1

Risk Management

2

Road Ahead

• Risk Management Process

• Cost and Schedule Risk

• Estimating Likelihood

• Mitigation

• Utility and Consequences

• Sensitivity Analysis

3

Reference Material

• Risk Management Guide for DOD Acquisition, 6th Ed, Ver 1.0, August 2006 http://www.dau.mil/pubs/gdbks/docs/RMG%206Ed%20Aug06.pdf.

• INCOSE Systems Engineering Handbook, Ver 3, INCOSE-TP-2003-002-03, June, 2006, Chapter 7.3

4

Why Do We Care?

• Every change from a current state to a future state occurs as a result of a process.

• The future behavior of any process is affected by uncertainties.

• Management of the process requires consideration of these uncertainties to minimize their influence on the desired final outcome of the process.

5

Elementary ProcessElementary Process

TransformationInput Output

Unknowns Uncertainty

6

Risk vs. Issues

• Risk refers to a future uncertain event.

• If an event has occurred it is no longer uncertain. It becomes an issue or a problem.

• Management of either requires expenditure of resources.

7

Risk Definitions

• Statistics: P[Undesirable Event]

• Risk Management: P[Undesirable Event] Plus the Consequences of the Event

• INCOSE considers both Undesirable and Desirable events

8

Consequences Can Impact:

• Technical Performance– Key Performance Parameters– Operational capability– Supportability

• Schedule

• Cost

• The above are not mutually independent

9

Risk Categories

• Technical Risk – A technical requirement may not be satisfied during the life cycle

• Cost Risk – Available budget may be exceeded

• Schedule Risk – May fail to reach Scheduled Milestones

• Programmatic Risk – Events beyond the control of the Program Manager

10

A Risk Reporting Matrix

1

2

3

4

5L

ikel

ihoo

d L

evel

1 2 3 4 5

Consequence Level

Low Risk

Moderate Risk

High Risk

11

Possible Risk Likelihood Criteria

Level Likelihood P[Occur]

1 Not Likely p< 0.10

2 Low Likelihood 0.10<=p<0.30

3 Likely 0.30<=p<0.70

4 Highly Likely 0.70<=p<0.90

5 Near Certainty 0.90<=p

12

Possible Consequence Level Criteria

Level Technical Performance

Schedule Cost

1 None to Minimal None to Minimal None to Minimal

2 Can be tolerated

Little program impact

Slip< ? mo C< 1% of Budget

3 Moderate, Limited program impact

Slip<? mo

Subsys slip>? Mo

1%<=C<5% of Budget

4 Significant degradation,

May jeopardize program

Critical path affected

5%<=C<10% of Budget

5 Severe degradation, Will jeopardize program success

Cannot meet key program milestones

10% of Budget

<= C

13

Risk Criteria Depend on Program

Risk Level Criteria

1 P<0.0001

2 P<0.02

3 P<0.05

4 P<0.25

5 P>0.25

14

Risk Reporting Illustration

1

2

3

4

5L

ikel

ihoo

d L

evel

1 2 3 4 5

Consequence Level

Risk Title (Category)CauseMitigation Approach

15

Risk Management

• A process to minimize the adverse effects of uncertain future events on the achievement of end state objectives.

• Basic Tasks:– Identify and characterize process properties.

– Decide on a course of action to minimize adverse affects of events on program objectives.

– Implement and control the course of action.

16

Risk Management Functions

• Planning

• Resourcing

• Staffing

• Controlling

17

Planning

• Define the strategy and process to be used• Establish a Risk Management Plan (RMP)

– Tasks– Schedules– Reviews– Reporting

• Define the resources required– People– Funds– Space and support resources

18

An Example RMP Format Summary

1. Introduction

2. Program Summary

3. Risk Management Strategy and Process

4. Responsible/Executing Organization

5. Risk Management Process and Procedures

6. Risk Identification

7. Risk Analysis

8. Risk Mitigation Planning

9. Risk Mitigation Implementation

10. Risk Tracking

19

Events May Cause RMP Update

• A change in acquisition strategy, • Preparation for a milestone decision, • Results and findings from event–based

technical reviews, • An update of other program plans, • Preparation for a Program Objective

Memorandum submission, or • A change in support strategy.

20

Identify Uncertain Events

EstimateP[Occur]

EstimateConsequences

Formulate AlternativeCourses of Action

EvaluateAlternatives

Choose Approach Execute andTrack

Working? OR

Continueor Stop

Yes

No

Risk Management Process

Planning

21

Risk (Uncertain Event) Identification

• What can go wrong?

