Transcript Slide 1

INTRODUCTION
TO
RISK MANAGEMENT
Defense Resources Management Institute
Naval Postgraduate School
Monterey, California
WHAT IS RISK?
DEFINITIONS I
• Arabic - Fortuitous and favorable.
• Greek - Fortuitous and neither favorable nor unfavorable.
• Latin (risicum) - the challenge that a barrier reef presents
to a sailor.
• French (risque) - mainly negative connotation, but
sometimes positive.
• Oxford Dictionary - “... the chance of hazard, bad
consequences, loss, etc....”
DEFINITIONS II
• Economic risk
- the chance of loss due to ….
• Business risk
- the chance of loss associated with …
• Market risk
- the chance that a portfolio of investments can
•
CHANCE
lose money because…..
Random
Occurrence
Inflation risk
- the danger that a general increase in prices ...
• Interest-rate risk - market risk due to interest rate fluctuations
• Credit risk
- the chance that of a default on a loan ...
difficulty in selling a fixed asset ...
BAD- the
CONSEQUENCE
Derivative risk - the chance of financial loss due to increased
Sense
volatility …. of Loss
• Liquidity risk
•
• Cultural risk
- the chance of loss because of product market …..
Hirschey & Pappas, Fundamentals of Managerial Economics Dryden Press, 1998
A
LITTLE BIT
OF
PROBABILITY
PROBABILITY
• It’s a number – it’s JUST A NUMBER!
• It’s a number between 0 and 1 ( 0 ≤ P ≤ 1)
• It quantifies the likelihood of an event
• It’s a function of experience, judgment, subjective
assessment, available data
• It’s uses all information you think is relevant to the
determination of the likelihood of occurrence of an
event
Probability Rules
Probability = 0 if “never/impossible”
Probability = 1 if “always/certain”
If we are collectively exhaustive and
mutually exclusive then
the probabilities over the outocmes
SUM to 1.
Probability Rules
• If mutually exclusive then :
P(A or B) = P(A) + P(B)
• If independent :
P(A and B) = P(A) x P(B)
Probability from Data
• Given data we can always derive approximate
probabilities using relative frequency.
• Relative frequency can be used as an estimate
of the probability of the observed value
• Taken all together, these can represent the
underlying PROBABILITY DISTRUBUTION
FUNCTION
1.99
2.17
2.24
1.73
2.50
2.21
2.38
2.57
3.80
1.78
1.67
2.49
1.52
3.17
4.88
2.96
2.37
1.61
6.46
1.79
5.39
4.12
3.26
1.96
1.53
2.36
1.53
2.79
6.53
5.34
2.30
2.58
3.48
2.00
7.76
2.34
3.77
1.53
3.87
4.33
3.14
1.76
2.09
3.14
7.90
4.12
4.39
1.52
1.87
1.51
3.92
1.51
6.89
6.48
1.94
1.88
1.54
2.68
2.29
2.36
2.46
2.44
1.58
4.09
3.90
2.71
1.59
1.83
2.17
3.99
1.88
1.82
4.33
3.83
2.89
1.86
1.89
2.12
4.24
4.30
5.08
5.16
4.52
5.48
2.10
2.91
2.16
1.90
2.89
1.90
5.34
4.86
2.36
1.73
2.55
3.15
3.24
1.58
1.79
2.05
2.16
1.64
3.70
3.63
3.75
5.71
2.73
1.69
2.33
1.71
1.59
3.70
1.79
3.94
5.13
1.92
5.02
1.72
2.85
3.17
3.95
6.37
3.30
7.10
2.10
1.85
2.52
2.50
1.75
2.17
4.96
4.18
3.12
3.32
2.24
2.00
3.23
3.60
1.86
1.71
2.07
5.00
4.09
1.97
1.64
1.68
2.45
3.06
4.77
2.33
3.83
3.71
4.86
1.64
2.30
2.43
1.88
1.98
1.78
1.86
1.74
1.66
2.25
2.19
2.26
6.79
4.92
7.11
2.64
3.16
2.81
1.69
2.08
