# 6.1: Cost-Volume-Profit Analysis for Single-Product Companies

Skills to Develop

• Perform cost-volume-profit analysis for single-product companies.

Question: The profit equation1 shows that profit equals total revenues minus total variable costs and 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 (first introduced in Chapter 5). What is the relationship between the profit equation and the 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:

$$\text{S = Selling price}\; per\; unit$$

$$\text{V = Variable cost}\; per\; unit$$

$$\text{S = Selling price}\; per\; unit$$

$$\text{F} = Total\; \text{fixed costs}$$

$$\text{Q = Quantity of units produced and sold}$$

Thus

$$\begin{split} \text{Profit} &= \text{Total sales − Total variable costs − Total fixed costs} \\ &= (S \times Q) - (V \times Q) - F \end{split}$$

Figure 6.1 clarifies the link between the contribution margin income statement presented in Chapter 5 and the profit equation stated previously. Study this figure carefully because you will encounter these concepts throughout the chapter.

Figure 6.1 - Comparison of Contribution Margin Income Statement with Profit Equation

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.

### Break-Even and Target Profit

Question: Companies such as Snowboard Company often want to know the sales required to break even, which is called the break-even point. What is meant by the term break-even point?

The break-even point can be described either in units or in sales dollars. The break-even point in units3 is the number of units that must be sold to achieve zero profit. The break-even point in sales dollars4 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.

##### Break-Even Point in Units

Question: How is the break-even point in units calculated, and what is the break-even point for Snowboard Company?

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

#### Profit Equation

Question: Let’s formalize this discussion by using the profit equation. How is the profit equation used to find a target profit amount in units?

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):

$$\begin{split} \text{Profit} &= (S \times Q) - (V \times Q) - F \\ \ 30,000 &= \ 250Q - \ 150Q - \ 50,000 \\ &= \ 100Q - \ 50,000 \\ \ 80,000 &= \ 100Q \\ Q &= 800\; \text{units} \end{split}$$

Thus Snowboard Company must produce and sell 800 snowboards to achieve $30,000 in profit. This answer is confirmed in the following contribution margin income statement: #### Shortcut Formula Question: Although using the profit equation to solve for the break-even point or target profit in units tends to be the easiest approach, we can also use a shortcut formula derived from this equation. What is the shortcut formula, and how is it used to find the target profit in units for Snowboard Company? Answer: The shortcut formula is as follows: $$Q = (F + \text{Target Profit}) \div (S - V)$$ If you want to find the break-even point in units, set “Target Profit” in the equation to zero. If you want to find a target profit in units, set “Target Profit” in the equation to the appropriate amount. To confirm that this works, use the formula for Snowboard Company by finding the number of units produced and sold to achieve a target profit of$30,000:

$$\begin{split} Q &= (F + \text{Target Profit}) \div (S - V) \\ &= (\ 50,000 + \ 30,000) \div (\ 250 - \ 150) \\ &= \ 80,000 \div \ 100 \\ &= 800\; \text{units} \end{split}$$

The result is the same as when we used the profit equation.

#### Break-Even Point in Sales Dollars

Question: Finding the break-even point in units works well for companies that have products easily measured in units, such as snowboard or bike manufacturers, but not so well for companies that have a variety of products not easily measured in units, such as law firms and restaurants. How do companies find the break-even point if they 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.

#### Contribution Margin per Unit

The contribution margin per unit6 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).

$$\text{Contribution margin per unit = S - V}$$

For Snowboard Company the contribution margin is $100: $$\begin{split} \text{Contribution margin per unit} &= \text{S - V} \\ \ 100 &= \ 250 - \ 150 \end{split}$$ Thus each unit sold contributes$100 to covering fixed costs and increasing profit.

#### Contribution Margin Ratio

The contribution margin ratio7 (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.)

$$\text{Contribution margin ratio = (S − V)} \div \text{S}$$

For Snowboard Company the contribution margin ratio is 40 percent:

$$\begin{split} \text{Contribution margin ratio} &= \text{S - V} \div \text{S} \\ 40 \% &= (\ 250 - \ 150) \div \ 250 \end{split}$$

Thus each dollar in sales contributes 40 cents ($0.40) to covering fixed costs and increasing profit. Question: With an understanding of the contribution margin and contribution margin ratio, we can now calculate the break-even point in sales dollars. How do we calculate the break-even point in sales dollars for Snowboard Company? Answer: The formula to find the break-even point in sales dollars is as follows. $$\text{Break-even point in sales dollars} = \frac{\text{Total fixed costs + Target profit}}{\text{Contribution margin ratio}}$$ For Snowboard Company the break-even point in sales dollars is$125,000 per month:

### Definitions

1. The number of units that must be sold to achieve zero profit.
2. The total sales measured in dollars required to achieve zero profit.
3. The number of units that must be sold to achieve a certain profit.
4. The amount each unit sold contributes to (1) covering fixed costs and (2) increasing profit.
5. 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; also called contribution margin percent.
6. The total sales measured in dollars required to achieve a certain profit.
7. The excess of expected sales over the break-even point, measured in units and in sales dollars.