Accounting: The Language of Business - Vol. 1 (Part 122)

"We have the most crude accounting tools. It`s tragic because our accounts and our national arithmetic doesn`t tell us the things that we need to know." - Susan George 

Cost Behavior and Cost-Volume-Profit Analysis (Part D)

by

Charles Lamson

Mathematical Approach to Cost-Volume-Profit Analysis

Accountants use various approaches for expressing the relationship of costs, sales (volume), and income from operations (operating profit). The mathematical approach is one approach that is used in practice.

The mathematical approach to cost-volume-profit analysis uses equations (1) to determine the units of sales necessary to achieve the break-even point in operations or (2) to determine the units of sales necessary to achieve a target or desired profit. We will next describe and illustrate these equations and their use by management in profit planning.

Break-Even Point

The break-even point is the level of operations in which a business's revenues and expired costs are exactly equal. At break-even, a business will have neither an income nor a loss from operations. The break-even point is useful in business planning, especially when expanding or decreasing operations.

To illustrate the computation of the break-even point, assume that the fixed costs [Fixed costs tend to be costs that are based on time rather than the quantity produced or sold by your business. Examples of fixed costs are rent and lease costs, salaries, utility bills, insurance, and loan repayments. Some kinds of taxes, like business licenses, are also fixed costs (Costs: Fixed Costs, Variable Costs, and Volume).]for Barker Corporation are estimated to be $90,000. The unit selling price, unit variable cost [Variable cost per unit refers to the costs of each unit of goods that a company produces, variable costs change as changes occur in the production level or activity level of the company. Unit Variable Cost is affected by changes in the business, extra cost is incurred when more units of goods are produced (thebusinessprofessor.com).], and unit contribution margin [Contribution margin (CM), or dollar contribution per unit, is the selling price per unit minus the variable cost per unit. "Contribution" represents the portion of sales revenue that is not consumed by variable costs and so contributes to the coverage of fixed costs (wikipedia.com).] for Barker Corporation are as follows:

The break-even point is 9,000 units, which can be computed by using the following equation:

Break-even sales (units) = Fixed costs / Unit contribution margin

Break-even sales (units) = $90,000 / $10 = 9,000 units

The following income statement verifies the preceding computation:

The break-even point is affected by changes in the fixed costs, unit variable costs, and the unit selling price. Next, we will briefly describe the effect of each of these factors on the break-even point.

Effect of Changes in Fixed Costs

Although fixed costs do not change in total with changes in the level of activity, they may change because of other factors. For example, changes in property tax rates or factory supervision salaries change fixed costs. Increases in fixed costs will raise the break-even point. Likewise, decreases in fixed costs will lower the break-even point.

To illustrate, assume that Bishop Co. is evaluating a proposal to budget an additional $100,000 for advertising. Fixed costs before the additional advertising are estimated at $600,000 and the unit contribution margin is $20. The break-even point before the additional expense is 30,000 units, computed as follows: 

Break-even sales (units) = Fixed costs / Unit contribution margin

Break-even sales (units) = $600,000 / $20 = 30,000 units

If the additional amount is spent, the fixed costs will increase by $100,000 and the break-even point will increase to 35,000 units, computed as follows:

Break-even sales (units) = Fixed costs / Unit contribution margin

Break-even sales (units) = $700,000 / $20 = 35,000 units

The $100,000 increase in the fixed costs requires an additional five thousand units ($100,000 / $20) of the sales to break even. In other words, an increase in sales of 5,000 units is required in order to generate an additional $100,000 of total contribution margin (5,000 units * $20) to cover the increased fixed costs.

Effect of Changes in Unit Variable Costs

Although unit variable costs are not affected by changes in volume of activity, they may be affected by other factors. For example, changes in the price of direct materials and the wages for factory workers providing direct labor will change unit variable costs. Increases in unit variable costs will raise the break-even point. Likewise, decreases in unit variable costs will lower the break-even point. For example, when fuel prices rise or decline, there is a direct impact on the break-even freight load for the Union Pacific Railroad.

To illustrate, assume that Park Co. is evaluating a proposal to pay an additional 2% commission on sales to its sales people as an incentive to increase sales. Fixed costs are estimated at $840,000, and the unit selling price, unit variable cost, and unit contribution margin before the additional 2% commission are as follows.

The break-even point is 8,000 units, computed as follows:

Break-even sales (units) = Fixed costs / Unit contribution margin

Break-even sales (units) = $840,000 / $105 = 8,000 units

If the sales commission proposal is adopted, variable costs will increase by $5 per unit ($250 x 2%). This increase in the variable costs will decrease the unit contribution margin by $5 from (from $105 to $100). Thus, the break-even point is raised to 8,400 units, computed as follows:

Break-even sales (units) = Fixed costs / Unit contribution margin

Break-even sales (units) = $840,000 / $100 = 8,400 units

At the original break-even point of 8,000 units, the new unit contribution margin of $100 would provide only $800,000 to cover fixed costs of $840,000. Thus, an additional 400 units of sales will be required in order to provide the additional $40,000 (400 units * $100) contribution margin necessary to break even.

Effect of Changes in the Unit Selling Price

Increases in the unit selling price will lower the break-even point, while decreases in the unit selling price will raise the break-even point. To illustrate, assume that Graham Co. is evaluating a proposal to increase the unit selling price of its product from $50 to $60. The following data have been gathered:

The break-even point on the current selling price is 30,000 units, computed as follows:

Break-even sales (units) = Fixed costs / Unit contribution margin

Break-even sales (units) = $600,000 / $20 = 30,000 units

If the selling price is increased by $10 per unit, the break-even point is decreased to 20,000 units, computed as follows:

Break-even sales (units) = Fixed costs / Unit contribution margin

Break-even sales (units) = $600,000 / $30 = 20,000 units

The increase of $10 per unit in the selling price increases the unit contribution margin by $10. Thus, the break-even point decreases by 10,000 units (from 30,000 units to 20,000 units).

Summary of Effects of Changes on Break-Even Point

The break-even point in sales (units) moves in the same direction as changes in the variable cost per unit and fixed costs. In contrast, the break-even point in sales (units) moves in the opposite direction to changes in the sales price per unit. A summary of the impact of these changes on the break-even point in sales (units) is shown below.

Target Profit

At the break-even point, sales and costs are exactly equal. However, the break-even point is not the goal of most businesses. Rather, managers seek to maximize profits. By modifying the break-even equation, the sales volume required to earn a target or desired amount of profit may be estimated. For this purpose, target profit is added to the break-even equation as shown below.

Sales (units) = Fixed costs + Target profit / Unit contribution margin

To illustrate, assume that fixed costs are estimated at $200,000, and the desired profit is $100,000. The unit selling price, unit variable cost, and unit contribution margin are as follows:

The sales volume necessary to earn the target profit of $100,000 is 10,000 units, computed as follows:

Sales (units) = Fixed costs + Target profit / Unit contribution margin

Sales (units) = $200,000 + $100,000 / $30 = 10,000 units

The following income statement verifies this computation:

*WARREN, REEVE, & FESS, 2005, ACCOUNTING, 21ST ED., PP. 833-837*

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