3.67
2.15
1.70
3.80
2.04
2.14
1.84
3.40
2.14
1.94
3.00
2.60
2.78
1.56
2.16
1.75
2.74
2.45
1.86
2.88
3.66
3.19
1.92
2.17
2.77
2.37
3.73
2.99
2.25
7.17
2.89
3.88
3.59
2.61
1.58
6.73
4.14
1.66
4.28
5.16
7.11
2.48
3.85
2.35
2.16
2.00
3.32
2.55
5.46
2.80
4.63
1.63
1.90
1.57
2.16
2.93
6.87
2.33
2.03
1.83
4.83
1.60
2.16
1.69
3.11
4.05
1.88
2.02
3.69
2.08
1.69
2.31
2.70
3.67
2.46
3.12
3.95
2.91
1.53
1.56
3.20
3.57
1.88
3.39
2.78
3.69
2.09
2.02
4.75
3.65
1.65
2.84
4.07
2.20
1.60
5.43
2.15
1.56
5.89
2.35
1.79
1.65
2.32
2.11
4.34
2.00
4.09
1.61
4.10
2.22
3.32
2.31
1.67
2.82
3.60
2.90
3.16
2.19
1.98
2.39
2.48
1.63
1.60
3.08
1.62
5.96
5.87
3.45
1.69
1.54
2.33
1.63
1.73
3.25
2.06
1.72
1.86
1.81
2.03
1.92
3.06
1.58
1.88
2.83
2.48
5.88
3.66
4.53
2.17
4.27
3.30
2.83
3.14
1.64
2.29
4.26
2.19
4.40
1.81
4.07
6.96
4.85
3.12
2.37
2.71
1.75
2.09
2.39
2.42
1.96
3.02
2.40
2.21
5.67
3.48
1.54
1.77
2.98
2.16
1.57
5.04
5.25
5.27
2.59
4.24
3.74
3.94
3.85
1.91
4.43
2.22
3.48
2.48
2.98
5.98
2.44
2.46
1.89
3.23
2.83
3.65
4.90
2.72
1.94
1.55
2.93
5.49
2.31
2.06
2.82
1.66
5.51
4.11
4.64
4.65
1.62
2.56
3.36
1.80
3.02
1.53
1.78
1.62
3.95
2.29
5.40
2.65
2.60
2.11
1.51
1.69
3.33
1.99
4.32
2.44
3.10
4.17
5.31
3.84
2.54
2.32
1.55
1.51
3.35
6.21
4.73
3.65
1.67
3.75
1.90
3.05
2.59
1.58
1.55
1.52
1.63
4.94
2.11
4.11
1.53
1.55
3.31
2.59
3.35
4.28
3.36
3.94
4.53
1.78
8.14
1.62
3.27
2.56
2.23
3.64
1.62
1.66
2.35
4.29
3.51
3.15
7.33
2.30
4.20
2.76
7.02
3.34
2.21
2.18
2.48
2.02
2.33
2.64
2.15
2.69
2.28
1.89
3.11
6.50
6.58
1.96
4.08
2.44
2.22
2.40
2.35
1.83
1.72
5.38
2.16
2.17
2.90
6.21
1.53
4.39
1.78
5.10
2.83
2.54
4.45
3.02
4.14
1.69
1.95
3.67
2.36
1.62
1.87
1.99
4.95
1.92
1.53
2.67
3.05
1.71
2.97
1.87
4.19
3.56
1.81
2.92
1.64
2.65
2.41
3.52
3.27
3.15
4.41
5.73
5.38
1.76
2.77
3.30
2.14
4.07
2.00
4.55
2.97
1.84
4.32
2.88
4.17
3.37
3.64
1.67
2.36
1.51
2.10
1.78
1.52
5.35
2.40
3.23
2.49
6.39
1.76
2.07
1.50
3.12
5.06
1.94
1.81
4.13
2.52
6.16
2.30
1.54
3.48
1.66
2.24
4.12
6.69
3.37
2.09
1.68
1.95
2.23
5.83
5.71
3.10
2.72
1.92
1.67
1.96
2.13
2.44
1.99
4.39
2.68
6.31
2.99
2.97
3.23
1.90
2.78
2.26
3.66
1.89
2.22
2.80
3.98
5.72
2.85
1.95
4.75
2.39
2.04
2.09
1.53
7.06
1.72
1.52
2.66
5.31
1.70
1.77
2.60
4.77
2.63
3.41
8.93
2.18
2.02
2.38
5.75
4.02
2.73
3.70
2.14
2.50
5.70
4.10
3.66
4.43
2.13
2.80
4.57
2.92
2.91
1.93
3.79
5.12
2.84
1.76
1.82
1.82
2.61
1.99
3.14
2.59
2.15
2.73
3.74
3.18
2.59
2.59
1.88
2.03
2.37
2.95
2.25
1.95
4.46
5.49
1.55
6.81
2.97
2.21
2.01
2.38
2.03
1.67
2.88
2.00
2.23
2.71
1.83
2.85
1.70
2.18
1.62
3.27
3.10
1.77
2.62
2.17
3.83
1.89
1.83
5.01
3.22
2.08
3.74
3.14
5.69
2.87
1.55
4.01
4.44
3.77
4.16
3.47
3.62
4.07
3.92
2.85
2.02
1.52
1.87
4.63
2.02
2.55
3.18
4.05
1.93
4.46
4.30
3.62
2.93
2.57
1.69
3.27
2.04
2.01
1.56
1.80
4.33
3.59
6.14
3.79
1.50
4.39
3.34
2.69
2.10
3.31
4.10
1.85
2.34
2.38
1.52
2.05
2.24
3.22
2.41
3.59
3.99
5.40
1.77
3.23
2.89
4.22
2.65
1.56
2.01
3.44
5.11
2.90
1.72
2.35
1.64
2.01
7.63
1.61
6.54
2.19
2.76
2.00
2.76
3.69
3.24
1.85
4.43
3.79
1.99
2.87
2.80
3.34
3.00
1.73
3.58
1.77
3.06
4.41
2.90
3.41
2.48
3.59
6.26
4.08
2.02
3.20
2.95
