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6 Cost Estimation and Cost-Volume-Profit Relationships

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Learning Objectives

After studying Chapter 6, 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 equation 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 understand 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 scattergraph approach, and the high-low method.

• Estimate cost functions using regression analysis, construct control limits, and apply multiple regression.

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Chapter Outline

6.1 Significance of Cost Behavior to Decision Making and Control Decision Making Planning and Control Trends in Fixed Costs

6.2 Cost-Volume-Profit (CVP) Analysis Basics of CVP Analysis A Desired Pretax Profit A Desired Aftertax Profit

6.3 Graphical Analysis The Break-Even Chart Curvature of Revenue and Cost Lines The Profit-Volume Graph

6.4 Analysis of Changes in CVP Variables Sales Volume Variable Costs Price Policy Fixed Costs Ethical Considerations When Changing CVP Variables

6.5 Measures of Relationship Between Operating Levels and Break- Even Points Operating Leverage Margin of Safety

6.6 The Sales Mix

6.7 Cost Estimation Account Analysis Engineering Approach Scattergraph and Visual Fit High-Low Method Ethical Considerations in Estimating Costs Other Issues for Cost Estimation

6.8 Regression and Correlation Analyses Linear Regression Control Limits Multiple Regression

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Section 6.1 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 15 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 of operating the cruise ships 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 shrinking 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 profitable. 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.

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 vol- ume for profit-planning purposes. These interactions are covered later in this chapter.

6.1 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 incremental 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 or adding another salaried employee.

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Section 6.1 Significance of Cost Behavior to Decision Making and Control

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 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 misleading price. For instance, if fixed costs total $2,000 and currently 2,000 units are being produced, then the fixed cost per unit is $1. However, if production is cut in half, then the fixed cost per unit increases to $2 ($2,000 / 1,000). Managers must know which costs are fixed, as well as antici- pated 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/output relation- ships identify changes in resources necessary to respond to the change in activity. 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 procedures can be trig- gered 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 investigate causes and elimi- nate 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 decision 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 deci- sion is the first point of control. After construction, control is implemented in using the build- ing 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. Many companies have laid off hourly employees, which is a variable cost, and replaced them with machinery that creates depreciation and other fixed costs. Thus, fixed costs are becoming a more significant part of total costs. Costs associated

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Section 6.2 Cost-Volume-Profit (CVP) Analysis

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 who were once hourly wage earners are now becoming salaried. With the use of more automated equip- ment, 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 monitor 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.

Contemporary Practice 6.1: Separating Variable and Fixed Costs

A survey of 148 German companies and 130 U.S. companies found that 46% of U.S. companies separate variable and fixed costs for each cost center, as compared to 36% of German companies.

Source: Krumwiede, K., & Suessmair, A. (2007, June). Getting down to specifics on RCA. Strategic Finance, 51–55.

6.2 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 (i.e., total revenues equal total costs) 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 specific

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 determine 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:

(Unit selling price)(Sales volume) – (Unit variable cost)(Sales volume) – Total fixed cost =

Pretax profit

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Section 6.2 Cost-Volume-Profit (CVP) Analysis

The basic CVP equation is merely a condensed income statement, in equation form, where total variable costs (Unit variable cost × Sales volume in units) and total fixed costs are deducted from total sales revenues (Unit selling price × Sales volume in units) to arrive at pretax profit. This equation, as well as other variations that will be discussed later, appears as Equation 1 in Figure 6.1.

Figure 6.1: Equations for CVP analysis

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 ratio (p – v) / p

= Aftertax profit

= Pretax profit

= Income tax rate

= Sales volume in units

CM%

Where:

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 operating 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 margin to total sales revenue (or, equivalently, the ratio of the contribution margin per unit to the selling price).

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Section 6.2 Cost-Volume-Profit (CVP) Analysis

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 (i.e., the normal range of expected 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. 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 discus- sion may appear to presume that CVP analysis is applicable only to companies that sell physi- cal 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 Equation 1 and by creating its derivations, as shown in Figure 6.1.

