# Lesson 21--Cost-Volume-Profit Concepts

## Cost Volume Profit Relationships

Chapter 21 is one of the most important chapters in all of managerial accounting. A manager uses Cost-Volume-Profit (CVP) analysis for a wide variety of decisions. By understanding a few formulas, the effects of different decisions can quickly be determined.

## Cost Behavior

A fundamental assumption behind this lesson is that every cost can be classified as either a variable cost or a fixed cost. A second assumption is that costs and revenues can be modeled using straight-line relationships. Your background in algebra will be helpful, in that this chapter uses the straight-line formula, Y=mx+b, where Y is the total cost, m is the slope, and b is the y intercept. The variable x is typically used in this context as the number of units produced and sold.

## Variable Costs

A variable cost varies in direct proportion to changes in an activity index. Mathematically, we portray the activity index as an independent variable, while total cost is considered the dependent variable. Some typical examples of variable costs are direct materials cost, direct labor cost, sales commissions, and fuel costs. The graph of a variable cost appears as follows:

Note the following characteristics of a variable cost: 1) The cost per unit stays constant. In the graph shown above, the wage rate is constant at \$10 per hour. 2) A proportional change in the activity index causes the same proportional change in the cost. In this example, if the number of hours doubles from 2 to 4, the wages would also double from \$20 to \$40. 3) A variable cost begins at the point (0,0). This makes sense, as zero hours of work generates \$0 of wages. 4) The variable cost function is a straight line. In reality, the actual cost relationship could be curvilinear, but over the relevant range of activity at which a company operates, a straight line relationship is assumed.

## Fixed Costs

A fixed cost stays constant over a relevant range of activity. Examples of fixed costs include rent, property taxes, insurance, management salaries, and depreciation. Here is an example of a fixed cost as it would be graphed.

In the diagram, we are assuming that a company pays \$2,000 rent per month on a factory. No matter whether the factory is producing 0 units or 600 units, the rent cost remains the same. It is to a company's advantage to get as much productivity as possible from a fixed costs. In the case of rent, the company might decide to operate two or even three employee shifts, in order to reduce the rent cost per unit. As you might imagine, there will be a point at which further production will require more production capacity, so a fixed cost does not stay fixed forever. At some maximum point of production, the company will need to increase its fixed costs in order to produce more units.

An interesting facet of a fixed cost is that, as the activity increases, the fixed cost per unit declines. In the example shown above, the fixed cost per unit at 200 units is \$10 per unit (\$2000/200). But at 400 units, the fixed cost per unit is \$5.00 per unit (\$2,000/400), and at 600 units, the fixed cost per unit is \$3.33. Another example: a yoga teacher gets paid \$50 to teach a class. If there are 5 students, the cost per student is \$10; if there are 10 students, the cost is \$5 per student, and if there are 25 students, the cost is \$2 per student.

## Mixed Costs

A mixed cost is a cost that has both a fixed and a variable cost component. For example, suppose a phone company charges \$20 per month plus \$.10 per call. The graph of such a cost would appear as follows:

Notice that a mixed cost looks similar to a variable cost, but intersects the Y-axis at the fixed cost level, rather than at the origin.

## The High-Low Method

Frequently, a manager is faced with a situation in which the relationship between activity and total cost is unknown. For such situations, a mathematical approach can divulge whether the cost pattern is variable, fixed, or mixed. Your textbook illustrates the "high-low"method.

The high-low method relies on the formula for calculating the slope of a straight line. Slope equals the change in cost divided by change in activity between two points. Using the highest and the lowest activity levels (two points), the slope is determined, which provides the variable component in the cost. Subtracting the variable cost estimate from total cost at either the high or low point will provide the fixed cost amount. In other words, with a straight line relationship between volume and cost, the high-low method provides us with the slope and Y intercept for that straight line.

In mathematical terms, a straight line can be described by the formula Y=mX+b, where Y is total cost, m is the slope, X is the number of units produced and sold, and b is the Y-intercept. The high-low method is a way to determine the values of m and b.

The situation portrayed above could be used to illustrate the high-low method. A manager keeps track of a certain cost for five months, recording both the volume and cost each month. For the high-low method, we concentrate on two points from the graph shown. These are the low point (10,59) and the high point (50,215). Note that it is the low X value that determines the low point, and the high X value that determines the high point. And, each point has an X and Y value.

Even though the points do not lie in a perfectly straight line, we will determine the equation of the straight line that connects the low point and the high point. The formula for a straight line is: Y=mX+b, where Y is the total cost, m is the slope, and b is the Y-intercept. This formula is derived as follows:

1. Compute the slope between the low and high point = (215-59) divided by (50-10). The slope is 3.9.

2. Use the slope and the original data to derive the fixed cost. The total cost at a volume of 50 units is \$215. The variable cost at 50 units would be 50*3.9=\$195. The fixed cost would be the difference between \$195 and \$215=\$20. You could arrive at the same figure by using the total cost at 10 units (\$59). Compare this cost with the variable cost of 10*3.9=\$39. The same fixed cost is arrived at: \$20.