2.61
3.77
2.76
1.63
2.64
2.17
6.42
3.23
3.26
4.85
1.76
6.03
4.07
2.43
2.76
5.89
1.59
3.74
2.16
5.86
3.69
6.17
1.75
3.20
1.86
3.10
1.68
1.53
3.56
1.73
1.92
3.59
1.79
2.57
4.11
3.55
1.53
3.55
3.54
2.68
1.92
2.17
1.71
1.71
3.11
5.83
2.13
2.28
2.60
2.03
3.64
2.12
1.74
2.71
3.86
3.53
1.59
2.25
2.82
1.54
3.57
2.01
1.60
1.92
2.50
1.84
1.84
3.95
3.04
1.83
2.86
2.33
2.97
4.72
2.46
4.98
2.78
5.52
1.81
3.50
3.97
1.84
1.62
3.38
2.48
4.95
3.46
2.51
4.52
2.80
2.44
2.68
2.61
2.45
1.95
3.60
3.36
1.93
5.08
2.37
2.51
1.58
3.29
2.26
2.35
1.56
2.28
1.84
2.13
5.59
2.05
3.74
2.35
3.45
Earthquakes
2.95
2.01
1.93
3.77
1.72
2.98
1.88
6.38
2.22
2.18
3.46
1.85
4.86
1.60
3.07
3.05
5.22
5.67
1.64
3.49
5.77
3.46
2.49
3.70
7.18
6.16
3.25
4.32
1.62
4.69
2.37
2.50
2.14
1.71
1.90
2.48
2.44
4.15
1.55
1.97
4.04
2.63
3.36
2.40
2.11
1.98
1.87
1.76
3.37
1.80
1.53
5.26
1.81
4.28
1.52
1.63
6.82
2.25
5.02
1.88
3.76
5.00
3.84
1.72
1.94
3.35
3.04
1.83
2.81
2.24
2.47
1.89
1.58
1.56
3.93
3.23
1.88
3.10
4.29
2.66
4.26
2.73
2.15
2.14
2.79
3.42
3.97
2.03
1.65
1.75
2.36
3.33
4.04
3.20
1.91
1.98
3.20
1.73
1.51
6.20
1.50
1.50
1.51
1.51
1.51
1.51
1.51
1.51
1.52
1.52
1.52
1.52
1.52
1.52
1.52
1.52
1.53
1.53
1.53
1.53
1.53
1.53
1.53
1.53
1.53
1.53
1.53
1.53
1.54
1.54
1.54
1.54
1.54
1.55
1.55
1.55
1.55
1.55
1.55
1.55
1.56
1.56
1.56
1.56
1.56
1.56
1.56
1.57
1.57
1.58
1.58
1.58
1.58
1.58
1.58
1.58
1.59
1.59
1.59
1.59
1.60
1.60
1.60
1.60
1.60
1.61
1.61
1.61
1.62
1.62
1.62
1.62
1.62
1.62
1.62
1.62
1.62
1.63
1.63
1.63
1.63
1.63
1.63
1.64
1.64
1.64
1.64
1.64
1.64
1.64
1.65
1.65
1.65
1.66
1.66
1.66
1.66
1.66
1.67
1.67
1.67
1.67
1.67
1.67
1.68
1.68
1.68
1.69
1.69
1.69
1.69
1.69
1.69
1.69
1.69
1.70
1.70
1.70
1.71
1.71
1.71
1.71
1.71
1.71
1.72
1.72
1.72
1.72
1.72
1.72
1.72
1.73
1.73
1.73
1.73
1.73
1.73
1.74
1.74
1.75
1.75
1.75
1.75
1.75
1.76
1.76
1.76
1.76
1.76
1.76
1.77
1.77
1.77
1.77
1.77
1.78
1.78
1.78
1.78
1.78
1.78
1.79
1.79
1.79
1.79
1.79
1.80
1.80
1.80
1.81
1.81
1.81
1.81
1.81
1.81
1.82
1.82
1.82
1.83
1.83
1.83
1.83
1.83
1.83
1.83
1.84
1.84
1.84
1.84
1.84
1.84
1.85
1.85
1.85
1.85
1.86
1.86
1.86
1.86
1.86
1.86
1.87
1.87
1.87
1.87
1.87
1.88
1.88
1.88
1.88
1.88
1.88
1.88
1.88
1.88
1.88
1.89
1.89
1.89
1.89
1.89
1.89
1.90
1.90
1.90
1.90
1.90
1.90
1.91
1.91
1.92
1.92
1.92
1.92
1.92
1.92
1.92
1.92
1.93
1.93
1.93
1.93
1.94
1.94
1.94
1.94
1.94
1.95
1.95
1.95
1.95
1.95
1.96
1.96
1.96
1.96
1.97
1.97
1.98
1.98
1.98
1.98
1.99
1.99
1.99
1.99
1.99
1.99
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.01
2.01
2.01
2.01
2.01
2.01
2.02
2.02
2.02
2.02
2.02
2.02
2.02
2.03
2.03
2.03
2.03
2.03
2.03
2.04
2.04
2.04
2.05
2.05
2.05
2.06
2.06
2.07
2.07
2.08
2.08
2.08
2.09
2.09
2.09
2.09
2.09
2.10
2.10
2.10
2.10
2.11
2.11
2.11
2.11
2.12
2.12
2.13
2.13
2.13
2.13
2.14
2.14
2.14
2.14
2.14
2.14
2.15
2.15
2.15
2.15
2.15
2.16
2.16
2.16
2.16
2.16
2.16
2.16
2.16
2.16
2.17
2.17
2.17
2.17
2.17
2.17
2.17
2.17
2.17
2.18
2.18
2.18
2.18
2.19
2.19
2.19
2.19
2.20
2.21
2.21
2.21
2.21
2.22
2.22
2.22
2.22
2.22
2.23
2.23
2.23
2.24
2.24
2.24
2.24
2.24
2.25
2.25
2.25
2.25
2.25
2.26
2.26
2.26
2.28
2.28
2.28