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 volume 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

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Section 6.2 Cost-Volume-Profit (CVP) Analysis

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 Equa- tions 3 and 4 in Figure 6.1. For Felsen Electronics, the break-even point is determined as follows:

Break-even point in units = Total fixed cost / Contribution margin per unit

= $100,000 / $10 per unit

= 10,000 units

A break-even point measured in sales dollars can be computed by directly using Equation 4 of Figure 6.1, as follows:

Total fixed cost / Contribution margin ratio = Break-even point in sales dollars

$100,000 / 0.40 = $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 number into the basic CVP equation. This yields the following equations, which appear as Equations 5 and 6 in Figure 6.1:

(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 Presi- dent, 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 + $200,000) / $10 = 30,000 calculators

($100,000 + $200,000) / 0.40 = $750,000

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Section 6.3 Graphical Analysis

A Desired Aftertax Profit The profit objective may be stated as a net income after income taxes. Rather than changing 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 Equations 2, 7, and 8 in Figure 6.1. Equation 7 works as follows:

Target aftertax profit in units =

(Fixed costs + (Aftertax profit / (1 – Tax rate))) / Contribution margin per unit

Equation 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%. We use the above equations to obtain the following volume and sales:

($100,000 + ($105,000 / (1 – 0.30))) / $10 = 25,000 calculators

($100,000 + ($105,000 / (1 – 0.30))) / 0.40 = $625,000

Contemporary Practice 6.2: Break-Even Attendance per Game 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).

6.3 Graphical Analysis Total sales dollars and total costs at different sales volumes can be estimated and plotted on a graph. The information shown on the graph can also be given in conventional 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.

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Section 6.3 Graphical Analysis

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 quanti- ties 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 6.2, a break-even chart has been prepared for Felsen Electronics using the following data associated with sales levels between 5,000 and 30,000 calculators.

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

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 depending 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 volume increases. For exam- ple, volume increases might cause a jump in supervision, equipment, and space costs. There- fore, it may be possible to have two break-even points, as shown in Figure 6.3.

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Section 6.3 Graphical Analysis

Figure 6.2: Break-even chart

Number of Calculators (in Thousands)

Variable cost

Fixed cost

5 10 15 20 25 30

100

200

300

400

500

600

700

$800

D ol

la rs

( in

T ho

us an

ds )

Loss area

Break-even point

Profit area

Total revenue line

Total cost line

Units of Product Sold

D ol

la rs

Break-even point

Tot al c

os t li

To tal

re ve

nu e li

Figure 6.3: Break-even chart with two break-even points

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Section 6.3 Graphical Analysis

The Profit-Volume Graph A profit-volume (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 6.2 have also been used in preparing the P/V graph shown in Figure 6.4. In general, profits and losses appear on the vertical scale; units of product, sales revenue, and/or percentages 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 hori- zontal line is stated as units of product and is the contribution margin ratio if the horizontal line is stated as sales revenue.

The break-even point is the point where the profit line intersects the horizontal line. Dollars of profit are measured on the vertical scale above the line, and dollars of loss are measured 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.

Number of Calculators (in Thousands)

Revenue

(Thousands of Dollars)

$125 $250 $375 $500 $625 $750

Lo ss

ro fit

(T ho

us an

ds o

f D ol

la rs

)

Break-even point

Profit line

5 10 15 20 25 30

100

$200

200

Figure 6.4: Profit-volume graph

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Section 6.4 Analysis of Changes in CVP Variables

6.4 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 sell- ing price, and total fixed cost are discussed in the following paragraphs. In all these cases, the starting point for analysis is the CVP equations in Figure 6.1.

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.

For an illustration, consider Adrian Grant & Company, which 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 ÷ $.30 = 600,000 units

For every unit sold in excess of 600,000, a $0.30 profit before tax is earned. In such a situation, 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 recap- tured 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%, 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 × Change in sales volume = 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 × 500 units).

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Section 6.4 Analysis of Changes in CVP Variables

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 manu- facturing, it may be possible to save on materials cost by using cheaper materials 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 struc- ture, the low-cost producer will become profitable faster as sales volume increases.

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 producer. 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 6.5.

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Section 6.4 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 vulnerable to losses.

