This chapter shows how information on both costs and sales behavior is useful to managers in performing cost-volume-profit analysis. This analysis is an important part of successful management and sound business decisions.
Cost-volume-profit analysis will allow us to answer many questions and make important decisions involving the relationships between the volume of activity and costs and revenues. Before we can answer these questions using cost-volume-profit analysis, we must first study cost behavior.
We begin our study of cost behavior with fixed costs. Your basic land-line telephone has a monthly connect charge that remains constant regardless of the number of local calls that you might make. The monthly charge that is independent of call activity is a fixed cost..
Fixed costs per unit decline as activity increases. Dividing your monthly connect fee by more local calls reduces the cost per call by spreading the fixed amount over a higher number of calls. For example, if your monthly connect charge is twenty dollars and you make forty local calls in a month, your cost per local call is fifty cents. If you make one hundred local calls in a month, your cost per local call is twenty cents.
Total variable costs increase as activity increases. For most people, the total land-line long distance telephone bill is based on the number of minutes talked. So there’s a direct relationship between the number of minutes talked and your total bill. You can see a graph of that relationship in the lower left-hand part of your screen.
The cost per land-line long distance minute talked is normally constant. For example, for your service, it may be seven cents per minute. Talking more or less minutes will not change the per minute charge. So, on a per unit basis, variable costs remain unchanged. You can see the graph of that in the lower right-hand side of your screen.
We know that some of the language we use to differentiate fixed and variable costs in total and per unit can be very confusing when you first see it. So we’ve prepared this chart to help you identify how those costs behave.
Many costs are mixed in nature. That is, they have both a fixed and variable component. Think about your utility bill. You have a fixed monthly charge for the hook-up, and the variable portion of your bill depends upon the number of kilowatt hours you consume. The more the kilowatt hours you use, the higher your total utility bill will be.
Here we see a graph with utility cost on the vertical axis and kilowatt hours on the horizontal axis. Notice that the fixed monthly charge is the same at all levels of kilowatt usage, even the zero level of usage. The variable cost, which rises as more kilowatt hours are used, is added to the fixed cost to obtain the total mixed cost.
Another type of cost is referred to as a step cost. Step costs remain constant in total within a relatively narrow range of activity.
Total step costs increase as the level of activity increases beyond the initial narrow range of activity.
Not all costs are linear as shown in our previous examples. Here we see a curvilinear cost where the cost increases at an increasing rate as activity increases. Although curvilinear costs might exist on occasion, we will limit our analysis to linear relationships.
When presented with a mixed cost, we will separate the variable portion of the cost from the fixed portion of the cost. There are number of ways to do this. We will use a scatter diagram and the high-low method. A more sophisticated method, the least squares regression model, is also available, but we will not use it here.
A scatter diagram is a plot of cost data points on a graph. It is almost always helpful to plot cost data to be able to observe a visual picture of the relationship between cost and activity.
We begin by plotting the data points on our graph. The vertical axis is cost and the horizontal axis is activity.
Next, we draw a straight line through the data points with about an equal number of observations above and below the line. We continue the line past the observed points until it intersects with the vertical axis. The intercept in this case is our fixed cost, which is estimated to be ten thousand dollars.
Next, we determine the slope of the line. The slope of the line is the change in cost divided by the change in activity. The slope, the amount of change in cost for a one unit change in activity, is the variable cost per unit of activity.
Now let’s look at the high-low method. In our example, we’re going to look at a company’s relationship between cost and sales activity. During the year, the company reports sales and costs on a monthly basis. The month with the high level of sales shows sales of sixty seven thousand five hundred dollars and a corresponding cost of $29,000, and the month with the low level of sales show sales of $17,500 with a corresponding cost of $20,500. We will use this information to compute the variable cost per dollar of sales and the total fixed cost.
To determine the variable costs per unit of activity, we divide the change in cost by the change in activity, sales dollars in this example. In our case, the change in cost is $8,500 and the change in sales dollars is $50,000. The result is a variable cost rate of $0.17 per dollar of sales.
Next, we calculate the fixed cost by subtracting the total variable cost from the total cost. Since total cost and total variable cost are different amounts at different sales levels, we must choose either the high level or the low level for our computations.
Let’s choose the high level of activity, sixty seven thousand five hundred dollars in sales. Our first step is to calculate the total variable cost. At sixty seven thousand, five hundred dollars of sales, the total variable cost is point one seven per dollar of sales times sixty seven thousand, five hundred dollars, resulting in a total variable cost of eleven thousand, four hundred seventy five dollars. Next, we subtract eleven thousand, four hundred seventy-five dollars from the total cost at the high sales activity, to get the fixed cost, seventeen thousand, five hundred twenty-five dollars. You will obtain the same result if you select the low level of activity to compute fixed cost. Why don’t you compute fixed cost using the low level of sales activity before advancing to the next screen.
If we have a large number of observations, we’ll probably want to use computer software that can do regression analysis to determine cost volume relationships. Excel is a wonderful tool to carry out this computation.
Now that we have improved our knowledge of cost behavior, we are ready to apply the concepts to break-even analysis.
