Essentials of CVP Analysis

Learning Objective 1 Explain the features of cost–volume–profit (CVP) analysis

• how operating income changes with changes in output level, selling prices, variable costs, or fixed costs

In Chapter 2, we discussed total revenues, total costs, and income. Managers use cost–volume–profit (CVP) analysis to study the behavior of and relationship among these elements as changes occur in the number of units sold, the selling price, the variable cost per unit, or the fixed costs of a product. Consider this example:

Emma Jones is a young entrepreneur who recently used GMAT Success, a test-prep book and software package for the business school admission test. Emma loved the book and program so much that after graduating she signed a contract with GMAT Success’s publisher to sell the learning materials. She recently sold them at a college fair in Boston and is now thinking of selling them at a college fair in Chicago. Emma can purchase each package (book and software) from the publisher for $120 per package, with the privilege of returning all unsold packages and receiving a full $120 refund per package. She must pay $2,000 to rent a booth at the fair. She will incur no other costs. Should she rent the booth or not?

Emma, like most managers who face such a situation, works through the series of steps introduced in Chapter 1 to make the most profitable decisions.

1. IDENTIFY THE PROBLEM AND UNCERTAINTIES. Every managerial decision involves selecting a course of action. The problem of whether or not to rent the booth hinges on how Emma resolves two important uncertainties: the price she can charge and the number of packages she can sell at that price. Emma must decide knowing that the outcome of the action she chooses is uncertain. The more confident she is about selling a large number of packages at a high price, the more willing she will be to rent the booth.
2. OBTAIN INFORMATION. To better understand the uncertainties, Emma obtains information, for example, about the type of individuals likely to attend the fair and other test-prep packages that might be sold at the fair. She also collects data from the Boston fair.
3. MAKE PREDICTIONS ABOUT THE FUTURE. Emma predicts she can charge $200 for the GMAT Success package. At that price, she is reasonably confident that she will be able to sell at least 30 packages and possibly as many as 60. Emma must be realistic and exercise judgment when making these predictions. If they are too optimistic, she will rent the booth when she should not. If they are too pessimistic, she will not rent the booth when she should.

1. Emma believes that her experience at the Chicago fair will be similar to her experience at the Boston fair 4 months earlier. Yet Emma is uncertain about several aspects of her prediction. Are the fairs truly comparable? For example, will attendance at the two fairs be the same? Have market conditions changed over the past 4 months? Are biases creeping into her thinking? Is her keenness to sell at the Chicago fair because of lower-than-expected sales in the last couple of months leading to overly optimistic predictions? Has she ignored some competitive risks? Will other test-prep vendors at the fair reduce their prices? If they do, should she? How many packages can she expect to sell if she does?
2. Emma rethinks her plan and retests her assumptions. She obtains data about student attendance and sales of similar products in past years. She feels confident that her predictions are reasonable, accurate, and carefully thought through.

4. MAKE DECISIONS BY CHOOSING AMONG ALTERNATIVES. Emma uses the CVP analysis that follows and decides to rent the booth at the Chicago fair.
5. IMPLEMENT THE DECISION, EVALUATE PERFORMANCE, AND LEARN. At the end of the Chicago fair, Emma compares actual performance to predicted performance to understand why things worked out the way they did. For example, Emma evaluates whether her predictions about price and the number of packages she could sell were correct. This helps her learn and make better decisions about renting booths at future fairs.

As we described in Chapter 1, machine learning and data analytics can help Emma in several of these steps. Emma can store information on multiple factors, such as details about individuals who have attended similar fairs, the number of test-prep packages sold at different prices at these fairs, and characteristics such as weather, market conditions, number of competitive vendors, and location. Using this information, a data-analytic model can predict how many packages Emma might expect to sell at different prices. The model is free of some human bias because lower-than-expected sales in the last 2 months is not a feature of the model. The experience from the Chicago fair (how well the model predicted actual outcomes) becomes an input into the model and helps to improve and refine it. The machine and the model learn from each experience.

But how does Emma use CVP analysis in Step 4 to make her decision? She begins by identifying which costs are fixed and which costs are variable and then calculates contribution margin.

Contribution Margin

The booth-rental cost of $2,000 is a fixed cost because it is the same no matter how many packages Emma sells. The cost of the packages is a variable cost because it increases in proportion to the number of packages sold. Emma returns whatever she doesn’t sell for a full refund.

