Cost-Volume-Profit Analysis Notes
Learning Objectives
After reading this chapter, you should be able to:
L.O.1 Use cost-volume-profit (CVP) analysis to analyze decisions.
L.O.2 Understand the effect of cost structure on decisions.
L.O.3 Use Microsoft Excel to perform CVP analysis.
L.O.4 Incorporate taxes, multiple products, and alternative cost structures into the CVP analysis.
L.O.5 Understand the assumptions and limitations of CVP analysis.
Cost-Volume-Profit Analysis
Managers are concerned about the impact of their decisions on profit. The decisions they make are about volume, pricing, or incurring a cost. Therefore, managers require an understanding of the relations among revenues, costs, volume, and profit. The cost accounting department supplies the data and analysis, called cost-volume-profit (CVP) analysis, that support these managers.
Cost-volume-profit (CVP) analysis: Study of the relations among revenue, cost, and volume and their effect on profit.
CVP analysis helps managers evaluate the impact of alternative product pricing strategies on profits. It can also be useful for evaluating competitors’ pricing strategies and efforts to grow market share
Profit Equation
The key relation for CVP analysis is the profit equation. Every organization’s financial operations can be stated as a simple relation among total revenues (), total costs (), and operating profit:
Operating profit = Total revenues - Total costs
Both total revenues and total costs are likely to be affected by changes in the amount of output. We rewrite the profit equation to explicitly include volume, allowing us to analyze the relations among volume, costs, and profit. Total revenue () equals average selling price per unit () times the units of output ():
Total revenue = Price * Units of output produced and sold
In our profit equation, total costs () may be divided into a fixed component that does not vary with changes in output levels and a variable component that does vary. The fixed component is made up of total fixed costs () per period; the variable component is the product of the average variable cost per unit () multiplied by the quantity of output (). Therefore, the cost function is
Total costs = (Variable costs per unit * Units of output) + Fixed costs
Substituting the expanded expressions in the profit equation yields a form more useful for analyzing decisions:
Therefore,
Collecting terms gives
We defined contribution margin in Chapter 2 as the difference between the sales price and the variable cost per unit. We will refer to this as the unit contribution margin to distinguish it from the difference between the total revenues and total variable cost, the total contribution margin. In other words, the total contribution margin is the unit contribution margin multiplied by the number of units (Price – Variable costs) * Units of output, or () X. It is the amount that units sold contribute toward (1) covering fixed costs and (2) providing operating profits. Sometimes we use the contribution margin, in total, as in the preceding equation. Other times, we use the contribution margin per unit, which is
Profit equation: Operating profit equals total revenue less total costs.
Unit contribution margin: Difference between revenues per unit (price) and variable cost per unit.
Total contribution margin: Difference between revenues and total variable costs.
An important distinction for decision making is whether costs are fixed or variable. That is, for decision making, we are concerned about cost behavior, not the financial accounting treatment, which classifies costs as either manufacturing or administrative. Thus, V is the sum of variable manufacturing costs per unit and variable marketing and administrative costs per unit; F is the sum of total fixed manufacturing costs and fixed marketing and administrative costs for the period; and X refers to the number of units produced and sold during the period.
CVP Example
When Jamaal first opened U-Develop, he offered one service only, developing prints. He charged an average price of $0.60. The average variable cost of each print was $0.36, computed as follows:
Cost of processing (materials and labor) . . . . . . . . . . . . . . . . . . . . . $0.30
Other costs (sales and support). . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.06
Average variable cost per print. . . . . . . . . . . . . . . . . . . . . . . . . . . $0.36
The fixed costs to operate the store for March, a typical month, were $1,500.
In March, U-Develop processed 12,000 prints. The operating profit can be determined from the company’s income statement for the month.
As a manager, Jamaal might want to know how many units (prints) he needs to sell in order to achieve a specified profit. Assume, for example, that Jamaal is hoping for sales to improve in July, when the weather will improve and people take vacations. Given the data, price = $.60, variable cost per unit = $.36 (therefore, contribution margin per unit = $.24), and fixed costs = $1,500, the manager asks two questions: What volume is required to break even (earn zero profits)? What volume is required to make an $1,800 operating profit?
Recall that in March, U-Develop processed 12,000 prints. Using the profit equation, the results for March, therefore, were:
Profit = Contribution margin – Fixed costs
= (P – V) X – F = ($0.60 – $0.36) * 12,000 prints – $1,500 = $1,380
which is equal to the operating profit shown on the income statement . To simplify the equation, we use the term “Profit” in the equation to mean the same thing as “Operating Profit” on income statements.
