Sensitivity and Breakeven
Analysis
Lecture No. 29
Professor C. S. Park
Fundamentals of Engineering Economics
Copyright © 2005
Chapter 10
Handling Project Uncertainty

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Origin of Project Risk
Methods of Describing
Project Risk
Probability Concepts
for Investment
Decisions
Risk-Adjusted Discount
Rate Approach




In Engineering economics
we predict cash flows
How do you know for sure
that what you are claiming
for interest rate, costs,
revenues remain true???
Well for some situations you
can be close enough to
consider your “single point”
analysis to be worthwhile.
For other you need to
consider what is called RISK

We use the term risk to describe an
investment project where cash flows are not
known in advanced with certainty.

What to do: Instead of single point
analysis, an array of outcomes and their
probabilities or odds are to considered.
Origins of Project Risk



Risk: (in essence) the
potential for loss
Project Risk: variability
in a project’s NPW
Risk Analysis: The
assignment of
probabilities to the
various outcomes of an
investment project
Methods of Describing Project Risk

Sensitivity Analysis: a means of identifying the
project variables which, when varied, have the
greatest effect on project acceptability.

Break-Even Analysis: a means of identifying the
value of a particular project variable that causes
the project to exactly break even.

Scenario Analysis: a means of comparing a
“base case” to one or more additional scenarios,
such as best and worst case, to identify the
extreme and most likely project outcomes.
Sensitivity Analysis – Example 10.1
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

Transmission-Housing Project by Boston Metal Company
New investment = $125,000
Number of units = 2,000 units
Unit Price = $50 per unit
Unit variable cost = $15 per unit
Fixed cost = $10,000/Yr
Project Life = 5 years
Salvage value = $40,000
Income tax rate = 40%
MARR = 15%
Example 10.1 - After-tax Cash Flow for BMC’s TransmissionHousings Project – “Base Case”
0
1
2
3
4
5
Revenues:
Unit Price
Demand (units)
Sales revenue
50
50
50
50
50
2,000
2,000
2,000
2,000
2,000
$100,000 $100,000 $100,000 $100,000 $100,000
Expenses:
Unit variable cost
$15
$15
$15
$15
$15
Variable cost
30,000
30,000
30,000
30,000
30,000
Fixed cost
10,000
10,000
10,000
10,000
10,000
Depreciation
17,863
30,613
21,863
15,613
5,575
Taxable Income
$42,137
$29,387
$38,137
$44,387
$54,425
16,855
11,755
15,255
17,755
21,770
$25,282
$17,632
$22,882
$26,632
$32,655
Income taxes (40%)
Net Income
(Example 10.1, Continued)
Cash Flow Statement
0
1
2
3
4
5
Operating activities
Net income
25,282
17,632
22,882
26,632
32,655
Depreciation
17,863
30,613
21,863
15,613
5,575
Investment activities
Investment
(125,000)
Salvage
40,000
Gains tax
(2,611)
Net cash flow
($125,500)
$43,145
$48,245
$44,745
$42,245
$75,619
A
B
1 Example 10.1 BMC's
2
13 Income Statement
14
0
15 Revenues:
16
Unit Price
17
Demand (units)
18
Sales Revenue
19 Expenses:
20
Unit Variable Cost
21
Variable Cost
22
Fixed Cost
23
Depreciation
24
C
D
E
F
G
4
5
Transmission-Housings Project
1
2
3
$
50 $
50 $
50 $
50 $
50
2000
2000
2000
2000
2000
$ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000
$
15
30,000
10,000
17,863
$
15
30,000
10,000
30,613
$
15
30,000
10,000
21,863
$
15
30,000
10,000
15,613
$
15
30,000
10,000
5,581
25 Taxable Income
26
27 Income Taxes (40%)
$ 42,137
16,855
$ 29,387
11,755
$ 38,137
15,255
$ 44,387
17,755
$ 54,419
21,768
28
29
30
31
32
33
34
35
36
37
38
$ 25,282
$ 17,632
$ 22,882
$ 26,632
$ 32,651
25,282
17,863
17,632
30,613
22,882
21,863
26,632
15,613
32,651
5,581
Net Income
Cash Flow Statement
Operating Activities:
Net Income
Depreciation
Investment Activities:
Investment
Salvage
Gains Tax
39 Net Cash Flow
40
(125,000)
$ (125,000) $ 43,145
40,000
(2,613)
$ 48,245
$ 44,745
$ 42,245
$ 75,619
Example 10.1 - Sensitivity Analysis for Five
Key Input Variables
Deviation
-15%
-10%
-5%
0%
5%
10%
15%
20%
$57
$9,999
$20,055
$30,111
$40,169
$50,225
$60,281
$70,337
$80,393
Demand
12,010
19,049
26,088
33,130
40,169
47,208
54,247
61,286
68,325
Variable cost
52,236
49,219
46,202
43,186
40,169
37,152
34,135
31,118
28,101
Fixed cost
44,191
43,185
42,179
41,175
40,169
39,163
38,157
37,151
36,145
Salvage
value
37,782
38,378
38,974
39,573
40,169
40,765
41,361
41,957
42,553
Unit price
-20%
Base
Sensitivity graph – BMC’s transmission-housings
project (Example 10.1)
$100,000
90,000
Unit Price
80,000
70,000
Demand
60,000
50,000
Salvage value
40,000
Fixed cost
Variable cost
Base
30,000
20,000
10,000
0
-10,000
-20%
-15%
-10%
-5%
0%
5%
10%
15%
20%
Example 10.2 - Sensitivity Analysis for Mutually
Exclusive Alternatives
Electrical
Diesel
Power
LPG
Gasoline
Fuel
7 year
7 years
7 years
7 years
Initial cost
$30,000
$21,000
$20,000
$25,000
Salvage value
$3,000
$2,000
$2,000
$2,200
260
260
260
260
32 kWh
12 gal
11 gal
7 gal
Fuel cost/unit
$0.05/kWh
$1.00/gal
$1.20/gal
$1.10/gal
Fuel cost/shift
$1.60
$12
$13.20
$7.7
$500
$1,000
$1,200
$1,500
$5
$6
$7
$9
Life expectancy
Maximum shifts per year
Fuel consumption/shift
Annual maintenance cost:
Fixed cost
Variable cost/shift
Capital (Ownership) Cost




