1. Cost-Volume-Profit Relationships

2. Learning Objectives
Explain the purpose of cost-volume-profit (CVP) analysis
Explain the contribution margin (CM) concept
Compute the break-even (BE) point by using graph, equation, and contribution margin methods
Compute profit (income) at a given level of sales and also target sales by using the methods above
Calculate the effect on profit of changes in variable costs, fixed costs, selling price, and volume
Perform CVP analysis for a multi-product company
Compute the margin of safety and operating leverage and explain their significance

3. Cost-Volume-Profit (CVP) Analysis
CVP analysis is an analysis of the relationships among activity level, revenue, costs and profit.
Classification of cost items into fixed and variable is paramount in CVP analysis.
Contribution margin (CM) concept facilitates CVP analysis.

4. The CM Concept
Assume the following budgeted (expected) annual data for ABC, a single-product company.
Total Per Unit Percent
Sales (500 units) $10, %
-Variable costs , %
Contribution margin $ 4, %
-Fixed costs ,000
Net income $ 1,000

5. The CM Concept
Assume the following budgeted (expected) annual data for ABC, a single-product company.
Total Per Unit Percent
Sales (500 units) $10, %
-Variable costs , %
Contribution margin $ 4, %
-Fixed costs ,000
Net income $ 1,000
Contribution Margin (CM) is the amount remaining from sales revenue after variable costs have been deducted.

6. CM goes to cover fixed costs.
The CM Concept
Assume the following budgeted (expected) annual data for ABC, a single-product company.
Total Per Unit Percent
Sales (500 units) $10, %
-Variable costs , %
Contribution margin $ 4, %
-Fixed costs ,000
Net income $ 1,000
CM goes to cover fixed costs.

7. After covering fixed costs, any remaining CM contributes to profit.
The CM Concept
Assume the following budgeted (expected) annual data for ABC, a single-product company.
Total Per Unit Percent
Sales (500 units) $10, %
-Variable costs , %
Contribution margin $ 4, %
-Fixed costs ,000
Net income $ 1,000
After covering fixed costs, any remaining CM contributes to profit.

8. The CM Concept
Assume the following budgeted (expected) annual data for ABC, a single-product company.
Total Per Unit Percent
Sales (500 units) $10, %
-Variable costs , %
Contribution margin $ 4, %
-Fixed costs ,000
Net income $ 1,000
CM per unit is the amount that each unit contributes to fixed costs and profit.
CM Per unit therefore measures the change in profit as a result of a one unit change in sales.

9. The CM Concept
Assume the following budgeted (expected) annual data for ABC, a single-product company.
Total Per Unit Percent
Sales (500 units) $10, %
-Variable costs , %
Contribution margin $ 4, %
-Fixed costs ,000
Net income $ 1,000
CM percentage is the amount that each dollar of sales contributes to fixed costs and profit.
CM percentage therefore measures the change in profit as a result of a one dollar change in sales.

10. Quick Test
Carver Company produces a product which sells for $30. Variable manufacturing costs are $15 per unit. Fixed manufacturing costs are $5 per unit based on the current level of activity, and fixed selling and administrative costs are $4 per unit. A sales commission of 10% of the selling price is paid on each unit sold. The contribution margin per unit is: a. $ 3 b. $15 c. $ 8 d. $12

11. Quick Test
Carver Company produces a product which sells for $30. Variable manufacturing costs are $15 per unit. Fixed manufacturing costs are $5 per unit based on the current level of activity, and fixed selling and administrative costs are $4 per unit. A sales commission of 10% of the selling price is paid on each unit sold. The contribution margin per unit is: a. $ 3 b. $15 c. $ 8 d. $12

12. Applications of CVP Analysis
Break-even analysis (calculation)
Profit at a given level of sales
Target sales
The effect on profit of changes in cost structure
The effect on profit of changes in selling price
The effect on profit of changes in costs and selling price
The effect on profit of changes in sales mix (for a multi-product company)