3. Our calculation of the mixed cost formula for this example is: Y=3.9X + \$20. If the manager wanted to predict the cost of 45 units, the predicted cost would be 3.9*45 + 20 = \$195.50.

The high-low method will usually provide a good approximation of the relationship between X and Y. However, if you have a program like Excel, you can achieve a more statistically accurate formula by using the regression formula. Additionally, a regression analysis can provide a measure of the strength of the relationship between X and Y.

## Cost Volume Profit Analysis

Cost Volume Profit Analysis relies on categorizing all costs as to whether they are variable or fixed. Note that we are talking about all costs--both manufacturing costs and period costs. Also, there may be cases in which there are mixed costs. For any mixed cost, we must separate out the variable and fixed components. So if we pay for water on the basis of \$20 of fixed cost per month and .02 per gallon variable cost, we would add the \$20 to the fixed cost total, and include the .02 per gallon as part of the variable costs.

Once all costs have been categorized as to variable and fixed, many interesting types of problems can be easily solved. For example, Kim Long sells posters. Each poster has a variable cost of \$21, and is sold at a price of \$35. The company must cover fixed costs of \$7,000. I suggest that you approach each CVP problems by setting up a unit-contribution table as shown below.

 Price Per Unit \$35 -VC per Unit \$21 Contribution Margin Per Unit \$14 CM Ratio 14/35=.40 Fixed Costs \$7,000

I have filled in the price per unit and the variable cost per unit.  The contribution margin per unit is the price minus the variable cost per unit.  The CM ratio is the contribution margin per unit divided by the price per unit.  I recommend setting up this unit table for most of the problems in Chapter 21. If some of the figures are missing from a problem, this format will help you spot them and determine their values.

Suppose we wish to compute the breakeven point for this situation. The breakeven point can be expressed two different ways: 1. breakeven in units or 2. breakeven in dollars of revenue. The breakeven point is the number of units we’d have to sell (or the revenue we’d have to generate) to make a profit of exactly zero. At breakeven, the revenue earned equals variable costs + fixed costs.

Here’s one way you could calculate breakeven in units: Let X = the number of units sold. \$35X - \$21X-7,000 = 0. So, X = 500 units. You could then solve for the revenue at this level, which would be 500 * \$35 = \$17,500. So, the company will break even if they sell 500 units. This could also be expressed as a breakeven revenue of \$17,500. Although this solution is valid, I suggest that you memorize the following two formulas, which will make the homework problems go a little faster.

1. Break-even point in number of units: Divide fixed costs by the unit contribution margin per unit. The unit contribution margin is equal to the price per unit minus the variable cost per unit. In our example, the breakeven in units can be quickly calculated as \$7000/\$14 = 500 units.

2. Break-even point in sales dollars: Divide fixed costs by the contribution margin ratio. The contribution margin ratio is calculated by dividing the unit contribution margin by the unit price. . In our example, the breakeven sales revenue can be calculated by dividing \$7000 by .40. The quotient is \$17500.

Think about your results. If we sell 500 units, the revenue would be \$17,500. The 500 units would have total variable cost of 500*\$21=10,500 and fixed costs of \$7,000, for total cost of \$17500. Revenue = total costs and operating income would be zero. So a breakeven point of 500 units or \$17,500 sounds correct.

## Target Net Income

My suggestion is that you use the two formulas given above until you have mastered them. You will then be able to adapt to any alteration of the problem to be solved. (Furthermore, use of these two formulas will be more efficient in the long run than the "Equation Approach" demonstrated earlier.) For example, if you were asked to find the number of units which must be sold to achieve a profit of \$10,000, you would alter formula 1 above by adding the \$10,000 to the fixed costs, and then dividing by the unit contribution margin. The \$10,000 is a "target profit" figure. If Kim Long wanted to earn income of \$10,000, how many units must she sell? Units = (\$7,000 of fixed costs + \$10,000 profit) / \$14 contribution margin = 1214.3 units. The revenue generated would be 1214.3 * \$35 = \$42,500.50.

Suppose the target profit is \$10,000 and you wish to know the dollar amount of sales required to achieve that profit. Add \$10,000 to the fixed costs and divide by the contribution margin ratio -- which is an alteration of formula 2 shown above. Revenue required for \$10,000 of profit would be (\$7,000 + \$10,000) / CM ratio of .4 = \$42,500. Note that this matches the revenue calculated in the previous paragraph.

A graphic approach to breakeven analysis is shown below. I suggest that you spend some time with this graph, as it provides a visual approach to CVP analysis. In my version of the graph, find three lines: a red horizontal line indicating fixed costs; a blue revenue line coming from the origin; and a total cost line in green. Note that the green line represents the sum of the fixed costs and variable costs. The point where the revenue line crosses the total cost line is the breakeven point. Note that there are two ways in which the breakeven point can be represented -- either in units on the X axis (500 units), or in dollars on the Y axis (\$17,500).

