Cost-volume-profit analysis (CVP) is a tool used to estimate how changes in cost, sales volume, and price affect a company's profit. CVP is one of the most versatile and widely applicable tools used by managerial accountants to help managers make better decisions. CVP is used to find important benchmarks, such as the break-even point. The break-even point is the point where total revenue is equal to total profit (the point of zero profit). Companies want to make a profit, however, they also want to make sure they don't have a loss. The break-even point shows the point where the company will not have a loss.

**Operating Income in CVP Analysis**

Operating income is the total revenue minus total expense. For the income statement, expenses are classified according to function (manufacturing, selling and administrative). For CVP analysis, it's more useful to classify the expenses into fixed and variable components. The income statement that is formatted based on the separation of costs into fixed and variable components is called the contribution margin income statement.

Sales | $XXX |

Less: Total Variable Expense | __(XXX)__ |

Total Contribution Margin | $XXX |

Less: Total Fixed Expense | __(XXX)__ |

Operating Income | $XXX |

Contribution margin is the difference between sales and variable expense. It is the amount of sales revenue that is left over after all the variable expenses are covered, that can be used to cover the fixed cost. Subtracting the fixed costs from the contribution margin will give you the operating income. The formula for this reads as:

Sales - Variable Expenses = Contribution Margin

Contribution Margin - Fixed Expenses = Operating Income

This also allows you to rewrite the formula for operating income as:

Operating Income = Sales - Variable Expenses - Fixed Expenses

This formula can be expanded by expressing sales revenues and variable expenses in terms of price per unit and units sold. Sales revenue is equal to the unit selling price times the number of units sold.

Operating Income = (Price per unit X Units Sold) - (Variable cost per unit X Units Sold) - Total Fixed Expenses

At the break-even point, the operating income will be at $0. This formula can be used to find the number of units needed to be sold to get to a break-even point.

**Break-Even Point in Units Sold**

Assume Calbreath Computers sells each computer at $400, and each has a variable cost of $325 per computer. Total fixed costs are $45,000. We put the numbers into our formula, with an operating income of $0.

($400 X units sold) - ($325 X units sold) - $45,000 = $0

($75 x units sold) - $45,000 = $0

$75 x units sold = $45,000

units sold = $45,000 / $75 = 600

In the example above, Calbreath Computers must sell 600 computers to reach their break-even point.

As you can see, the break-even point in units can best be summarized as:

Break-even units = Total Fixed Costs / (Price - Variable Cost per unit)

Remember that sales minus variable expenses is the contribution margin. So this can also be expressed as:

Break-even units = Total Fixed Costs / Contribution Margin

**Break-Even Point in Sales Dollars**

Sometimes, CVP analysis will be used to calculate the sales revenue as the measure of sales activity instead of units sold. Any answer expressed in units sold can be easily converted to one expressed in sales revenues. When using CVP to calculate the break-even point in dollars, total variable costs are defined as a percentage rather than as a dollar amount. For example, a company has a product that is selling at $10 per unit. The variable cost is $6 (thus making the contribution margin $4). The variable cost and contribution margin can then be shown as a variable cost ratio and contribution margin ratio.

Variable Cost Ratio = Variable Cost / Sales

Contribution Margin Ratio = Contribution Margin / Sales

With the above example, the two would be shown as:

Variable Cost Ratio = $6 / $10 = 60%

Contribution Margin Ratio = $4 / $10 = 40%

Now let's look at the equation for calculating the break-even point in sales dollars using these ratios. Going back to the computers that Calbreath Computers was selling, we can determine the ratios as follows:

Variable Cost Ratio = $325 / $400 = 0.8125 = 81.25%

Contribution Margin Ratio = $75 / $400 = 0.1875 = 18.75%

Sales - Total variable expenses - Total fixed expenses = Operating Income

Break-even sales - (0.8125 X Break-Even Sales) - $45,000 = $0

Break-even sales = $45,000 / (1 - 0.8125)

Break-even sales = $240,000

So, Calbreath Computers has to have sales of $240,000 to reach break-even point.