2.29
2.29
2.29
2.30
2.30
2.30
2.30
2.31
2.31
2.31
2.32
2.32
2.33
2.33
2.33
2.33
2.33
2.33
2.34
2.34
2.35
2.35
2.35
2.35
2.35
2.35
2.35
2.36
2.36
2.36
2.36
2.36
2.36
2.37
2.37
2.37
2.37
2.37
2.37
2.38
2.38
2.38
2.38
2.39
2.39
2.39
2.40
2.40
2.40
2.40
2.41
2.41
2.42
2.43
2.43
2.44
2.44
2.44
2.44
2.44
2.44
2.44
2.45
2.45
2.45
2.46
2.46
2.46
2.46
2.47
2.48
2.48
2.48
2.48
2.48
2.48
2.48
2.48
2.49
2.49
2.49
2.50
2.50
2.50
2.50
2.50
2.51
2.51
2.52
2.52
2.54
2.54
2.55
2.55
2.55
2.56
2.56
2.57
2.57
2.57
2.58
2.59
2.59
2.59
2.59
2.59
2.59
2.60
2.60
2.60
2.60
2.61
2.61
2.61
2.61
2.62
2.63
2.63
2.64
2.64
2.64
2.65
2.65
2.65
2.66
2.66
2.67
2.68
2.68
2.68
2.68
2.69
2.69
2.70
2.71
2.71
2.71
2.71
2.72
2.72
2.73
2.73
2.73
2.73
2.74
2.76
2.76
2.76
2.76
2.76
2.77
2.77
2.78
2.78
2.78
2.78
2.79
2.79
2.80
2.80
2.80
2.80
2.80
2.81
2.81
2.82
2.82
2.82
2.83
2.83
2.83
2.83
2.84
2.84
2.85
2.85
2.85
2.85
2.86
2.87
2.87
2.88
2.88
2.88
2.89
2.89
2.89
2.89
2.90
2.90
2.90
2.90
2.91
2.91
2.91
2.92
2.92
2.93
2.93
2.93
2.95
2.95
2.95
2.96
2.97
2.97
2.97
2.97
2.97
2.98
2.98
2.98
2.99
2.99
3.00
3.00
3.02
3.02
3.02
3.04
3.04
3.05
3.05
3.05
3.06
3.06
3.06
3.07
3.08
3.10
3.10
3.10
3.10
3.10
3.11
3.11
3.11
3.12
3.12
3.12
3.12
3.14
3.14
3.14
3.14
3.14
3.15
3.15
3.15
3.16
3.16
3.17
3.17
3.18
3.18
3.19
3.20
3.20
3.20
3.20
3.20
3.22
3.22
3.23
3.23
3.23
3.23
3.23
3.23
3.23
3.24
3.24
3.25
3.25
3.26
3.26
3.27
3.27
3.27
3.27
3.29
3.30
3.30
3.30
3.31
3.31
3.32
3.32
3.32
3.33
3.33
3.34
3.34
3.34
3.35
3.35
3.35
3.36
3.36
3.36
3.36
3.37
3.37
3.37
3.38
3.39
3.40
3.41
3.41
3.42
3.44
3.45
3.45
3.46
3.46
3.46
3.47
3.48
3.48
3.48
3.48
3.49
3.50
3.51
3.52
3.53
3.54
3.55
3.55
3.56
3.56
3.57
3.57
3.58
3.59
3.59
3.59
3.59
3.59
3.60
3.60
3.60
3.62
3.62
3.63
3.64
3.64
3.64
3.65
3.65
3.65
3.66
3.66
3.66
3.66
3.67
3.67
3.67
3.69
3.69
3.69
3.69
3.70
3.70
3.70
3.70
3.71
3.73
3.74
3.74
3.74
3.74
3.74
3.75
3.75
3.76
3.77
3.77
3.77
3.77
3.79
3.79
3.79
3.80
3.80
3.83
3.83
3.83
3.84
3.84
3.85
3.85
3.86
3.87
3.88
3.90
3.92
3.92
3.93
3.94
3.94
3.94
3.95
3.95
3.95
3.95
3.97
3.97
3.98
3.99
3.99
4.01
4.02
4.04
4.04
4.05
4.05
4.07
4.07
4.07
4.07
4.07
4.08
4.08
4.09
4.09
4.09
4.10
4.10
4.10
4.11
4.11
4.11
4.12
4.12
4.12
4.13
4.14
4.14
4.15
4.16
4.17
4.17
4.18
4.19
4.20
4.22
4.24
4.24
4.26
4.26
4.27
4.28
4.28
4.28
4.29
4.29
4.30
4.30
4.32
4.32
4.32
4.33
4.33
4.33
4.34
4.39
4.39
4.39
4.39
4.40
4.41
4.41
4.43
4.43
4.43
4.44
4.45
4.46
4.46
4.52
4.52
4.53
4.53
4.55
4.57
4.63
4.63
4.64
4.65
4.69
4.72
4.73
4.75
4.75
4.77
4.77
4.83
4.85
4.85
4.86
4.86
4.86
4.88
4.90
4.92
4.94
4.95
4.95
4.96
4.98
5.00
5.00
5.01
5.02
5.02
5.04
5.06
5.08
5.08
5.10
5.11
5.12
5.13
5.16
5.16
5.22
5.25
5.26
5.27
5.31
5.31
5.34
5.34
5.35
5.38
5.38
5.39
5.40
5.40
5.43
5.46
5.48
5.49
5.49
5.51
5.52
5.59
5.67
5.67
5.69
5.70
5.71
5.71
5.72
5.73
5.75
5.77
5.83
5.83
5.86
5.87
5.88
5.89
5.89
5.96
5.98
6.03
6.14
6.16
6.16
6.17
6.20
6.21
6.21
6.26
6.31
6.37
6.38
6.39
6.42
6.46
6.48
6.50
6.53
6.54
6.58
6.69
6.73
6.79
6.81
6.82
6.87
6.89
6.96
7.02
7.06
7.10
7.11
7.11
7.17
7.18
7.33
7.63
7.76
7.90
8.14
8.93
Frequency Table
How Big?
