# Cost - Volume - Profit Analysis - Explained

What is Cost Volume Analysis?

# What is Cost-Volume-Profit Analysis for Single-Product Company?

The profit equation shows that profit equals total revenues minus total variable costs and total fixed costs.

Profit = Revenue - Total Variable Costs - Total Fixed Costs

This profit equation is used extensively in cost-volume-profit (CVP) analysis, and the information in the profit equation is typically presented in the form of a contribution margin income statement.

Recall that the contribution margin income statement starts with sales, deducts variable costs to determine the contribution margin, and deducts fixed costs to arrive at profit.

We use the term “variable cost” because it describes a cost that varies in total with changes in volume of activity.

We use the term “fixed cost” because it describes a cost that is fixed (does not change) in total with changes in volume of activity.

To allow for a mathematical approach to performing CVP analysis, the contribution margin income statement is converted to an equation using the following variables:

Key Equation

S = Selling price per unit

V = Variable cost per unit

F = Total fixed costs

Q = Quantity of units produced and sold

Thus Profit = Total sales − Total variable costs − Total fixed costs = (S×Q) − (V×Q)−F

Recall that when identifying cost behavior patterns, we assume that management is using the cost information to make short-term decisions.

Variable and fixed cost concepts are useful for short-term decision making.

The short-term period varies, depending on a company’s current production capacity and the time required to change capacity. In the long term, all cost behavior patterns are likely to change.

## Determining the Break-Even Point and Target Point

Companies often want to know the sales required to break even, which is called the break-even point.

The break-even point can be described either in units or in sales dollars.

The break- even point in units is the number of units that must be sold to achieve zero profit.

The break- even point in sales dollars is the total sales measured in dollars required to achieve zero profit.

If a company sells products or services easily measured in units (e.g., cars, computers, or mountain bikes), then the formula for break-even point in units is used.

If a company sells products or services not easily measured in units (e.g., restaurants, law firms, or electricians), then the formula for break-even point in sales dollars is used.

## How to Determine the Break-Even Point in Units?

The break-even point in units is found by setting profit to zero using the profit equation.

Profit = 0 = (S×Q) − (V×Q)−F

Once profit is set to zero, fill in the appropriate information for selling price per unit (S), variable cost per unit (V), and total fixed costs (F), and solve for the quantity of units produced and sold (Q).

## How to Determine the Target Profit in Units

Finding a target profit in units simply means that a company would like to know how many units of product must be sold to achieve a certain profit.

Finding the target profit in units is similar to finding the break-even point in units except that profit is no longer set to zero.

Instead, set the profit to the target profit the company would like to achieve. Then fill in the information for selling price per unit (S), variable cost per unit (V), and total fixed costs (F), and solve for the quantity of units produced and sold (Q):

Profit in \$ = (S×Q)−(V×Q)−F

You can confirm the finding by following contribution margin income statement.

## Break-Even Point in Sales Dollars

Often, break even cannot easily measure sales in units?

For these types of companies, the break-even point is measured in sales dollars.

That is, we determine the total revenue (total sales dollars) required to achieve zero profit for companies that cannot easily measure sales in units.

Finding the break-even point in sales dollars requires the introduction of two new terms: contribution margin per unit and contribution margin ratio.

The shortcut formula is as follows:

Q = (F + Target Profit) ÷ (S − V)

## Contribution Margin per Unit

The contribution margin per unit is the amount each unit sold contributes to (1) covering fixed costs and (2) increasing profit.

We calculate it by subtracting variable costs per unit (V) from the selling price per unit (S).

Key Equation:

Contribution margin per unit = S − V

## Contribution Margin Ratio (Margin of Safety)

The contribution margin ratio (often called contribution margin percent) is the contribution margin as a percentage of sales. It measures the amount each sales dollar contributes to (1) covering fixed costs and (2) increasing profit.

The contribution margin ratio is the contribution margin per unit divided by the selling price per unit.

(Note that the contribution margin ratio can also be calculated using the total contribution margin and total sales; the result is the same.)

With an understanding of the contribution margin and contribution margin ratio, we can now calculate the break-even point in sales dollars.