• If EVENT happens then CONSEQUENCE results

• Find everywhere Mr. Murphy can rear his ugly head!

22

What to Look At

• Current and proposed staffing, process, design, suppliers, concept of operation, resources, interfaces, interactions, etc.

• Test results and failures (especially readiness results)

• Potential Shortfalls• Trends• External Influences (programmatic, political)

23

Potential Root Causes of Risk Events

• Need (Threat)• Requirements• Technical Baseline• Test & Evaluation• Modeling & Simulation• Technology• Logistics• Production/Facilities

• Concurrency• Industrial Capability• Cost• Management• Schedule• External Factors• Budget• Earned Value Realism

24

Possible Risk Management Actions

• Avoid– Redesign– Change requirements

• Accept• Control

– Expand resources– Reduce likelihood and/or consequences

• Transfer– By mutual agreement to party more qualified to

mitigate

25

Interactions and Consequences

• Schedule affects costC = A +BT

• Budget constrains costForces B vs T tradeoff

• Schedule constraint affects costCauses B increase

• Technology maturity and workforce skills affect all terms

26

Example:Program to modify an existing guided submunition

Current CharacteristicsWeight 20 kg

Length 50 cm

Diameter 15 cm

Required CharacteristicsWeight 15 kg

Length 40 cm

Diameter 10 cm

27

1 2 3 4 5 6 7 8 9 10 11 12Weight Reduction

Length Reduction

Diameter Reduction

Month

Example Schedule

28

Uncertain Events?

If ---------------- Then ---------------------

29

ExampleGuided Antitank Submunition

• Effectiveness:– Dictates minimum number/carrier

• Carrier Vehicle– Constrains length, diameter, and weight

30

Technical Performance Vector

Element Required Current

Length (L) 40 cm 50 cm

Diameter (D) 10 cm 15 cm

Weight (W) 15 kg 20 kg

31

Actions to Meet RequirementsElement Discrepancy

Cause

Corrective

Action

LengthUse of discrete components

Special chips

Diameter Seeker antenna New detector

WeightDiscrete components,

battery

Reduce power requirements, use integrated circuits

32

Activities Required to Reduce LengthID Description Time

(Mo)Preceded by Expected

Length

A1 Analysis & Design 1 --- 50

A2 Breadboard Fabrication 2 A1 50

A3 Test 1 A2 50

A4 Modify Breadboard 0.5 A3 50

A5 Retest 0.5 A4 43

A6 Admin Lead Time 1.5 A1 ---

A7 Develop Chips 3 A5, A6, C5 43

A8 Test Chips 1 A7 40

33

A1 A2 A3 A4 A5

A6

A7 A8

1 2 1 0.5 0.5 3 1

1.5

From C5

1 3 4 4.5 5

2.5

8 9

Length Reduction Program

34

Length Reduction, No Interaction

0

10

20

30

40

50

60

0 1 2 3 4 5 6 7 8 9 10

Time, Months

Le

ng

th, c

m

35

Activities to Reduce Diameter

ID Description Time (Mo

Preceded By Expected Diameter

B1 Trade Offs 2 --- 15

B2 Subcontractor Lead Time 2.5 --- 15

B3 Prototype design & Fab 5 B1, B2 13

B4 Prototype Test 1 B3 13

B5 Redesign 3 B4, C5 13

B6 Retest 0.5 B5 10

36

B1

B2

B3 B4 B5 B6

2

2.5

5 1 3 0.5

2

2.5

7.5 8.5 11.5 12

From C5

Diameter Reduction Program

37

Diameter Reduction, No Interaction

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14

Time, Months

dia

me

ter,

cm

38

Activities Required to Reduce WeightID Description Time (Mo) Preceded By Expected

