount of profit change from 2014 to 2015:
Additional contribution margin 5 20 additional installations 3 $14,000 contribution margin per unit 5 $280,000
Additional fixed cost 5 $240,000
Additional profit 5 $280,000 2 $240,000 5 $40,000
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CHAPTER 2Key Terms
Key Terms
account analysis A method of estimat- ing fixed and variable costs by classifying accounts into one of these two categories.
break-even analysis An approach that examines the interaction of factors that influence the level of profits.
break-even chart A graph showing cost and revenue functions together with a break-even point.
break-even point The point where total revenues equal total costs.
confidence interval A range with a speci- fied probability of including the parameter of interest.
contribution margin ratio Contribution margin divided by sales revenues.
correlation analysis Involves the goodness of fit in the relationship between costs and activity as identified by the regression line.
cost estimation Determining expected or predicted costs.
cost–volume–profit analysis An approach that examines the interaction of factors that influence the level of profits.
dependent variable The item one is attempting to estimate or predict from one or more other variables.
engineering approach An approach that uses analysis and direct observation of processes to identify the relationship between inputs and outputs and then quantifies an expected cost behavior.
high-low method An approach for esti- mating costs that uses only two data points.
independent variable The item that is being used to predict or estimate another item.
least squares method A statistical method for estimation that derives a regression line by minimizing the squared distances between the line and the data points.
linear regression A method of regression analysis that uses a straight line to fit the data.
margin of safety The difference between sales revenue and the break-even point.
multicollinearity Existence of high cor- relation between two or more independent variables.
multiple regression Regression analysis using more than one independent variable.
normal distribution Data which can be plotted as a smooth, continuous, sym- metrically bell-shaped curve with a single peak in the center of the data.
operating leverage Using fixed costs to obtain higher percentage changes in profits as sales change.
profit-volume (p/v) graph A break-even chart that uses profits for the vertical axis.
regression analysis A statistical approach that fits a line to the cost and activity data using the least squares method.
regression line The cost function obtained from using regression analysis.
sales mix A combination of sales propor- tions from the various products.
scattergraph and visual fit An approach to estimating fixed and variable costs by visually obtaining a line to fit data points.
standard error of the estimate A measure of the deviation between the actual obser- vations of the dependent variable and the values predicted by the regression equation.
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CHAPTER 2Exercises
Questions for Review and Discussion
1. Identify the interrelated factors that are important to profit planning. 2. If the total fixed cost and the contribution margin per unit of product are given,
explain how to compute the number of units that must be sold to break even. 3. If the total fixed cost and the percentage of the contribution margin to sales rev-
enue are given, explain how to compute the sales revenue at the break-even point. 4. Can two break-even points exist? If so, describe how the revenue and cost lines
would be drawn on the break-even chart. 5. Is it possible to compute the number of units that must be sold to earn a certain
profit after income tax? Explain. 6. What does the slope of the P/V graph represent? 7. Define margin of safety. How is a margin of safety related to operating leverage? 8. Why is cost estimation so important? 9. Describe the two major steps involved in the account analysis method of cost
estimation. 10. Describe the steps for preparing an estimate of fixed and variable costs using the
scattergraph and visual fit method. 11. Describe the high-low method of cost estimation. 12. What is r2? What range of values can it take? 13. In a normal distribution of data, what proportion of the data should lie within
plus and minus one standard deviation from the mean? 14. What is multicollinearity? How does it pose problems in multiple regression
analysis?
Exercises
2-1. Break-even Point and Change in Fixed Costs. Spira Spirits Store sells each bottle of a particular whiskey for $50. The variable costs are 60 percent of revenue. The fixed costs amounted to $120,000 per year. Last year the store sold 7,500 of these bottles. During the current year, Spira plans to sell 8,700 bottles, and fixed costs will increase by $36,000.
