Chapter 3 Cost-Volume-Profit (CVP) Analysis

The Cost-Volume-Profit model examines the relationship between firm cost structure (i.e., relative proportion of fixed and variable costs) and sales volume and the effects of this relationship on the profitability of a firm. The model can be used by managers for the purposes of planning and decision making.

This basic model combines four important variables — volume of sales, costs, revenue, and profits. The basic model can be extended to assess the impact of price, cost, and volume changes, along with changes in product mix and income taxes.

Following are some applications of CVP analysis.

x What are the total sales (either in units or dollars) that the company needs to generate in order to break-even or to attain a desired level of profit?

x What effects will changes in operating activities (such as changes in selling price or operating  costs)  have  on  the  company’s  profit?  For  example: o What effect will an increase in fixed costs such as rent or advertising will have

on our BEP and our profits? o What  effect  will  increase  or  decrease  in  sale  price  have  on  the  company’s  profit? o Should the company buy or lease a new machine? o What effect will adopting a new technology that leads to increase in fixed costs by

a certain percentage but at the same time leads a reduction of variable costs by a certain  percentage  will  have  on  BEP  or  the  company’s  profit?

x Should we produce more of product A and less of product B or vice versa?

CVP Analysis Single Product Case Multiple Products Case

x Mathematical Solution Mathematical Solution Graphical Solution x Effect of Income Tax x Sensitivity Analysis x Operating Leverage x Margin of Safety

It is important to note that the CVP analysis is performed at the firm wide level.

The Basics of CVP Analysis

The Basic Assumptions of CVP Model:

The CVP model is simplified by the following assumptions: 1. Both the revenue function and the cost function are linear. 2. The selling prices, total fixed costs, and unit variable costs are known with certainty in

advance and will remain unchanged during the period. 3. The number of units produced equals the number of units sold. This suggests that there

no changes in the level of inventory during the period. 4. The productivity of workers is constant. 5. For multiple-product analysis, the sales mix is assumed to be known in advance and

remains constant during the period. The Concept of Contribution Margin (CM):

x Contribution margin is the amount remaining from sales revenue after variable expenses have been deducted. This amount contributes towards covering fixed costs and then towards making profit.

x Contribution margin is the net summary of the changes in that operating income. As the quantity of units sold increases, both total variable costs and total revenues increase at the same rate. If revenues increase due to volume increases, the contribution margin increases.

x Understanding contribution margin enables the manager to quickly note that an increase in selling price without a corresponding change in variable cost will increase the “contribution”  to  cover  fixed  cost  and  make  income.    Or  a  decrease  in  variable  cost  without a corresponding decrease  in  selling  price  will  “contribute”  more  to  income  and/or  the coverage of fixed costs.

x Using  revenues  and  variable  costs  as  per  unit  measures,  the  “contribution”  per  unit  of  product  sold  can  provide  a  shortcut  to  breakeven  calculations  or  “what-if”  questions.    Each  unit  of  product  sold  contributes  that  amount  as  it  “walks  out  the  door.”  

How to Calculation of Contribution margin: There several ways to calculate and express the CM

a. Unit CM = Unit SP – Unit VC b. Total CM = Total Sales Revenues – Total Variable costs c. Total CM = Unit CM x Q d. CM% = Total CM/ Total Sales Revenues e. CM% = Unit CM/Unit SP f. CM% + VC% = 100%

I. SINGLE PRODUCT CASE A. MATHEMATICAL SOLUTION

X (units) = the number of units that should be produced and sold to achieve the desired level of profit X ($) = the total sales revenues that should be generated to achieve the desired level of profit NI = the desired (target) level of profit SP = selling price per unit VC = variable cost per unit TR = total revenues TVC = total Variable Costs = FC = total fixed costs

NI = TR – TVC – FC

= X(SP) – X(VC) – FC

NI = X(SP-VC) – FC

X (SP-VC) = FC +NI X (units) = FC + NI = FC + NI (SP-VC) CM/unit

Unit sales to attain target profits = marginon contributiUnit

profitsTarget + expenses Fixed (1)