Richter Scale
1.5 ≤ R < 2.5
2.5 ≤ R < 3.5
3.5 ≤ R < 4.5
4.5 ≤ R < 5.5
5.5 ≤ R < 6.5
6.5 ≤ R < 7.5
7.5 ≤ R < 8.5
8.5 ≤ R < 9.5
How Many?
Number of Earthquakes
474
240
158
65
38
20
4
1
Relative Frequency Table
How Big?
Richter Scale
1.5 ≤ R < 2.5
2.5 ≤ R < 3.5
3.5 ≤ R < 4.5
4.5 ≤ R < 5.5
5.5 ≤ R < 6.5
6.5 ≤ R < 7.5
7.5 ≤ R < 8.5
8.5 ≤ R < 9.5
How Many?
Number of Earthquakes
0.474
0.240
0.158
0.065
0.038
0.020
0.004
0.001
Relative Frequency Histogram
0.50
0.474
0.45
0.40
0.35
0.30
0.240
0.25
0.20
0.158
0.15
0.10
0.065
0.038
0.05
0.020
0.004
0.001
0.00
[1.5 - 2.5)
[2.5 - 3.5)
[3.5 - 4.5)
[4.5 - 5.5)
[5.5 - 6.5)
[6.5 - 7.5)
[7.5 - 8.5)
[8.5 - 9.5)
Prob. of an event = proportion of observations that
corresponds to the event
= percent of observations that
corresponds to the event
= portion of area of histogram that
corresponds to the event
AN INVESTMENT DECISION
• Planning for retirement
• Two options for investment
• Each has a track record, the historical rates-ofreturn over a specified time period
• Each can be used to compute various statistics;
e.g., average rate-of-return, etc.
AN INVESTMENT DECISION
Expected Value
Std. Dev.
Variance
A1
5.00%
1.25%
1.5625
A2
5.70%
2.75%
7.5625
Alternative 1
14
12
r
10
n
8
6
4
2
0
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
n
-2
r
time
Alternative 2
14
12
10
8
6
4
2
0
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
-2
time
Alternative 1
1200
What’s the likelihood of
r<0 ?
1000
800
600
400
200
I don’t want
a rate of
return < 0!
0
-0.30
-0.23
-0.16
-0.09
-0.02
0.05
0.12
0.19
0.26
0.33
0.40
Alternative 2
I want a rate
of return > 0!
1200
What’s the likelihood of
r<0 ?
1000
800
600
400
200
0
-0.30
-0.23
-0.16
-0.09
-0.02
0.05
0.12
0.19
0.26
0.33
0.40
THE FREQUENCY HISTOGRAM
( The “KEY to it ALL’’ )
• The relative frequency histogram over the
outcomes contains all relevant
information.
• This information allows us to quantify risk.
• This is provides our most powerful tool for
risk management.
A
QUANTITATIVE
DEFINITION
OF RISK
A QUANTITATIVE DEFINITION OF RISK
Risk is a COMBINATION of the answers to three questions:
(1) “What can go wrong?”
(2) “How likely is it to go wrong?”
(3) “If it does go wrong, what are the consequences?”
Adapted from S. Kaplan and B. John Garrick, “On the Quantitative Definition of Risk”, Risk Analysis, Vol.1, no.1, 1981
EXAMPLE: Hinterland Illegal Immigration
What can go wrong?
recession;
depression;
economic collapse
How likely is it to go wrong?
chances are 1 in a 10;
a 10% chance;
PF = .10
If it does go wrong, what happens to Drmecia?
large numbers of illegal
immigrants ;
increasing crime;
failing social services;
social unrest;
A QUANTITATIVE DEFINITION OF RISK
What can go wrong?
F
Future
scenario
How likely is it to go wrong?
PF
Probability
of F
If it does go wrong, what are the consequences?
