Cost-Volume-Profit Analysis
Calculating The Break-Even Point
P
x
= Selling Price Per Unit
= Units Produced and Sold
V
= Variable Cost Per Unit
• Sales Revenue = Px
• Total Costs = Vx + F
• When Firm Breaks Even (BE):
F
= Total Fixed Costs
OP = Operating Profits-Before Tax
t
Sales Revenue – Total Costs = 0
Sales Revenue = Total Costs
= Tax rate
Revenue = Total Costs
Px = Vx + F
Px - Vx = F
x(P - V) = F
__F__
x =
(P -V)
(FORMULA "A")
• Denominator also called CMU
(Contribution Margin Per Unit)
x =
__F__
CMU
(FORMULA "A")
1
Formula A:
x = F/(P-V)
E.g., Assume Firm has following
costs, revenues and tax rates:
P = $200
V = $120
F = $2,000
x
=
__2,000__
(200 - 120)
x
=
2,000
x
= 25 units
_F__
x = (P-V)
Px
= 25 units
`
___F__
Px = (P-V)/P
=
80
• This is common sense
• Firm makes $80 every time it
sells a unit.
2,000
80
x
Px =
Px
=
__F__
CMR
_FP_
(P-V)
PX = F/((P-V)/P)
____2,000____
Px =
• Denominator also called CMR
(Contribution Margin Ratio)
=
__F__
(FORMULA "B")
(P-V)/P
Formula B:
(Formula “B”)
(Formula “A”)
(200 - 120) /200
Px =
2,000
Px =
.40
$5,000
(FORMULA "B")
2
• Why do we need Formula B?
– 25 units x $200 = $5,000
• E.g., What is BE Point for Co.?
Revenue:
Less VCs:
CM:
Less FCs:
OP:
$100K
-30K
$ 70K
-50K
$ 20K
(Px)
(Vx)
(Px – Vx)
(F)
(Px - Vx – F)
• We can calculate the CMR
– So we can use Formula “B”
• We know CM/Rev, which is the CMR
CM
=
Sales Revenue
=
(P-V)
(P-V)x
=
Px
P
Px - Vx
Px
• What is BE?
– We do not know the # of units sold
– We do not know P or V
– We cannot use Formula “A”
x =
_$50K_
(P-V)
• The CMR is .70 (70,000/100,000)
• The BE Point in Sales Revenue is:
Px = F/CMR = 50,000/.70 =
$71,428.57
= CMR
Revenue - Costs = OP
Targeted Operating Profits
Px - Vx – F = OP
Px - Vx = F + OP
x(P - V) = F + OP
x = (F + OP)
(P - V)
Modified
Formula
“A”
x = (F + OP)
CMU
3
x
= (F + OP)
(P - V)
Px = (F + OP)P
(P - V)
Px = _(F + OP)_
(P - V)/ P
Px = (F + OP)
CMR
Modified
Formula “A”
Modified
Formula “B”
• OP do not include tax expense
• OP – Tax Expense = Net Income
• What if given Targeted NI?
– Convert your Net Income figure into the
Operating Profits that will result in the Net
Income that you desire
x
=
x
=
x
=
($2,000 + $40,000)
($200 - $120)
$42,000
$80
525 units
• E.g., Given target NI (after-tax) of
$50K & tax rate of 40%
• Convert $50K NI into OP:
OP - Taxes
OP - .4 (OP)
.6 (OP)
OP
OP
=
=
=
=
=
NI
50,000
50,000
50,000/ .6
83,334
What if Target OP is given as % of Revenue?
E.g., OP = 10% of Px
Px = (F+OP) / (P-V)/P
• You can check this:
OP:
Taxes (40%):
NI:
• Same E.g.