Price Policy One of the ways to improve profit is to get more sales volume; to stimulate sales volume, man- agement may decide to reduce prices. Bear in mind, however, that if demand for the product is inelastic (i.e., not affected by price changes) 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 or increases in variable costs will decrease the unit contribution margin. On a unit basis, price decreases may appear to be insignificant, but when the unit differential is multi- plied 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

Figure 6.5: Effects of variable cost changes

Units of Product (in Thousands)

D ol

la rs

( in

T ho

us an

ds )

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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Section 6.4 Analysis of Changes in CVP Variables

stimulating sales volume; to do so, it plans to reduce the unit price by 10%. 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 rev- enues 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% price cut, unit sales must increase by 25%:

(75,000 – 60,000) ÷ 60,000 = 25% increase

Similarly, revenue must increase to $337,500 (75,000 units × $4.50 per unit). This represents a 12.5% increase, as follows:

($337,500 – $300,000) ÷ $300,000 = 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 6.6.

Units of Product (in Thousands)

D ol

la rs

( in

T ho

us an

ds )

Fixed cost line

Total revenue line, price $4.50

To ta

l r ev

en ue

lin e,

p ric

e $5

30 60 90 120 150

100

200

300

400

$500

Tot al c

ost lin

Figure 6.6: Effects of price reduction

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Section 6.4 Analysis of Changes in CVP Variables

The present pretax income of $100,000 can still be earned if 125,000 units are sold. This is obtained using Equation 5 of Figure 6.1:

($150,000 + $100,000) ÷ $2 = 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 following general relationship holds:

(Contribution margin per unit × Change in sales volume) – Change in fixed cost =

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 × 2,500) – $30,000].

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 6.7 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 originally. To break even or maintain the same profit as before, Felsen Electronics must sell 3,000 more calculators.

Units of Product (in Thousands)

D ol

la rs

( in

T ho

us an

ds )

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

Figure 6.7: Effects of an increase in fixed cost

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Section 6.5 Measures of Relationship Between Operating Levels and Break-Even Points

Decreases in fixed cost would cause the total cost line to shift downward. The total contribu- tion 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 com- panies have downsized in recent years by eliminating jobs and even closing plants. Managers must be careful not to cut costs in an unethical manner, such as the use of inferior quality materials in a product, which may risk injury to the product users. 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.”

6.5 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 oper- ating 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 operat- ing well away from the break-even point, management will be more aggressive 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 rela- tively 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) = 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.

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Section 6.5 Measures of Relationship Between Operating Levels and Break-Even Points

Suppose Felsen Electronics is currently selling 15,000 calculators. With its unit contribution margin of $10 and fixed costs of $100,000, its operating leverage factor is 3, computed as follows:

At a sales volume of 15,000, if sales volume can be increased by an additional 10%, profit can be increased by 30%:

Increase in sales volume × Operating leverage factor = Increase in pretax profit

10% × 3 = 30%

A 10% increase in sales volume will increase sales from 15,000 units to 16,500 units. Operat- ing leverage suggests that the pretax profit should be $65,000 ($50,000 × 1.3). Indeed, when we deduct the $100,000 fixed cost from the new contribution margin of $165,000 (16,500 × $10), we obtain a pretax profit of $65,000.

A company with high fixed costs will have to sell in large volume to recover the fixed costs. How- ever, 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 equations for calculating margin of safety are:

Margin of safety in dollars = Actual (or expected) sales – Break-even sales

Margin of safety in percentage form = (Margin of safety in dollars) / (Actual (or expected) sales)

For our purposes, margin of safety is the percentage form. Therefore, unless otherwise speci- fied, 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:

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 lever- age factor.

(15,000)(10)

(15,000)(10) – 100,000

150,000

50,000 = 3=

[(15,000)($25) – $250,000]

(15,000)($25)

$125,000

$375,000 = 33% or 1/3=

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Section 6.6 The Sales Mix

6.6 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 mar- gin, 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 specifically 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 averaging 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.