Questions:
1. What was the break-even point last year? 2. This year, the break-even point will increase by how many bottles?
2-2. Break-Even Analysis and Cost Changes. In Houston Mill Company, the follow- ing costs are known for 10,000 units:
Selling price $13 per unit
Variable costs 5 per unit
Fixed costs 6 per unit
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CHAPTER 2Exercises
The operations manager, Wendy Solon, is thinking of buying new equipment to automate certain operations. The new equipment will add $20,000 to fixed costs and cut variable costs by $2 per unit.
Questions:
1. What is the current break-even point in units? 2. By what number of units will the break-even point change and in what direction?
Why would you or would you not add this new equipment?
2-3. Changes to Break-Even Variables. Brasch Company sees its profit picture chang- ing. Variable cost will increase from $4 per unit to $4.50. Competitive pressures will allow prices to increase only by $0.30 per unit to $8. Fixed costs ($200,000) and sales volume (60,000 units) likely will remain constant.
Questions:
1. What will likely happen to the break-even point? 2. To maintain the same profit level, fixed costs will have to change to what
amount? 3. To maintain the same profit level, volume will have to change to what amount?
2-4. Break-Even Point and Variable Cost Increase. Lydia Perl, the owner of Schloss Hardware Supplies in Frankfurt, Germany, is concerned about increased costs to purchase a hardware item that is sold by the company through one of its retail outlets. This year the variable cost per unit of product was £28. Next year the variable cost is expected to increase to £32 per unit. The selling price per unit, however, cannot be increased and will remain at £39 per unit. The fixed costs amount to £90,000.
Questions:
1. What was the break-even point in units of product this year? 2. What was the break-even point in revenues for this year? 3. How many units of product must be sold next year to break even? 4. How much revenue will generate break-even volume for next year?
2-5. Effects of Changes in Volume and Costs. Marnie’s Boutique sells umbrellas that have a contribution margin ratio of 35 percent of $440,000 annual sales (40,000 units). Annual fixed expenses are $95,000.
Questions:
1. Calculate the change in net income if sales were to increase by 850 units. 2. The store manager, Laura Romain, believes that if the advertising budget were
increased by $18,000, annual sales would increase by $75,000. Calculate the addi- tional net income (or loss) if the advertising budget is increased.
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CHAPTER 2Exercises
3. Laura Romain believes that the present selling price should be cut by 15 percent and the advertising budget should be raised by $12,000. She predicts that these changes would boost unit sales by 30 percent. Calculate the predicted additional net income (or loss) if these changes are implemented.
2-6. Break-Even Point and Profits. Sheila Hershey has noticed a demand for small tables for personal computers and printers. Retail office furniture outlets are charging from $400 to $600 for a table. Sheila believes that she can manufacture and sell an attractive small table that will serve the purpose for $210. The cost per table of materials, labor, and variable overhead is estimated at $110. The fixed costs consisting of rent, insurance, taxes, and depreciation are estimated at $25,000 for the year. She already has orders for 180 tables and has established contacts that should result in the sale of 150 additional tables.
Questions:
1. How many tables must Sheila make and sell to break even? 2. How much profit can be made from the expected production and sale of 330 tables? 3. How many tables are needed for a profit objective of $11,000?
2-7. Target Profit and Taxes. Fiszon, Inc. had the following results for October:
Fixed Variable Total
Sales ($200 per unit) $600,000
Cost of sales $180,000 $360,000 (540,000)
Net income before taxes $ 60,000
Taxes (40%) 24,000
Net income after taxes $ 36,000
Question:
For net income to be $60,000 after taxes in November, what will Fiszon’s sales in units need to be?
2-8. Computation of Fixed Cost and Price. Matlin’s Ranch offers horseback riding lessons for children. “Wild Bill” Matlin has figured that break-even revenues are $560,000, based on a contribution margin ratio of 35 percent and variable costs per rider of $6.50.
Questions:
1. Compute the total fixed costs. 2. Compute the fee charged per rider.
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CHAPTER 2Exercises
2-9. Contribution Margin Ratio. In 2013, Simon’s Barber Shop had a contribution margin ratio of 22 percent. In 2014, fixed costs will be $41,000, the same as in 2013. Revenues are predicted to be $250,000, a 25 percent increase over 2013. Simon’s wishes to increase pretax profit by $8,000 in 2014.