Weight

C1 Trade Offs 2.5 --- 20

C2 Administrative Lead Time 1 C1, A1, B1, B2 20

C3 Design 1 C2 20

C4 Refine & Deliver 1 C3, A3, B4 20

C5 Integrate & Test 1.5 C4 17

C6 Administrative Lead Time 1 C5 17

C7 Delivery 2 C6 17

C8 Integrate & Test 2 C7 15

39

C1 C2 C3 C4 C5 C6 C7

2.5 1 1 1 1.5 1 2

2.53.5

4.55.5 7 8 10

C82 12

Weight Reduction Program

From A1From A3

From B1 From B2 From B4

40

Weight Reduction, No Interaction

0

5

10

15

20

25

0 2 4 6 8 10 12

Time, Months

We

igh

t, k

g

41

C1 C2 C3 C4 C5 C6 C7

2.5 1 1 1 1.5 1 2

2.53.5

4.59.5 11 12 14

C82 16

B1

B2

B3 B4 B5 B62

2.5

5 1 3 0.5

2

2.5

7.5 8.5 14 14.5

D2 D4 D5

8.52.5

11

15 13 10

20 17

15

A1 A2 A3 A4 A5

A6

A7 A8

1 2 1 0.5 0.5 3 1

1.5

1 3 4 4.5 5

2.5

14 15

D1 D3D6

1 4

11

50 43 40

Development Program

42

Length Reduction Comparison

0

10

20

30

40

50

60

0 2 4 6 8 10 12 14 16

Time, Months

Le

ng

th, c

m

43

Diameter Reduction, With Interaction

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16

Time, Months

Dia

me

ter,

cm

44

Weight Reduction, with Interaction

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18

Time, Months

We

igh

t, k

g

45

Effect of Variability on Schedule

• For the length reduction program:– Average time to achieve 43 cm = 5 months– 95% confidence band: 4.49 –5.51 months

• Assumes Normal Distribution of activity time and 10% coefficient of variation

• Activity time distributions are usually triangular (a,a,c)– Moves mean and right tail to the right

46

Time

Tech Parameter

95 % Confidence Regions

Parameter – Time Relationship

47

Cost

C = A + B*T

C = Activity cost

A = Fixed Cost

B = Expenditure RateT = Elapsed Time

48

1 2 3 4 5 6 7 8 9 10 11 12Activity RN Tbar Std dev T A B ES T LF Sum T C Sum C

1 A1 -0.10534 1 0.1 5 12 0 0.989466 0.989466 0.989466 16.87359 16.873592 A2 1.623725 2 0.2 5 12 0.989466 2.324745 3.314211 3.314211 32.89694 49.770533 A3 -0.64564 1 0.1 5 12 3.314211 0.935436 4.249646 4.249646 16.22523 65.995764 B1 -2.33173 2 0.2 5 12 0 1.533655 1.5336556 B2 -1.16468 2.5 0.25 5 12 0 2.20883 2.20883 2.20883 63.01192 63.011925 B3 1.512863 5 0.5 5 12 2.20883 5.756431 7.965261 7.965261 74.07718 137.08917 B4 0.928476 1 0.1 5 12 7.965261 1.092848 9.058109 9.058109 18.11417 155.20338 C1 -0.37803 2.5 0.25 5 12 0 2.405491 2.405491 2.405491 33.8659 33.86599 C2 -1.02047 1 0.1 5 12 2.405491 0.897953 3.303444 3.303444 15.77543 49.6413310 C3 -0.98917 1 0.1 5 12 3.303444 0.901083 4.204527 4.204527 63.24298 112.884311 C4 0.450058 1 0.1 5 12 9.058109 1.045006 10.10311 10.10311 17.54007 130.424412 C5 0.614575 1.5 0.15 5 12 10.10311 1.592186 11.6953 11.6953 24.10624 154.530613 C6 0.471922 1 0.1 5 12 11.6953 1.047192 12.74249 12.74249 17.56631 172.096914 C7 -0.66776 2 0.2 5 12 12.74249 1.866448 14.60894 14.60894 27.39738 199.494315 C8 -0.24237 2 0.2 5 12 14.60894 1.951527 16.56047 16.56047 28.41832 227.912616 B5 -0.11804 3 0.3 5 12 11.6953 2.964587 14.65989 14.65989 40.57504 195.778317 B6 3.841706 0.5 0.05 5 12 14.65989 0.692085 15.35197 15.35197 13.30502 209.083318 A4 -1.01906 0.5 0.05 5 12 4.249646 0.449047 4.698694 4.698694 10.38857 76.3843219 A5 -0.90231 0.5 0.05 5 12 4.698694 0.454885 5.153578 5.153578 10.45862 86.8429420 A6 -0.02406 1.5 0.15 5 12 0 1.496391 1.49639121 A7 -0.6049 3 0.3 5 12 11.6953 2.818529 14.51383 14.51383 38.82235 125.665322 A8 0.391379 1 0.1 5 12 14.51383 1.039138 15.55297 15.55297 17.46966 143.1349

Max time A Time B Time C Time A Cost B Cost C Cost Total Cost16.56047 15.55297 15.35197 16.56047 143.1349 209.0833 227.9126 580.1309

