Chapter 20: Cost-Volume-Profit Analysis

Cost Behavior (Learning Objective 1)

• Cost behavior is the manner in which a cost changes as a related activity changes.
• The behavior of a cost depends on:
  • Identifying activity bases (or activity drivers) that cause the cost to change.
  • Specifying the relevant range of activity.
• Costs are classified as:
  • Variable costs.
  • Fixed costs.
  • Mixed costs.

Classifying Costs

• Important for predicting cost behavior, i.e., how a cost is affected by changes in activity.
• Understanding cost structure.
• Four types of costs:
  • Fixed
  • Variable
  • Step
  • Mixed

Fixed and Variable Costs

• Variable Costs:
  • Affected by a driver (base).
  • Increase in direct proportion to output.
• Fixed Costs:
  • Can be committed or discretionary (managed).
• Both occur within a relevant range.
• Example: Gym membership
  • Membership: 60/month (Fixed)
  • Personal trainer: 20 per session (Variable)

Summary of Cost Behavior for Fixed Costs (FC) and Variable Costs (VC)

• Variable Cost
  • In Total: Increase and decrease in proportion to changes in the activity level.
  • Per Unit: Remains constant.
• Fixed Cost
  • In Total: Is not affected by changes in the activity level within the relevant range.
  • Per Unit: Decreases as the activity level rises and increases as the activity level falls.

Variable Costs

• Vary in proportion to changes in the activity base.
• Direct materials and direct labor are typically classified as variable costs when the activity base is units produced.
• Example: Jason Sound Inc. produces stereo systems (Model JS-12), relevant range: 5,000 to 30,000 units.
  • Cost per unit remains the same regardless of changes in activity base.
  • Total cost changes in proportion to changes in activity base.
• Exhibit 1 - Variable Cost Graphs.
• Exhibit 2 - Variable Costs and Their Activity Bases
  • VC in total increase as output increases; if there's no activity, no cost is incurred.

Fixed Costs

• Remain the same in total dollar amount as the activity base changes.
• Many factory overhead costs, like straight-line depreciation, are classified as fixed costs when the activity base is units produced.
• Example: Minton Inc. manufactures, bottles, and distributes perfume, relevant range: 50,000 to 300,000 bottles.
  • Fixed cost per bottle decreases as units produced increase.
• Exhibit 3 - Fixed Cost Graphs.
• Exhibit 4 - Fixed Costs and Their Activity Bases.

Mixed Costs

• Have characteristics of both variable and fixed costs.
• Sometimes called semi-variable or semi-fixed costs.
• Graphically:
  • X-axis: Independent variable.
  • Y-axis: Dependent variable.
  • Slope: Variable cost per unit.
  • Y-intercept: Total Fixed Costs (TFC).

Estimating Mixed Costs

• Assumption: Cost is linear within the relevant range.
• Have a fixed and variable component. Example: Cable bill (e.g., 20/month + 5 per movie rented).
• Total Cost Equation
• Methods:
  • Visual Fit Method: Create a scattergraph.
  • High-Low Method: Estimate using highest and lowest output observations.
  • Least-Squares Regression: Statistical approach.
• The book focuses on the high-low method.

High-Low Method

• Pick highest and lowest activity driver, NOT dollar amount.
• Formula: (\$H - \$L) / (QH – QL). This results in dollars per unit; it solves for Variable Cost (VC).
• Plug into the cost equation at either the high or low point to solve for Fixed Costs (FC).
• Question: What if quarter 3:1 was 24,000 sold and shipping expense was $240,000? What is the estimated shipping expense if we plan on selling 27,000 units?

Mixed Costs

• Costs that have characteristics of both variable and fixed costs.
• Sometimes called semi-variable or semi-fixed costs.
• Example: Simpson Inc. manufactures sails, using rented machinery.
  • Rental charges for various hours used within the relevant range of 8,000 hours to 40,000 hours.
• Exhibit 5 - Mixed Costs.

The High-Low Method

• A cost estimation method that separates mixed costs into fixed and variable components.
• Uses the highest and lowest activity levels and their related costs to estimate the variable cost per unit and the fixed cost.
• Example: Equipment Maintenance Department of Kason Inc.
  • The number of units produced is the activity base, and the relevant range is the units produced between June and October.