Let’s consider cost-volume-profit analysis with a sales mix. Suppose that Bijan’s Exotic Cater- ing 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:

Product Lines

Sales Volume (Meals)

Price per Meal

Unit Variable Cost

Contribution Margin

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 contribution margin as follows:

Product Lines Sales Mix

Proportions

Contribution Margin per

Meal

Weighted Contribution

Margin

Duck 50% × $30 = $15.00

Venison 25% × 20 = 5.00

Shark 25% × 10 = 2.50

Weighted contribution margin $22.50

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Section 6.6 The Sales Mix

(Fixed cost) / (Weighted contribution margin) = $500,000 / $22.50 = 22,222 total units

The detailed composition of this overall break-even point of 22,222 units (and contribution margins at this level) is as follows:

Product Lines

Sales Mix Proportions

Total Units

No. of Meals

Margin per

Meal Contribution

Margin

Duck 50% × 22,222 = 11,111 × $30 = $333,330

Venison 25% × 22,222 = 5,556 × 20 = 111,120

Shark 25% × 22,222 = 5,556 × 10 = 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 earlier in the chapter. Simply divide the weighted contribution margin ratio into the fixed costs. Indi- vidual product line revenues will be total revenues multiplied by individual sales mix propor- tions. 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%, and revenue at the break-even point is $1,111,111.

(Total contribution margin) / (Total revenues) = $900,000 / $2,000,000 = 45%

(Fixed cost) / (Weighted contribution margin ratio) = 500,000 / (45%) = $1,111,111

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Section 6.6 The Sales Mix

Another way to obtain the number of meals needed to break even is by using an equation 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 + $20v + $10s = $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 + $20(.5d) + $10(.5d) = $500,000

Solving this equation, we obtain d = 11,111, and therefore, v = s = .5(11,111) = 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 antici- pated. The difference was due to a changed sales mix. Assume the following actual results:

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 commissions 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.

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Section 6.7 Cost Estimation

6.7 Cost Estimation Cost estimation is the process of determining a cost relationship with activity for an individ- ual 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—a + 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 = a + 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 five techniques: (1) account analysis, (2) engineering approach, (3) scattergraph and visual fit, (4) high-low method, and (5) regression analysis.

Account Analysis In account analysis, accountants estimate variable and fixed cost behaviors of a particular 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 according to the prepon- derant 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 Joe Carmela & Associates, 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

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Section 6.7 Cost Estimation

The fixed costs total $17,200, while the variable cost per mile is 8.62 cents [($3,200 + $1,800) ÷ 58,000]. Hence, the cost function would be expressed as:

Y = $17,200 + $.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 estimated auto expenses should total $22,803 [$17,200 + ($.0862 × 65,000)]. The managing partner intends to investigate any significant deviation from this total cost.

Account analysis is fairly accurate for determining cost behavior in many cases. Vendor invoices, for instance, imply that direct materials have a variable cost behavior; leasing costs are fixed. A cell phone bill is a semi-variable cost. One portion is fixed for the minimum monthly charge, and the remainder may be variable with data or minutes charges.

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 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 behavior. A basic issue in manufacturing a product is determining the amount of direct materials, direct labor, and overhead required to run a given process. For a service, the question relates pri- marily 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 specifi- cations and vendor information can be used to quantify the units of the various materials 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.

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Section 6.7 Cost Estimation

A major advantage of the engineering approach is that it details each step required to per- form 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 weaknesses in the process. Another advantage is that it does not require historical accounting data. It is, there- fore, a useful approach in estimating costs of new products and services. The major disadvan- tage is the expensive nature of the approach. For example, time and motion studies require in-depth examinations of tasks and close observations of individuals 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 scatter- graph 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 observa- tions 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 vertical axis and activity or cost driver on the horizontal axis. The second step is to fit visually 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 6.8 shows a graph of maintenance cost and hours of operation for ten weeks of activity at Bleich Industries. 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

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Section 6.7 Cost Estimation

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 indicated by the line at 40 hours is approximately $110. Compute variable costs as follows:

Figure 6.8: A visually fitted cost function

Hours

M ai

nt en

an ce

C os

2010 30 40 50 60

120

$150

Total costs at 40 hours of operation $110

Less fixed costs 30

Variable costs $80

$80 variable costs ÷ 40 hours = $2 per hour of operation

This analysis yields the following cost function:

Y = $30 + $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

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Section 6.7 Cost Estimation

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 activity 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 differ- ence 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 several 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 – 50), and the difference in cost is $180 ($397 – $217). The variable supplies cost per hour is computed below:

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 = Total cost at highest activity – (Variable cost per unit × Highest activity)

Total fixed cost = Total cost at lowest activity – (Variable cost per unit × Lowest activity)

Cost at highest activity – Cost at lowest activity

Highest activity – Lowest activity

$180

45 = $4 Variable cost per hour=

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Section 6.7 Cost Estimation

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 = $397 – ($4 Variable cost per hour × 95 hours)

= $397 – $380

= $17

Total fixed cost = $217 – ($4 Variable cost per hour × 50 hours)

= $217 – $200

= $17

The cost function that results from the high-low method is:

Y = $17 + $4 (X)

If Edlin’s manager forecasts that a particular job would require 75 hours, the bid would include an estimate of $317 [$17 + ($4 × 75)] 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 dis- advantage 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 6.10.