Question:
Determine the contribution margin ratio needed in 2014 for Simon’s Barber Shop to achieve its goal.
2-10. Multiple Product Analysis. The Lane Division of Stefan Products, Inc. manufac- tures and sells two grades of canvas. The contribution margin per roll of Lite- Weight canvas is $25, and the contribution margin per roll of Heavy-Duty canvas is $75. Last year, this division manufactured and sold the same amount of each grade of canvas. The fixed costs were $675,000, and the profit before income taxes was $540,000.
During the current year, 14,000 rolls of Lite-Weight canvas were sold, and 6,000 rolls of Heavy-Duty canvas were sold. The contribution margin per roll for each line remained the same; in addition, the fixed cost remained the same.
Questions:
1. How many rolls of each grade of canvas were sold last year? 2. Assuming the same sales mix experienced in the current year, compute the num-
ber of units of each grade of canvas that should have been sold during the cur- rent year to earn the $540,000 profit that was earned last year.
2-11. Cost Segregation by High-Low Method. Betty Grus sells various ceramics and crafts at flea markets in the area. She uses a motor home for transportation and lodging. She recognizes that travel costs with the motor home are relatively high and would like to estimate costs so that she can decide how far she can travel and still operate at a profit.
Records from one round trip of 150 miles show that the total cost was $320. On another round trip of 340 miles, the total cost was $472. A local round trip of 50 miles cost $240. She is convinced that the time and cost for trips of more than 300 miles are too high unless the sales potential is very high.
Question:
Calculate the variable cost per mile and the fixed cost per trip by the high-low method.
2-12. Cost Prediction with Account Analysis. Muhammed Alley, a bowling alley lo- cated in Louisville, had the following costs during a recent month:
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CHAPTER 2Exercises
Expenses Total amount Classification
Wages $44,500 Variable
Supplies 9,200 Variable
Rent 12,000 Fixed
Advertising 3,700 Fixed
Utilities 20,000 Variable
Depreciation 11,300 Fixed
During the month, 1,593 customers patronized the bowling alley. Next month, 1,780 cus- tomers are expected to come.
Question:
Estimate Muhammed Alley’s total cost for next month.
2-13. Visual Fit and the Extremes. The cost of utilities at Harpaz Supply Store varies according to hours of operation, but a portion of the cost is fixed. Hour and cost data for several months are given as follows:
Hours Total Cost
100 $ 800
200 700
300 800
400 900
500 1,000
600 1,100
700 1,200
800 1,300
900 1,600
Questions:
1. Fit a line to the data by visual fit. 2. Using your line, compute the variable cost per hour and the fixed cost per month. 3. Determine the variable cost per hour and the fixed cost per month by the high-
low method. 4. In which of the foregoing two variable costs per hour and two fixed cost numbers
do you have more confidence for making business decisions? Explain.
2-14. Cost Estimation and Break-Even Point. Water is supplied to Lake Tillem Town- ship by pumping water from a lake to a storage tank at the highest elevation in town, from which it then flows to the customers by gravity. The town council
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CHAPTER 2Exercises
notes that the costs to pump water vary to some extent by the number of gallons pumped, but fixed costs are also included in the pumping costs. A record of gal- lons consumed per month and total pumping cost per month is as follows:
Gallons consumed (000)
Pumping cost
Gallons consumed (000)
Pumping cost
1,750 $29,100 1,800 $29,700
1,900 30,800 2,300 35,900
2,150 34,000 2,000 31,800
2,050 32,600 1,500 25,500
In addition to pumping costs, 1.1 cents in variable costs and $75,000 in fixed costs are incurred to supply water to the residents. Lake Tillem charges its residents 4.6 cents per gallon consumed.
Questions:
1. Use the high-low method to obtain a cost function for Lake Tillem Township’s pumping costs.
2. At what level of water consumption would Lake Tillem Township break even?
2-15. Confidence Interval. The Greenbelt Fire Department wants to examine the costs of operating their fire engines (fuel, maintenance, etc.). Seventeen months of data were obtained on these costs as well as on the number of trips made (actual fires, practice drills, and false alarms). A regression of costs on the number of trips yielded the following results:
Mean of X 22
Constant 11,000
r2 .59
X coefficient 950
Se 3,125
Question:
Develop a 95% confidence interval for the predicted cost if 18 trips are made.
