# Formula to Calculate Profit Level

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```					COST VOLUME PROFIT
Richard E. McDermott Ph.D.
Costs Volume Profit
   In this chapter we study what happens to both cost
and profit as volume changes.
   The tools we will use will be helpful in the
development of flexible budgets.
   I am going to begin by giving you some basic
definitions.
   I will then give you what I feel are the best formulas
to use in working the problems you will be assigned.
Definitions
   Variable Costs: Costs that vary in total, directly and
proportionately, with changes in production (also
called activity level).
   If the activity level increases by 50%, variable costs
increase by 50%.
Examples of Variable Costs
   Direct labor
   Direct materials
Definitions
   Mixed Costs: costs that contain both a fixed and a
variable element.
   Mixed cost change in total, but not proportionately,
with changes in the activity level.
   An example of a mixed cost might be maintenance
cost on a taxi.
 Maintenance    costs increase as miles increase. Even if
the truck is never driven, however, it is a good idea to
change the oil every three months to keep it from
degenerating.
Separating Fixed and Variable Costs

   If you will look at a general ledger, you will never
find accounts labeled ―variable labor‖ or ―fixed
   In order to perform cost volume profit analysis,
therefore, it is usually necessary to separate fixed
and variable costs.
   There are three methods to do this:
 Scatter graph method
The best method is the least
 High-low method
squares method, but since
 Least squares method
the author does not teach it
we will not cover it either.
Definitions
   Fixed Costs: Costs that do not change with increases or
decreases in production volume.
   It is important to add, there is usually a relevant range.
   For example, a plant may be built to manufacture 1 to
10,000 shoes per month.
   Fixed costs would not change within this ―relevant
range.‖
   If a company wanted to manufacture 12,000 shoes per
month, however, there of course would be a new
relevant range and the fixed costs might increase.
Examples of Fixed Costs
   Rent on a factory
   Depreciation – straight-line method
   Heating and air-conditioning expense
   Housekeeping

It is important to emphasize that what might be a fixed
cost in one factory, could be a variable cost in another,
depending upon the way the firm does business.
Other Definitions
   Contribution Margin: Revenue (or unit price) minus
total variable costs (or unit variable costs).
   Example:
Contribution Margin Income
Statement                  Note that
contribution
Sales                   \$100,000
margin is what
Less variable costs       60,000        is left after
Contribution margin      \$40,000        paying
Less fixed costs          30,000        variable costs.
Income from              \$10,000
operations
Other Definitions

Contribution Margin Income
Statement             After fixed
costs are paid,
Sales                   \$100,000
where does the
Less variable costs       60,000   contribution
Contribution margin      \$40,000   margin go? To
Less fixed costs          30,000   the bottom line!
Income from              \$10,000
operations
Notation
   I will use the following notation when referring to
contribution margin:
 CMu   = contribution margin per unit of produce,
calculated as follows: Price – Variable Cost Per Unit =
Cmu.
 CMt = contribution margin total calculated as follows:
Total Sales – Total Variable Costs = CMt
 CMr = contribution margin ratio calculated as follows:
CMt/Total Sales or CMu/Price.
Tip for Examination
   In any problem where the only change is volume
(i.e. variable costs per unit, and fixed costs do not
change), then the impact of the change on
operating income can be calculated simply by
calculating the change in total contribution margin.
   The change in total contribution margin will equal
the change in operating income.
   We will illustrate this in a moment.
Here Are Some Formulas I Would like
You to Learn
   PX – VX – F = P
    where
   P = unit price
   X = volume of units sold
   V = variable cost per unit
   F = total fixed price
   P = operating income or profit
Breakeven
   Breakeven is defined as the point where the firm
neither makes nor loses money.
   The formula for breakeven is:
   PX – VX – F = 0
   Let us illustrate the solution for breakeven with an
example.
Example
   Morris Electronics makes calculators.
   The market dictates a price of \$30 for a model
with their particular features.
   Variable costs per unit are \$16, total fixed costs
are \$200,000.
   What is the breakeven point in units for sales of the
calculators?
Solution
   PX - VX – F = 0                    Income statement
   30X – 16X – 200,000        Sales                   \$428,571
=0                         Variable cost            228,571
   14X = 200,000              Contribution            \$200,000
margin
   X = 14,285.714286
Fixed cost               200,000
calculators
Operating                          \$0
income