49

Max time A Time B Time C Time A Cost B Cost C Cost Total CostAVG 15.97062 15.01999 14.49016 15.97062 143.1657 204.0246 223.8933 571.0836SD 0.565504 0.629229 0.547394 0.565504 5.05307 7.084639 4.971504 12.03786

Estimated 16 15 14.5 16 143 204 220 567F(Est) 0.52072 0.487328 0.507168 0.52072 0.486923 0.498614 0.216778 0.36722Risk 0.47928 0.512672 0.492832 0.47928 0.513077 0.501386 0.783222 0.63278

Schedule and Cost Risk

50

Summary

• All activities affecting the desired end result and their interactions must be considered.

• Network representation takes care of this

• Activities must be considered at a low enough level to permit reasonable accurate time estimates

51

Summary (cont’d)

• Cost for development programs is a function of time

• There is variability in everything

• Variability can cause the critical path to change

• Plan for the occurrence of bad outcomes

52

Decision Environments

• Certainty

• Uncertainty

• Risk

53

Probability Refresher

54

Classical Interpretation

• N possibilities– Equally likely– One must occur– S of N possibilities = event “success”

• P[success] =S/N

55

Frequency Interpretation

• P[event] = proportion of the time event occurs over the long run

• Not very practical for situations that result in only one trial

56

Probability Axioms

Consider sample space S for events A, B, C … in S.

0 ≤ P[A] ≤ 1 for all A in S

P[S] = 1

If A and B are mutually exclusive,

P[A B] = P[A or B or both] = P[A] + P[B]

57

Theorem

If A is an event in finite S, and Ei, i = 1, 2, …n are events comprising A, then

nn EPEPEPEEEPAP ...... 2121

58

Theorem

If A and B are two events in S then

where is the common part of A and B

BAPBPABAP

BA

59

Total Probability

If Bi , i = 1, 2, …n, are mutually exclusive events, then

][][][1

n

i ii BPBAPAP

60

Conditional Probability

BPBAP

BAP

61

Bayes’ Theorem

• If Bi , i = 1, 2, …n, are mutually exclusive events, then

• P[Bi] are prior, or “a priori” probabilities and must be determined prior to some experiment that results in event A based on the nature of the problem, data, experience, or subjectively based on experience.

n

i ii

rrr

BPBAP

BPBAPABP

1

62

Expected Value

ni

i ii xpxxEV1

)()(

Discrete Distribution

dxxxfxEV )()(

Continuous Distribution

63

Subjective Probability

• Consider the following Events– Flip a coin and let it hit the floor. Before looking at the

coin, what is the probability it is “heads”?

– What is the probability that the coin flip before the 2009 Auburn/Alabama football game resulted in “tails”?

• Both events have occurred.The outcome is certain. The only uncertainty is in your mind; i.e., in your “degree of belief”

64

Subjective Probability Problems

• Same phrase has different connotations with different people

• Interpretation is context dependent

• People are uncomfortable doing it

• There is no “correct” value

65

Example

• You and a friend disagree on P[Your team will win its next game]. Discussion of factors, home advantage, injuries, weather, etc., does not result in agreement.

• How do you resolve the disagreement?

66

Assessing Subjective Probabilities

• Direct Inquiry– Just make an estimate based on knowledge and

perception– May not be able to come up with a value– May not have much confidence in the result

• Analysis of bets

• Comparison of Lotteries

67

Analysis of Bets

• Compare two bets• Look for indifference point• Example

– LA Lakers vs Boston Celtics– What do you think is P[Lakers Win]

• Bets– Win X$ if LA wins, lose Y$ if LA loses– Lose X$ if LA wins, win Y$ if LA loses

68

Analysis of Bets (Cont’d)

Bet for LA

Bet Against LA

LA wins

LA loses

LA loses

LA wins

X

-Y

-X

Y

Procedure:

Change X and Y until willing to take either bet.

69

Analysis of Bets (Cont’d)

• Expected Value

• At indifference EV should be the same

i ii ZPZZEV

YX

Yp

YpXpYpXp

YpXpAgainstLAEV

YpXpForLAEV

w

wwww

ww

ww

))(1())(1(

))(1()(

))(1()(

70

Example

10$,1$

1$,10$

YX

YX

8$,2$

2$,8$

YX

YX

5$,4$

4$,5$

YX

YX

9

4

45

4

wp

For LA

Against LA

For LA

Against LA

Indifferent

Round 1

Round 2

Round n

.

.

.

