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ENGM 661 Engineering Economics for Managers Risk Analysis Risk Analysis
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ENGM 661Engineering Economics
for Managers
Risk AnalysisRisk Analysis
Class Problem
Suppose we have the following cash flow diagram (MARR = 15%).
1 2 3
A1 A2 A3
10,000
Ai
p
p
p
3 000 1 4
4 000 1 2
5 000 1 4
, /
, /
, /
Determine if the project is worthwhile.
Solution Methodologies
Bounding C.L.T. (Assume Normality) Analytic Simulation
Bounding
1 2 3
A1 A2 A3
10,000
Ai
p
p
p
3 000 1 4
4 000 1 2
5 000 1 4
, /
, /
, /
NPW P A
10 000 3 000 15 3
10 000 3 000 2 2832
3150
, , ( / , , )
, , ( . )
,
Lower BoundLower Bound
Bounding
1 2 3
A1 A2 A3
10,000
Ai
p
p
p
3 000 1 4
4 000 1 2
5 000 1 4
, /
, /
, /
NPW P A
10 000 5 000 15 3
10 000 5 000 2 2832
1 416
, , ( / , , )
, , ( . )
,
Upper BoundUpper Bound
Bounding
1 2 3
A1 A2 A3
10,000
Ai
p
p
p
3 000 1 4
4 000 1 2
5 000 1 4
, /
, /
, /
Upper & Lower BoundsUpper & Lower Bounds
3150 1 416, ,NPW
Central Limit Theorem
1 2 3
A1 A2 A3
10,000
Ai
ppp
3 000 1 44000 1 25 000 1 4
, /, /, /
PreliminaryPreliminaryE Ai[ ] , (. ) , (. ) , (. )
,
3 000 25 4 000 5 5 000 25
4 000
2 2 2
2 2 2 23 000 25 4 000 5 5 000 25 4 000
500 000
707
A i ii E A E A
[ ] [ ]
, (. ) , (. ) , (. ) ,
,
Central Limit Theorem
1 2 3
A1 A2 A3
10,000
Ai
ppp
3 000 1 44000 1 25 000 1 4
, /, /, /
Distribution of NPWDistribution of NPWE NPW E A P A[ ] , [ ]( / , , )
, , ( . )
10 000 15 3
10 000 4 000 2 2832
867
Central Limit Theorem
1 2 3
A1 A2 A3
10,000
Ai
ppp
3 000 1 44000 1 25 000 1 4
, /, /, /
Distribution of NPWDistribution of NPW
E NPW[ ] 867
2 2 2 4 6707 115 115 115
500 000 17602
880 100
938
NPW
( ) [( . ) ( . ) ( . ) ]
, ( . )
,
Central Limit Theorem
1 2 3
A1 A2 A3
10,000
Ai
ppp
3 000 1 44000 1 25 000 1 4
, /, /, /
Distribution of NPWDistribution of NPW
E NPW[ ] 867
2 938NPW
N(-867, 938)
-3,681 1,947
Central Limit Theorem
1 2 3
A1 A2 A3
10,000
Distribution of NPWDistribution of NPW
N(-867, 938)
-3,681 1,947
P NPW PNPW
P Z
P Z
{ }( )
{ . }
{ . }
( . )
.
RST
UVW
00 867
938
0 92
1 0 92
1 08212
0178
A i = 3,000 4,000 5,000
P(A i) = 0.25 0.50 0.25
P{NPW} = P(A1)P(A2)P(A3)MARR = 0.15
No. t=0 A1 A2 A3 NPW P{NPW}
1 (10,000) 3,000 3,000 3,000 (3,150) 0.0162 (10,000) 3,000 3,000 4,000 (2,493) 0.0313 (10,000) 3,000 3,000 5,000 (1,835) 0.0164 (10,000) 3,000 4,000 3,000 (2,394) 0.0315 (10,000) 3,000 4,000 4,000 (1,737) 0.0636 (10,000) 3,000 4,000 5,000 (1,079) 0.0317 (10,000) 3,000 5,000 3,000 (1,638) 0.0168 (10,000) 3,000 5,000 4,000 (981) 0.0319 (10,000) 3,000 5,000 5,000 (323) 0.01610 (10,000) 4,000 3,000 3,000 (2,281) 0.03111 (10,000) 4,000 3,000 4,000 (1,623) 0.06312 (10,000) 4,000 3,000 5,000 (966) 0.03113 (10,000) 4,000 4,000 3,000 (1,525) 0.06314 (10,000) 4,000 4,000 4,000 (867) 0.12515 (10,000) 4,000 4,000 5,000 (210) 0.06316 (10,000) 4,000 5,000 3,000 (768) 0.03117 (10,000) 4,000 5,000 4,000 (111) 0.06318 (10,000) 4,000 5,000 5,000 547 0.03119 (10,000) 5,000 3,000 3,000 (1,411) 0.01620 (10,000) 5,000 3,000 4,000 (754) 0.03121 (10,000) 5,000 3,000 5,000 (96) 0.01622 (10,000) 5,000 4,000 3,000 (655) 0.03123 (10,000) 5,000 4,000 4,000 2 0.06324 (10,000) 5,000 4,000 5,000 660 0.03125 (10,000) 5,000 5,000 3,000 101 0.01626 (10,000) 5,000 5,000 4,000 759 0.03127 (10,000) 5,000 5,000 5,000 1,416 0.016