• Assume Co wants Target OP of $40K
$83,334
-33,334
$50,000
Px =
(2,000 + .1 Px)
[(200 - 120)/200]
Px =
(2,000+.1Px) / .40
.4 Px =
.4Px - .1 Px =
.3 Px =
2,000 + .1Px
2,000 + .1Px
2,000
Px =
2,000/.3
Px =
$6,667
4
Multiple Products
• Firm has >1 product
• Same analysis but use composite:
– CMU (Formula A) or
– CMR (Formula B)
• If using CMR (Formula B)
• The easiest way to calculate
BE is to calculate CMR for entire
Company
– Remember:
• (Px – Vx)/Px = (P-V)/P
– CMR = CM/Sales
• We did this earlier
Revenue:
Less VCs:
CM:
Less FCs:
OP:
$100K
-30K
$ 70K
-50K
$ 20K
(Px)
(Vx)
(Px – Vx)
(F)
(Px - Vx – F)
• If using CMU (Formula A)
• Calculate composite CMU
– Basket (Package) Method
– Weighted Average CM Method
• CMR = 70K/100K = .7
E.g., Co sells 3 Bananas for each
Orange that it sells (75% vs. 25%)
Bananas Oranges
Price:
$2
$4
VC per
-$1
-$2
Unit:
CMU:
$1
$2
Common Fixed Costs: $2,000
• Basket Method: (75%:25%)
• Basket  3 Bs & 1 Or:
CMB = 3 CMban + 1CMor
CMB = 3 (1) + 1 (2)
CMB = 5
5
• Now, plug CMB into BE formula
• Gives you BE Point in Baskets:
Bananas: 3 x 400 Baskets = 1200
Oranges:
1 x 400 Baskets = 400
Baskets = F/CMB
Baskets = 2,000/ 5
Baskets = 400 Baskets
• Weighted Average Method
CMWA
CMWA
CMWA
= .75 CMban + .25 CMor
= .75 (1) + .25 (2)
= .75 + .5 = $1.25
• The BE Point in units is:
x
x
x
= F/CMWA
= 2,000/ 1.25
= 1600 units
Margin of Safety
Bananas:
Oranges:
.75 (1600) = 1200
.25 (1600) = 400
• Margin Of Safety  Amount that
actual sales (dollars or units)
exceed BE Point
– If Co actually sells 40 units
($8,000) & its BE Point is 25 units
(or $5,000)
– Margin Of Safety is 15 units (or
$3,000)
6
Operating Leverage
• If we increase our CM by a given
percentage, what will be our new
OP?
• OL gives you the answer quickly
Rev:
VCs:
CM:
FCs:
OP:
Projected
$12,000
-$7,200
$4,800
-$2,000
$2,800
(200 x 60 units)
(120 x 60 units)
(80 x 60 units)
Rev:
VCs:
CM:
FCs:
OP:
Current
$ 8,000
-$4,800
$ 3,200
-$2,000
$1,200
(200 x 40 units)
(120 x 40 units)
(80 x 40 units)
• Operating Leverage (OL)
– Divide CM by OP
• OL x (% increase in Co’s CM) 
– Gives you % increase in Co’s OP
• There is a 50% increase in:
• Units
• Revenue
• CM
• Operating Leverage:
Current CM
OP
=
$3,200
= 2.67
$1,200
•
•
•
•
Remember that CM increases by 50%
OL x CM↑% = OP↑%
2.67 x 50% = 133%
OP of $1,200 will increase by $1,600
(1.33 x $1,200)
– New OP 
• $1200 + $1600 = $2,800
7
Formula B
Px = (F/CMR)
• The CMR is for all of your products
• The easiest way to do this is to
prepare an income statement for
the whole business and then divide
the CM by the Sales
• BUT, some people try to figure it
out using the weighted-average
method.
Price:
VC per Unit:
CMU:
Apples
$2
-$1
$1
Pears
$4
-$3
$1
CMR [(p-v)/p]:
1/2
1/4
Sales (Units)
60
40
Common Fixed Costs: $2,000
Income Statement
• The Sales Revenue is:
• 60x(PA) + 40x(PP)
• 60x(2) + 40x(4) = 120+160 = 280
• The Variable Costs are:
• 60x(VA) + 40x(VP)
• 60x(1) + 40x(3) = 60+120 = 180
• BUT, If you are going to calculate the
weighted average CMR using a formula,
then remember, the product mix is in
Sales dollars:
• Formula A:
– Calculates needed UNITS
– CMU Weights based on UNITS
• Formula B:
– Calculates need DOLLARS
– CMR Weights based on DOLLARS
Sales
$280K
Less: Variable Costs:
-180K
Contribution Margin:
$100K
Less: Fixed Costs
-2K
Operating Profits
$98K
• CMR= CM/Sales
• CMR= $100K/$280K
• CMR= 35.71%
• What is the Relative Sales in DOLLARS?
• Revenue From Each Product:
• Apples: 60 x $2 = $120
• Pears: 40 x $4 = $160
• Total Revenue = $120 + $160 = $280
• Relative Revenue From Each Product:
• Apples  $120/$280 = 42.86%
• Pears  $160/$280 = 57.14%
• You must use these percentages – not the
relative unit percentages (60% Apples &
40% Pears)
8
• Weighted Average Method
CMRWA = .4286 CMRa + .5714 CMRp
CMRWA = .4286 (.5) + .5714 (.25)
CMRWA = .2143 + .14285 = .35715
• Please note that this is the same
weighted average CMR that we got
from the income statement
Apples: .4286 ($5600) = $2400
Pears: .5714 ($5600) = $3200
Apples: $2400/$2 = 1200 apples
Pears: $3200/$4 = 800 pears
• Using Formula B
• The BE Point in Sales Dollars is:
Px
Px
Px
= F/CMRWA
= 2,000/ .35715
= $5600
Let’s Check To See if Correct Answer:
Sales
(1200 x 2) + (800 x 4)
VC
(1200 x 1) + (800 x 3)
$5,600
-3,600
Contribution Margin
$2,000
Fixed Costs
-2,000
Operating Profits
0
THE END
© Dr. Michael Constas 2013
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