71

Reference Lottery

• Lottery 1– Win A if LA wins– Win B if LA loses

• Lottery 2 (Reference)– Win A with probability p– Win B with probability 1-p

72

Reference Lottery (Cont’d)

Lottery 1

Lottery 2Reference

LA wins

LA loses

1-p

p

A

B

A

B

Procedure:

Change p until willing to take either bet.

Outcome of lottery 2 is determined by a random process.

Choose p, generate a random variate, x, from U(0,1) distribution.

If x p then win A; otherwize, win B

73

Example

1.0

9.0

p

p

2.0

8.0

p

p

3.0

7.0

p

p

Round 1

Round 2

Round 3

Prefer lottery 2

Prefer lottery 19.01.0 p}

Prefer lottery 2

Prefer lottery 2

Prefer lottery 2

Prefer lottery 1 }

}

08.2.0 p

3.02.0 p

Home in slowly.Check for consistency with probability axioms and theorems.

74

Continuous Probability Distributions

• Strategies– Direct assessment of p– Fractile assessment of X

ix

ii dxxfxxPxF )()()(

Probability Distribution Function(Cumulative probability)

75

Direct Assessment

• Estimate range of values for x, xmin, xmax

• Pick value xi : xmin < xi < xmax

• Estimate • Repeat for a number of points (for 3 points pick

mid range, then mid range of the two segments unless distribution is strongly skewed)

• Use Reference lottery approach to find indifference

][ ixxP

76

Reference Lottery

Lottery 1

Lottery 2Reference

x<xi

x>xi

1-F(x)

F(x)

$1000

$0

$1000

$0

77

Example

Estimate time to get home from work

Min 20 min F(t)=0.00 25 F(t)=0.50 30 F(t)=0.70 35 F(t)=0.85Max 40 F(t)=0.95

Distribution Function

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30 35 40 45

Time, Min

F(t

)

78

Fractile Method

• Pick range values for x: xmin, xmax

• Pick a number of fractiles– F(x)=0.05 Min– F(x)=0.95 Max– F(x)=0.50– F(x)=0.25– F(x) =0.75

• Note order• Use Reference lottery to find indifference

79

Reference Lottery

Lottery 1

Lottery 2Reference

x<xi

x>xi

1-p

p

$1000

$0

$1000

$0

80

Heuristics and Biases

• Representativeness

• Availability

• Anchoring and adjusting

• Motivational bias

81

Representativeness

• Judge that something or someone belongs to a particular category– Stereotyping– Insensitive to base rates and prior probabilities– Unreliable information– Failure to account for inherent uncertainty

• Misunderstanding random processes– Regression to the mean– Extreme outcomes likely to be followed by one closer

to the mean

82

Availability

• Judge P[event] according to ease of recalling similar events

• Influenced by unbalanced reporting

• Illusory correlation – pair of events perceived as happening together frequently

83

Anchoring and Adjusting

• Choose initial point then adjust about it• Affects continuous distributions more that

discrete probabilities• If initially estimate median will tend to

underestimate extremes so distribution is too narrow

• Estimate extremes first- the “worst” extreme before the other

84

Motivational Bias

Incentives often exist that motivate person to report forecasts or probabilities that do not reflect their true beliefs– Salesman forecasting sales– Weather forecasters forecasting rain– Program managers predicting lower cost,

shorter schedule, and low risk

85

Decomposition

• Break problem down to make probability assessment easier– Permits using people with subject matter knowledge

– More likely to get a realistic estimate

• Look at fault tree analysis and hazard analysis• Use laws of probability to reconstruct the problem• Very important concept for planning and

constructing mitigation efforts

86

Coherence

All assessed probabilities MUST obey the laws of probability

87

Using Test Results

• Requirements are often placed on characteristics that are stochastic– Reliability

– Accuracy

• Can use probability to estimate probability of not satisfying the requirement if test data exist

• Example: Impact accuracy of a projectile specified by the standard deviation of miss distance

88

Example• Requirement:• Data yield sample estimate S, sample size n where

• Recall

• Then

• Look up alpha in a table or use the chidist(chi,dof) function in Excel to find Alpha=0.067

• Alpha is the risk probability

3Re q

2

22

,1

)1(

Snn

10,4 nS

169

16*92,9

89

Chi Squared Probability Distribution Function9 DOF

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25

X

1-al

pha

90

Value of Information

• Information reduces uncertainty– Toss a pair of dice and do not look at result– Estimate the probability that it is 7– If told it is not 5, does this change your

estimate?