Similarly, the basic equation can also be expressed in terms of sales dollars using the variable expense ratio to drive the formula for the dollar Sales needed to achieve target income as follows:

Net Income = Sales - (Variable expense ratio u Sales) - Fixed expenses

 

(1 � Variable expense ratio) u Sales = Fixed expenses + Net Income

 

Contribution margin ratio* u Sales = Fixed expenses + Net Income

 

Sales = CM%

NI+ FC

* 1 � Variable expense ratio = 1�Sales

expenses Variable

= Sales

expenses Variable-Sales

= Sales

marginon Contributi = CM%

Sales $ to achieve target profits = CM%

ProfitTarget +FC (2)

Example 1: The following information is extracted from ABC Co. Sales price per unit $ 36.00 Per unit variable costs Production costs 18.60 Sales commissions 5.40 Total per unit variable costs $ 24.00 Total fixed costs per period Advertising $ 24,000 Rent 30,000 Salaries 126,000 Total fixed costs $180,000

Required: 1. How many units to break even? 2. How many units to achieve net income of $200,000? 3. Repeat 1 and 2 above for the amounts rather than number of units.

Example 2: Steve Bendo owns car service station in Hamilton. Steve is considering leasing a machine that will allow him to offer customers the mandatory Ontario emissions test. Every car in Ontario must be tested every two years. The machine costs $6,000 per month to lease. The variable cost per test (i.e., per car inspected) is $10. The amount that Steve can charge each customer is set by the Province law, and is currently $40. Required:

1. How many inspections would Steve have to perform monthly to break even from this part of his business?

2. How many inspections would Steve have to perform monthly to generate a profit of $3,000 from this part of his business?

Example 3: Alice Waters (age 9) runs a lemonade stand in the summer in Dundas, Ontario. Her daily fixed costs are $20. Her variable costs are $2 per glass of ice-cold, refreshing, lemonade. Alice sells an average of 100 glasses per day. Required: 1. What price would Alice have to charge per glass, in order to break-even per day? 2. What price would Alice have to charge per glass, in order to generate profit of $20 per day? 3. Refer to the information about Alice, but now assume that Alice wants to charge $3 per glass of

lemonade, and at this price, Alice can sell 110 glasses of lemonade daily. What would the variable cost per glass have to be, in order to generate profits of $200 per day?

Example 4: Sunshine Co. sells a single product. The company's most recent income statement is given below. Sales (4,000 units) $120,000 Less variable expenses (68,000) Contribution margin 52,000 Less fixed expenses (40,000) Net income $ 12,000 Required: a. Contribution margin per unit is $ _______ per unit b. If sales are doubled to $240,000, total variable costs will equal $ _______________ c. If sales are doubled to $240,000, total fixed costs will equal $ _______________ d. If Sunshine is past the breakeven point and 10 more units are sold, profits will increase by $ _______________ e. Compute how many units must be sold to break even. # ______________ f. Compute how many units must be sold to achieve a profit of $20,000. # _______________ MC: 1.. Nantucket Company has the following cost-volume-profit (CVP) relationships:

What is the variable cost per unit? A) $515.00 B) $562.50 C) $625.00 D) $655.25 2. AAA Company produced a product which had a selling price of $20 and a variable cost which

amounted to 60% of sales. Given a fixed cost of $60,000, the breakeven sales will be a. 5,000 units b. 5,500 units c. 6,000 units d. 7,000 units e. 7,500 units

3. AAA Company produced a product which had a selling price of $20 and a variable cost which amounted to 40% of sales. The company wants a profit before tax of $15,000. The tax rate is 20% and fixed costs amount to $60,000. AAA must sell a. 6,250 units b. 7,396 units c. 9,375 units d. 9,844 units e. None of the above

4. AAA currently has a profit of $15,000 at a sales volume of 9375 units and a fixed cost which amounts

to $65,625 and a selling price of $20 per unit. Variable cost per unit should be a. $12.6 b. $12.0 c. $11.4 d. $11.0 e. None of the above