Electrical power:
CR(10%) = ($30,000 - $3,000)(A/P, 10%, 7) + (0.10)$3,000
= $5,845
LPG:
CR(10%) = ($21,000- $2,000)(A/P, 10%, 7) + (0.10)$2,000
= $4,103
Gasoline:
CR(10%) = ($20,000-$2,000)(A/P, 10%, 7) + (0.10) $2,000
= $3,897
Diesel fuel:
CR(10%) = ($25,000 -$2,200)(A/P, 10%, 7) +(0.10) $2,200
= $4,903
Annual O&M Cost




Electrical power:
$500 + (1.60 + 5)M = $500 + 6.6M
LPG:
$1,000 + (12 + 6)M = $1,000 + 18M
Gasoline:
$800 + (13.2 + 7)M = $800 + 20.20M
Diesel fuel:
$1,500 + (7.7 + 9)M = $1,500 + 16.7M
Annual Equivalent Cost




Electrical power:
AE(10%) = 6,345
LPG:
AE(10%) = 5,103
Gasoline:
AE(10%) = 4,697
Diesel fuel:
AE(10%) = 6,403
+ 6.6M
+ 18M
+ 20.20M
+ 16.7M
12000
8000
Electrical
LPG
Gasoline
6000
Disel Fuel
4000
2000
Number of Shifts (M)
260
240
220
200
180
160
140
120
100
80
60
40
20
0
0
Annual Equivalent Cost ($)
10000
Break-Even Analysis