Finding Break-Even and Target Volumes
We can use the profit equation to answer Jamaal’s questions about volumes needed to break even or achieve a target profit by developing the formulas discussed here. We start with the answer to the first question, which we call finding a break-even volume. Managers might want to know the break-even volume expressed either in units or in sales dollars. If the company makes many products, it is often much easier to think of volume in terms of sales dollars; if we are dealing with only one product, it’s easier to work with units as the measure of volume.
Break-Even Volume in Units
We can use the profit equation to find the break-even point expressed in units:
If Profit = 0, then
Break-even volume (in units) =
Fixed costs / Unit contribution margin = $1,500 / $0.24 = 6,250 prints
To show this is correct, if U-Develop processes 6,250 prints, its operating profit is
Profit = TR – TC = PX – VX – F
= ($0.60 * 6,250 prints) – ($0.36 * 6,250 prints) – $1,500 = $0
Break-Even Volume in Sales Dollars
To find the break-even volume in terms of sales dollars, we first define a new term, contribution margin ratio. The contribution margin ratio is the contribution margin as a percentage of sales revenue. For example, for U-Develop, the contribution margin ratio can be computed as follows:
Contribution margin ratio =
Unit contribution margin / Sales price per unit = $0.24 / $0.60 = 0.40 (or 40%)
Using the contribution margin ratio, the formula to find the break-even volume follows:
Break-even volume sales dollars =
=
Break-even point: Volume level at which profits equal zero.
Contribution margin ratio: Contribution margin as a percentage of sales revenue.
For U-Develop, the break-even volume expressed in sales dollars is
Break-even sales dollars = $1,500 / 0.40 = $3,750
Note that $3,750 of sales dollars translates into 6,250 prints at a price of $0.60 each. We get the same result whether expressed in units (6,250 prints) or dollars (sales of 6,250 prints generates revenue of $3,750).
Target Volume in Units
To find the target volume, we use the profit equation with the target profit specified. The formula to find the target volume in units is
Target volume (units) =
Using the data from U-Develop, we find the volume that provides an operating profit of $1,800 as follows:
Target volume =
(Fixed costs + Target profit) / Contribution margin per unit = ($1,500 + $1,800) / $0.24 = 13,750 prints
U-Develop must sell 13,750 prints per month to achieve the target profit of $1,800. Each additional print sold increases operating profits by $0.24.
Target Volume in Sales Dollars
To find the target volume in sales dollars, we use the contribution margin ratio instead of the contribution margin per unit. The formula to find the target volume follows:
Target volume (sales dollars) =
For U-Develop the target volume expressed in sales dollars is
Target volume (sales dollars) =
($1,500 + $1,800) / 0.40
Note that sales dollars of $8,250 translates into 13,750 prints at $0.60 each. We get the same target volume whether expressed in units (13,750 prints) or dollars (sales of 13,750 prints generates revenue of $8,250).
Graphic Presentation
The total revenue () line relates total revenue to volume (for example, if U-Develop sells 12,000 prints in a month, its total revenue would be $7,200, according to the graph). The slope of TR is the price per unit, P (for example, $0.60 per print for U-Develop).
The total cost () line shows the total cost for each volume. For example, the total cost for a volume of 12,000 prints is $5,820 ( = [12,000 * $0.36] + $1,500). The intercept of the total cost line is the fixed cost for the period, F, and the slope is the variable cost per unit, V.
The break-even point is the volume at which TR = TC (that is, where the TR and TC lines intersect). Volumes lower than break even result in an operating loss because TR < TC; volumes higher than break even result in an operating profit because TR > TC. For U-Develop, 6,250 prints is the break-even volume.
The amount of operating profit or loss can be read from the graph by measuring the vertical distance between TR and TC. For example, the vertical distance between TR and TC when X = 12,000 indicates Profit = $1,380 ( = $7,200 – $5,820).
Profit-Volume Model
Instead of considering revenues and costs separately, we can analyze the relation between profit and volume directly. This approach to CVP analysis is called profit-volume analysis. A graphic comparison of profit-volume and CVP relationships is shown in Exhibit 3.4. The cost and revenue lines are collapsed into a single profit line. Note that the slope of the profit-volume line equals the unit contribution margin. The intercept equals the loss at zero volume, which equals fixed costs. The vertical axis shows the amount of operating profit or loss.
Profit-volume analysis: Version of CVP analysis using a single profit line.
Use of CVP to Analyze the Effect of Different Cost Structures
An organization’s cost structure is the proportion of fixed and variable costs to total costs. Cost structures differ widely among industries and among firms within an industry.
Cost structure: Proportion of an organization’s fixed and variable costs to its total costs.