I have also plotted a profit line in purple, which indicates that, if there is no revenue, the loss is \$7,000--the amount of the fixed costs. That purple line crosses the X-axis at the breakeven point of 500 units. To make a \$10,000 profit, the units would have to be just over 1200 according to the graph.

## The CVP Income Statement

CVP analysis relies on categorizing all costs. Each cost that a company incurs is classified as either a variable cost or a fixed cost. In fact, the entire income statement can be recast as a CVP Income Statement. The general format for a contribution income statement is: Sales - Variable Costs = Contribution Margin; Contribution Margin - Fixed Costs = Net Income. A contribution income statement would not be published for stockholders or other parties. Rather, it would be used internally for planning and decision-making purposes. Suppose Kim sold 1500 units. Here is a CVP or Contribution Income Statement at that level of activity.

 Kim Long Products Contribution Income Statement For Year Ended xx/xx/xx Calculations Total Revenue \$52,500 1,500*\$35 -VC per Unit \$31,500 1500*\$21 Contribution Margin \$21000 1500*\$14 Fixed Costs \$7,000 Operating Income \$14,000 \$21000-\$7,000 CM Ratio .40 \$21000/\$52500

Note: A subtlety of the Contribution Income Statement is that you can calculate the CM ratio by dividing the contribution margin by the total sales figure.  This will result in the same mathematical result as calculating the CM ratio at the unit level. In the case of Ms. Long's poster business, we can calculate the CM ratio as \$21,000/\$52,500, that is, dividing the total CM by the total Sales Revenue. You will get the same figure, 40%, that we calculate by using the unit data earlier.

## The Operating Leverage Factor

We can do one other thing with Kim's income statement. We can calculate the Operating Leverage Factor (OLF). The calculation itself is very simple. Divide the Contribution Margin by the Operating Income. In the income statement shown above, the OLF = 21,000/14,000 = 1.5. What does this tell us?

The operating leverage factor is a multiplier. If someone asked you, "suppose that Kim's revenue went up 10%. By what percentage would the operating income increase?" You might be tempted to say, "if revenue goes up 10%, then operating income goes up 10%--but this would be a wrong answer. Why? Because not all costs are variable; some are fixed. In Kim's case, a 10% increase would lead to a 1.5 * 10% or 15% increase in operating income. Multiply the OLF times the proposed increase in revenue, 1.5*10%=15%. Let's prove that this is the case.

In the income statement above, the units sold was 1500. A 10% increase would mean that 1650 units were sold. Here's the contribution income statement at 1650 units.

 Kim Long Products Contribution Income Statement For Year Ended xx/xx/xx Calculations Total Revenue \$57,750 1,650*\$35 -VC per Unit \$34,650 1650*\$21 Contribution Margin \$23,100 1650*\$14 Fixed Costs \$7,000 Operating Income \$16,100 23,100-7,000 CM Ratio .40 23,100/57750

Note that the operating income is 16,100, and increase of 2100 compared to the previous operating income. The rate of increase is 2100/14,000 is .15 or 15%.

If we compute the OLF at this point, it will have changed. It is now 23,100/16,100 or 1.435. From this point, an increase of 10% in revenue would lead to an income increase of 14.35%.

## CVP Assumptions

One of these is that a company selling multiple products will maintain a constant sales mix over a period of time. Other assumptions pertinent to CVP are that: 1) revenue is linear; 2) the behavior of total costs is linear; 3) inventory levels at the beginning and end of the year are equal. In your business life, you will run into situations in which our straight line model is a perfect fit, and other situations in which it's not so perfect.

## A Note on Sales Mix

Your textbook discusses a situation that is likely to occur in most companies--the sales mix. Suppose that a company sells two products--product A with price of \$10, variable cost per unit of \$6, and product B with a price of \$15 and variable cost per unit of \$12. The fixed costs are \$88,400. Furthermore, the company's sales are 40% product A and 60% product B. What is the breakeven point in units?

Solution: calculate a combined contribution margin by weighting the individual contribution margins. In other words, the contribution margin would be calculated as \$4 times 40% + \$3 times 60%= \$1.60 + \$1.80 = \$3.40. Then divide the fixed costs of \$88,400 by \$3.40 = 26,000 units. (Using shortcut formula 1). We know that the sales mix is 40/60, so multiply .4*26,000= 10,400 units of product A and .6*26,000=15,600 units of product B.

## Appendix--Variable Costing

The chapter concludes with a discussion of variable costing. This is an interesting approach which considers the product cost to be the sum of direct materials + direct labor + variable manufacturing overhead. Fixed manufacturing overhead, on the other hand, is treated as a period cost and expensed when incurred. Although this philosophy could never be used for formal financial statements, (where absorption costing is required), it is an interesting approach for management purposes. If a company is in a position to produce a batch of products for a special order, it might be the case that the only incremental costs of the order are variable costs. The fixed costs, including salaries, depreciation, rent, and supervisory costs may not change at all in producing that special order. In such a case, the fixed costs may be irrelevant to the decision.