The High-Low Method

• For Kason, the differences between the units produced and the total costs at the highest and lowest levels of production are as follows:
  • The 20,250 difference in the total cost is the change in the total variable cost. Dividing this difference of 20,250 by the difference in production is an estimate of the variable cost per unit.
• Formula to estimate variable cost per unit
  • For Kason, this estimate is 15, computed as follows:
• To estimate fixed costs:

The High-Low Method

• The fixed cost is the same at the highest and lowest levels of production.
• Using the variable cost per unit and the fixed cost, the total equipment maintenance cost can be computed for various levels of production.
• The estimated total cost of 2,000 units of production is 60,000.
• Exhibit 6 - Variable and Fixed Cost Behavior.
• Exhibit 7 - Variable, Fixed, and Mixed Costs.

Knowledge Check Activity 1

• Question: Which of the following costs remain the same regardless of activity level within the relevant range?
  • (b) Total fixed cost and unit variable cost
• Cost behavior varies based on changing activity levels. Total variable cost and unit fixed costs will either increase or decrease. However, total fixed cost and unit variable cost will remain the same regardless of activity level within the relevant range.

Cost-Volume-Profit Relationships (Learning Objective 2)

• Cost-volume-profit analysis is a management tool used to examine the relationships among selling prices, sales and production volume, costs, expenses, and profits and is useful for managerial decision making.
• Ways cost-volume-profit analysis may be used include:
  • Analyzing the effects of changes in selling prices on profits
  • Analyzing the effects of changes in costs on profits
  • Analyzing the effects of changes in volume on profits
  • Setting selling prices
  • Selecting the mix of products to sell
  • Choosing among marketing strategies

What CVP Tells us

• Relationship between cost, volume (quantity), and profit.
• Affected by:
  • Selling Price
  • Sales Volume (Q)
  • Variable Cost per Unit
  • Total Fixed Costs
  • Mix of Products Sold
• Basic Equation: Revenue – (Variable Cost * Quantity) – Fixed Costs = Profit.
• Rev - (VC * Q) - FC = Profit
• Know this equation

CVP Assumptions

• Selling price is constant (price does not change when quantity changes).
• Costs are linear and can be divided into fixed and variable costs.
• The costs per unit (variable cost) and total fixed cost do not change within the relevant range.
• Constant sales mix in constant CVP analysis.
• Inventories do not change for manufacturers and units produced equals units sold.

Cost-Volume-Profit (CVP) Basics

• Math related to the Variable Costing (or Contribution Format) Income statement.
• Key Definitions:
  • Contribution Margin:
    • The amount we have left over to contribute to fixed costs.
• Formula: Revenue - Variable Costs
  • Break Even Point (BEP):
    • The quantity sold when profit is equal to 0.
  • Target profit quantity:
    • The quantity to be sold when profit is at a specified level.

CVP Relationship in Graphical Form

• Total Revenue Line
• Total Cost Line (and the formula for this line)
• Fixed Cost
• Variable Cost per unit
• Break-Even Point
• Loss Area
• Profit (or Gain) Area

Contribution Margin

• Contribution margin is the excess of sales over variable costs.
• Is the amount of revenue that is available for covering fixed costs and earning a profit.
• Example: Lambert Inc.
• Can calculate per unit or in total.
• Think of as how much a company can "contribute" to fixed costs.

The CM represents:

• Portion of sales not consumed by variable costs.
• The dollar amount we can contribute to fixed costs.
• The point where our revenues = costs (Fixed Costs + Variable Costs).
• Look at our equation and put profit equal to 0.
• Exhibit 8 - Contribution Margin Income Statement Format

Contribution Margin Ratio

• The contribution margin as a percentage of sales. Formula: CM / Sales
  • Can be in total amounts or per unit; the result is the same.
• Example: for every dollar increase in sales, we have __ to contribute to fixed costs.
• CM Ratio: At Q = 200 or At Q = 300
• On a per-unit basis
  Pprice/unit = $200, VC/unit = $30, FC = $42,500

Contribution Margin Ratio

• Contribution margin can also be expressed as a percentage.
• Contribution margin ratio (profit-volume ratio) indicates the percentage of each sales dollar available to cover fixed costs and to provide operating income.
• CM Ratio is the same whether you calculate per unit or in total as long as calculations for sales and VC are constant on a per unit basis.