13. Break-even (BE) Analysis
The objective of BE analysis is to determine the quantity or dollar amount of sales that generates zero profit.
There are basically three methods for BE analysis:
Graph
Equation
Contribution margin

14. BE Analysis – Graphic Method
# of units produced
and sold, X
$, Y
Total cost =FC + VC
VC/unit
Total VC (12 * X)
VC/unit
Total FC ($3,000) FC

15. BE Analysis – Graphic Method
Total revenue (20 * X)
# of units produced
and sold, X
$, Y
Total cost =FC + VC FC
BE in units (375)
BE in $
(7,500)

16. BE Analysis – Equation Method
Revenue = (Unit SP) * Volume
Total cost = (Unit VC) * Volume + Total FC
Profit = Revenue - Total cost
= (Unit SP) * Volume - (Unit VC) * Volume - Total FC
= (Unit SP - Unit VC) * Volume - Total FC
At break-even, profit is zero. Thus:
Volumebe = (Total FC) / (Unit SP - Unit VC)
Dollarbe = Volumebe * Unit SP

17. BE Analysis – CM Method
CM per unit (CM ratio) is the amount that each unit (dollar of sales) contributes toward recovering fixed costs and then toward earning a profit for the period.
The volume and dollar sales at break-even then are:
Volumebe = (Total FC) / (Unit CM) = $3,000 / $ 8
Dollarbe = (Total FC) / (CM ratio) = $3,000 / .40
CM formulas can be derived from the equation method.

18. Profit at a Given Level of Sales
The same three methods as in BE analysis can be used. See next slide for graphic solution.
Using equation method, Profit = Revenue - Total cost
= (Unit SP) * Volume - (Unit VC) * Volume - Total FC
= (Unit SP - Unit VC) * Volume - Total FC
Using CM method, Profit = (CM/unit) * Volume - Total FC or, (CM ratio) * Revenue - Total FC

19. Profit Calculation – Graphic Method
# of units produced
and sold, X
$, Y
Total cost =FC + VC FC
Total revenue
-FC
CM/unit
Pretax profit=Total revenue
minus Total cost

20. Target Sales
The objective here is to determine the level of sales that has to be achieved to make a given amount of profit.
The same three methods as in BE analysis can be used. Using contribution margin method:
Volumets = (Fixed cost + Profit) / (Unit CM)
Dollarts = (Fixed cost + Profit) / (CM ratio)
Assuming ABC desires a profit of $1,000, then
Volumets = ($3,000 + $1,000) / $8 = 500 units

21. The Effect of Changes in Parameters
Selling price and costs in the computations above are estimates and thus may not materialize.
It may also be possible to change the proportions of fixed and variable costs, e.g., through automation or use of higher quality material.
Thus, managers often perform an (a sensitivity) analysis to determine the impact of changes in estimates or cost components on BE point and profit.

22. The Effect of Change in Cost Elements
For example, NBC produces a surge protector.
SP = $30; VC = $18; total FC = $15,000; and sales volume = 2,000.
A proposed automation decreases VC by $3 per unit, but increases FC by $5,000.
What is the impact on profit?
Increase in CM ($3 * 2,000) $ 6,000
Increase in FC (5,000)
Net effect on profit $ 1,000

23. The Effect of Change in Selling Price
For example, XYZ produces a spray paint.
SP = $20; VC = $8; total FC = $10,000; and sales volume = 1,000.
Sales people insist that reducing the selling price by $4 per unit increases sales volume by 20%.
What is the impact on profit?
Proposed CM (8 * 1,200) $ 9,600
Current CM (12 * 1,000) 12,000
Net effect on profit $(2,400)

24. The Effect of Change in Sales Volume and Costs
For example, XYZ produces a spray paint.
SP = $20; VC = $8; total FC = $10,000; and sales volume = 1,000.
An advertising campaign costing $3,000 increases sales volume by 20%.
What is the impact on income?
Increase in CM (12 * 200) $ 2,400
Increase in FC (3,000)
Net change in profit $ (600)

25. Multi-Product Case
A multi-product case can be converted into a single-product case by use of the weighted-average CM per unit or a composite unit.
Example: Sam Corp. sells two products with the following SP and VC: A B SP/unit $ $ 20 VC/unit
Sam has been selling two units of A for every three units of B.
Total FC is $ 20,000.
What is the BE point?