Key Equation:

Contribution margin ratio = (S − V) ÷ S

The formula to find the break-even point in sales dollars is as follows.

Break-even point in sales dollars=Total fixed costs + Target profitContribution margin ratio

## Target Profit in Sales Dollars

Finding a target profit in sales dollars simply means that a company would like to know total sales measured in dollars required to achieve a certain profit.

Finding the target profit in sales dollars is similar to finding the break-even point in sales dollars except that “target profit” is no longer set to zero. Instead, target profit is set to the profit the company would like to achieve.

Key Equation:

Target profit in sales dollars = (Total fixed costs + Target profit) / Contribution margin ratio

Instead of setting the target profit to \$0, set it to a target dollar profit.

This results in an answer of the amount of monthly sales needed to achieve this profit.

## Cost-Volume-Profit Analysis for Multiple-Product and Service Companies

The following information is required to find the break-even point for companies with more than one product:

Monthly fixed costs total .

Percent of total sales volume that each product represents

The unit selling price and variable cost information for the products.

## Finding the Break-Even Point and Target Profit in Units for Multiple-Product Companies

First, we must expand the profit equation presented earlier to include multiple products.

The following terms are used once again.

However, subscript r identifies one product, and subscript s identifies the second product.

(Note: Sr stands for the 1st model’s selling price per unit.)

CM is new to this section and represents the contribution margin.

Key Equation:

S = Selling price per unit

V = Variable cost per unit

F = Total fixed costs

Q =  Quantity of units produced and sold

CM = Contribution margin

Thus,

Profit = Total sales − Total variable costs − Total fixed costs [(Sr×Qr)+(Ss×Qs)] −[(Vr×Qr )+ (Vs × Qs)] − F

Without going through a detailed derivation, this equation can be restated in a simplified manner, as follows:

Profit = (Unit CM for r × Quantity of r) + (Unit CM for s × Quantity of s) − F

There actually are many different break-even points, because the profit equation has two unknown variables, Qr and Qs.

## Break-Even Point in Units and the Weighted Average Contribution Margin per Unit

Because most companies sell multiple products that have different selling prices and different variable costs, the break-even or target profit point depends on the sales mix.

The sales mix is the proportion of one product’s sales to total sales.

In calculating the break-even point, we must assume the sales mix for the r and s models will remain consistent with historical sales, respectively, at all different sales levels.

The formula used to solve for the break-even point in units for multiple-product companies is similar to the one used for a single-product company, with one change.

Instead of using the contribution margin per unit in the denominator, multiple-product companies use a weighted average contribution margin per unit.

The formula to find the break-even point in units is as follows.

Weighted Average Contribution margin = (Unit CM for r × Sales Mix % of r) + (Unit CM for s × Sales Mix % of s)

Break Even in Units = Fixed Costs / Weighted Average Contribution Margin

When a company assumes a constant sales mix, a weighted average contribution margin per unit can be calculated by multiplying each product’s unit contribution margin by its proportion of total sales.

## Target Profit in Units

We now know how to calculate the break-even point in units for a company with multiple products.

Finding the target profit in units for a company with multiple products is similar to finding the break-even point in units except that profit is no longer set to zero.

Instead, profit is set to the target profit the company would like to achieve.

## Break-Even Point in Sales Dollars and the Weighted Average Contribution Margin Ratio

For companies that have unique products not easily measured in units, it requires extra steps to find the break-even point.

Rather than measuring the break-even point in units, a more practical approach for these types of companies is to find the break-even point in sales dollars.

We can use the formula that follows to find the break-even point in sales dollars for organizations with multiple products or services.

Note that this formula is similar to the one used to find the break-even point in sales dollars for an organization with one product, except that the contribution margin ratio now becomes the weighted average contribution margin ratio.

Key Equation:

Break-even point in sales dollars = (Total fixed costs + Target profit) / Weighted average contribution margin ratio

The contribution margin ratio differs for each department:

If we have the contribution margin ratio for each department, we will need it for the company as a whole.

The contribution margin ratio for the company as a whole is the weighted average contribution margin ratio.