Ethical Considerations in Estimating Costs All cost estimation methods involve some degree of subjectivity. With account analysis, man- agers judgmentally classify costs. In the engineering method, the manager must often subjec- tively adjust data obtained from near-optimal working conditions to normal working condi- tions. 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.

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Section 6.8 Regression and Correlation Analyses

The subjectivity inherent in cost estimation can lead to biased cost estimates. Indeed in cer- tain instances, incentives exist for managers to bias cost estimates. In developing budgets or cost-based prices, managers may want to overestimate costs. In developing proposals 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 rel- evant 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 repre- sented 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.

6.8 Regression and Correlation Analyses Regression and correlation analyses are statistical techniques that can be used to estimate and examine cost functions. Regression analysis fits a line to the cost and activity data using the least squares method. Correlation analysis deals with the “goodness of fit” in the rela- tionship 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 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 approximation 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 determined so that the algebraic sum of the squared deviations from that line is at a minimum.

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Section 6.8 Regression and Correlation Analyses

We will explain the application of linear regression through an illustration. Suppose that the buildings manager of the Atlanta Juvenile Community Center (AJCC) is asked to assist in bud- geting 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 application of linear regression. By entering data on a spreadsheet and using a few function commands, 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

X coefficient(s) 10

The information necessary to obtain a cost function can be read from the output. The con- stant is the fixed cost of $200. The X coefficient is our variable cost per hour, $10. Hence, our cost function is:

Y = $200 + $10X

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Section 6.8 Regression and Correlation Analyses

This cost function tells us that the total electricity cost is $200 plus $10 for each hour of elec- tricity. The number of observations is given as a check. The remaining terms in the regression output will be discussed later.

The buildings manager of the AJCC can now use the derived cost function to budget the elec- tricity 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 + ($10 × 35)].

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 cor- relation of the regression line to the cost and activity data and the “reliability” of the cost estimates, as discussed below.

Goodness of Fit The relationship between cost and activity is called correlation. At times costs may be ran- domly distributed and are not at all related to the cost driver used in defining the relationship. This is illustrated in Figure 6.9.

Figure 6.9: No correlation

Hours

C os

At the other extreme, the relationship may be so close that the data can almost be plotted on a line, as shown in Figure 6.10.

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Section 6.8 Regression and Correlation Analyses

Figure 6.10: Positive correlation

Hours

C os

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 6.11.

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 desig- nated 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 elec- tricity cost estimation. Recall that the regression output showed an r2 of 0.886. Thus, 88.6% of the variation in electricity costs could be explained by hours of activity.

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Reliability Because a regression equation will not result in a perfect fit on the data observations, a mea- sure of variability in the data is necessary with respect to the regression equation. The stan- dard error of the estimate (Se) is a measure of the deviation between the actual observa- tions 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 distribution 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 6.12 of differences between the actual cost and predicted cost.

Hours of Activity

E le

ct ric

ity C

os t

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

Figure 6.11: Explanation of correlation

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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 r2, 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 esti- mate, 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 imperfections 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. Under- standing 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 following three categories: (1) major errors in the original data, (2) errors in keying data, and (3) inappropriate mea- sures of activity.

Major errors in the original data are minimized through (1) reviewing the cutoff procedures 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 calcula- tion, 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 available in the data, try other cost drivers to locate one with a higher r2 value.

Figure 6.12: Graph of differences

Hours of Activity

A ct

ua l C

os t M

in us

P re

di ct

ed C

os t

10 20 30 40 50 60 70 80

–100

–50

100

$150

+ $67.08

– $67.08

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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 activ- ity, 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 lim- its 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 stan- dard 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:

If more tolerance is permitted for control purposes, the limits may be extended. For instance, a 95% 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% 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 × $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 investiga- tion and the higher the likelihood that waste and inefficiencies will go uncorrected.

