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Cost-Volume-Profit Relationships
Learning Objectives
1. Explain how changes in activity affect contribution margin and net operating income.
2. Prepare and interpret a cost-volume-profit (CVP) graph.
3. Use the contribution margin ratio (CM ratio) to compute changes in contribution margin and net operating income resulting from changes in sales volume.
4. Show the effects on contribution margin of changes in variable costs, fixed costs, selling price, and volume.
5. Compute the break-even point in unit sales and sales dollars.
6. Determine the level of sales needed to achieve a desired target profit.
7. Compute the margin of safety and explain its significance.
8. Compute the degree of operating leverage at a particular level of sales, and explain how the degree of operating leverage can be used to predict changes in net operating income.
9. Compute the break-even point for a multiple product company and explain the effects of shifts in the sales mix on contribution margin and the break-even point.
Chapter Overview
A. The Basics of Cost-Volume-Profit (CVP) Analysis. Cost-volume-profit (CVP) analysis is a key step in many decisions. CVP analysis involves specifying a model of the relations among the prices of products, the volume or level of activity, unit variable costs, total fixed costs, and the sales mix. This model is used to predict the impact on profits of changes in those parameters.
1. Contribution Margin. Contribution margin is the amount remaining from sales revenue after variable expenses have been deducted. It contributes towards covering fixed costs and then towards profit.
2. Unit Contribution Margin. The unit contribution margin can be used to predict changes in total contribution margin as a result of changes in the unit sales of a product. To do this, the unit contribution margin is simply multiplied by the change in unit sales. Assuming no change in fixed costs, the change in total contribution margin falls directly to the bottom line as a change in profits.
3. Contribution Margin Ratio. The contribution margin (CM) ratio is the ratio of the contribution margin to total sales. It shows how the contribution margin is affected by a given dollar change in total sales. The contribution margin ratio is often easier to work with than the unit contribution margin, particularly when a company has many products. This is because the contribution margin ratio is denominated in sales dollars, which is a convenient way to express activity in multi-product firms.
B. Some Applications of CVP Concepts. CVP analysis is typically used to estimate the impact on profits of changes in selling price, variable cost per unit, sales volume, and total fixed costs. CVP analysis can be used to estimate the effect on profit of a change in any one (or any combination) of these parameters. A variety of examples of applications of CVP are provided in the text.
C. CVP Relationships in Graphic Form. CVP graphs can be used to gain insight into the behavior of expenses and profits. The basic CVP graph is drawn with dollars on the vertical axis and unit sales on the horizontal axis. Total fixed expense is drawn first and then variable expense is added to the fixed expense to draw the total expense line. Finally, the total revenue line is drawn. The total profit (or loss) is the vertical difference between the total revenue and total expense lines. The break-even occurs at the point where the total revenue and total expenses lines cross.
D. Break-Even Analysis and Target Profit Analysis. Target profit analysis is concerned with estimating the level of sales required to attain a specified target profit. Break-even analysis is a special case of target profit analysis in which the target profit is zero.
1. Basic CVP equations. Both the equation and contribution (formula) methods of break-even and target profit analysis are based on the contribution approach to the income statement. The format of this statement can be expressed in equation form as:
Profits = Sales - Variable expenses - Fixed expenses
In CVP analysis this equation is commonly rearranged and expressed as:
Sales = Variable expenses + Fixed expenses + Profits
a. The above equation can be expressed in terms of unit sales as follows:
Price x Unit sales = Unit variable cost x Unit sales + Fixed expenses + Profits
Unit contribution margin x Unit sales = Fixed expenses + Profits
Unit sales = 
b. The basic equation can also be expressed in terms of sales dollars using the variable expense ratio:
Sales = Variable expense ratio x Sales + Fixed expenses + Profits
(1 - Variable expense ratio) x Sales = Fixed expenses + Profits
Contribution margin ratio* x Sales = Fixed expenses + Profits
Sales = 
* 1 - Variable expense ratio = 1 - 
= 
=Contributionmargin/Sales
=Contributionmarginratio
2. Break-even point using the equation method. The break-even point is the level of sales at which profit is zero. It can also be defined as the point where total sales equals total expenses or as the point where total contribution margin equals total fixed expenses. Break-even analysis can be approached either by the equation method or by the contribution margin method. The two methods are logically equivalent.
a. The Equation Method—Solving for the Break-Even Unit Sales. This method involves following the steps in section (1a) above. Substitute the selling price, unit variable cost and fixed expense in the first equation and set profits equal to zero. Then solve for the unit sales.
b. The Equation Method—Solving for the Break-Even Sales in Dollars. This method involves following the steps in section (1b) above. Substitute the variable expense ratio and fixed expenses in the first equation and set profits equal to zero. Then solve for the sales.
3. Break-even point using the contribution method. This is a short-cut method that jumps directly to the solution, bypassing the intermediate algebraic steps.
a. The Contribution Method—Solving for the Break-Even Unit Sales. This method involves using the final formula for unit sales in section (1a) above. Set profits equal to zero in the formula.
Break-even unit sales =  = 
b. The Contribution Method—Solving for the Break-Even Sales in Dollars. This method involves using the final formula for sales in section (1b) above. Set profits equal to zero in the formula.
Break-even sales =  = 
4. Target profit analysis. Either the equation method or the contribution margin method can be used to find the number of units that must be sold to attain a target profit. In the case of the contribution margin method, the formulas are:
Unit sales to attain target profits = 
Dollar sales to attain target profits = 
Note that these formulas are the same as the break-even formulas if the target profit is zero.