2-16. “Goodness of Fit” and Confidence Interval. Dan White manages a dormitory cafeteria at Tradition University. He has analyzed the actual costs of serving din- ners on Tuesday and Thursday nights for the past three years. For a midterm exam night, attendance is lower than normal. Tonight, he estimates that costs will be $830 for the 90 students expected. But with about 95 percent confidence, he thinks that costs will not be higher than $880 nor lower than $780 for that number of students eating. On a more normal night, about 110 students eat with a cost estimate of $970.
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CHAPTER 2Problems
Questions:
1. What cost function underlies Dan White’s cost estimates? 2. How did he get the $880 and $780 estimates? (Describe rather than calculate.) 3. Dan White tells us that the r2 using number of students eating is .75 and that the
r2 using the total weight of the students eating is .88. What does this say about his approach to budgeting dinner costs above?
Problems
2-17. CVP with Changes in Prices and Costs. Data with respect to a basic product line sold by Carson Music Stores are as follows:
Selling price per unit $50
Cost per unit 30
Contribution margin per unit $20
The fixed costs for the year are $360,000. The income tax rate is 40 percent.
Questions:
1. Determine the number of units that must be sold in order to break even. 2. If a profit before income taxes of $270,000 is to be earned, how many units of
product must be sold? 3. If a profit after income taxes of $180,000 is to be earned, how many units of prod-
uct must be sold? 4. If the selling price per unit is reduced by 10 percent, how many units must be
sold to earn a profit of $8 per unit before income taxes? 5. Assume that the selling price remains at $50. How many units must be sold
to earn a profit of $8 per unit before income taxes if the variable cost per unit increases by 10 percent?
6. Why does a 10 percent decrease in the selling price have more effect on the con- tribution margin than a 10 percent increase in the variable unit cost?
2-18. Determining a Selling Price. Eileen Stewart and her brother, Ralph, would like to make extra money when they have time away from their studies at Pepper Pike University. They are skilled in carpentry and plan to build and sell rustic lawn chairs. Estimates of the cost to make and sell each chair are as follows:
Lumber and other materials ..................................................... $30
Labor (wages to student helpers) .............................................. 15
Commission to stores selling the chairs ........ 8% of selling price
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CHAPTER 2Problems
Radio and direct mail advertising is estimated at $5,000. A pickup truck to transport the chairs can be rented for $1,000. Eileen and Ralph each plan to earn a profit of $4,500 for their efforts. This amount was calculated by considering what they could earn if they used their time in another way.
Eileen believes that 800 units can be made and sold. Ralph is more optimistic and believes that 900 units can be made and sold. Both agree that the price must be less than the com- mercial price of $98 per chair.
Questions:
1. Compute a selling price to obtain the desired profit if 800 chairs are sold. 2. Compute a selling price to obtain the desired profit if 900 chairs are sold. 3. Assume that the selling price is based on the sale of 800 chairs but that 900 chairs
are actually sold. How much additional profit will each of them make? 4. Assume that the selling price is based on the sale of 800 chairs but that only 700
chairs are actually sold. Will Eileen and Ralph achieve their profit objectives? Explain.