Y
Result due
to F
THE ANSWER TO THE FIRST QUESTION
1. It all starts with the future scenario, F.
2. The F is uncertain so we need probability, PF.
3. F causes a result, an outcome of concern, Y.
4. Y is a function of F. We need to know this relation!
The relation between Y and F is uncertain!!!
BEGINNING – MIDDLE – END
F →X→Y
F1
F2
F3
…
FK
X
X2 then….. XM
THE
“SYSTEM”
1 then
Y1
Y2
Y3
…
YN
F1, F2,… → X1 then X2 then… → Y1, Y2,…
EXAMPLE: Hinterland Illegal Immigration
Illegal immigration is proportional to the ratio of per capita GDP.
F
GDPD/popD
illegal immigration =
=
GDPH/popH
Y
Annual Illegal Immigration
Relative GDP per capita
1.6E+05
10.0
9.0
1.4E+05
8.0
1.2E+05
7.0
1.0E+05
6.0
5.0
8.0E+04
4.0
6.0E+04
3.0
2.0
4.0E+04
1.0
2.0E+04
0.0
1975
1980
1985
1990
1995
2000
Year
2005
2010
2015
2020
2025
0.0E+00
1980
1985
1990
1995
2000
2005
2010
2015
2020
2025
G. H. Hanson (2009), “The Economics and Policy of Illegal Immigration in the U.S.”, Washington, D.C.: Migration Policy Institute
THE ANSWER TO THE SECOND QUESTION
PF
X
X2 then….. XM
THE
1 then“SYSTEM”
PY
SIMULATION ProbabilityforDistribution
→ Math Model →Outcomes of Interest
MODELING
Probability Distribution
for
Future Scenarios
THE ANSWER TO THE SECOND QUESTION
PY
Total Illegal Immigration (Baseline)
7.0
6.0
Values x 10^-6
5.0
4.0
3.0
2.0
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Illegal Immigration [Millions]
0.7
0.8
0.9
1
THE ANSWER TO THE THIRD QUESTION
What number of illegal immigrants do you most want
to avoid? 10000; 100000; 1000000; 10000000;
20000000.
HOW YOU FEEL
(about the possible Y)
=prefer to avoid: minor
What outcome do you most
economic strain; substantial strain; or collapse of
government social/educational services?
PREFERENCE
THE ANSWER TO THE THIRD QUESTION
1. It all starts with the future scenario, F.
2. The F is uncertain so we need probability, PF.
3. F causes a result, an outcome of concern, Y.
4. Y is a function of F. Given PF we can derive PY
5. How do you feel about the probable outcomes?
Do you prefer to avoid some Y more than other Y?
THE ANSWER TO THE THIRD QUESTION
• Preferences < = > value function < = > v(Y)
(1) v(Y) > 0 if Y is “good”
(2) v(Y) < 0 if Y is “bad”
• Value Function Charcteristics
(1) reference point [defining GAINS from LOSSES]
(2) loss aversion [losses MORE IMPORTANT than GAINS]
(3) decreasing marginal values
v(Y)
Gains ( + )
concave
Illegal Immigration
Reference Point
Losses ( - )
convex
A QUANTITATIVE DEFINITION OF RISK
1. It all starts with the future scenario, F.
2. The F is uncertain so we need probability, PF.
3. F causes a result, an outcome of concern, Y.
4. Y is a function of F. Given PF we can derive PY
5. Your preference info, v(Y), defines
is the LAST
thePIECE!
consequences!
Probability
Distribution
AND
(Outcome)
Total Illegal Immigration (Baseline)
Decision Maker
Preferences
v(Y)
7.0
6.0
Values x 10^-6
5.0
AND
4.0
3.0
2.0
Illegal Immigration, Y
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Illegal Immigration [Millions]
PY
AND
v(Y)
Hinterland Economy Collapse
Prob. of Economic Collapse
PF
F
g(F)
Number of Illegal
Immigrants
g(F)
PY
Y
Prob. Dist. Illegal
Immigrants
Risk
v(Y)
How does Drmecia “feel” about the Y?
SPECIAL CASE OF PREFERENCE
v(Y)
Y
SPECIAL CASE OF PREFERENCE
v(Y)
Y
In the limit the weight we assign
to all outcomes <=> a loss
tends to - ∞.
In this case risk is very simple to
quantify risk.
“I can’t bear the thought of
experiencing loss! “
“Experiencing loss would be a
catastrophe!”
ASSESSING THE RISK
Total Illegal Immigration (Collapse)
6.0
5.0
PY
Values x 10^-6
4.0
3.0
2.0
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.8
0.9
1
Values in Millio ns
2
1
0
-3.5
0.1
-2.5
0.2
0.3
-1.5
0.4
-0.5
0.5
0.5
0.6
0.7
1.5
-1
v(Y)
+
-2
-3
-4
-5
-6
-7
-8
-
2.5
1.0
3.5
ASSESSING THE RISK
Total Illegal Immigration (Collapse)
6.0
5.0
PY
Values x 10^-6
4.0
3.0
2.0
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.8
2.5
0.9
1
Values in Millio ns
-3.5 0.1
-2.5
0.2
0.3 -1.5 0.4
-0.5
0.5
-5
-25
v(Y)
-45
-65
-85
-105
0.5
0.6
0.7
1.5
1.0 3.5
ASSESSING THE RISK (SPECIAL
CASE)
Total Illegal Immigration Collapse)
6.0
5.0
Values x 10^-6
4.0
3.0
2.0
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Values in Millio ns
RISK = P{ Y correspond to loss }
RISK = P{ Y ≥ reference point }
RISK = P{ unacceptable Y }
RISK = P{ Y you prefer to avoid }
Who uses this stuff?.......