Note that at breakeven, contribution margin equals fixed costs.
New Problem
   Most companies are not in the business to
breakeven.
   Assume that Morris Electronics wants to make
\$50,000.
   How many calculators will they have to sell to
achieve that objective?
   We are solving for ―target sales‖
Solution
   PX - VX – F = P
   30X – 16X – 200,000 = 50,000
   14X = 250,000
   X = 17,857.14285 calculators
   What if we want to know the sales dollar volume to
break even?
   17,857.14285 x 30 = \$535,714
Deriving an Alternate Formula
   PX – VX – F = P
   Simplifying the equation
   X(P – V) – F = P
   But we know that price per units minus variable cost per unit
equals the contribution margin
   Since (P – V) = CMu
   (CMu)(X) = F + P
   X = (F + P)/CMu when we want to know target sales in units
for a specific level of profits
   If we are solving for target sales at breakeven, P = 0 so
   X = F/CMu

Formulas for target sales with a profit and no profit (breakeven)
Example Problem
   Let us use the same data from Morris Electronics,
where the company wants to earn a \$50,000 profit.
   X = (F + P)/CMu
   X = (200,000 + 50,000)/14
   X = 250,000/14
   X = 17,857.142858 units breakeven
   Or
   Sale at breakeven = 17,857.142858 x 30 =
\$535,714
Now Finding Breakeven Sales Dollar
Volume
   We can either multiply 17,857.142858 times the sales
price of \$30, or use this formula
   S = (F + P)/CMr where S = sales dollars, CMr stands
for contribution margin ratio, and the contribution
margin ratio is defined thus:
   CMr = Total Contribution Margin/Sales
   Or CMr = Unit Contribution Margin/Price
   CMr in this problem is 14/30 = .466667
   So sales in dollars = (200,000 + 50,000)/.466667 =
   \$535,714
Therefore the Formula For Breakeven
Would Be:
   S = (F + P)/CMr
   Or
   S = (F + 0)/CMr
   Or
   S = F/CMr
   Two formulas to get the same answers!
   To get breakeven in dollars use F/CMr where CMr
is the contribution margin ration (contribution
margin divided by sales).
Let Us Solve Another Problem
   Assume that Morris Electronics can only manufacture
15,000 units a year.
   The variable cost is \$16 per unit, and fixed costs
are \$200,000 per year.
   Assuming that they want to earn \$50,000 per year,
what must the price be to obtain their net profit
objective?
Solution
   PX – VX – F = 
   P(15,000) – 16(15,000) – 200,000 = 50,000
   15,000P – 240,000 – 200,000 = 50,000
   15,000P = 490,000
   P = \$32.6667
   To check:
    \$490,000 – 240,000 – 200,000 = \$50,000
Alternate Formulas
   We could have used:
   X = (F + )/CMu to solve for breakeven in units
where of course  is 0
   Or
   S = (F + )/CMr to solve for breakeven in sales
dollars to get the same results (again at breakeven
 is 0).
 In   this case, S is of course sales dollars.
Income Statements

Traditional income statement          CVP Income Statement

Sales                          \$xxx   Sales                  \$xxx
Cost of Goods Sold              xxx   Variable Costs          xxx
Gross Margin                    xxx   Contribution Margin     xxx
Administrative and              xxx   Fixed Costs             xxx
Selling Expense                       Income                   xx
Income                           xx

A better income statement to use for internal reporting,
especially when management wants to do CVP analysis, is the
CVP income statement shown above.
Income Statements

It is better, because it gives us the figures needed for CVP
analysis.

FASB still requires the traditional income statement for external
reporting, however.
One More Concept . . .
   Margin of Safety: The difference between actual or
expected sales and sales at the break even point.
   The formula is:
 Actual (expected) sales – breakeven sales = margin of
safety in dollars.
Problem
   Johnson Foundry currently sells \$1,500,000 of
product a year.
   Their breakeven point is \$1,250,000.
   What is their margin of safety?
   \$1,500,000 – \$1,250,000 = \$250,000
Separating Fixed and Variable Costs

   One method (a very inaccurate method) of separating
fixed and variable costs is the high-low method.
   The method is inaccurate because it depends upon only
two data points, a high and low point.
   The small sample size, plus the fact that high and/or
low points are often outliers caused by inaccurate
   Nevertheless, since the book teaches it, so will I (sigh).
Steps

   Use the following formula to determine variable
costs.
(high costs  low costs)
 variable costs per unit
(high activity  low activity )
Steps
   Calculate total variable cost and total costs at high
or low activity.
   Subtract total variable costs from total costs (at high
or low activity) to determine total fixed costs.
Example
   Community Hospital’s controller has provided you with
the following information. Using the high-low method,
divide payroll into fixed and variable costs.
Chest     Labor
X-rays    Dollars
January     1,200    16,400
February    1,400    18,800     High
March         900    12,800
April         875    12,500     Low
May         1,300    17,600
June        1,350    18,200
Solution