Did you notice also that the next to last statement of the formula showed another way of looking at this?

Break-even sales = $45,000 / (1 - 0.8125)

Break-even sales = Total fixed costs / (1 - variable cost ratio)

Since the variable cost ratio and contribution margin ratio add up to 100% of the sales, the formula can be simplified as the following:

Break-even sales = Total fixed expenses / Contribution Margin Ratio

**Units and Sales Dollars Needed To Achieve a Target Income**

So far we've been looking at a break-even point, where the target income is $0. Most companies, however, want to make a profit. For them, their target income is higher than $0. The exact same formulas are used as above, except that we have a target operating income of a number over $0.

Target Income units = (Total Fixed Costs + Target Income) / Contribution Margin

Target Income sales = (Total fixed Costs + Target Income) / Contribution Margin Ratio

For example, if Calbreath Computers wanted to have a target income of $37,500, then the formulas would be rewritten as follows:

**Units to sell for a target income**

($400 X units sold) - ($325 X units sold) - $45,000 = $37,500

($75 x units sold) - $45,000 = $37,500

$75 x units sold = $45,000 + $37,500

units sold = $82,500 / $75 = 1,100

**Sales dollars for a target income**

Sales - Total variable expenses - Total fixed expenses = Operating Income

Target income sales - (0.8125 X Target income sales) - $45,000 = $37.500

Target income sales = ($45,000 + $37,500) / (1 - 0.8125)

Target income sales = $82,500 / 0.1875

Target income sales = $440,000

**Graphs**

**Profit-Volume Graph**

A profit-volume graph visually portrays the relationship between profits (operating income) and units sold. In this graph, operating income is the dependent variable, and units is the independent variable. Units are plotted along the horizontal axis. Operating income is plotted along the vertical axis. For example, a company has fixed costs of $100, variable costs per unit of $5 and a selling price of $10.

The point on the graph where the line crosses the x-axis is the break-even point (at $20). If no units are sold, the company will record a loss of $100.

**Cost-Volume-Profit Graph**

The cost-volume-profit graph visually portrays the relationships among cost, volume and profits. This is done by plotting two separate lines: the total revenue line and the total cost line. The vertical axis is measured in dollars. The horizontal axis is measured in units sold.

**Assumptions of CVP Analysis**

Both graphs have certain assumptions that must be made in order for them to be accurate.

- Revenue and Cost Functions are linear
- Price, total fixed costs, and unit variable costs can be identified and remain constant over relevant range
- All units produced are sold - there are no change in inventory levels
- Sales mix is constant
- Selling prices and costs are known with certainty

Linear Cost and Revenue Functions - CVP assumes that cost and revenue functions are linear. In other words, they are straight lines.

Production is Equal to Sales - CVP assumes that all units produced are sold, and inventory levels do not change over the period. CVP focuses on current costs by excluding inventory costs of previous periods.

Constant Sales Mix - In a single product analysis, sales mix is obvious - 1 product accounts for 100 percent of sales. With multiple products, it is virtually impossible to predict with certainty the sales mix. CVP assumes that the sales mix stays constant. With two products, for example, for every 3 units of product A sold, 2 units of product B are sold.

Certainty of Prices and Costs - Firms seldom know prices, variable costs, and fixed costs with certainty.

**Multiple Product Analysis**

Multiple product analysis is computed in the same way as the single product analysis above. However, the products have to be put together into a package to determine the variable costs per unit, and the selling price per unit.

Calbreath Computers sells computers, and also sells computer servers. The fixed costs for the company total $96,250. The costs for each product are as follows:

Product | Selling Price | Variable Cost |

Computer | $400 | $325 |

Server | $800 | $600 |

The expected sales mix is 3 computers for every 2 servers sold.

Product | Price | Unit Variable Cost
| Unit Contribution Margin | Sales Mix | Package Unit Contribution |

Computer | $400 | $325 | $ 75 | 3 | $225 |

Server | $800 | $600 | $200 | 2 | $400 |

Package Total | $625 |

For every 1 unit of this sales package, the contribution margin is $625.