The break-even point is the level of sales where a company’s income is exactly equal to zero. At breakeven, total costs equal total revenues.
We’re going to concentrate exclusively on the contribution format income statement for our break-even analysis. Contribution margin is the amount remaining after we deduct all our variable expenses from sales revenue. Contribution margin can be expressed as a total amount, thirty thousand dollars in this example, or as an amount per unit, twenty dollars in this example. Each unit sold contributes twenty dollars toward covering fixed costs and providing for profits.
Contribution margin goes to cover our fixed costs. If all our fixed costs are covered, the company will operate in the profit area. If we fail to cover our fixed expenses, we will operate in the loss area. How much contribution must this company have to cover its fixed costs?
Fixed costs are twenty-four thousand dollars, so this company must generate twenty-four thousand dollars in contribution margin to cover its fixed costs. When contribution margin is exactly twenty-four thousand dollars, the company’s sales are at breakeven as its income will be zero.
This company is earning thirty-six thousand dollars income by selling two thousand units. The breakeven point will obviously occur at a sales volume less than two thousand units. If each unit contributes thirty dollars to covering fixed costs, can you compute the number of units that must be sold to cover the thirty thousand dollars in fixed costs and allow the company to breakeven?
We compute the break-even sales volume in units by dividing fixed costs by the unit contribution margin.
The results of the previous question can be expressed in equation form as seen on your screen. The break-even point in units is equal to fixed costs divided by the unit contribution margin.
The break-even point in sales dollars is equal to fixed costs divided by the contribution margin ratio. The contribution margin ratio is equal to the the unit contribution divided by the unit sales price. In the earlier example, the contribution margin ratio is thirty percent, resulting from dividing the thirty dollars unit contribution margin by the one hundred dollars unit sales price. You might want to refer back to the example to verify these numbers. The contribution margin ratio tells us that thirty cents of each sales dollar contributes to covering fixed costs and providing for income.
Let’s look at a couple of questions to see if we have these concepts mastered.
Here’s your first question.
The unit contribution margin is two dollars. The break-even point in units is equal to two hundred thousand dollars in fixed costs divided by the two dollars unit contribution margin.
Here’s your second question using the same information.
The contribution margin ratio is equal to the two dollars unit contribution divided by the five dollars unit sales price. The break-even point in sales dollars is equal to two hundred thousand dollars in fixed costs divided by the forty percent contribution margin ratio.
In this graph, we have plotted costs and revenues on the vertical axis and volume in units on the horizontal axis. The total cost line has a slope equal to the variable cost per unit and intercepts the vertical cost axis at the fixed cost.
When we add the sales line to our graph, we see the break even point where the sales line crosses the total cost line. The sales line begins at the origin and has a slope equal to the unit sales price. The sales line is steeper, that is increases at a faster rate than the total cost line, because because the unit sales price is greater than the unit variable cost.
There are some basic assumptions related to cost volume profit analysis that we are studying in this chapter. Some of these assumptions may be very restrictive. First, costs and revenues are assumed to be linear in nature, meaning that the selling price is assumed to be constant, the unit variable cost is assumed to be constant, and total fixed costs are assumed to be constant. Also, for manufacturing companies, inventories don’t increase or decrease during the period. All units produced, are sold.
We have seen what it takes for a company to breakeven, but we are not in business just to breakeven. Hopefully our business will earn an income. The break-even relationships that we have studied can be slightly altered to include income.
Income is equal to sales less total costs. Subtracting Rydell’s one hundred five thousand dollars variable cost and its twenty four thousand dollars fixed cost from one hundred fifty thousand dollars in sales results in a pretax income of twenty one thousand dollars Work through the numbers and see if you agree.
We can adjust the break-even formulas that we used earlier to incorporate income. Recall that we calculated breakeven by dividing fixed costs by contribution. When we incorporate income, contribution must cover the fixed cost as well as provide for income. To adapt the break-even formulas for income, we add the desired amount of income to the numerator.
Let’s see if we can use these formulas to answer a question.
Here’s your question.
The unit contribution margin is two dollars. ABC Company must sell one hundred twenty thousand units to first cover its two hundred thousand dollars in fixed costs and then provide for the forty thousand dollars target income.
Our previous formulas allowed us to solve for sales necessary to earn a target income used pretax income. Pretax income which has two components, net income (after tax) and the income taxes paid on the pretax income are shown on your screen.
If our target income is stated as after-tax net income, we can covert to pretax income by dividing the target after-tax net income by one minus the tax rate. Let’s work an example to see how income taxes affect cost-volume-profit problems.
Rydell’s target after tax net income is eighteen thousand dollars and the tax rate is twenty five percent. First, we need to convert the eighteen thousand dollars after-tax net income to before-tax income, and then multiply the before-tax income by the twenty five percent tax rate to find the income tax expense. Work through the computations before advancing to the next screen.
Divide the eighteen thousand dollars after-tax income by one minus the twenty five percent tax rate to convert to before-tax income. Now that we have the twenty four thousand dollars before-tax income, we can multiply it by twenty five percent to find that the income tax expense is six thousand dollars.