1. Calculate variable costs using this formula.
(high costs  low costs)
 variable costs per unit
high activity  low activity

\$18,800  \$12,500
 \$12 / xray
1, 400 875
Solution
   2. Calculate variable costs at high or low (we will use
high here).
   \$12 variable cost per x-ray x 875 x-rays = \$10,500
   3. Subtract variable costs at high or low (we will use
low here) from total costs to get fixed costs.
   \$12,500 total costs – 10,500 variable costs =
\$2,000 fixed costs
   Summary: variable costs = \$12 per unit and fixed
costs = \$2,000 per period (per month in this case)
Question
   Given this data what would be the total cost at
1,145 x-rays?
   Variable costs = \$12 x 1,145 = \$13,740
   Fixed costs = \$2,000
   Total costs at 1,145 x-rays = \$15,740

The best method, the least squares method can be
worked using Excel or a financial calculator. If you
want to learn how to do it on a financial calculator,
look in the hp 10bII instruction book under
―regression).
Brief Exercise 5-1
   Monthly costs for two levels of production are given
below. From this information determine which costs
are variable, fixed, and mixed, and give the reason

Cost                     3000 units      6000 units
Indirect labor             \$10,000         \$20,000
Supervisory salaries        \$5,000          \$5,000
Maintenance                   4,000          7,000
Brief Exercise 5-1

Cost                                3000 units          6000 units
Indirect labor                        \$10,000             \$20,000
Supervisory salaries                   \$5,000              \$5,000
Maintenance                             4,000               7,000

Indirect labor is obviously variable, as it varies proportionally
with volume. When volume doubles, costs double.
Brief Exercise 5-1

Cost                              3000 units          6000 units
Indirect labor                      \$10,000             \$20,000
Supervisory salaries                 \$5,000              \$5,000
Maintenance                           4,000               7,000

Supervisory salaries are obviously fixed, since they stay the same
At different levels of production.
Brief Exercise 5-1

Cost                             3000 units          6000 units
Indirect labor                     \$10,000             \$20,000
Supervisory salaries                \$5,000              \$5,000
Maintenance                          4,000               7,000

Maintenance salaries are mixed, since they increase,
but not proportional to increase in production volume.
Brief Exercise 5-2
   For Loder Company, the relevant range of
production is 40% to 80% of capacity.
   At 40% of capacity, variable cost is \$4000 and
fixed cost \$6,000.
   Diagram the behavior of each costs within the
relevant range assuming the behavior is linear.
Brief Exercise 5-2

To create these graphs I used Excel.
Brief Exercise 5-3
   For Hunt Company, a mixed cost is \$20,000 plus
\$16 per direct labor hour.
   Diagram the behavior of the cost using increments
of 500 hours up to 2,500 hours on the horizontal
access, and increments of \$20,000 up to \$80,000
on the vertical axis

Note to students: I think the question would have been clearer if
the author had said “fixed cost is \$20,000 plus \$16 per direct
labor hour.”
Brief Exercise 5-3
Exercise 5-2
   Kozy Enterprises is considering manufacturing a new
product.
   It projects the costs of direct materials and rents for
a range of output as shown on the following slide.
Output in Units   Rent Expense   Direct Materials
1000            5000              4000
2000            5000              6000
3000            5000              7800
4000            7000              8000
5000            7000            10,000
6000            7000            12,000
7000            7000            14,000
8000            7000            16,000
9000            7000            18,000
10,000          10,000            23,000
11,000          10,000            28,000
12,000          10,000            36,000
Exercise 5-2