Sum = 1.000
A i = 3,000 4,000 5,000
P(A i) = 0.25 0.50 0.25
P{NPW} = P(A1)P(A2)P(A3)MARR = 0.15
No. t=0 A1 A2 A3 NPW P{NPW}
1 (10,000) 3,000 3,000 3,000 (3,150) 0.0162 (10,000) 3,000 3,000 4,000 (2,493) 0.0313 (10,000) 3,000 3,000 5,000 (1,835) 0.0164 (10,000) 3,000 4,000 3,000 (2,394) 0.0315 (10,000) 3,000 4,000 4,000 (1,737) 0.0636 (10,000) 3,000 4,000 5,000 (1,079) 0.0317 (10,000) 3,000 5,000 3,000 (1,638) 0.0168 (10,000) 3,000 5,000 4,000 (981) 0.0319 (10,000) 3,000 5,000 5,000 (323) 0.01610 (10,000) 4,000 3,000 3,000 (2,281) 0.03111 (10,000) 4,000 3,000 4,000 (1,623) 0.06312 (10,000) 4,000 3,000 5,000 (966) 0.03113 (10,000) 4,000 4,000 3,000 (1,525) 0.06314 (10,000) 4,000 4,000 4,000 (867) 0.12515 (10,000) 4,000 4,000 5,000 (210) 0.06316 (10,000) 4,000 5,000 3,000 (768) 0.03117 (10,000) 4,000 5,000 4,000 (111) 0.06318 (10,000) 4,000 5,000 5,000 547 0.03119 (10,000) 5,000 3,000 3,000 (1,411) 0.01620 (10,000) 5,000 3,000 4,000 (754) 0.03121 (10,000) 5,000 3,000 5,000 (96) 0.01622 (10,000) 5,000 4,000 3,000 (655) 0.03123 (10,000) 5,000 4,000 4,000 2 0.06324 (10,000) 5,000 4,000 5,000 660 0.03125 (10,000) 5,000 5,000 3,000 101 0.01626 (10,000) 5,000 5,000 4,000 759 0.03127 (10,000) 5,000 5,000 5,000 1,416 0.016
Sum = 1.000
P{NPW >0} = .031 .063.031.016.031.016.188
A i = 3,000 4,000 5,000
P(A i) = 0.25 0.50 0.25
P{NPW} = P(A1)P(A2)P(A3)MARR = 0.15
No. t=0 A1 A2 A3 NPW P{NPW}
1 (10,000) 3,000 3,000 3,000 (3,150) 0.0162 (10,000) 3,000 3,000 4,000 (2,493) 0.0313 (10,000) 3,000 3,000 5,000 (1,835) 0.0164 (10,000) 3,000 4,000 3,000 (2,394) 0.0315 (10,000) 3,000 4,000 4,000 (1,737) 0.0636 (10,000) 3,000 4,000 5,000 (1,079) 0.0317 (10,000) 3,000 5,000 3,000 (1,638) 0.0168 (10,000) 3,000 5,000 4,000 (981) 0.0319 (10,000) 3,000 5,000 5,000 (323) 0.01610 (10,000) 4,000 3,000 3,000 (2,281) 0.03111 (10,000) 4,000 3,000 4,000 (1,623) 0.06312 (10,000) 4,000 3,000 5,000 (966) 0.03113 (10,000) 4,000 4,000 3,000 (1,525) 0.06314 (10,000) 4,000 4,000 4,000 (867) 0.12515 (10,000) 4,000 4,000 5,000 (210) 0.06316 (10,000) 4,000 5,000 3,000 (768) 0.03117 (10,000) 4,000 5,000 4,000 (111) 0.06318 (10,000) 4,000 5,000 5,000 547 0.03119 (10,000) 5,000 3,000 3,000 (1,411) 0.01620 (10,000) 5,000 3,000 4,000 (754) 0.03121 (10,000) 5,000 3,000 5,000 (96) 0.01622 (10,000) 5,000 4,000 3,000 (655) 0.03123 (10,000) 5,000 4,000 4,000 2 0.06324 (10,000) 5,000 4,000 5,000 660 0.03125 (10,000) 5,000 5,000 3,000 101 0.01626 (10,000) 5,000 5,000 4,000 759 0.03127 (10,000) 5,000 5,000 5,000 1,416 0.016
Sum = 1.000
AnalyticAnalytic P{NPW > 0} = 0.188
C.L.T.C.L.T. P{NPW > 0} = 0.178
Simulation
1 2 3
A1 A2 A3
10,000
Ai
ppp
3 000 1 44000 1 25 000 1 4
, /, /, /
Cumulative (CDF)
0.00
0.25
0.50
0.75
1.00
2000 3000 4000 5000 6000
x
P{A
<x
}
Simulation
1 2 3
A1 A2 A3
10,000
Ai