Contribution Margin Ratio

• In Exhibit 8, Lambert Inc. earned 100,000 of operating income on sales of 1,000,000. Assume that Lambert Inc. increases its sales by 80,000 (4,000 units × 20), from 1,000,000 to 1,080,000 (54,000 units).
• Lambert Inc.’s operating income will increase by 32,000, from 100,000 to 132,000.

Unit Contribution Margin

• Formula:
  Unit\, Contribution \,Margin = Sales\, Price\, per \,Unit - Variable\, Cost\, per\, Unit

• Selling an additional 15,000 units increases Lambert Inc.’s operating income by 120,000, from 100,000 to 220,000.

• Thus, Lambert could spend up to 120,000 for special advertising or other product promotions to increase sales by 15,000 units and still increase income by 100,000.

Mathematical Approach to Cost-Volume-Profit Analysis (Learning Objective 3)

• Uses equations to determine:
  • Sales necessary to break even
  • Sales necessary to make a target or desired profit.
• The break-even point is the level of operations at which a company’s revenues and expenses are equal.
• Exhibit 9 - Break-Even Point.

Important Formula

• Generalized Formula:
  Rev – VC – FC = π
  (P/unit * Q) – (VC/unit * Q) – FC = π

• At BE, π = 0
  (P/unit * Q) – (VC/unit * Q) – FC = 0

Target Profit Equation

• What is the total amount of units we would have to sell to make 68,000? What are the sales dollars we would need to make that much profit?
• Target profit question can be asked in units or sales dollars.
  Rev – VC – FC = π
  (P/unit * Q) – (VC/unit * Q) – FC = π

Break-Even Point Example

• Example: Baker Corporation

• Break-even point for Baker is 9,000 units.

Break-Even Point Example

• As shown in Baker’s income statement, the break-even point is 225,000 (9,000 units × 25) of sales, and the break-even point in sales dollars can also be determined directly.
• The contribution margin ratio for Baker is 40%.
• The break-even sales dollars for Baker of 225,000 can be computed as:

Derivation of the book equation

• Can always solve with algebra or can use the following formula:
  Break\, Even\, Sales\, in \,Units = \frac{Fixed\, Expenses}{CM\, per\, Unit}

Derivation of Break Even in Sales Dollars Formula

• BE\, Sales\, Dollars = \frac{Fixed\, Costs}{CM\, Ratio}

Effect of Changes in Fixed Costs

• The break-even point is affected by changes in the fixed costs, unit variable costs, and unit selling price.
• Fixed costs do not change in total with changes in the level of activity but may change because of other factors such as changes in advertising, property taxes, or supervisor salaries.
• Assume Bishop Co. is evaluating a proposal to budget an additional 100,000 for advertising

Effect of Changes in Fixed Costs

• Bishop’s break-even point before the additional advertising expense of 100,000 is 30,000 units.
  Break-Even \,Point \,=\frac{Fixed\, Costs}{Unit \,Selling\, Price - Unit\, Variable \,Costs}

• The 100,000 increase in advertising (fixed costs) requires an additional 5,000 units (35,000 – 30,000) of sales to break even.

• Exhibit 10 - Effect of Change in Fixed Costs on Break-Even Point

Effect of Changes in Unit Variable Costs

• Unit variable costs do not change with changes in the level of activity, but variable costs may be affected by changes in the cost per unit of direct materials, the wage rate for direct labor, or in the sales commission paid to salespeople.
• Assume Park Co. is evaluating a proposal to pay an additional 2% commission on sales to its salespeople as an incentive to increase sales:

Effect of Changes in Unit Variable Costs

• Park’s break-even point before the additional 2% commission is 8,000 units:
• If the 2% sales commission proposal is adopted, unit variable costs will increase by 5 (250 × 2%), from 145 to 150 per unit.
• This increase in unit variable costs will decrease the unit contribution margin from 105 to 100 (250 – 150).
• Park’s break-even point after the additional 2% commission is 8,400 units.
• An additional 400 units of sales will be required to break even.

Effect of Changes in Unit Selling Price

• Changes in the unit selling price affect the unit contribution margin and, thus, the break-even point.
• Assume Graham Co. is evaluating a proposal to increase the unit selling price of its product from 50 to 60.
• Graham’s break-even point before the price increase is 30,000 units.
• Graham’s break-even point after the price increase is 20,000 units.