26. Multi-Product – WA Approach
W.A. CM/unit = 2/5 * $7 + 3/5 * $12 = $ 10
Now we can assume that we have one product with a CM of $ 10 per unit.
BE in units = total FC / CM per unit = $20,000 / $10 = 2,000 units
Units of A to be sold = 2/5 * 2,000 = 800
Units of B to be sold = 3/5 * 2,000 = 1,200

27. Multi-Product – Composite Unit
We introduce a composite unit, Z, consisting of 2 units of A and 3 units of B.
CM per unit of Z = 2 * $7 + 3 * $12 = $ 50
Now we can assume that we have one product (Z) with CM of $ 50 per unit.
BE in units = total FC / CM per unit = $20,000 / $50 = 400 units of Z
Units of A to be sold = 2 * 400 = 800
Units of B to be sold = 3 * 400 = 1,200

28. Assumptions in CVP Analysis
Linearity of revenues and costs, i.e., efficiency, productivity, and selling price do not change
Accurate classification of costs into variable and fixed (i.e., only one cost driver, unit)
Constancy of sales and production mix
Constancy of the inventory level, i.e., sales = production
Equality of revenues and expenses with cash flows
Ignoring time value of money and non-quantitative information

29. Margin of Safety
Margin of safety is the excess of budgeted sales over the break-even sales. It is the amount by which sales can drop before losses are incurred.
Margin of safety = Budgeted sales - Break-even sales
Let’s calculate the margin of safety for ABC.

30. Margin of Safety
ABC has a break-even sales of $7,500; budgeted sales are $10,000. The margin of safety is $2,500.
Budgeted BE Sales
Sales $10, $7,500
-Variable costs , ,500
Contribution margin $ 4, $3,000
-Fixed costs , ,000
Net income $ 1, $ 0

31. Margin of Safety
The margin of safety can be expressed as 25 percent of sales.
($2,500 ÷ $10,000)
Budgeted BE Sales
Sales $10, $7,500
-Variable costs , ,500
Contribution margin $ 4, $3,000
-Fixed costs , ,000
Net income $ 1, $ 0

32. Operating Leverage
Operating leverage measures the percentage change in current profit as a result of a given percentage change in sales.
It is a measure of how sensitive net income is to change in sales.
Contribution margin
Net income
Degree of
operating leverage =

33. Operating Leverage $4,000 = 4 $1,000 Budgeted Sales
Sales $10,000 -Variable costs ,000 Contribution margin $ 4,000 -Fixed costs ,000 Net income $ 1,000
$4,000
$1,000
= 4

34. Operating Leverage
With an operating leverage of 4, if ABC increases its sales by 10%, net income would increase by 40%.

35. Mixed Costs – Definition
Mixed costs are costs that contain both variable and fixed components, e.g., utilities.
Costs aggregated in various ways are also mixed, e.g., overhead or machining.
The fixed component represents the minimum cost of having a service available for use; the variable component represents the cost of actual consumption.
Graphically, a mixed cost is represented by a straight line that intersects the vertical (Y) axis.

36. Variable Utility charge
Mixed Costs
Variable Utility charge
Total mixed cost
Fixed Utility charge

37. Variable Utility charge
Mixed Costs
Variable Utility charge
Total mixed cost
Fixed Utility charge

38. Analysis of Mixed Costs (Cost Estimation)
Involves estimating the fixed and the variable components of a mixed (or total) cost, i.e., generally estimating a linear cost formula Y= a + bX that can be used to estimate cost at various levels of activity.
Y is the total mixed cost (the dependent variable – it is affected by the level of activity).
a is the total fixed cost (the constant).
b is the variable cost per unit (the multiplier for X).
X is the level of activity (the independent variable).