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. Or, tele- phone 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 forwarding. 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 controlling costs. In simple

Upper Limit (+1 Standard Error)

Lower Limit (−1 Standard Error)

Regression prediction [$200 + ($10 × 40)] $600.00 $600.00

Standard error of the estimate +67.08 −67.08

$667.08 $532.92

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regression only one factor is considered, but in multiple regression several factors are con- sidered in combination. The basic form of the multiple regression model is:

Y = a + b1X1 + b2X2 + . . . . . + 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. Out- put 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

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 coefficients given for the three independent variables. These represent b1, b2, and b3.

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 independent 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 multicollinearity. For example, direct labor hours and direct labor costs would be highly correlated. Multicol- linearity 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 coefficients can be used in pric- ing decisions and cost-volume-profit analyses.

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.

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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 substan- tial 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 prod- uct line is known as the sales mix. To maximize profit, management will try to maximize 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.

Five methods for identifying variable and fixed cost behavior are account analysis, the engi- neering approach, the scattergraph and visual fit method, the high-low method, and regres- sion analysis. 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 historical activity of the cost. The engineering approach uses analysis and direct observa- tion 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. Regression analysis fits a line to the cost and activity data using a statistical approach.

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.

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break-even chart A graph showing cost and revenue functions together with a break-even point.

break-even point The point where total revenue equals total costs.

confidence interval A range with a speci- fied probability of including the parameter of interest.

contribution margin ratio Contribution margin divided by total sales revenue.

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 (CVP) 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 pro- cesses to identify the relationship between inputs and outputs and then quantifies an expected cost behavior.

high-low method An approach for estimat- ing costs that uses only two data points—the highest and the lowest activity levels.

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 that can be plot- ted as a smooth, continuous, symmetrically 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 visu- ally 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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Problem for Review Rodbell Roofing Co. installs residential roofs throughout Alabama. In 2018, 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 2019, 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% increase in number of installations for 2019; however, the average price charged to cus- tomers is not planned to change.

The 2020 forecast is for 20 more installations than in 2019, and fixed costs are expected to jump by $240,000 more than in 2019. The average price and variable cost is not expected to change from 2019.

Questions:

1. For 2018, calculate the number of installations needed to break even. 2. For 2019, calculate the operating leverage factor. 3. What is expected to be the amount of profit change from 2019 to 2020?

Solution:

1. For 2019, the break-even point:

2. For 2019, the operating leverage factor:

Number of installations in 2019 = 1.2 × 145 = 174

Unit contribution margin in 2019 = $30,000 – $16,000 = $14,000

Total contribution margin in 2019 = 174 × $14,000 = $2,436,000

Pretax profit in 2019 = $2,436,000 – $1,350,000 = $1,086,000

Operating leverage factor in 2019 = $2,436,000 ÷ $1,086,000 = 2.24

3. Amount of profit change from 2019 to 2020:

Additional contribution margin = 20 additional installations ×

$14,000 contribution margin per unit = $280,000

Additional fixed cost = $240,000

Additional profit = $280,000 – $240,000 = $40,000

$1,200,000 (Fixed cost)

$12,000 (Unit contribution margin) = 100 installations

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Questions for Review and Discussion

1. When the total contribution margin is equal to the total fixed cost, is the company operating at a profit or at a loss? Explain.

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. Describe the components of a break-even chart. 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. What is multiple regression? When would it be used in cost estimation? 14. What is multicollinearity? How does it pose problems in multiple regression

analysis?

Exercises

6-1. Break-Even Point and Change in Fixed Costs. Schloss Spirits sells each bottle of a particular whiskey for $50. The variable costs are 60% of revenue. The fixed costs amounted to $120,000 per year. Last year the store sold 7,500 of these bottles. Dur- ing the current year, Schloss 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?

6-2. “What if” Prices. The following budget numbers from Cann Corporation were pro- vided by the controller, Sue Ann Leonard:

Sales (12,000 units) $540,000

Variable costs (360,000)

Fixed costs (180,000)

Net income $ 0

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Questions:

1. If Cann can reduce variable costs by $4 per unit, what price must be charged to earn $90,000? Assume that 12,000 units will still be sold.