• Information has a cost and a worth• Should never pay more than it is worth

91

Example*

A new manufacturing process has been developed. It will cost $0.5M to implement. It will save $0.5M over investment if implemented by your current vender. There are two other vendors willing to implement the process, but both require some relief from EPA regulations in order to provide savings, which could be least double those offered by the current vendor. No relief offers savings, but not as great as keeping the current vendor. It is known the EPA is re-examining these regulations and could either provide relief, make no change, or increase the requirements. The current vendor is not impacted regardless of any changes. An increase will cause a loss by both of the other vendors which you must reimburse.

* Based on an example in Making Hard Decisions, An introduction to Decision Analysis; PWS Kent, 1991; Robert T. Clemen

92

Payoffs

EPA Action

Vendor

Relief

P= 0.5

Same

P= 0.3

Increase

P=0.2

V1 $0.5 $0.5 $0.5

V2 $1.0 $0.2 -$0.1

V3 $1.5 $0.1 -$1.0

93

Decision Tree Structure

V1

V2

V3Relief

Relief

Same

Same

Increase

Increase

0.5

0.5

0.3

0.3

0.2

0.2

$0.5M

$1.0M

$0.2M

-$0.1M

$1.5M

$0.1M

-$1.0M

(0.5M)

($0.54M)

($0.58M)

(xxx)=EMV

94

Dilema

• Highest expected saving is obtained by choosing Vendor 3

• This choice also involves the highest potential loss– Risk probability is 0.2– Risk consequence is -$1.0 M– High Yellow

• How to mitigate?

95

96

Possible Actions

• Obtain Information– Re-estimate probabilities– Hire knowledgeable consultant

• Knows for sure

• Probably knows

• Questions:– What is information worth?– How reliable is the consultant’s result?

97

Perfect Information

(xxx)=EMV

$0.5

$1.0

$1.5

$0.5

$0.2

$0.1

$0.5

-$0.1

-$1.0

Relief0.5

Same0.3

Increase0.2

Perfect Information

NoInformation

($

($0.58MFrom previous analysis)

($1.0M)

EMVPI = $1.0M - $0.58M = $0.42M

V1

V1

V1

V2

V2

V2

V3

V3

V3

98

Imperfect InformationRelief

Same

Increase

Relief

Same

Increase

ReliefSame

Increase

Relief

Same

Increase

Same

Increase

Relief

Same

Increase

No Consultant

HireConsultant

“Relief”

“Same”

“Increase”

V3

V3

V3

V2

V2

V2

V1

V1

V1

$1.5M

$1.5M

$1.5M

$0.1M

$0.1M

$0.1M

-$1.0M

-$1.0M

-$1.0M

$1.0M

$0.2M

-$0.1M

$1.0M

$0.2M

-$0.1M

$1.0M

$0.2M

-$0.1M

$0.5M

$0.5M

$0.5M

($0.58M)

(xxx)=EMV

]Re"Re[ liefliefP ?

Relief

99

Problem

• Need to find conditional probabilities for chance events

• Solution– Total probability law– Bayes’ theorem

100

Solution

Total Probability

Where:R=ReliefS=SameI=Increase“R” denotes predict Relief, etc.

][]""[][]""[][]"["]"[" IPIRPSPSRPRPRRPRP

101

Bayes’ Theorem

]"["

][]""[]""[

RP

RPRRPRRP

102

Prior Conditional Probabilities

True EPA ActionPrediction Relief Same Increase P[prediction]"Relief" 0.80 0.15 0.20 0.485"Same" 0.10 0.70 0.20 0.300"Increase" 0.10 0.15 0.60 0.215Sum 1.00 1.00 1.00 1.00P[Action] 0.50 0.30 0.20

Posterior Conditional Probabilities

True EPA ActionPrediction Relief Same Increase P[prediction]"Relief" 0.8247 0.0928 0.0825 1.0000"Same" 0.1667 0.7000 0.1333 1.0000"Increase" 0.2326 0.2093 0.5581 1.0000

103

Imperfect InformationRelief

Same

Increase

Relief

Same

Increase

ReliefSame

Increase

Relief

Same

Increase

Same

Increase

Relief

Same

Increase

No Consultant

HireConsultant

“Relief”

“Same”

“Increase”

V3

V3

V3

V2

V2

V2

V1

V1

V1

$1.5M

$1.5M

$1.5M

$0.1M

$0.1M

$0.1M

-$1.0M

-$1.0M

-$1.0M

$1.0M

$0.2M

-$0.1M

$1.0M

$0.2M

-$0.1M

$1.0M

$0.2M

-$0.1M

$0.5M

$0.5M

$0.5M

($0.58M)