39. Methods of Cost Estimation
Engineering Method
It is based on a study of input-output relationship.
The cost of all inputs are added to estimate the cost of the output.
This method is used only when input-output relationship remains stable over-time and indirect costs are a small portion of total cost; it is also used when there is no past data to analyze.
Analysis of Past Data

40. Analysis of Past Data Analysis of Past Data High-low Method
Scatter-graph
Simple Ordinary Least-Square (OLS) Regression
Note: all of these methods assume linearity and one independent variable (one cost driver).

41. Cost Estimation Example
Wise Co. recorded the following machine hours and Overhead cost for the last six months.
Machine Hours Overhead Cost
8 19

42. High-low Method
Graphically, a line that connects the observations with the highest and lowest value for the dependent variable.

43. High-low Method
Mathematically, choose the observations with the highest and lowest value for the dependent variable to estimate a and b in the cost formula Y = a + bX
b = rise/run = change in cost / change in activity = ( ) / (22 - 8) = 2.14
Go to the high or low observation only:
Y = a + bX a = Y - bX
a = * a = 1.92
Cost formula: Y = X

44. Notes on High-low Method
The 1.92 is not the estimated fixed overhead at zero level of activity. It is the estimated fixed overhead within the relevant range. Similarly, 2.14 is the estimated variable overhead per machine hour within the relevant range.
The method ignores all but two extreme observations. Thus, the accuracy of cost estimates depends on how representative the two points are of the entire set of observations.

45. Scatter-graph
Plot the data points on a graph and draw a line through those points that best fits the data.

46. Scatter-graph To estimate a and b in the cost formula:
Read the intercept and the slope from the graph.
Alternatively, use the coordinates of any two points on the line and the formulas as in the high-low method.
The method uses all the observations, but it is subjective (thus the reason for low usage and the quick-and-dirty label).

47. Simple Ordinary Least-Squares Regression
Estimate a and b in the cost formula in such a way that the sum of the squared errors (vertical distances between observations and regression line) is minimized.
Regression analysis is more appealing because we can measure the goodness of fit of the regression equation, and more importantly, because we can make probability statements about the estimated coefficient (b) and the estimated costs (Y).

48. Simple Ordinary Least-Squares Regression
Many computer programs provide estimates of a and b in the cost formula and regression related statistics; one only needs to understand the output.
Alternatively, for a simple regression, you can solve the following system of equations:
 Y = na + b  X
 XY = a  X + b  X2
where, n is the number of observations

49. Measures of Goodness of Fit for OLS
R2 , the % of variation in Y that is explained by X. R2 is between 0 and 1.
Standard error of regression (standard error of Y estimate), S(e), is an estimate of the standard deviation of the error term.
Note that the regression equation is Y = a + bX + e (e is the random variation, error term), and the estimated regression equation is: Y = a + bX

50. Measures of Goodness of Fit for OLS
Standard error of b, S(b), is an estimate of the standard deviation of b.
t = b/S(b)
As a rule of thumb, if t > 2, the estimated coefficient (b) is significantly different from zero, i.e., X explains a significant portion of the variation in Y.
Computer programs report the p-value, i.e., the significance level, for each variable.

51. Cost Estimation Example – Summary
Fixed Cost Variable Cost per period (a) per unit (b)
High-Low
Scatter-graph
Regression
Estimated cost for the next month at 17 MH:
High-Low = $ $2.14 * 17 = $38.30
Scatter-graph = $ $2.12 * 17 = $38.04
Regression = $ $2.23 * 17 = $39.32

52. Data Analysis Steps
Before using any of the three methods above, a few data analysis steps should be taken.
Review alternative activity bases
activity base chosen should be closely related to cost item (Y)
Plot the data
catch outliers and omit them
Examine data correspondence and period of accumulation
Examine the constancy of production process