Break-Even Point = Fixed Costs / Contribution Margin

Break-Even Point = $96,250 / $625

Break-Even Point = 154 packages

Since there are 3 computers and 2 servers in each package, the break-even point for Calbreath Computers is at (154 X 3) 462 computers and (154 X 2) 308 servers.

Fixed costs above were calculated as one total number (Total fixed costs). When working with multiple products, the individual fixed costs will fall into one of two categories: direct fixed expenses and common fixed expenses. Direct fixed expenses are the fixed costs that can be traded to each segment and would be avoided if the segment did not exist. An example is the supervisor that watches over a specific product line. Common fixed expenses are the fixed costs that are not traceable to the segments and would remain even if one of the segments was eliminated. An example is the CEO's salary.

The break-even point in dollars is calculated the same way as with single product analysis. The formula is still:

Break-even sales = Total fixed expenses / Contribution Margin Ratio

To compute the contribution ratio, we have to remember that there are three computers and two servers in each unit.

Contribution Margin Ratio = Contribution Margin / Sales

Contribution Margin Ratio = $625 / (($400 X 3) + ($800 X 2))

Contribution Margin Ratio = $625 / $2800

Contribution Margin Ratio = 22% (rounded)

Break-even sales = $96,250 / .22

Break-even sales = $437,500

**CVP Analysis: Risk and Uncertainty**

The break-even point, and the sales mix, can be affected by changes in price, unit contribution margin, or fixed costs. An important assumption of CVP is that prices and costs re known with certainty, which is seldom accurate. Because of this, managers must use sensitivity, or what-if analysis, to find an area of sales that are break-even. Rather than having a break-even point, they will have a "break-even band".

One crude area of risk is the margin of safety. This is the units sold or the revenue earned above the break-even volume. Calbreath Computers plans to sell 1,000 computers at $400 each. Remember, we calculated earlier that the variable costs are $325, fixed costs are $45,000 and break-even units is at 600.

Margin of safety in units = Sales in units - Break-even units

Margin of safety in units = 1,000 - 600

Margin of safety in units = 400

Margin of safety in dollars = sales - Break-even sales

Margin of safety in dollars = ($400 X 1,000) - ($400 X 600)

Margin of safety in dollars = $400,000 - $240,000

Margin of safety in dollars = 160,000

Operating leverage is the use of fixed costs to extract higher percentage changes in profits as sales activity changes. Higher proportions of fixed costs to the amount of variable costs create higher operating leverage. The greater the degree of operating leverage, the larger the effect on operating income when sales change. The degree of operating leverage (DOL) can be measured for a given level of sales by taking the ratio of contribution margin to operating income.

Degree of operating leverage = Contribution Margin / Operating Income

Calbreath Computers has an operating income at 1,000 computers of $30,000. The DOL would be:

Degree of operating leverage = Contribution Margin / Operating Income

Degree of operating leverage = (($400 - $325) X (1,000)) / $30,000

Degree of operating leverage = 2.5

The greater the degree of operating leverage, the more that changes in sales will affect operating income. Because of this, the mix of fixed and variable costs an organization chooses can have a considerable influence on operating risk and profit level. A company's mix of fixed costs relative to variable costs is referred to as its cost structure.

The degree of operating leverage can be used to calculate the change in operating income that would result from a given percentage change in sales. The percentage change in operating income would be calculated, and then added to the current operating income to find the expected operating income. For example, if Calbreath Computers were expecting an increase of 20% in sales next year, then the affect on operating income would be:

Percentage change in operating income = DOL X Percent change in sales

Percentage change in operating income = 2.5 X 20%

Percentage change in operating income = 50%

Expected operating income = Operating income + (Percentage change in operating income X Operating income)

Expected operating income = $30,000 + (0.5 X $30,000)

Expected operating income = $30,000 + $15,000

Expected operating income = $45,000