Q 1 2 3 4 A b e a c

Breakeven point in units sold 2,000Sales price per unit $ 625Total fixed costs $125,000

THE EFFECT OF INCME TAX x Organizations making profit must pay income taxes. A business only gets to keep income after

taxes (net income). Thus, the CVP analysis is more informative if it shows what it will take to generate a target amount of net income.

x In order to calculate the sales volume needed to achieve a specified amount of net income, a tax rate as a percentage of operating income is assumed. Target profit must be adjusted by incorporating income tax.

x Mathematically, the amount of net income can be expressed as a percentage of operating income. Therefore, the target after-tax profit must be converted to its before-tax equivalent using the following formula:

Target before - tax profitTarget after - tax profit

1 - Tax rate

Or

unitpermarginonContributi

rate)]Tax–profit)/(1tax(After[costFixed=(units) X � (1')

x Note that the BEP unaffected by income taxes because no tax is no operating income

Use the following data to answer the next two questions: Kelvin Co. produces and sells socks. Variable costs are $4 per pair, and fixed costs for the year total $90,000. The selling price is $6 per pair. 1. The sales units required to make an after-tax profit of $15,000, given an income tax rate of forty percent, are

calculated to be: A) 56,000 units. B) 56,500 units. C) 57,000 units. D) 60,000 units. E) 57,500 units.

2. The sales dollars required to make an after-tax profit of $15,000, given an income tax rate of forty percent,

are calculated to be: A) $336,000. B) $339,000. C) $342,000. D) $360,000. E) $345,000.

Q 1 2 A e e

-

Sensitivity analysis x Sensitivity analysis:  “what-if”  technique  managers  use  to  examine  how  a  result  will  change  if  

original predicted data not achieved or if an underlying assumption changes

x Uncertainty: possibility that an actual amount will deviate from an expected amount

x Used before committing costs: Perform analysis of changes in operating income for changes underlying assumptions

Example 1: Selling price per unit $500 Variable cost per unit $300 Total monthly fixed cost $200,000 Units sold per month 1,600

Required: 1. How many units do we need to sell to break even? 2. What effect will a 10% discount on sales price have on BEP in units? 3. New equipment will increase FC by 30%, but decrease VC per unit by 30%. What effect will it

have on BEP in units? 4. Purchase of higher quality raw materials will increase VC by $25 per unit, but decrease FC by

$17,500. How many units do we need to sell to maintain operating income of $120,000? MC 1. AAA currently has a profit of $15,000 at a sales volume of 6250 and a variable cost of $8 and a

selling price of $20. If variable costs increase to $9, by how much can the fixed costs change to still maintain the same profit?

a. $6,000 decrease b. $6,250 decrease c. $6,000 increase d. $6,250 increase

2. AAA Company produced a product which had a selling price of $20 and a variable cost which

amounted to 60% of sales. The company wants a profit after tax of $15,000. The tax rate is 20% and fixed costs amount to $60,000. AAA must sell

a. 6,250 b. 7,396 c. 9,375 d. 9,844

3. The Beta Mu Omega Chi (BMOC) fraternity is looking to contract with a local band to perform at its

annual mixer. If BMOC expects to sell 250 tickets to the mixer at $10 each, which of the following arrangements with the band will be in the best interest of the fraternity?

a. $2500 fixed fee b. $1000 fixed fee plus $5 per person attending c. $10 per person attending d. $25 per couple attending

Q 1 2 3 A b d b

CVP RELATIONSHIPS IN GRAPHIC FORM. x Graphs of CVP relationships can be used to gain insight into the behavior of expenses and profits.

The basic CVP graph is drawn with dollars on the vertical axis and volume in units on the horizontal axis. Total fixed expense is drawn first, then variable expense is added to the fixed expense in order to draw the total expense line. Finally, the total revenue line is drawn. The total profit (or loss) is the vertical difference between the total revenue and total expense lines.

x The cost-volume-profit graph depicts the relationships among cost, volume, and profits.

x The point where the total revenue line and the total cost line intersect is the break-even point.