We will divide the fixed costs plus income by the contribution margin ratio to find the sales revenue necessary to earn after-tax income of eighteen thousand dollars. Be careful with the income that you add to fixed costs in the numerator of your computation. Is it after-tax income or pretax income? Refer to the previous discussion a couple of screens back if you have doubts.
The first step in working this problem is finding the contribution margin ratio.
Work through the computations before advancing to the next screen.
We subtract the seventy dollars unit variable cost from the one hundred dollars unit selling price to get the thirty dollars unit contribution. The thirty percent contribution margin ratio is equal to the thirty dollars unit contribution divided by the one hundred dollars unit selling price. Now we are ready to solve for the sales revenue needed to achieve an eighteen thousand dollars after-tax target net income.
Note that the numerator contains the eighteen thousand dollars after-tax target net income and the six thousand dollars income tax expense. The sum of these two amounts is twenty four thousand dollars, the before-tax target income.
We can also solve for the number of units that we must sell to achieve the after-tax target net income. The only difference is that we use the thirty dollars unit contribution margin in the denominator of our computation.
The margin of safety is the excess of expected sales (or actual sales) over the breakeven sales. It’s the amount by which expected sales can drop before the company begins to incur losses. We can also express the margin of safety as a percent of sales. The margin of safety percentage is equal to the margin of safety in dollars divided by the expected sales in dollars.
Here’s a question for you dealing with margin of safety. Calculate the margin of safety first and then calculate the margin of safety percentage.
The margin of safety is equal to actual sales of one hundred thousand dollars less the break-even sales of eighty thousand dollars. The margin of safety percentage is equal to the twenty thousand dollars margin of safety divided by actual sales of one hundred thousand dollars.
Our basic assumptions related to cost volume profit analysis such as the selling price is assumed to be constant, the unit variable cost is assumed to be constant, and total fixed costs are assumed to be constant, can be restrictive. Let’s look at an example where we change some of the costs and see what happens.
The new machine is more efficient, reducing unit variable costs by ten dollars, but the monthly fixed costs for the new, more efficient machine are six thousand dollars higher. Will the break-even point change?
We use our same formula to determine the break-even point, but with the dollar amounts for the new machine. The revised level of fixed costs for the new machine is thirty thousand dollars. The revised unit contribution margin is the one hundred dollars unit selling price minus sixty dollars variable cost for the new machine. The revised contribution margin ratio is the forty dollars revised unit contribution margin divided by the one hundred dollar unit selling price.
To this point, we’ve assumed that a company sells a single product. We can extend the cost-volume-profit relationships to cover multiproduct companies. Instead of unit contribution margin for one unit, we will have a composite unit contribution for all units. The composite unit contribution margin is dependent on the sales mix of the products sold.
Note that the break-even formula looks the same for a multiproduct company. The only difference is the denominator. The unit contribution margin for one unit is replaced by a composite unit contribution for all units. A composite unit is composed of specific numbers of each product in proportion to the product sales mix. Next, we will see how sales mix is used to compute the contribution per composite unit.
Sales mix is the ratio of the volumes of the various products. In this case, the sales mix is four basics cuts sold for each budget cut, and two ultra cuts sold for each budget cut.
The four-two-one mix means that if we sell five hundred budget cuts, then we will sell one thousand ultra cuts and two thousand basic cuts.
The first thing we do in computing the contribution margin for a composite unit is to multiply the unit contribution for each product times the sales mix number for each product. The resulting amounts are called weighted unit contributions because they are weighted by the sales mix numbers in the computation.
The second thing we do in computing the contribution margin for a composite unit is to add the weighted unit contributions. The resulting number, thirty two dollars in this example, is the contribution margin per composite unit.
We calculate our breakeven point in composite units by dividing our total fixed cost by the contribution margin per composite unit that we have just calculated. Go ahead and solve for the breakeven point in composite units before advancing to the next screen.
How did you do? We can see from the computations on this screen that we must sell three thousand composite units to break even.
Now that we know the number of composite units that must be sold to break even, we can solve for the number of each product that we must sell to break even. We do this by multiplying the sales mix number for each product times three thousand composite units. Notice that the resulting twelve thousand basic cuts, six thousand ultra cuts, and three thousand budget cuts remains in the same relative sales mix of four-two-one.
We can verify the results of our break-even computations by preparing an income statement for the three products. You might want to review the original given information for this example before you work through this income statement.
Operating leverage is an important concept for managers to understand. It’s a measure of how sensitive operating income is to changes in sales. When operating leverage is high, a small percentage increase in sales can result in a much larger percentage increase in operating income. The degree of operating leverage is equal to contribution margin divided by net income. Let’s look at an example.
At Rydell, the operating leverage is three computed by dividing the thirty-six thousand dollars contribution margin by the twelve thousand dollars income. We multiply the operating leverage times the percentage increase in sales to find the percentage increase in income. If Rydell increases sales by ten percent, what will be the percentage increase in income?
With an operating leverage of two, a ten percent increase in sales will produce a thirty percent increase in income. We multiply the percentage increase in sales times the degree of operating leverage to determine the percentage increase in profit.
Now that we have mastered some of the basic concepts and principles of managerial accounting, we are ready to put this knowledge to work.