ppp
3 000 1 44000 1 25 000 1 4
, /, /, /
Cumulative (CDF)
0.00
0.25
0.50
0.75
1.00
2000 3000 4000 5000 6000
x
P{A
<x
}
SimulationCumulative (CDF)
0.00
0.25
0.50
0.75
1.00
2000 3000 4000 5000 6000
x
P{A
<x
}
MARR = 0.15t Rand Ai0 (10,000)1 0.488 4,0002 0.955 4,0003 0.802 3,000
NPV = (1,525)t Rand Ai0 (10,000)1 0.908 4,0002 0.399 3,0003 0.634 3,000
NPV = (2,281)
t Rand Ai0 (10,000)1 0.642 5,0002 0.248 5,0003 0.874 4,000
NPV = 759
Simulation
Cumulative (CDF)
0.00
0.25
0.50
0.75
1.00
2000 3000 4000 5000 6000
x
P{A
<x
}
Rep. No. NPW1 (1,525)2 (2,281)3 7594 1,4165 5476 (867)7 28 (2,493)9 (867)10 (210)11 (2,394)12 (210)13 1,41614 (867)15 (210)16 (1,835)17 (966)18 (655)19 (1,638)20 (210)
Simulation
Cumulative (CDF)
0.00
0.25
0.50
0.75
1.00
2000 3000 4000 5000 6000
x
P{A
<x
}
Rep. No. NPW1 (1,525)2 (2,281)3 7594 1,4165 5476 (867)7 28 (2,493)9 (867)10 (210)11 (2,394)12 (210)13 1,41614 (867)15 (210)16 (1,835)17 (966)18 (655)19 (1,638)20 (210)
P{NPW > 0} = 5/20 = 0.25
Simulation
Cumulative (CDF)
0.00
0.25
0.50
0.75
1.00
2000 3000 4000 5000 6000
x
P{A
<x
}
Rep. No. NPW1 (1,525)2 (2,281)3 7594 1,4165 5476 (867)7 28 (2,493)9 (867)10 (210)11 (2,394)12 (210)13 1,41614 (867)15 (210)16 (1,835)17 (966)18 (655)19 (1,638)20 (210)
AnalyticAnalytic P{NPW > 0} = 0.188C.L.T.C.L.T. P{NPW > 0} = 0.178SimulationSimulation P{NPW > 0} = 0.25
@Risk Distribution for /C9
0
0.05
0.1
0.15
0.2
0.25
0.3
PR
OB
AB
ILIT
Y
@Risk
AnalyticAnalytic P{NPW > 0} = 0.188
C.L.T.C.L.T. P{NPW > 0} = 0.178
SimulationSimulation P{NPW > 0} = 0.25
@Risk@Risk P{NPW > 0} = 0.20
Class Problem
You are given the following cash flow diagram. The Ai are iid shifted exponentials with location parameter a = 1,000 and scale parameter = 3,000. The cumulative is then given by
7,000
A1 A2 A3
F x e x( ) ( , )/ , 1 1 000 3 000 , x > 1,000
Class Problem
You are given the first 3 random numbers U(0,1) as follows:
P1 = 0.8
P2 = 0.3
P3 = 0.5
You are to compute one realization for the NPW.MARR = 15%.
7,000
A1 A2 A3
F x e x( ) ( , )/ , 1 1 000 3 000
Class Problem
P1 ex 10003000
ex 10003000
1 P
x 10003000
ln(1 P)
x1,000 3,000 ln(1 P)
Class Problemx1,000 3,000ln(1 P)
A1 = 1,000 - 3000 ln(1 - .8)
= 5,828
A2 = 1,000 - 3000 ln(1 - .3)
= 2,070
A3 = 1,000 - 3000 ln(1 - .5)
= 3,079
Class Problem
7,000
5,8282,0703,079
NPW = -7,000 + 5,828(1.15)-1 + 2,070(1.15)-2 + 3,079(1.15)-3
= 1,657
Class Problem
You are given the following cash flow diagram. The Ai are iid gammas with shape parameter = 4 and scale parameter = 3,000. The density function is given by 7,000
A1 A2 A3
f x x e x( )( )
/
1 , x > 0
Class Problem
You are given the first 3 random numbers U(0,1) as follows:
P1 = 0.8
P2 = 0.3
P3 = 0.5
You are to compute one realization for the NPW.MARR = 15%.
7,000
A1 A2 A3
Class ProblemFor = integer, the cumulative distribution function is given by
Set P = F(x), solve for x
0,!
)/(1)(
1
0
/
Xj
xexF
j
jx
Class Problem
For general (not integer),
F(x) = not analytic
Class Problem
For general (not integer),
F(x) = not analytic
No Inverse
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