2-19. Break-Even Comparisons. Dan and Elaine Miller, owners of Evening Star Motel, want to know potential maximum profits and the break-even occupancy for the operation. Evening Star Motel is a low-cost operation to attract business people and families traveling on low budgets. A study of costs shows a difference between summer and winter operations. Swimming pool maintenance adds to summer costs while utilities (heat and light) add to winter costs. Variable costs have been determined on the basis of cost per room occupied per day and are as follows:
Cost per room
Laundry ................................................... $ 1.90
Heat and light (summer) ....................... 1.10
Heat and light (winter) .......................... 2.20
Repairs ..................................................... .75
Supplies ................................................... 1.60
Taxes and insurance ............................... 3.60
Maintenance ............................................ 1.50
Pool maintenance (summer only) ........ .60
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CHAPTER 2Problems
Fixed costs per month have been estimated as follows:
Housekeeping .................................................................... $14,000
Management ...................................................................... 17,000
Desk service ....................................................................... 2,700
Repairs and maintenance ................................................. 1,600
Taxes .................................................................................... 1,430
Insurance ............................................................................ 1,120
Heat and light .................................................................... 1,000
Depreciation—motel ......................................................... 26,000
Depreciation—furnishings............................................... 12,500
Pool maintenance and personnel (summer only) ........ 1,800
Evening Star has 300 rooms and charges $40 per room per night. Summer is relatively short and is defined as June, July, and August. All other months are designated as winter months. A month consists of 30 days for making calculations. Maximum capacity for a month would be 9,000 room days (300 rooms x 30 days).
Questions:
1. Compare the maximum operating net incomes that can be expected for a sum- mer month versus a winter month.
2. How do the break-even points (in terms of room days) compare for summer ver- sus winter? Also, state the break-even points as percentages of total capacity.
3. Based on advance reservations and normal expectations, Evening Star Motel plans for 5,000 room days in August. Determine the estimated operating income for August. Also, determine the percentage of capacity expected for August.
2-20. CVP with a Sales Mix. Sharon Arts Festival rents four types of booths to exhibi- tors, providing them with space, tables, chairs, and some pre-festival publicity. Stan Harris, the festival organizer, has indicated that the booth rental fee is three times the amount of variable costs associated with the particular type of booth. Harris expects that the festival will incur fixed costs of $9,200. The number of booths which are expected to be rented this year, as well as the related variable costs, are:
Booth size Variable cost
Per booth Number of
booths
8’ x 10’ $25 15
10’ x 12’ 28 10
10’ x 15’ 30 20
15’ x 15’ 35 5
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CHAPTER 2Problems
Question:
Assuming that booths are rented according to the expected mix, determine the number of each type of booth that needs to be rented for the festival to break even.
2-21. Account Analysis. The following is a partial list of account titles appearing in the chart of accounts for Lee Caplan Industries:
(a) Direct Materials (b) Supervisory Salaries—Factory (c) Heat, Light, and Power—Factory (d) Depreciation on the Building (e) Depreciation on Equipment and Machinery (units-of-production method) (f) Janitorial Labor (g) Repair and Maintenance Supplies (h) Pension Costs (as a percentage of employee wages and salaries) (i) FICA Tax Expense (employer’s share) (j) Insurance on Property (k) Sales Commission (l) Travel Expenses—Sales (m) Telephone Expenses—General and Administrative (n) Magazine Advertising (o) Bad Debt Expense (p) Photocopying Expense (q) Audit Fees (r) Dues and Subscriptions (s) Depreciation on Furniture and Fixtures (double-declining-balance method) (t) Group Medical and Dental Insurance Expense
Questions:
1. Discuss each account title in terms of whether the account represents a variable, fixed, or semivariable cost.
2. For accounts designated as variable or semivariable, indicate the most likely cost driver with which the cost varies.
3. Explain the problems associated with using the account analysis approach to establish cost behavior patterns.
2-22. High-Low and Visual Fit. Kathleen Otwell, an insurance claims adjuster for Shoe- nfield Casualty Company, notes that the cost to process a claim has both fixed and variable components. She believes that she can estimate costs more accurately if she can separate the costs into their variable and fixed components. The monthly record of the number of claims and the costs for the past year is as follows:
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CHAPTER 2Problems
Month
Number
of Claims Cost
January 120 $20,600
February 134 20,670
March 142 20,710
April 156 20,780
May 160 20,800
June 220 21,100
July 250 21,250
August 330 21,650
September 114 20,570
October 280 21,400
November 274 21,370
December 230 21,150
Questions:
1. Estimate the variable costs per claim and the fixed costs per month by the high- low method.
2. Estimate the variable costs per claim and the fixed costs per month by the scatter- graph and visual fit method.
3. Explain the differences between the two methods in the variable costs per claim and the fixed costs per month.
2-23. Account Analysis and Cost Prediction. A listing of accounts prepared by Cenker Video Arcade revealed the following data for October, during which time 9,500 video games were played:
Account Costs
Rent $1,600
Depreciation on equipment 70
Wages for part-time help $1,650
Insurance 120
Prizes 200
Supplies 85
Manager’s salary 2,000
Electricity 500
Telephone 100
Heat 250
Advertising 150
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CHAPTER 2Problems
Question:
Estimate the cost for November, when 11,000 video games are expected to be played.
2-24. Control Limits. Harry Czinn, supervisor of the Maintenance Department at Severn Airport, has estimated that the department’s power cost varies at the rate of $0.80 per hour and that the fixed cost for the month is $500. The standard er- ror of the estimate from the line of regression is $60. Czinn investigates the cost of any month that is more or less than one standard deviation from the line of regression. Actual monthly hours and costs for the last year are as follows:
Month Hours Cost
January 600 $ 980
February 550 970
March 600 960
April 650 1,050
May 550 940
June 500 980
July 700 1,180
August 800 1,150
September 750 1,050
October 900 1,330
November 850 1,180
December 450 900
Questions:
1. Using the formula Y 5 a 1 b (X), what should be the cost for each month? 2. Calculate the difference between the actual cost and the predicted cost for each
month. 3. Plot the differences in part (2) on a graph, and draw lines that represent the plus
or minus one standard error of the estimate. 4. For which month or months should the cost be investigated? Why?
2-25. Varying Control Limits. For the last five years, the management of Shulman Fas- teners of Singapore has followed the practice of investigating variations if a cost differs by more than one standard error of estimate from the line of regression. One standard error of estimate is S$205 (Singapore dollars).
A new supervisor, hired to manage the machining operation, asks, “Did you ever con- sider that you may be overdoing it by investigating every cost that is over one standard error from the line of regression? After all, you still have a probability of about 1 in 3
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CHAPTER 2Problems
that the variation will be random. Then you have gone to a lot of bother for nothing. For example, last March in this operation you investigated the cost of lubrication and found nothing wrong.”
The supervisor of the fabrication operation replies, “I’ll grant that we may whip a few dead horses, but your idea of investigating anything over 1.96 standard error would have missed a very important variance for one month that was brought under control.”
The cost of lubrication is estimated to vary at S$6 per hour in the machining operation with a fixed cost of S$3,500 each month. The actual costs for last year that the supervisors were discussing are given as follows:
Month Hours Cost
January 1,200 S$10,850
February 1,400 12,000
March 1,000 9,800
April 1,100 10,000
May 1,000 9,650
June 600 7,600
Month Hours Cost
July 500 S$6,350
August 700 7,700
September 900 9,100
October 1,000 9,850
November 1,200 10,600
December 1,300 11,450
Questions:
1. Calculate the difference between the actual cost and the predicted cost for each month.
2. Identify the months which were investigated with the control limit set at one standard error of estimate from the line of regression.
3. Identify the months which would have been investigated if the control limit had been set at 1.96 standard error of estimate from the line of regression.
4. The supervisor of fabrication operation has stated that one very important month would have been missed using the rule suggested by the supervisor of the ma- chining operation. Which month would have been missed? What was the vari- ance for that month?
2-26. Linear Regression. Larry Beck has a night job at a ballpark. He manages the late night clean-up crew at Nantahalla Field. As a student of the game and of garbage, he has selected a sample of 2013 games for study. The attendance fig- ures and clean-up costs, as well as output from a regression analysis, are given as follows.