Operating leverage describes the extent to which an organization’s cost structure is made up of fixed costs. Operating leverage can vary within an industry as well as between industries.
Operating leverage: Extent to which an organization’s cost structure is made up of fixed costs.
Operating leverage is high in firms with a high proportion of fixed costs and a low proportion of variable costs and results in a high contribution margin per unit. The higher the firm’s fixed costs, the higher the break-even point. Once the break-even point has been reached, however, profit increases at a high rate.
Suppose that both companies experience a 10 percent increase in sales. Lo-Lev Company’s profit increases by $25,000 ($0.25 * $100,000), and Hi-Lev Company’s profit increases by $75,000 ($0.75 * $100,000). Of course, if sales decline, the fall in Hi-Lev’s profits is much greater than the fall in Lo-Lev’s profits. In general, companies with lower fixed costs have the ability to be more flexible to changes in market demands than do companies with higher fixed costs and are better able to survive tough times.
Margin of Safety
The margin of safety is the excess of projected (or actual) sales over the break-even sales level. This tells managers the margin between current sales and the break-even point. In a sense, margin of safety indicates the risk of losing money that a company faces, that is, the amount by which sales can fall before the company is in the loss area. The margin of safety formula is
Sales volume – Break-even sales volume = Margin of safety
Margin of safety: The excess of projected or actual sales over the break-even volume.
If U-Develop sells 8,000 prints and its break-even volume is 6,250, then its margin of safety is
Sales – Breakeven = 8,000 – 6,250 = 1,750 prints
Sales volume could drop by 1,750 prints per month before it incurs a loss, all other things held constant. In practice, the margin of safety also may be expressed in sales dollars or as a percent of current sales.
The excess of the projected or actual sales volume over the break-even volume expressed as a percentage of actual sales volume is the margin of safety percentage.
Margin of safety percentage: The excess of projected or actual sales over the break-even volume expressed as a percentage of actual sales volume.
If U-Develop sells 8,000 prints and the break-even volume is 6,250 prints, the margin of safety percentage is 22 percent ( = 1,750 / 8,000). This means that volume can fall by 22 percent, a relatively large amount, before U-Develop finds itself operating at a loss.
CVP Analysis with Spreadsheets
A spreadsheet program such as Microsoft Excel ® is ideally suited to doing CVP routinely. Once the data are entered, an analysis tool such as Goal Seek can be used to find the volume associated with a given desired profit level.
Extensions of the CVP Model
The basic CVP model that we have developed can be easily extended to answer other questions or modified to incorporate complications. For example, we can use the model to determine the fixed costs required to achieve a certain profit for a given volume. We can incorporate the effects of income taxes by modifying the profit equation to include taxes. Making some simplifying assumptions, we can extend the analysis to firms that make multiple products. Finally, we can incorporate more complicated cost structures (for example, step fixed costs) by incorporating these complications in the profit equation.
Income Taxes
Assuming that operating profits before taxes and taxable income are the same, income taxes may be incorporated into the basic model as follows:
After-tax profit = [(P – V)X – F] * (1 – t)
where t is the tax rate.
Rearranging, we can find the target volume as follows;
Target volume (units) =
Notice that taxes affect the analysis by changing the target profit. That is, to determine the volume required to earn a target after-tax income, you first determine the required before-tax operating income ( = target after-tax income / [1 – tax rate]) and then solve for the target volume using the required before-tax income as before.
For example, suppose that the owner of U-Develop wants to find the number of prints required to generate after-tax operating profits of $1,800. Recall that P = $0.60, V = $0.36, the contribution margin per unit = $0.24, and F = $1,500. We assume the tax rate t = 0.25; that is, U-Develop has a 25 percent tax rate. To find the target volume, first determine the required before-tax income, which is $2,400 ( = $1,800 / [1 – 0.25]). Now, we can use the formula to determine the volume required to earn a target profit of $2,400:
Target volume (units) =
(Fixed costs + [Target profit / (1 – t)]) / Unit contribution margin = ($1,500 + $2,400) / $0.24 = 16,250 prints
Multiproduct CVP Analysis
When U-Develop started, it provided only one service, print processing. After a short time, a second service, enlargements of photos, was offered. The prices and costs of the two follow:
Prints | Enlargements | |
|---|---|---|
Selling price | $0.60 | $1.00 |
Variable cost | $0.36 | $0.56 |
Contribution margin | $0.24 | $0.44 |
When these two services were offered, monthly fixed costs totaled $1,820.
Without some assumptions, there is an infinite number of combinations of the two services that would achieve a given level of profit. To simplify matters, managers often assume a particular product mix and compute break-even or target volumes using either of two methods, a fixed product mix or weighted-average contribution margin, both of which give the same result.
