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Cost-Volume-Profit Analysis:
A Managerial Planning Tool
Learning Objectives
1. Determine the break-even point in number of units and in
total sales dollars.
2. Determine the number of units that must be sold, and the amount
of revenue required, to earn a targeted profit.
3. Prepare a profit-volume graph and a cost-volume-profit graph and
explain the meaning of each.
4. Apply cost-volume-profit analysis in a multiple-product setting.
5. Explain the impact of risk, uncertainty, and changing variables
on cost-volume-profit analysis.
Lecture Notes
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, the unit variable costs, the total fixed
costs, and the mix of products sold. 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. When there is a single product,
the unit contribution margin can be used to predict changes in the
contribution margin and in profits (assuming there is no change in
fixed costs) as a result of changes in unit sales. To do this, the
unit contribution margin is simply multiplied by the change in unit
sales.
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. Managers often find the contribution margin ratio easier
to work with than the unit contribution margin, particularly when
a company has multiple 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 profits 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. 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 as 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:
Price x Unit sales = Unit variable cost x Unit sales + Fixed expenses
+ Profits
Unit contribution margin x Unit sales = Fixed expenses + Profits
Unit sales = (Fixed expenses + Profits) / (Unit contribution
margin)
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
ß
(1 - Variable expense ratio) x Sales = Fixed expenses + Profits
ß
Contribution margin ratio* x Sales = Fixed expenses + Profits
ß
Sales = (Fixed expenses + Profits) / (Contribution margin ratio)
* 1 - Variable expense ratio = 1- (Variable expenses) / (Sales)
= (Sales - Variable expenses) / Sales
= (Contribution Margin / Sales
= Contribution margin ratio
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 revenue 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. Simply
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 as indicated.
b. The Equation Method-Solving for the Break-Even Sales in Dollars.
This method involves following the steps in section (1b) above.
Simply substitute the variable expense ratio and fixed expenses
in the first equation and set profits equal to zero. Then solve
for the sales as indicated.
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. Simply set profits equal to zero in the formula.
Break-even unit sales = (Fixed expenses + 0) / Unit
contribution margin
= Fixed expenses / Unit contribution margin
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. Simply set profits equal to zero in the formula.
Break-even sales = (Fixed expenses + 0) / Contribution margin ratio
= Fixed expenses / Contribution margin ratio
4. Target profit analysis. Managers frequently desire to know
the number of units that must be sold to attain a target profit. Either
the equation method or the contribution margin method discussed previously
can be used. In the case of the contribution margin method, the formulas
are:
Unit sales to attain target profits = (Fixed expenses + Target profits)
/ Unit contribution margin
Sales to achieve target profits = (Fixed expenses + target profits
/ Contribution margin ratio
Note that these formulas are the same as the break-even formulas
if the target profit is zero.
D. CVP Relationships in Graphic Form. Graphs of CVP relationships
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 volume
in units on the horizontal axis. Total fixed expense is drawn first,
then variable expense is added to the fixed expense in order 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.
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 = Margin of safety in dollars / Total
sales
F. Cost Structure. Cost structure refers to the relative proportion
of fixed and variable costs in an organization. The best cost structure
for a company depends on many factors, including the long-run trend
in sales, year-to-year fluctuations in the level of sales, and the attitudes
of owners and managers towards risk. 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 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:
Degree of operating leverage = Contribution margin / Net income
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:
Degree of operating leverage x Percentage change in sales =
(Contribution margin/Net income) x ((New sales - Sales)/ Sales)
=
(Contribution margin/Sales) x ((New sales - Sales)/ Net Income)
=
Contribution Margin ratio x ((New sales - Sales)/ Net Income) =
((CM ratio x New sales) - (CM ratio * Sales) )/ Net Income =
(New contribution margin - Contribution margin) / Net Income =
Change in net income / Net Income
= Percentage change in net income
Thus, providing that there is no change in fixed expenses 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 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. 316-329, 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 two companies are being compared that are
identical except for the proportions of fixed and variable costs in
their cost structures.
5. Automation and CVP analysis. The move toward greater automation
and other trends in the economy have resulted in a shift toward greater
fixed costs relative to variable costs. In turn, this shift in cost
structure has had an impact on the break-even point, CM ratios, and
other CVP factors. These impacts are summarized in Exhibit 6-3 in
the text.
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 lower profits. For example, would a firm make more money if
its salespeople steered customers toward Model A or Model B as described
below?
|
Model A |
Model B |
| Price ..............$100 |
Price ...............$150 |
| Variable cost ......75 |
Variable cost .....130 |
| Unit CM .........$ 25 |
Unit CM ...........$ 20 |
Which model will salespeople push hardest if they are paid a
commission of 10% of sales revenue?
I. Sales Mix. (The term sales mix means 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
mix in which the various products are sold.
Different products typically have different selling prices, costs,
and contribution margins.
1. Constant sales mix assumption. The assumption is
usually made in CVP analysis that the sales mix will not change. Under
the constant sales mix assumption, the break-even level of sales dollars
can be computed using the overall contribution margin (CM) ratio. In
essence, the assumption is made that the firm 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 = Total contribution margin/ Total sales
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 firm. For a multi-product firm the formulas
for break-even sales dollars and the sales required to attain a target
profit are:
Break-even sales = Fixed expenses / Overall CM ratio
Sales to achieve target profits = (Fixed expenses
+ Target profits) / Overall CM ratio
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 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 firms than in
single product firms.
J. Limiting 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 there is usually an inverse relationship
between price and unit volume. 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 and there are no major changes
in worker efficiency. It also implies that the fixed costs are really
fixed. When there are large changes in volume, this assumption becomes
tenuous. However, if a manager is able to estimate the effects of
a decision on fixed costs, these estimates can be explicitly taken
into account in CVP analysis. Again, 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 in order to use the simple break-even and target
profit formulas in multi-product firms. 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 upon 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 income and the profits calculated
using the contribution approach. This topic is covered in detail in
the next chapter.
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