We calculate it by dividing the total contribution margin by total sales.

Weighted-Average Contribution Margin Ratio = Total Contribution Margin / Total Sales

This assumes that the sales mix remains the same at all levels of sales. (The sales mix here is measured in sales dollars for each department as a proportion of total sales dollars.)

Now that you know the weighted average contribution margin ratio, it is possible to calculate the break-even point in sales dollars.

## Target Profit in Sales Dollars

Finding the target profit in sales dollars for a company with multiple products or services is similar to finding the break-even point in sales dollars except that profit is no longer set to zero.

Instead, profit is set to the target profit the company would like to achieve.

Break- even point in sales dollars = (Total fixed costs + Target profit) / Weighted average contribution margin ratio

## Important Assumptions

Several assumptions are required to perform break-even and target profit calculations for companies with multiple products or services.

These assumptions are as follows:

• Costs can be separated into fixed and variable components.
• Contribution margin ratio remains constant for each product, segment, or department.
• Sales mix remains constant with changes in total sales.

However, these assumptions may not be realistic, particularly if significant changes are made to the organization’s operations.

When performing CVP analysis, it is important to consider the accuracy of these simplifying assumptions.

## Income Taxes and Cost-Volume-Profit Analysis

Some organizations, such as not-for-profit entities and governmental agencies, are not required to pay income taxes.

How do we find the target profit in units or sales dollars for organizations that pay income taxes?

Three steps are required:

Step 1. Determine the desired target profit after taxes.

Step 2. Convert the desired target profit after taxes to the target profit before taxesThe formula used to solve for target profit before taxes is as follows.

Target profit before taxes = Target profit after taxes ÷ (1 − tax rate)

Step 3. Use the target profit before taxes in the appropriate formula to calculate the target profit in units or sales dollars.

The formula used to solve for target profit in units is

(Total fixed costs + Target profit) / (Selling price per unit − Variable cost per unit)

## Using Cost-Volume-Profit Models for Sensitivity Analysis

We can use the cost-volume-profit (CVP) financial model described in this chapter for single-product, multiple-product, and service organizations to perform sensitivity analysis, also called what-if analysis.

Sensitivity analysis shows how the CVP model will change with changes in any of its variables (e.g., changes in fixed costs, variable costs, sales price, or sales mix).

The focus is typically on how changes in variables will alter profit.

Management makes a best guess of  units in monthly sales. This is called the “base case.”

A sensitivity analysis can answer the following questions for management

1. How will profit change if the sales price increases by \$___ per unit?

2. How will profit change if sales volume decreases by ___ units?

3. How will profit change if fixed costs decrease by \$_______ and variable cost increases \$______ per unit (10 percent)?

The CVP model answers these questions.

The best approach is to create a table representing the various scenarios (generally in Excel).

Each column represents a different scenario, with the first column showing the base case and the remaining columns providing answers to the three questions posed by management.

The top part shows the value of each variable based on the scenarios presented previously, and the bottom part presents the results in contribution margin income statement format.

Although the focus of sensitivity analysis is typically on how changes in variables will affect profit, accountants also use sensitivity analysis to determine the impact of changes in variables on the break-even point and target profit.

## Impact of Cost Structure on Cost-Volume-Profit Analysis

Cost structure is the term used to describe the proportion of fixed and variable costs to total costs.

Operating leverage refers to the level of fixed costs within an organization.

Companies with a relatively high proportion of fixed costs have high operating leverage.

Businesses that rely on direct labor and direct materials tend to have higher variable costs than fixed costs.

Operating leverage is an important concept because it affects how sensitive profits are to changes in sales volume.

This is best illustrated by comparing two companies with identical sales and profits but with different cost structures.

High Operating Leverage Company (HOLC) has relatively high fixed costs, and Low Operating Leverage Company (LOLC) has relatively low fixed costs.

One way to observe the importance of operating leverage is to compare the break-even point in sales dollars for each company.

Higher operating leverage can lead to higher profit. However, high operating leverage companies that encounter declining sales tend to feel the negative impact more than companies with low operating leverage.

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