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