2. If Cann can increase sales revenues by spending $20,000 more on advertising, what price must be charged to earn $120,000? Assume that 12,000 units will still be sold.

6-3. Changes to Break-Even Variables. Gerald Ruby & Co. 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?

6-4. Break-Even Point and Target Profit. The Marla & Kerry Theatre Group operates five discount movie theatres in the Chicago area. All tickets are priced at $1.75. Variable costs are $1.10 per patron, and the fixed costs per year total $900,000. The fixed costs include concessions, but the variable costs do not. The concessions earn an average contribution margin of $0.60 per patron.

Questions:

1. How many patrons are required to break even? 2. If 800,000 patrons are expected for the year, what ticket price would be necessary to

earn a profit of $150,000?

6-5. Effects of Changes in Volume and Costs. Marnie’s Boutique sells umbrellas that have a contribution margin ratio of 35% 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 Stein, 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.

3. Laura Stein believes that the present selling price should be cut by 15% and the advertising budget should be raised by $12,000. She predicts that these changes would boost unit sales by 30%. Calculate the predicted additional net income (or loss) if these changes are implemented.

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6-6. Break-Even Point and Profits. Sheila Hershey has noticed a demand for small tables for personal computers and printers. Retail office furniture outlets are charg- ing 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 consist- ing 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?

6-7. Target Profit and Taxes. Gris, 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 the sales in units need to be?

6-8. 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% and variable costs per rider of $6.50.

Questions:

1. Compute the total fixed costs. 2. Compute the fee charged per rider.

6-9. Contribution Margin Ratio. In 2018, Simon’s Barber Shop had a contribution margin ratio of 22%. In 2019, fixed costs will be $41,000, the same as in 2018. Rev- enues are predicted to be $250,000, a 25% increase over 2018. Simon’s Barber Shop wishes to increase pretax profit by $8,000 in 2019.

Question:

Determine the contribution margin ratio needed in 2019 for Simon’s Barber Shop to achieve its goal.

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6-10. Multiple Products. Bettsak Security Systems installs three types of auto alarm sys- tems. Last year, fixed costs totaled $315,000, and target pretax profit was $190,000. Additional information for the past year was as follows:

Type of Alarm Unit Contribution Margin Number Sold

Best $50 1,000

Better 45 1,100

Good 38 1,300

Question:

Determine the sales volumes (number of alarm systems of each type) that were needed to achieve the target profit for the past year.

6-11. Multiple Product Analysis. The South Division of Euclid 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 number

of units of each grade of canvas that should have been sold during the current year to earn the $540,000 profit that was earned last year.

6-12. Fixed Cost Increase and Margin of Safety. Joey Fink, the owner of Kay Manufac- turing Company, is concerned about increased fixed manufacturing costs. Last year, the fixed manufacturing costs were $600,000. This year, the fixed manufacturing costs increased to $750,000. The fixed selling and administrative costs of $540,000 were the same for both years. The company operated in both years with an average contribution margin ratio of 30%. It earned a profit before income taxes of $910,000 last year.

Questions:

1. What was the sales revenue last year? 2. How much would sales revenue have had to increase this year for the company to

have earned the same profit as it did last year? 3. Calculate the margin of safety for each year. Which year was better?

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6-13. 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.

6-14. Cost Prediction With Account Analysis. Muhammed Alley, a bowling alley located in Louisville, had the following costs during a recent month:

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 customers are expected to come.

Question:

Estimate Muhammed Alley’s total cost for next month.

6-15. 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:

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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.

6-16. 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 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 gallons consumed per month and total pumping cost per month is as follows:

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

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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 pump- ing costs.

2. At what level of water consumption would Lake Tillem Township break even?

6-17. Confidence Interval. The Greenbelt Fire Department wants to examine the costs of operating its 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 fol- lowing 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.

6-18. Scattergraph and Visual Fit. Lewyn Financial Advisors uses the scattergraph and visual fit method to estimate its utilities costs. After plotting data for the past 15 weeks, the controller drew a cost function and saw that it hit the vertical axis at $6,500. He then picked a point on the cost function corresponding to $9,700 and 800 hours of activity for further analysis.

Question:

What amount of utilities cost would the controller predict for 920 hours of activity?