(xxx)=EMV

0.8247

0.1667

0.2326

0.485

0.300

0.215

0.8247

0.1667

0.2326

0.0928

0.0928

0.700

0.700

0.2093

0.2093Relief

0.0825

0.0825

0.1333

0.1333

0.5581

0.5581

($1.164M)

($0.835M)

($0.187M)

($0.293M)

-($0.188)

($0.219M)

($0.822M)

EMVI=$0.822M-$0.580M=$0.242M

104

Impact on Risk

• Probability of $1M loss reduced from 0.2 to 0.0825

• Risk changed from high yellow to low level

• Can pay consultant up to $0.242M

105

Problem• Using EMV may lead to solutions that may not be intuitively appealing• Example:

– A1 Win $30, p=0.5 EMV = $14.50

Lose $1, p=0.5– A2 Win $2000, p=0.5 EMV = $500

Lose $1000, p=0.5

• Choose A2 based on EMV

• What about Risk and Consequence?– Would probably rather have A1

106

Reason

• EMV is valid for the long run; i.e., multiple occurrences of the chance event

• This is a one time event

• Ignores the range of possible outcomes

• Play 10 times– Max loss A1 = $10– Max loss A2 = $20000

107

Solution

• Find a transformation of consequence into a utility measure, U

• Must accommodate attitude toward the risk/consequence combination– Risk averse– Risk taking– Risk neutral

108

Utility Functions

• Tabular

• Math functions

• Graph

)()(

1)(

)log()(

5.0xXU

exU

xxU

R

x

109

Risk AttitudesUtility Functions

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100 120

Consequence, $

Uti

lity

Averse

NeutralSeeking

110

Scenario

• Forced gamble– Win $500, p=0.5– Lose $500, p=0.5

• Would you pay $x to get out of this gamble?– If so, you are risk averse

• Examples– Insurance– Ransom

111

Comment

• Not everyone is risk averse

• Many people are risk seeking over some range and risk averse over others

• Depends on wealth level (range of consequences)

• Risk attitudes are important in analyzing mitigation alternatives

112

Mixed Risk Attitude

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60 70 80 90 100

Consequence, $

Uti

lity

Risk Seeking

Risk Averse

113

Transformation• Utility function transforms wealth (consequence) into a

measure that accounts for risk attitude• Certainty equivalent (CE)

– Gamble: Win $2000, p=0.5 EMV=$990 Lose $20, p=0.5

– Offered $300 for the gamble. Ask $301, won’t accept $299

• Then CE=$300 for this gamble• If CE ~ EMV then

– U(CE)=EMV($)– Can substitute U(x) for $x

• EMV – CE = Risk Premium

114

Utility Function AssessmentCE Method

1. Fix min and max values of wealth

2. Set U(min)=0, U(max)=1

3. Structure a lotterymin

max

CE1

0.5

0.5

4. Find CE1. Then

5.0)10(5.0(max)05.(min)5.0)( 1 UUCEUExample:

5.0)30($

30$,100$max,10$min

U

CE

115

CE Method (Cont’d)

5. Pick another range, say CE1 – max

6. Structure a new lottery

CE1

max

CE2

0.5

0.5

75.0)15(.5.0(max)5.0)(5.0)( 12 UCEUCEU

7. Set range 0 –CE1

8. Repeat 6. To obtain U(CE3) = 0.25

9. Stop or further subdivide the intervals

10. Either draw the curve or fit a function

116

Probability Equivalent (PE) Method1. Fix max and min of wealth2. Set U(min) = 0, U(max) = 13. Pick CE: min < CE < max4. Structure lottery

min

max

CE

1-p

p

5. Find p: CE ~ p(min) + (1-p)(max)

Then:

pppppUCEU )1)(1()0()(max)1((min))(

6. Repeat steps 3. – 5. For other values of CE

117

Risk Tolerance

Consider the utility function

max

1)(

011)0(

1)(

xx

as

xU

U

exU R

x

R determines shape

Larger R -> Flatter function

Smaller R -> more concave (risk averse)

Hence, R depends on risk attitude

118

Determination of R

Consider a lottery

Y

Y/2

0

0.5

0.5A1

A2

Find largest value of Y for which A1 > A2

R = YY

x

exU

1)(

R

x

eUE

1)(

To find CE, Find E(U) for the decision. Solve

For x = CE

119

Caveats

• Utilities do not add U(A+B) = U(A)+U(B)• Utilities do not express strength of preference.