Target Profit

• By modifying the break-even equation, the sales required to earn a target or desired amount of profit may be computed:
  Sales = \frac{Fixed\, Costs + Target\, Profit}{Unit\, Contribution\, Margin}
• Assume the following data for Waltham Co.:
• The sales necessary for Waltham to earn the target profit of 100,000 would be 10,000 units.

Knowledge Check Activity 2

• Cathy’s Confectionery has fixed costs of 78,000 and would like to earn a target profit of 90,000. Its product sells for 15 and has a unit variable cost of 12. How many units of the product must be sold to attain the target profit?
• Answer: a. 56,000 units 2.

Graphic Approach to Cost-Profit-Volume Analysis (Learning Objective 4)

• A cost-volume-profit chart, sometimes called a break-even chart, graphically shows sales, costs, and the related profit or loss for various levels of units sold.
• The cost-volume-profit chart in Exhibit 14 is based on the following data for Munoz Co.

Steps to Construct a Cost-Profit-Volume Chart

• Step 1: Volume in units of sales is indicated along the horizontal axis. Dollar amounts of total sales and total costs are indicated along the vertical axis.
• Step 2: A total sales line is plotted by connecting the point at zero on the left corner of the graph to a second point on the chart. The second point is determined by multiplying the maximum number of units in the relevant range, which is found on the far right of the horizontal axis, by the unit sales price.
• Step 3: A total cost line is plotted by beginning with total fixed costs on the vertical axis. A second point is determined by multiplying the maximum number of units in the relevant range, which is found on the far right of the horizontal axis, by the unit variable costs and adding the total fixed costs.
• Step 4: The break-even point is the intersection point of the total sales and total cost lines. A vertical dotted line drawn downward at the intersection point indicates the units of sales at the break-even point.

Cost-Volume-Profit (Break-Even) Chart

• Changes in the unit selling price, total fixed costs, and unit variable costs can be analyzed by using a cost-volume-profit chart.
• Using the data in Exhibit 14, assume that Munoz is evaluating a proposal to reduce fixed costs by 20,000.
• The total fixed costs would be 80,000 (100,000 – 20,000).
• The revised cost-volume-profit chart in Exhibit 15 indicates that the break-even point for Munoz decreases to 200,000 and 4,000 units of sales.

Profit-Volume Chart

• Plots only the difference between total sales and total costs (or profits).
• Allows managers to determine the operating profit (or loss) for various levels of units sold.
• Changes in the unit selling price, total fixed costs, and unit variable costs on profit can be analyzed using a profit-volume chart.
• Using the data in Exhibit 16, consider the effect that a 20,000 increase in fixed costs will have on profit.
• At the maximum sales of 10,000 units, the maximum operating profit would be 80,000.

Steps to Construct a Profit-Volume Chart

• Step 1: Volume in units of sales is indicated along the horizontal axis. Dollar amounts indicating operating profits and losses are shown along the vertical axis.
• Step 2: A point representing the maximum operating loss is plotted on the vertical axis at the left. This loss is equal to the total fixed costs at the zero level of sales.
• Step 3: A point representing the maximum operating profit within the relevant range is plotted on the right.
• Step 4: A diagonal profit line is drawn connecting the maximum operating loss point with the maximum operating profit point.
• Step 5: The profit line intersects the horizontal zero operating profit line at the break-even point in units of sales. The area indicating an operating profit is identified to the right of the intersection, and the area indicating an operating loss is identified to the left of the intersection.

Use of Spreadsheets and Assumptions in Cost-Volume-Profit Analysis

• With spreadsheets and graphs, managers can vary assumptions regarding selling prices, costs, and volume and can observe the effects of each change on the break- even point and profit.
• This analysis is called a “what if” analysis or sensitivity analysis.
• Cost-volume-profit analysis depends primarily on several assumptions:
  • Total sales and total costs can be represented by straight lines.
  • Within the relevant range of operating activity, the efficiency of operations does not change.
  • Costs can be divided into fixed and variable components.
  • The sales mix is constant.
  • There is no change in the inventory quantities during the period.