Fixed Product Mix
Using the fixed product mix method, managers define a package or bundle of products in the typical product mix and then compute the break-even or target volume for the package. For example, suppose that the owner of U-Develop is willing to assume that the prints and enlargements will sell in a 9:1 ratio; that is, of every ten “units” of service sold, nine will be prints and one will be an enlargement. Defining X as a package of nine prints and one enlargement, the contribution margin from this package is
Prints . . . . . . . . . . . . . . . . . 9 * $0.24 = $2.16
Enlargements . . . . . . . . . . . 1 * $0.44 = 0.44
Contribution margin = $2.60
Now the break-even point is computed as follows
X = Fixed costs / Contribution margin
$1,820 / $2.60 = 700 packages
where X refers to the break-even number of packages. This means that the sale of 700 packages of nine prints and one enlargement per package, totaling 6,300 prints and 700 enlargements, is required to break even.
Weighted-Average Contribution Margin
The weighted-average contribution margin also requires an assumed product mix, which we continue to assume is 90 percent prints and 10 percent enlargements. The problem can be solved by using a weighted-average contribution margin per unit. When a company assumes a constant product mix, the contribution margin is the weighted-average contribution margin of all of its products.
For U-Develop, the weighted-average contribution margin per unit can be computed by multiplying each product’s proportion by its contribution margin per unit
(0. 90 * $0.24) + (0.10 * $0.44) = $0.26
The multiple product breakeven for U-Develop can be determined from the break-even formula:
X = $1,820 / $0.26 = 7,000 units of service
where X refers to the break-even number. The product mix assumption means that U-Develop must sell 6,300 ( = 0.90 * 7,000) prints and 700 ( = 0.10 * 7,000) enlargements to break even.
Find Breakeven in Sales Dollars
To find the breakeven in sales dollars, divide the fixed costs by the weighted-average contribution margin percent. The weighted-average contribution margin percent is the ratio of the weighted-average contribution margin (which is $0.26 in our example) divided by the weighted-average revenue.
To find the weighted-average revenue, multiply the proportion of sales (90 percent prints and 10 percent enlargements) by the sales prices per unit. Prints sell for $0.60 per unit and enlargements sell for $1.00 per unit. Therefore, the weighted-average revenue can be found as follows:
(0. 90 * $0.60) for prints + (0.10 * $1.00) for enlargements = $0.64
Now, the weighted-average contribution margin percent is found as follows:
$0. 26 weighted-average contribution margin / $0.64 weighted-average revenue = 40.625%
The break-even sales amount in dollars is:
$1, 820 fixed costs / 0.40625 weighted-average contribution margin percent = $4,480
Alternative Cost Structures
The cost structures we have considered so far have been relatively simple. We have separated costs into fixed and variable and we have assumed that the variable cost per unit is the same for all levels of volume. In Chapter 2, we defined other cost behavior patterns, including semivariable costs and step costs.
We illustrate how more complicated cost structures can be analyzed by assuming that the fixed costs of U-Develop include the rental of equipment for photo developing and that the capacity of these machines is limited. Suppose, for example, that the fixed costs of $1,500 (from Exhibit 3.1) are sufficient for monthly volumes less than or equal to 5,000 prints. For every additional 5,000 prints, another machine, renting monthly for $480, is required. Now what is the break-even volume for U-Develop?
We know from our analysis earlier in the chapter that for a fixed cost of $1,500, the break-even point is 6,250 prints. But 6,250 prints cannot be developed without the additional machine. At a volume of 6,250 prints, U-Develop’s profit will be
Profit = ($0.60 * $0.36) * 6,250 – ($1,500 + $480) = ($480)
which is less than breakeven.
If we are going to have to sell more than 5,000 prints to break even, we are going to have to rent the additional machine. Therefore, to break even, our monthly fixed costs will be (at least) $1,980 ( = $1,500 + $480). At this level of fixed costs, the break-even point is
Break-even volume =
Fixed costs / Unit contribution margin = $1,980 / $0.24 = 8,250
which is less than 10,000 prints. Therefore, U-Develop can break even at a volume of 8,250 prints. If we had found that the new break-even point was greater than 10,000 prints, we would have repeated the analysis, adding another $480 for an additional machine.
Assumptions and Limitations of CVP Analysis
As with all methods of analysis, CVP analysis relies on certain assumptions and these assumptions might limit the applicability of the results for decision making. It is important to understand, however, that the limitations are due to the assumptions that the cost analyst makes; that is, they are not inherent limitations to the method of CVP analysis itself.