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CHAPTER 2Case: Mendel Paper Company
Game date Attendance Clean-up costs
April 28 25,216 $5,677
May 10 32,566 6,381
May 11 19,945 4,219
June 6 26,009 5,422
June 10 39,856 6,722
June 31 34,561 6,106
July 15 19,723 4,846
July 19 25,924 5,533
July 25 42,119 6,311
July 29 38,923 6,722
August 21 26,912 4,909
August 23 21,902 4,771
September 5 36,205 6,508
September 7 26,931 5,456
September 22 19,636 4,323
Regression output:
Constant 2642.63
Std err of Y est 339.05
r2 0.85125
No. of observations 15
Degrees of freedom 13
X coefficient(s) 0.10143
Questions:
1. What amount of clean-up cost would Beck budget for the 2014 opening day game, which is expected to have an estimated crowd of 50,000 fans?
2. How would you caution Beck about using the estimate obtained in part (1)?
Case: Mendel Paper Company Mendel Paper Company produces four basic paper product lines at one of its plants: com- puter paper, napkins, place mats, and poster board. Materials and operations vary accord- ing to the line of product. The market has been relatively good. The demand for napkins and place mats has increased with more people eating out, and the demand for the other lines has been growing steadily. (continued)
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CHAPTER 2Case: Mendel Paper Company
Case: Mendel Paper Company (continued) The plant superintendent, Marlene Herbert, while pleased with the prospects for increased sales, is concerned about costs:
“We hear talk about a paperless office, but I haven’t seen it yet. The computers, if anything, have increased the market for paper. Our big problem now is the high fixed cost of production. As we have automated our operation, we have experienced increases in fixed overhead and even variable overhead. And, we will have to add more equipment since it appears that we need even more plant capacity. We are operating over our normal capacity as it is.
“The place mat market concerns me. We may have to discontinue printing the mats. Our specialty printing is driving up the variable overhead to the point where we may not find it profitable to continue with that line at all.”
Cost and price data for the next fiscal quarter are as follows:
Computer paper Napkins Place mats Poster board
Estimated sales volume in units
30,000 120,000 45,000 80,000
Selling prices ............... $14.00 $7.00 $12.00 $8.50
Materials costs ........... 6.00 4.50 3.60 2.50
Variable overhead includes the cost of hourly labor and the variable cost of equipment operation. The fixed plant overhead is estimated at $420,000 for the quarter. Direct labor, to a large extent, is salaried; the cost is included as a part of fixed plant overhead. The superintendent’s concern about the eventual need for more capacity is based on increases in production that may reach and exceed the practical capacity of 60,000 machine hours.
In addition to the fixed plant overhead, the plant incurs fixed selling and administrative expenses per quarter of $118,000.
“I share your concern about increasing fixed costs,” the supervisor of plant operations replies. “We are still operating with about the same number of people we had when we didn’t have this sophisticated equipment. In reviewing our needs and costs, it appears to me that we could cut fixed plant overhead to $378,000 a quarter without doing any vio- lence to our operation. This would be a big help.”
“You may be right,” Herbert responds. “We forget that we have more productive power than we once had, and we may as well take advantage of it. Suppose we get some hard figures that show where the cost reductions will be made.” (continued)
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CHAPTER 2Case: Mendel Paper Company
Case: Mendel Paper Company (continued) Data with respect to production per machine hour and the variable cost per hour of pro- ducing each of the products are given as follows:
Computer paper Napkins Place mats Poster board
Units per hour 6 10 5 4
Variable overhead per hour
$9.00 $6.00 $12.00 $8.00
“I hate to spoil things,” the vice-president of purchasing announces. “But the cost of our materials for computer stock is now up to $7. Just got a call about that this morning. Also, place mat materials will be up to $4 a unit.”
“On the bright side,” the vice-president of sales reports, “we have firm orders for 35,000 cartons of computer paper, not 30,000 as we originally figured.”
Questions:
1. From all original estimates given, prepare estimated contribution margins by product line for the next fiscal quarter. Also, show the contribution margins per unit.
2. Prepare contribution margins as in part (1) with all revisions included. 3. For the original estimates, compute each of the following: (a) Break-even point for the given sales mix. (b) Margin of safety for the estimated sales volume. 4. For the revised estimates, compute each of the following: (a) Break-even point for the given sales mix. (b) Margin of safety for the estimated sales volume. 5. Comment on Herbert’s concern about the variable cost of the place mats.
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