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Problems

6-19. CVP With Changes in Prices and Costs. Data with respect to a basic product line sold by Yozef 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%.

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 prod-

uct must be sold? 3. If a profit after income taxes of $180,000 is to be earned, how many units of product

must be sold? 4. If the selling price per unit is reduced by 10%, 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%?

6. Why does a 10% decrease in the selling price have more effect on the contribution margin than a 10% increase in the variable unit cost?

6-20. 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

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 commercial price of $98 per chair.

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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.

6-21. Break-Even Comparisons. Dan and Elaine Miller, owners of Spira Motel, want to know potential maximum profits and the break-even occupancy for the operation. The 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

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

Fixed costs per month have been estimated as follows:

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The motel 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 × 30 days).

Questions:

1. Compare the maximum operating net incomes that can be expected for a summer month versus a winter month.

2. How do the break-even points (in terms of room days) compare for summer versus winter? Also, state the break-even points as percentages of total capacity.

3. Based on advance reservations and normal expectations, the 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.

6-22. CVP With a Sales Mix. Sharon Arts Festival rents four types of booths to exhibitors, providing them with space, tables, chairs, and some pre-festival publicity. Stan Har- ris, 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 expected to be rented this year, as well as the related variable costs, are:

Booth Size Variable Cost per Booth Number of Booths

8' × 10' $25 15

10' × 12' 28 10

10' × 15' 30 20

15' × 15' 35 5

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.

6-23. Pricing to Break Even. Senkfor Township is inaugurating a youth soccer program. Estimates are that 70 children between the ages of 7 and 9 will enroll and that six teams will be formed. The program will be staffed by volunteer coaches, referees, and administrative personnel, so the only out-of-pocket costs involve the following:

Shirts $4.10 per shirt plus $20 screening charge

Balls $85 (total for all teams)

Goals $120 (total for all teams)

Cones $70 (total for all teams)

Shin guards $9.50 per pair (i.e., $9.50 per player)

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The price obtained for the shirts presumes two colors (shirt color and lettering). The program director, Leslie Stewart, would like very colorful shirts. She has learned that a “rainbow” pattern having several additional colors would cost $1 extra per shirt plus an additional $100 screening charge. The township’s recreation director has asked Leslie to determine how much to charge each child so that the soccer program fully covers its costs.

Question:

If the “rainbow” pattern shirt is selected, what fee per child should Leslie suggest to the rec- reation director?

6-24. High-Low and Visual Fit. Kathleen Otwell, an insurance claims adjuster for Shoenfield 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:

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

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Question:

Estimate the cost for November, when 11,000 video games are expected to be played.

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.

6-25. 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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Questions:

1. Using the equation Y = a + 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?

6-27. Cost Estimation With Multiple Regression. Saylush Painters has gathered data on a random sample of 20 paint jobs. The company’s accountant, Fred Blair, feels that labor costs are related to the surface area painted (X1 = sq. ft. of surface area) and the length of the trim work needed to complete a paint job (X2 = edge trim, in feet). A multiple regression analysis was performed, and the results are as follows:

6-26. 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 error 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

R2 .83

Constant 102.50

Coefficient of X1 .127

Coefficient of X2 .214

Standard error of estimate 77.23

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Questions:

1. Does the regression line provide a good fit to the data? Briefly explain. 2. How much does it cost to paint an extra 400 sq. ft. of surface? 3. How much does it cost to get set up for a painting job? 4. The cost for a new job needs to be estimated. It will have 8,800 sq. ft. of surface area

and 2,900 ft. of edge trim. What is your estimate?

6-28. Linear Regression and Data Concerns. 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 2018 games for study. The attendance figures and clean-up costs, as well as output from a regression analysis, are given as follows.

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

X coefficient(s) 0.10143

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Case: Mendel Paper Company

Mendel Paper Company produces four basic paper product lines at one of its plants: computer paper, napkins, place mats, and poster board. Materials and operations vary according 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.

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:

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

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

Questions:

1. What amount of clean-up cost would Beck budget for the 2019 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)—i.e., what concerns would you have about the data?

(continued)

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Case: Mendel Paper Company (continued)

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 violence 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.”

Data with respect to production per machine hour and the variable cost per hour of producing each of the products are given as follows:

“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.

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

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