Example 1: ABC Company has a target profit after tax of $12,000 for 2007. The company pays tax at a rate of 25%. Use the next graph to answer the following questions:

1. Calculate the SP per unit, VC per Unit, and CM per Unit. 2. How many units to produce and sell so as to achieve the target NI for 2007. 3. If the company produced and sold 3,500 units during 2007, how much is the NI after tax.

TC, 0, 12,000

TC, 1,000, 16,000 TC, 2,000, 20,000

TC, 3,000, 24,000

TC, 4,000, 28,000

TSR, 1,000, 10,000

TSR, 3,000, 30,000

TSR, 4,000, 40,000

Dollars

Units Sold

Total Revenue

Profit

Total Cost

Break-Even Point

Loss

Example 2:

Use the above graph to answer the following questions:

Required: 1. What is the variable cost per unit? 2. What is the net income if the company sold 900 units? 3. What is the net income if the company sold 200 units? 4. If fixed costs increased by $3000, what is the new breakeven point.

Example 3

The following graph portrays the CVP activities for ABC Co. for the year 2006. ABC produced and sold 4000 units during 2006.

Required: During 2006, ABC produced and sold 4000 units. Complete the income statement for ABC Co. for 2006

Income Statement for the period ended December 31 2006

Sales Revenues

- Total Variable Costs

= Contribution Margin

- Total Fixed Costs

= Net Income

Margin of Safety The margin of safety is the excess of budgeted (or actual) sales over the break-even volume of sales. It is

the amount by which sales can drop before losses begin to be incurred. The margin of safety can be

computed in three different ways:

CVP Chart for ABC Co.

$48,000

$36,000

$0

$20,000

$40,000

$60,000

$80,000

0 1,000 2,000 3,000 4,000 5,000 6,000

Sales Volume

Margin of safety (in units) = Expected sales (in units) – Sales at break-even (in units) Margin of safety ($) = Expected sales ($) – Sales at break-even ($)

Margin of safety percentage = sales Total

dollarsin safety ofMargin

x The margin of safety ratio is a useful measure of comparing the relative risk among alternative products or for assessing the riskiness in any given product. A relatively low margin of safety ratio for a product is usually an indication that the product is riskier than higher margin of safety products.

Use the following to answer the next three questions: OutlyTech Corp. expected to sell 24,000 telephone switches. Fixed costs were $12,150,000, unit sales price was $4,190, and unit variable costs were $1,440. 1. OutlyTech's margin of safety in units is calculated to be: A) 18,276.37. B) 19,581.82. C) 20,887.27. D) 16,970.91. E) 22,192.73. 2. OutlyTech's margin of safety in sales dollars is calculated to be: A) $76,577,990. B) $87,517,661. C) $82,047,826. D) $71,108,113. E) $92,987,539. 3. OutlyTech's margin of safety ratio is calculated to be: A) 81.59%. B) 87.03%. C) 70.71%. D) 92.47%. E) 76.15%.

Operating Leverage & Cost Structure Cost Structure Cost structure refers to the relative proportion of fixed and variable costs in an organization. The optimal cost structure for a company depends on many factors, including the long-run trend in sales, year-to-year fluctuations in the level of sales, and the attitudes of owners and managers towards  risk.  Understanding  a  company’s  cost  structure  is  important  for  decision  making  as  well  as for analysis of performance. Operating leverage Operating leverage is  named  for  the  “lever”  effect  that  comes  from  the  use  of  fixed  costs  to  generate more profit. If the choice exists to incur fixed or variable cost, and fixed is chosen, then variable cost would be less, yielding a larger contribution margin and the possibility of larger profit. Once the fixed costs are recovered, the contribution margin is profit. This effect can be seen on a breakeven graph. The intersecting revenue and total cost lines create equal and opposite angles at the intersection point. One can note that the risk (downside) is equal to the reward (upside). The larger the fixed cost, the wider the intersection angles usually: the greater the opportunity for reward, the greater the possibility of loss. How to Measure operating leverage Operating leverage is a measure of how sensitive net income is to a given percentage change in sales volume. It is affected by the relative mix of fixed and variable costs. This trade-off affects the amount by which profits will change as sales fluctuate and is measured by the degree of operating leverage (CM/Profit) using a given level of sales as the reference point. Once the degree of operating leverage (DOL) is computed, the percentage change in profits is computed by multiplying the percentage change in profits by the DOL. The degree of operating leverage at a given level of sales is computed as follows:

incomeNet marginon Contributi=leverage operating of Degree

How to Use operating leverage A company with high operating leverage would have high FC, low VC, and high CM. Such a company would experience a large change in operating income for a small change in sales, as compared with a company which have low operating leverage.

The DOL can be used to measure the percentage of change in profit that results from a percentage of change in sales. The higher the degree of operating leverage, the greater the change in profit when sales change.

Percentage change in profit = DOL × Percentage change in sales

Thus, given that there is no change in fixed costs, the degree of operating leverage provides a quick way to predict the percentage effect on profits of a given percentage increase in sales. The higher the degree of operating leverage, the larger the increase in net income.

x Note that the degree of operating leverage is not constant. It changes as the level of sales changes. For example, at the break-even point the degree of operating leverage is infinite since the denominator of the ratio is zero. Therefore, the degree of operating leverage should be used with some caution and should be recomputed for each level of starting sales.

Example: The following data are related to two identical companies with different cost structure:

Comp A Comp B Sales 300,000 300,000 Variable cost 255,000 120,000 CM 45,000 180,000 Fixed cost 15,000 150,000 Operating income 30,000 30,000 Both companies sell same product at same price. While Company A is more labour intensive, Company B is more capital intensive.

Required: 1. Calculate the BEP in dollar for both companies. 2. Calculate the degree of operating leverage for both companies. 3. What effect will a 20% increase in sales  have  on  both  companies’  operating  income?

MULTIPLE-PRODUCT ANALYSIS

Sales Mix: x The  term  sales  mix  means  the  relative  proportions  in  which  a  company’s  products  are  sold.  

Most companies produce a number of different products with different selling prices, costs, and contribution margins. Thus, changes in the sales  mix  can  cause  variations  in  a  company’s  profits. As a result, the break-even point in a multi-product company is dependent on the mix in which the various products are sold.

Constant sales mix assumption x CVP analysis with multiple products assumes that sales will continue at the same mix of

products, expressed in either sales units or sales dollars. In essence, the assumption is made that the firm has only one product that consists of a basket (package) of its various products in a specified proportion.

x This assumption is essential, because a change in the product mix will probably change: A) the weighted-average sales price, B) the weighted-average variable cost, and C) the weighted-average contribution margin.

x The sales mix can be expressed either in terms of units or dollars of sales revenues. Thus, we can calculate the sales revenues (in units or dollars) to achieve a certain level of target profit in one of three different equations as follows: 1. # of packages that the company should produced and sell to achieve an overall target

profit. 2. # of units of each product that the company should produced and sell to achieve an overall

target profit. 3. Total sales revenues ($) that the company should generate from all products to achieve an

overall target profit. 1. # of packages that the company should produce and sell to achieve an overall target

profit.

Total Packages to sold to achieve target profits = /packageCM

profitsTarget + expenses Fixed (3)

2. Total # of all units that the company should sell to achieve an overall target profit.

Total Sales Units from all products to achieve target profits = /unitWACM

profitsTarget + expenses Fixed (4)

Where: WACM/unit = Weighted Average contribution margin per unit WACM/unit = ($CM/unit of A x SM% units A ) + ($CM/unit of B x SM% units B )+  …   Or

WACM/unit = Total CM per package/ total # of units in a package from all products

3. Total sales revenues ($) that the company should generate from all products to achieve an overall target profit.

Example 1: California Co. Produces two products A & B. Total fixed costs per month for the company is $200,000. Following information is extracted from the company's records.