To understand how operating income will change with different quantities of packages sold, Emma calculates operating income for sales of 5 packages and 40 packages.

The only numbers that change as a result of selling different quantities of packages are total revenues and total variable costs. The difference between total revenues and total variable costs is called contribution margin. That is,

Contribution margin explains why operating income increases by $2,800 from a loss of $(1,600) to income of $1,200 as the number of units sold increases from 5 packages to 40 packages. The contribution margin when Emma sells 5 packages is $400 ($1,000 in total revenues minus $600 in total variable costs); the contribution margin when Emma sells 40 packages is $3,200 ($8,000 in total revenues minus $4,800 in total variable costs), an increase of $\mathrm{\$}\text{2,800}\mathrm{\hspace{0.17em}}(\mathrm{\$}\text{3,200}-\mathrm{\$}400)\mathrm{.}$ When calculating the contribution margin, be sure to subtract all variable costs. For example, if Emma incurs variable selling costs from commissions paid to salespeople for each package sold at the fair, variable costs would include the cost of each package plus the sales commission paid on it.

Contribution margin per unit is a useful tool for calculating contribution margin and operating income. It is defined as:

In the GMAT Success example, the contribution margin per package, or per unit, is $\$200-\$120=\$80.$ Contribution margin per unit recognizes the tight coupling of selling price and variable cost per unit. Unlike fixed costs, Emma will only incur the variable cost per unit of $120 when she sells a package of GMAT Success.

Contribution margin per unit provides a second way to calculate contribution margin:

For example, when Emma sells 40 packages, contribution  $\text{margin}=\mathrm{\$}80\text{per}\text{unit}\times$ $40\text{units}=\$\text{3,200}.$

Even before she gets to the fair, Emma incurs $2,000 in fixed costs. Because the contribution margin per unit is $80, Emma will recover $80 for each package that she sells at the fair. Emma hopes to sell enough packages to fully recover the $2,000 she spent renting the booth and to then make a profit.

To get a feel for how operating income will change for different quantities of packages sold, Emma can prepare a contribution income statement as in Exhibit 3-1. The income statement in Exhibit 3-1 is called a contribution income statement because it groups costs into variable costs and fixed costs to highlight contribution margin.

Each additional package sold from 0 to 1 to 5 increases contribution margin by $80 per package and helps Emma recover more and more of her fixed costs and reduce her operating loss. If Emma sells 25 packages, contribution margin equals $\mathrm{\$}\text{2,000}\mathrm{\hspace{0.17em}}(\mathrm{\$}80\text{per}\text{package}\times 25\text{packages})\mathrm{.}$ This quantity exactly recovers her fixed costs and results in $0 operating income. If Emma sells 40 packages, contribution margin increases by another $\mathrm{\$}\text{1,200}\mathrm{\hspace{0.17em}}(\mathrm{\$}\text{3,200}-\mathrm{\$}\text{2,000})\mathrm{,}$ all of which becomes operating income. As you look across Exhibit 3-1 from left to right, you see that the increase in contribution margin exactly equals the increase in operating income (or the decrease in operating loss).

When companies such as Samsung and Prada sell multiple products, calculating contribution margin per unit is cumbersome. Instead of expressing contribution margin in dollars per unit, these companies express it as a percentage called contribution margin percentage (or contribution margin ratio):

Consider a sales level such as the 40 units sold in Exhibit 3-1:

Contribution margin percentage is the contribution margin per dollar of revenue. Emma earns 40% for each dollar of revenue (40¢) she takes in. Contribution margin percentage is a handy way to calculate contribution margin for different dollar amounts of revenue. Rearranging terms in the equation defining contribution margin percentage, we get

To derive the relationship between operating income and contribution margin percentage, recall that

Substituting for contribution margin in the above equation:

For example, in Exhibit 3-1, if Emma sells 40 packages,

When there is only one product, as in our example, we can divide both the numerator and denominator of the contribution margin percentage equation by the quantity of units sold and calculate contribution margin percentage as follows:

In our example,

Contribution margin percentage is a useful tool for calculating how a change in revenues changes contribution margin. As Emma’s revenues increase by $3,000 from $5,000 to $8,000, her contribution margin increases from $2,000 to $3,200 (by $1,200):

Contribution margin analysis is a widely used technique. For example, managers at Home Depot use contribution margin analysis to evaluate how sales fluctuations during a recession will affect the company’s profitability.