Special Cost-Volume-Profit Relationships (Learning Objective 5)

• Many companies sell more than one product at different selling prices.
• Break-even analysis can still be performed by considering the sales mix.
• The sales mix is the relative distribution of sales among the products sold by a company.
• Assume Cascade Company sold Products A and B during the past year.

Multiple Product Sales Mix

• The sales mix for Products A and B is expressed as a percentage of total units sold.
• For Cascade, a total of 10,000 (8,000 + 2,000) units were sold during the year; therefore, the sales mix is 80% (8,000 ÷ 10,000) for Product A and 20% for Product B (2,000 ÷ 10,000).

Multiple Product Sales Mix Example

• For Cascade Company, this “mix” of company products is labeled M.
• The unit selling price of M equals the sum of the unit selling prices of each product multiplied by its sales mix percentage.
• The unit variable cost and unit contribution margin of M equal the sum of the unit variable costs and unit contribution margins of each product multiplied by its sales mix percentage:

Multiple Product Sales Mix Example

• Cascade has total fixed costs of 200,000.
• The break-even point of 8,000 units of M can be determined using the unit selling price, unit variable cost, and unit contribution margin of M:
• Because the sales mix for Products A and B is 80% and 20%, respectively, the break-even quantity of A is 6,400 units (8,000 units × 80%) and B is 1,600 units (8,000 units × 20%)
• The effects of changes in the sales mix on the break-even point can be determined by assuming a different sales mix, and the break-even point of M can then be recomputed.

Operating Leverage

• The relationship between a company’s contribution margin and operating income is measured by operating leverage:
  Operating\, Leverage = \frac{Contribution\, Margin}{Operating\, Income}
• The difference between contribution margin and operating income is fixed costs; thus, companies with high fixed costs will normally have high operating leverage.

DOL Example

• Using the original Devos Inc. 300 units of sales example (Price = 200, VC/Unit = 30, Q = 300, FC = 42,500):
• What is the effect on Net Income if Sales increase by 20%?
• What about decrease by 20%?
• When Q =300
  DOL = \frac{CM}{π}

Operating Leverage

• Jones and Wilson have the same sales, the same variable costs, and the same contribution margin, but Jones has larger fixed costs than Wilson and, thus, a higher operating leverage.
• Operating leverage can be used to measure the impact of changes in sales on operating income:
  Change\, in\, Operating\, Income \approx Operating\, leverage \times Change\, in\, Sales

Operating Leverage

• Assume that sales increased by 10%, or 40,000 (400,000 × 10%), for Jones and Wilson:
• Jones’s operating income increases by 50%, while Wilson’s operating income increases by only 20%:

Margin of Safety

• Indicates the possible decrease in sales that may occur before an operating loss results and may be expressed in dollars of sales, units of sales, and percent of current sales.
• Assume the following data:
• The margin of safety in dollars of sales is 50,000 (250,000 – 200,000), and the margin of safety in units is 2,000 units (50,000 ÷ 25).
• The margin of safety expressed as a percent of current sales is 20%:
  Margin\, of \,Safety\, %= \frac{Sales - Break \,even \,Sales}{Sales}

Knowledge Check Activity 3

• Whitman Industries had sales of 800,000 at the end of the year. Its break-even point for sales was 680,000. What is the margin of safety expressed as a percentage of sales?
• Answer: d. 15%
• The formula to express the margin of safety as a percentage of sales is:

Analysis for Decision Making: Cost-Volume-Profit Analysis for Service Companies (Learning Objective 6)

• The break-even point is as relevant in a service company as it is in a manufacturing company.
• Cost-volume-profit relationships in a service company are measured with respect to customers and activities, rather than units of product:

Analysis for Decision Making: Cost-Volume-Profit Analysis for Service Companies

• Example: Consider the break-even number of students for a noncredit course in pottery with a 500 tuition:
  BreakEven\, Point \,in\, Students = \frac{Fixed \,Costs}{Tuition \,per\, Student - Variable\, Costs\, per\, student}

Additional Use of the Shortcut

• Example 3: If not told how many units, just the following: If Devos Inc. increases their advertising by 20,000, their sales will increase by 26,000. What is the effect on profit?
• Example 4: Times are tough and Devos Inc. figures if they decrease their advertising by 10,000, their sales will only decrease by 5,000. Should they undertake this cost-cutting measure?