Product Mix Price/Unit VC/Unit

A 3 $10 $6 B 2 8 5

Required:

1. How many units of each products to breakeven 2. How much is total sales revenues to breakeven

Total Sales $ from all products to achieve target profits = ratio WACM

profitsTarget + expenses Fixed (5)

Where: WACM% = Weighted Average contribution margin percentage

WACM% = (CM% A x SM% $ A ) + (CM% B x SM% $ B ) +  ……….   Or

WACM ratio = sales Total

marginon contributi Total

Use the following data to answer the next three questions: The following annual information is for Barnett Corporation: Product X Product Y Revenue per unit: $10.00 $15.00 Variable cost per unit: $ 2.50 $ 5.00 Total fixed costs: $50,000 1. If the sales mix consists of two units of Product X and one unit of Product Y, what is the weighted

revenue per unit of composite product? a. $10.00 b. $11.66 c. $13.33 d. $15.00 2. If the sales mix consists of two units of Product X and one unit of Product Y, what is the break-even

point in units for a year? a. 1,000 units of Y and 2,000 units of X b. 1012.5 units of Y and 2,025 units of X c. 2012.5 units of Y and 4,025 units of X d. 2,000 units of Y and 4,000 units of X 3. What is the operating income for a year, assuming actual sales total 150,000 units, and the sales mix is

two units of Product X and one unit of Product Y? a. $1,200,000 b. $1,250,000 c. $1,750,000 d. None of the above is correct. 5. ABC  Company  sells  three  products  with  exactly  the  same  price  of  $20  a  unit.    However,  A’s  variable  cost  

is  at  40%,  B’s  at  50%,  and  C’s  at  60%. Sales mix for A, B, and C is at 500, 1500, and 3000 units respectively. Fixed costs amount to $18,000. Breakeven sales for B should be a. 600 b. 1,200 c. 1,800 d. 2,000

7. ABC Company sells three products with exactly the same price of $20 a unit. However, A’s  variable  cost  is  at  40%,  B’s  at  50%,  and  C’s  at  60%.    Fixed  costs  amount  to  $18,000.    An  additional  $9,000  needs  to  be  spent on advertising to boost sales. Sales mix is at 500, 1500, and 3000 units for A, B, and C respectively. Sales in dollars for C at breakeven amounts to a. $18,000 b. $27,000 c. $36,000 d. $45,000

8. ABC’s  sales  mix  has  drastically  changed  due  to  market  conditions  to  3000,  1500,  and  500  units  for  A,  B,  and  C  respectively. Fixed costs have increased to $22,000 per period. The selling price is at $20 a unit for all products with a variable cost of 40%, 50%, and 60% for A, B, and C respectively. Breakeven units for A will be a. 300 b. 600 c. 900 d. 1,200

Use the following data to answer the next two questions:

Crown Co. can produce two types of lamps, the Enlightner and Foglighter. The data on the two lamp models are as follows: Enlightner Foglighter Sales volume in units 500 400 Unit sales price $300 $400 Unit variable cost (200 ) (240 ) Unit contribution margin $100 $160 It takes one machine hour to produce each product. Total fixed costs for the manufacture of both products is $90,000. Demand is high enough for either product to keep the plant operating at maximum capacity. 9. Assuming that sales mix remains constant in units, the breakeven point in total units is: A) $ 687. B) $ 710. C) $ 805. D) $ 945. E) $1,006.

Answer: B

10. Assuming that sales mix remains constant in dollars, the breakeven point in units is: A) $ 687. B) $ 710. C) $ 805. D) $ 945. E) $1,006.

Answer: A 11. Twin Products Company produces and sells two products. Product M sells for $12 and has variable

costs of $6. Product W sells for $15 and has variable costs of $10. Twin predicted sales of 25,000 units of M and 20,000 of W. Fixed costs are $60,000 per month. Assume that Twin achieved its sales goal of $600,000 for September, but fell short of its expected operating income of $190,000. Which of the following descriptions best describes the actual results reported of revenue of $600,000 and operating income of less than $190,000?

a. Twin sold 50,000 of M and no product W. b. Twin sold more of both products M and W than expected. c. Twin sold more of product W and less of product M than expected. d. Twin sold more of product M and less of product W than expected.