OVERALL C-RATING System for Readiness:
C-1 = MAE > 89%
P{not capable} ≤ 0.11
C-2 = MAE 80-89% 0.11 ≤ P{not capable} ≤ 0.20
C-3 = MAE 70-79% 0.21 ≤ P{not capable} ≤ 0.30
C-4 = MAE 50-69% 0.31 ≤ P{not capable} ≤ 0.50
C-5 = MAE < 50% 0.50 ≤ P{not capable}
Senate Armed Services Committee, terminology used in arguments before the committee, Feb. 1997
AR 220 – 1 (2010), AFI 10-201 (2006), SORTS (US Department of Defense)
A
QUANTITATIVE
APPROACH TO
RISK MANAGEMENT
The History of Risk Management
• 1950 B.C. – Code of Hamurabi – formalization of bottomry
contracts containing a risk premium for chance of loss of ships
and cargo.
• 750 B.C. – Greece – the use of bottomry contracts.
• 1285 A.D. – King Edward - forbids use of soft coal in kilns to
manage air pollution in London.
• 1583 A.D. – 1st life insurance policy issued in England.
• 19th and 20th century – water and garbage sanitation, building
codes, fire codes, boiler inspections, railroads, steamboats,
autos.
• 1959 A.D. – H. Markowitz, stock portfolio diversification.
RISK MANAGEMENT PROCESS
Identify
Risks
What can go wrong F?
What is F and PF?
Assess
Risks
What are the outcomes [Y, and PY]?
What are the consequences, v(Y)?
What is the risk [quantified]?
Prevent
Implement
Monitor
Mitigate
Negotiate
Management
Action
Tools of Risk Management
• Prevention.
• Mitigation.
• Hedging.
• Diversification.
ASSESSING THE RISK
Definition depends on a reference point.
National policy often specifies a reference point.
Not everyone has the same reference point.
CURVE
Why not plotTHE
P{ Y RISK
≥ y* } versus
y*, for any y* ?
Determining the Outcome
Distribution
• theoretical derivation
• direct assessment
• simulation
Generating your own data
EARTHQUAKES
Number of
earthquakes
Earthquake cost
Size of
earthquake
VARIABLE
Cost per
earthquake
Total cost
Program Cost
FIXED
Earthquake
policy
PY Total Earthquake Cost (Baseline)
0.1
0.08
0.06
0.04
0.02
1.00
70
60
50
40
30
0.11Cost
0.012 ≥0.002
P{0.652
Total
y* } 0.00
20
10
0
0
THE RISK CURVE
1.00
0.80
P{Y>y*}
0.60
0.40
0.20
y*
70
60
50
40
30
20
10
0
0.00
0.10
0.08
0.06
A1 : Do Nothing
0.04
0.02
30
40
50
60
70
30
40
50
60
70
30
40
50
60
70
20
20
20
10
10
10
0
0
0
0.00
0.10
0.08
0.06
A2 : New Building Codes
0.04
0.02
0.00
0.10
0.08
0.06
A3 : Retro-Fit & New Codes
0.04
0.02
0.00
P{ Total Cost ≥ x }
1.00
0.80
The
0.60
RISK CURVES
0.40
compared
0.20
70
60
50
40
30
20
10
0
0.00
1.00
0.80
0.652
A1 (red)
0.60
A2 (blue)
0.40
0.328
A3 (green)
0.20
0.065
70
60
50
40
30
20
10
0
0.00
ACCEPTABLE RISK
“The perennial question free people ask with regard to defense is:
‘How much is enough?’ To this there can be no precise answer.
A country’s security is a function of the DEGREE OF RISK A
COUNTRY IS WILLING TO ACCEPT.”
Hitch & McKean, The Economics of Defense in the Nuclear Age, Atheneum, 1986
ACCEPTABLE RISK
Risk
Too Risky
Acceptable
Risk
Proposed Budget
Required Budget
Cost
Who uses this stuff?.......
Ultimately, policy makers must decide how much the United
States is willing to pay to lower the risks associated with deploying forces abroad. But some might argue that defense
planners occasionally focus on absolute requirements – the
minimum number of forces that they believe will meet
DoD’s military needs – without fully weighing the
relative risks and costs of alternative levels.
Moving U.S. Forces: Options for Strategic Mobility
Congressional Budget Office, Feb. 1997
Who uses this stuff?.......
“Our armed forces remain capable, within an acceptable
level of risk, of meeting the demands of our strategy.”
Maj. Gen. John J. Maher,
Vice Director for Operations, Joint Staff:
testimony before House National Security readiness subcommittee,
Feb. 1997
Who uses this stuff?.......
“Computer security is basically risk management.”
“…. Managers have to decide what they are trying to protect
and how much they are willing to spend, both in cost and
convenience, to defend it.”