– They only provide a numerical scale for ordering preferences

• Utility functions are not the same person to person– They are subjective and express personal preferences

120

Example Revisited

V1

V2

V3Relief

Relief

Same

Same

Increase

Increase

0.5

0.5

0.3

0.3

0.2

0.2

$0.5M

$1.0M

$0.2M

-$0.1M

$1.5M

$0.1M

-$1.0M

(0.5M)

($0.54M)

($0.58M)

$ U

1.5M 1.00

1.0M 0.86

0.5M 0.65

0.2M 0.52

0.1M 0.46

-0.1M 0.33

-1.0M 0.00

121

Utility Axioms

• Ordering and transitivityConsider events A1, A2, A3. Then

A1> A2, A2> A1, or A1~ A2

If A1> A2 and A2> A3, then A1> A3

• Reduction of compound uncertain eventsA DM is indifferent between compound uncertain events (a complicated mix of gambles and lotteries) and a simple uncertain event as determined using standard probability manipulations.

122

Example

A1

A2

E1

E2

E3

E4

E5

E6

E7

E8

E9

2

5

10.3

5

0

10.3

5

0

0.17

0.2

0.5

0.3

0.2

0.5

0.3

0.5

0.33

123

Example Continued

A1

A2

E1

E4

E5

E6

E7

E8

E9

2

5

10.3

5

0

10.3

5

0

0.17

0.15=0.5*0.3

0.066=0.33*0.2

0.165=0.33*0.5

0.099=0.33*0.3

0.1=0.5*0.2

0.25=0.5*0.5

124

Example Concluded

A1

A2

2

10.3

5

0

0.166=0.10+0.066

0.585=0.17+0.25+0.165

0.249=0.15+0.99

125

Axioms Continued

• ContinuityA DM is indifferent between outcome A and an uncertain event with outcomes A1 and A2 where A1>A>A2. Hence we can construct a reference gamble with p(A1) and (1-p)(A2) such that the DM is indifferent between A and the gamble

• SubstitutabilityA DM is indifferent between an uncertain event A and one found by substituting for A an equivalent uncertain event (gamble can be substituted for CE)

126

Axioms Continued

• MonotonicityGiven two reference gambles having the same possible outcomes, a DM will prefer the one with higher probability of winning the preferred outcome.

• InvarianceOnly outcome payoffs and probabilities are needed to determine DM’s preferences

• BoundednessNo outcomes are infinitely bad or infinitely bad

127

Comments

• There are some controversies and paradoxes regarding some of the axioms

• If you accept them then– There exist U1, U2, … , Un (utilities) with

associated payoffs such that the overall preference for uncertain events A and B can be determined by E(U)

– You should be using E(U) to make decisions (rational behavior)

128

Sensitivity Analysis

• To manage risk we need both probabilities and consequences

• Probabilities, utilities, and maybe consequences are likely subjective estimates

• Need to find out how much change will impact the decision

129

Example Problem Again

V1

V2

V3Relief

Relief

Same

Same

Increase

Increase

0.5

0.5

0.3

0.3

0.2

0.2

$0.5M

$1.0M

$0.2M

-$0.1M

$1.5M

$0.1M

-$1.0M

(0.5M)

($0.54M)

($0.58M)

$ U

1.5M 1.00

1.0M 0.86

0.5M 0.65

0.2M 0.52

0.1M 0.46

-0.1M 0.33

-1.0M 0.00

0.65

0.86

0.52

0.33

1.00

0.46

0.00

0.65

0.652

0.638

130

Change Utility for $1.0M

856.05.0

33.0*2.052.0*3.065.0

33.0*2.052.0*3.0*5.065.0

33.0*2.052.0*3.0*5.0)]2([

65.0)]1([

x

x

xVUE

VUE

131

Change in Probabilities

006.034.0

33.0*2.052.0*3.086.0*5.065.0

)52.086.0(33.0*2.052.0*3.086.0*5.065.0

33.0*2.0*52.052.0.*3.0*86.086.0*5.065.0

65.033.0*2.052.0*)3.0(86.0*)5.0()]2([

d

d

dd

ddVUE

132

Summary

• Risk management process• Cost and schedule risk

– Models– Monte carlo simulation

• Estimation of likelihoods• Analysis of mitigation alternatives• Value of information• Estimation of utility metrics• EMV vs Expected utility• Sensitivity analysis