Relevant Range
Exercise 5-2

Relevant Range
Exercise 5-2
   The relevant range is 4,000 – 9,000 units of output
since a straight-line relationship exists for both
direct materials and rent within this range.
Exercise 5-4
   Identify each of the following costs as variable, fixed,
or mixed.
   Wood used in production of furniture—variable
   Fuel used in delivery trucks– variable
   Straight-line depreciation on factory buildings– fixed
   Screws used in production of furniture– variable
   Sales staff salaries– fixed
   Sales commissions– variable
   Property taxes– fixed
Exercise 5-4
   Insurance on buildings– fixed
   Hourly wages of furniture craftsman– fixed
   Salaries of factory supervisor– fixed
   Utilities expense– mixed
   Telephone bill– mixed
Brief Exercise 5-8
   Larisa Company has a unit selling price of \$520,
variable costs per unit of \$286, and fixed cost of
\$187,200.
   Compute the breakeven in units using the
mathematical equation and contribution margin per
unit.
Mathematical Equation
   PX – VX – F = 
   520X – 286X – 187,200 = 0
   234X = 187,200
   X = 800
Contribution Margin Per Unit
   Breakeven = F/CMu
   187,200/(520 – 286)
   187,200/234
   X = 800 units
Brief Exercise 5-9
   Turgro Corp. had total variable cost of \$180,000
total fixed costs of \$160,000, and total revenues of
\$300,000.
   Compute the required sales in dollars to break
even.
Brief Exercise 5-9
   We know that at breakeven, fixed costs =
contribution margin.
   Since fixed costs are \$160,000 then the
contribution margin must be \$160,000
   We know that CMr is CMt/Sales (CMt = total
contribution margin from income statement)
   From the data given in the problem we know the
CMr is (300,000 – 180,000)/300,000 = 40%
   So when CMt is \$160,000, then sales must be
\$160,000/.40 = \$400,000
Brief Exercise 5-10
   For MeriDen Company, variable costs are 60% of
sales and fixed costs are \$195,000.
   Management’s net income goal is \$75,000.
   Compute the required sales in dollars needed to
achieve management’s target net income if
\$75,000.
   Use the contribution margin approach.
Brief Exercise 5-10
   S (target sales) = (F + )/CMr
   S = (195,000 + 75,000)/.40
 How   did I get .40 for CMr? One trick to remember is
that contribution margin ratio is (1 – variable expense
ratio)
 Thus CMr is 1 - .60 = .40

   S = \$270,000/.40
   S = \$675,000
Exercise 5-8
   Green with Envy provides environmentally friendly
lawn services for homeowners.
   Its operating costs are as follows:
 Depreciation = \$1,500/month
 Insurance = \$2,000/month
 Weed and feed materials = \$13/lawn
 Direct labor = \$12/lawn
 Fuel = \$2/lawn

   Compute breakeven in units and dollars.
Exercise 5-8
   Fixed costs are \$1,500 + \$200 + \$2,000 =
\$3,700 per month
   Variable costs are \$13 + \$12 + \$2 = \$27 per
lawn
   PX - VX - F = 0
   60X – 27X – 3,700 = 0
   33X = 112.12 lawns
   Breakeven sales dollars = 112.12 x 60 =
\$6,727.27
Exercise 5-10 (Not assigned)
   During the month of March, New Day Spa services
570 clients at an average price of \$120. During the
month, fixed costs were \$21,000 and variable costs
were 65% of sales.
   What was the contribution margin . . .
 Indollars
 Per unit

 And as a ratio?
Contribution Margin
In dollars . . .

Sales (570 x \$120)               \$68,400
Variable costs (.65 x \$68,400)    44,460
Contribution margin              \$23,940

Per unit . . .

\$23,940/570= \$42

As a ratio . . .

42/120 = .35
Breakeven Point
   Breakeven in \$ = F/CMr
   Breakeven in \$ = \$21,000/.35 = \$60,000
   Breakeven in Units = F/CMu
   Breakeven in Units = \$21,000/\$42
   Breakeven in Units = 500
Exercise 5-14
   Lynn Company had \$150,000 of net income in
2008 when selling price per unit was \$150.
   Variable costs were \$90.
   Fixed costs were \$570,000.
   Management expects per-unit data and total fixed
cost to remain the same in 2009.
   Management is under pressure to increase net
income by \$60,000 in 2009.
Exercise 5-14
   Compute the number of units sold in 2008
   PX – VX – F = P
   150X – 90X – 570,000 = 150,000
   60X = 720,000
   X = 12,000 units
Exercise 5-14
   Question: How many units would have had to have
been sold in 2009 to reach the stockholder’s
desired profit level?
   Target sales in units = P
   PX – VX – F = P
   150X – 90X - \$570,000 = (\$150,000 + \$60,000)
   60X = \$570,000 + \$150,000 + \$60,000
   60X = \$780,000
   X = 13,000 units
Exercise 5-14
   Assume Lynn Company sells he same number of units
in 2009 as it did in 2008.
   What would the selling price have to be in order to
reach the stockholder’s desired profit level?
   PX – VX – F = P
   12,000P – (90)(12,000) – 570,000 = 210,000
   12,000P = 1,860,000
   P = \$155
The End!

```
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