Excel using a Goal Seek function

Analytical Approach
Excel Using a Goal Seek Function
?
Goal Seek
Set cell:
X
NPW
$F$5
0
To value:
By changing cell:
Ok
Breakeven Value
$B$6
Cancel
Demand
A
Goal Seek
Function
Parameters
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
B
C
D
E
F
G
Example 10.3 Break-Even Analysis
Input Data (Base):
Unit Price ($)
Demand
Var. cost ($/unit)
Fixed cost ($)
Salvage ($)
Tax rate (%)
MARR (%)
Output Analysis:
$
$
$
$
50
1429.39
15
10,000
40,000
40%
15%
Output (NPW)
0
Income Statement
Revenues:
Unit Price
Demand (units)
Sales Revenue
Expenses:
Unit Variable Cost
Variable Cost
Fixed Cost
Depreciation
1
2
$0
3
4
5
$
50 $
50 $
50 $
50 $
50
1429.39
1429.39
1429.39
1429.39
1429.39
$ 71,470 $ 71,470 $ 71,470 $ 71,470 $ 71,470
$
15
21,441
10,000
17,863
$
15
21,441
10,000
30,613
$
15
21,441
10,000
21,863
$
15
21,441
10,000
15,613
$
15
21,441
10,000
5,581
26 Taxable Income
27
28 Income Taxes (40%)
$
22,166
8,866
$
9,416
3,766
$
18,166
7,266
$ 24,416
9,766
$
34,448
13,779
29
30
31
32
33
34
35
36
37
38
39
$
13,299
$
5,649
$
10,899
$ 14,649
$
20,669
10,899
21,863
14,649
15,613
Net Income
Cash Flow Statement
Operating Activities:
Net Income
Depreciation
Investment Activities:
Investment
Salvage
Gains Tax
40 Net Cash Flow
41
13,299
17,863
5,649
30,613
20,669
5,581
(125,000)
40,000
(2,613)
$ (125,000) $
31,162
$ 36,262
$
32,762
$ 30,262
$
63,636
Analytical Approach
Unknown Sales Units (X)
0
1
2
3
4
5
Cash Inflows:
Net salvage
37,389
X(1-0.4)($50)
30X
30X
30X
30X
30X
7,145
12,245
8,745
6,245
2,230
-X(1-0.4)($15)
-9X
-9X
-9X
-9X
-9X
-(0.6)($10,000)
-6,000
-6,000
-6,000
-6,000
-6,000
21X +
1,145
21X +
6,245
21X +
2,745
21X +
245
21X +
33,617
0.4 (dep)
Cash outflows:
Investment
Net Cash Flow
-125,000
-125,000
 PW of cash inflows
PW(15%)Inflow= (PW of after-tax net revenue)
+ (PW of net salvage value)
+ (PW of tax savings from depreciation
= 30X(P/A, 15%, 5) + $37,389(P/F, 15%, 5)
+ $7,145(P/F, 15%,1) + $12,245(P/F, 15%, 2)
+ $8,745(P/F, 15%, 3) + $6,245(P/F, 15%, 4)
+ $2,230(P/F, 15%,5)
= 30X(P/A, 15%, 5) + $44,490
= 100.5650X + $44,490
 PW of cash outflows:
PW(15%)Outflow
= (PW of capital expenditure_
+ (PW) of after-tax expenses
= $125,000 + (9X+$6,000)(P/A, 15%, 5)
= 30.1694X + $145,113
 The NPW:
PW (15%)
= 100.5650X + $44,490
- (30.1694X + $145,113)
=70.3956X - $100,623.
 Breakeven volume:
PW (15%)
Xb
= 70.3956X - $100,623 = 0
=1,430 units.
PW of
inflow
PW of
Outflow
NPW
X
100.5650X
- $44,490
30.1694X
+ $145,113
70.3956X
-$100,623
0
$44,490
$145,113
100,623
500
94,773
160,198
65,425
1000
145,055
175,282
30,227
1429
188,197
188,225
28
1430
188,298
188,255
43
1500
195,338
190,367
4,970
2000
245,620
205,452
40,168
2500
295,903
220,537
75,366
Demand
Break-Even Analysis Chart
$350,000
300,000
Break-even Volume
200,000
Profit
Outflow
150,000
Xb = 1430
PW (15%)
250,000
Loss
100,000
50,000
0
-50,000
-100,000
0
300
600
900
1200
1500
Annual Sales Units (X)
1800
2100
2400
Scenario Analysis
Variable
Considered
WorstCase
Scenario
Most-LikelyCase
Scenario
Best-Case
Scenario
Unit demand
1,600
2,000
2,400
Unit price ($)
48
50
53
Variable cost ($)
17
15
12
Fixed Cost ($)
11,000
10,000
8,000
Salvage value ($)
30,000
40,000
50,000
PW (15%)
-$5,856
$40,169
$104,295