E. Margin of Safety. The margin of safety is the excess of budgeted (or actual) sales over the break-even volume of sales. It is the amount by which sales can drop before losses begin to be incurred. The margin of safety can be computed in terms of dollars:
Margin of safety in dollars = Total sales – Break-even sales
or in percentage form:
Margin of safety percentage = 
F. Cost Structure. Cost structure refers to the relative proportion of fixed and variable costs in an organization. Understanding a company’s cost structure is important for decision-making as well as for analysis of performance.
G. Operating Leverage. Operating leverage is a measure of how sensitive net operating income is to a given percentage change in sales.
1.Degree of operating leverage. The degree of operating leverage at a given level of sales is computed as follows:
2.The math underlying the degree of operating leverage. The degree of operating leverage can be used to estimate how a given percentage change in sales volume will affect net income at a given level of sales, assuming there is no change in fixed expenses. To verify this, consider the following:
 = 
= 
= 
= 
= 
= 
= Percentage change in net operating income
Thus, providing that fixed expenses are not affected and the other assumptions of CVP analysis are valid, the degree of operating leverage provides a quick way to predict the percentage effect on profits of a given percentage increase in sales. The higher the degree of operating leverage, the larger the increase in net operating income.
3.Degree of operating leverage is not constant. The degree of operating leverage is not constant as the level of sales changes. For example, at the break-even point the degree of operating leverage is infinite since the denominator of the ratio is zero. Therefore, the degree of operating leverage should be used with some caution and should be recomputed for each level of starting sales.
4. Operating leverage and cost structure. Richard Lord, “Interpreting and Measuring Operating Leverage,” Issues in Accounting Education, Fall 1995, pp. 31xx-229, points out that the relation between operating leverage and the cost structure of the company is contingent. It is difficult, for example, to infer the relative proportions of fixed and variable costs in the cost structures of any two companies just by comparing their operating leverages. We can, however, say that if two single-product companies have the same profit, the same selling price, the same unit sales, and the same total expenses, then the company with the higher operating leverage will have a higher proportion of fixed costs in its cost structure. If they do not have the same profit, the same unit sales, the same selling price, and the same total expenses, we cannot safely make this inference about their cost structure. All of the statements in the text about operating leverage and cost structure assume that the companies being compared are identical except for the proportions of fixed and variable costs in their cost structures.
H. Structuring Sales Commissions. Students may have a tendency to overlook the importance of this section due to its brevity. You may want to discuss with your students how salespeople are ordinarily compensated (salary plus commissions based on sales) and how this can lead to dysfunctional behavior. For example, would a company make more money if its salespeople steered customers toward Model A or Model B as described below?
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Model A |
Model B |
| Price |
$100 |
$150 |
| Variable cost |
75 |
130 |
| Unit CM |
$ 25 |
$ 20 |
Which model will salespeople push hardest if they are paid a commission of 10% of sales revenue?
I. Sales Mix. Sales mix is the relative proportions in which a company’s products are sold. Most companies have a number of products with differing contribution margins. Thus, changes in the sales mix can cause variations in a company’s profits. As a result, the break-even point in a multi-product company is dependent on the sales mix.
1.Constantsales mix assumption. In CVP analysis, it is usually assumed that the sales mix will not change. Under this assumption, the break-even level of sales dollars can be computed using the overall contribution margin (CM) ratio. In essence, it is assumed that the company has only one product that consists of a basket of its various products in a specified proportion. The contribution margin ratio of this basket can be easily computed by dividing the total contribution margin of all products by total sales.
Overall CM ratio = 
2. Use of the overall CM ratio. The overall contribution margin ratio can be used in CVP analysis exactly like the contribution margin ratio for a single product company. For a multi-product company the formulas for break-even sales dollars and the sales required to attain a target profit are: Break-even sales = 
Sales to achieve target profits = 
Note that these formulas are really the same as for the single product case. The constant sales mix assumption allows us to use the same simple formulas.
3. Changes in sales mix. If the proportions in which products are sold change, then the overall contribution margin ratio will change. Since the sales mix is not in reality constant, the results of CVP analysis should be viewed with more caution in multi-product companies than in single product companies.
J. Assumptions in CVP Analysis. Simple CVP analysis relies on simplifying assumptions. However, if a manager knows that one of the assumptions is violated, the CVP analysis can often be easily modified to make it more realistic.
1. Selling price is constant. The assumption is that the selling price of a product will not change as the unit volume changes. This is not wholly realistic since unit sales and the selling price are usually inversely related. In order to increase volume it is often necessary to drop the price. However, CVP analysis can easily accommodate more realistic assumptions. A number of examples and problems in the text show how to use CVP analysis to investigate situations in which prices are changed.
2. Costs are linear and can be accurately divided into variable and fixed elements. It is assumed that the variable element is constant per unit and the fixed element is constant in total. This implies that operating conditions are stable. It also implies that the fixed costs are really fixed. When volume changes dramatically, this assumption becomes tenuous. Nevertheless, if the effects of a decision on fixed costs can be estimated, this can be explicitly taken into account in CVP analysis. A number of examples and problems in the text show how to use CVP analysis when fixed costs are affected.
3. The sales mix is constant in multi-product companies. This assumption is invoked so as to use the simple break-even and target profit formulas in multi-product companies. If unit contribution margins are fairly uniform across products, violations of this assumption will not be important. However, if unit contribution margins differ a great deal, then changes in the sales mix can have a big impact on the overall contribution margin ratio and hence on the results of CVP analysis. If a manager can predict how the sales mix will change, then a more refined CVP analysis can be performed in which the individual contribution margins of products are computed.
4. In manufacturing companies, inventories do not change. It is assumed that everything the company produces is sold in the same period. Violations of this assumption result in discrepancies between financial accounting net operating income and the profits calculated using the contribution approach. This topic is covered in detail in the chapter on variable costing.
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