Stephen H. Wildstrom,
review of the book “Secrets and Lies by Bruce Schneier,
Businessweek
Sept. 2000
Who uses this stuff?.......
“…we continue to believe the federal government can benefit
from risk management.”
“…. An effective risk management approach includes a
threat assessment, a vulnerability assessment and a
criticality assessment ...”
Raymond J. Decker,
Director, Defense Capabilities and Management, GAO,
Testimony before the Senate Committee on Governmental Affairs
Oct. 2001
RISK MANAGEMENT
Risk
old
new
Cost
RISK MANAGEMENT
Risk
new
Proposed Budget
Cost
RISK MANAGEMENT
Risk
Acceptable
Risk
new
Cost
APPLICATION I
ENTERPRISE BUSINESS
RISK
Error
P = .02
P(uncorrected) = 0.1
Reconciliation
Check
P(corrected) = 0.9
P(error) = 0.2
Data
Entry
Correct
P = .18
Correct
P = .80
P(correct) = 0.8
SECNAV M-5200.35 March 2007
End User
Define Needs,
prepare Purchase
Requisition Form
Direct or
Indirect/Reimb.?
DIR.
Forward
to ASA
INDIR.
Forward
to SPFA
Purchase
Agent
Sponsored
Program
Admin.
Support
NO
YES
ASA reviews PR,
confirms funds,
obtains approval
SPFA
Reviews PR
Funds
Available, etc.?
NO
Funds
Available, etc.?
YES
SPFA assigns
PR number and
form to PA
End User
ASA /OA will
Assign req.,
number and
task Purchaser
Purchase
Agent
Sponsored
Program
Admin.
Support
Clarify
requirements
with end user
NO
Purchaser
reviews for
completeness
of
documentation
YES
All required info.
present and adequate to
make procurement?
Screen request
for mandatory
sources of
supply,
prohibited or
special items,
and authority to
buy
Buy from
mandatory source or
go open market?
End User
Receive
ordered items
and sign
acknowledging
Purchase
Agent
Sponsored
Program
Admin.
Support
STOP
YES
Place order with
source, direct
delivery point,
and provide
estimated
delivery date
Receive
order (if
delivery
point)
NO
Order complete
and accurate?
Reconcile
with vendor
What can go wrong?
P = .6561
0.9
How likely is it to go wrong?
NO
ERROR
Receipt
Review
ERROR
P
= .1
0.1
0.9
ASA
0.1
P =ERROR
.0729
Screen
Request
0.9
0.9
0.1
Reimburse
Or
Direct Funds
Purchaser
P =ERROR
.081
0.9
0.1
P =ERROR
.09
SPFA
0.1
P ERROR
= .1
P = .3439
P = .69255
0.9
NO
ERROR
Receipt
Review
ERROR
P
= .05
0.05
0.9
ASA
0.1
P = .07695
ERROR
Screen
Request
0.9
0.95
Reimburse
Or
Direct Funds
0.1
Purchaser
P = .0855
ERROR
0.95
0.1
P =ERROR
.095
SPFA
0.05
P =ERROR
.05
P = .30745
P = .8145
0.95
NO
ERROR
Receipt
Review
ERROR
P
= .05
0.05
0.95
ASA
0.05
P =ERROR
.04287
Screen
Request
0.95
0.95
Reimburse
Or
Direct Funds
0.05
P =ERROR
.0451
Purchaser
0.95
0.05
P =ERROR
.0475
SPFA
0.05
P =ERROR
.05
P = .1855
P = .9606
0.95
P = .01
NO
ERROR
Receipt
Review
ERROR
0.01
0.99
ASA
0.01
P ERROR
= .009703
Screen
Request
0.99
0.99
Reimburse
Or
Direct Funds
0.01
P ERROR
= .009801
Purchaser
0.99
0.01
ERROR
P = .0099
SPFA
0.01
ERROR
P = .01
P = .0394
APPLICATION II
COST RISK
ASSESSMENT
ESTIMATING SHIPBOARD HELICOPTER
O&M COSTS
Life-cycle cost estimates for the helicopter are needed. The cost analysis staff
is organized into four groups, one each for the four main components of the
life-cycle cost: (1) R&D; (2) Procurement; (3) Operations and Maintenance; and
(4) Salvage/Residual. As leader of the O&M cost estimating group you have
decided to use a factor cost estimates since:
1. Relevant O&M cost data produce reliable CERs for the three components of
the O&M cost [POL, Parts, and “Other”] as functions of the procurement cost.
2. The helicopter is a recently developed model and procurement cost is
expected to be $3.7 million (+/- 3%).
3. Why not just use an O&M cost factor approach: annual O&M cost = 10% of
acquisition cost?
POL ($/hr) vs Acquisition ($M): Summary Output
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
Intercept
Acquisition cost
0.736
0.542
0.522
28.224
25.000
POL[$/hr] = 112.84 + 30.16 × ACQ + error
Coefficients Standard Error
112.845
22.447
30.155
5.778
t Stat
5.027
5.219
P-value
0.000
0.000
Parts ($/hr) vs Acquisition ($M): Summary Output
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.922
0.851
0.844
24.425
25.000
Coefficients
Intercept
Acquisition cost
-84.956
57.215
Parts[$/hr] = -84.96 + 57.21 × ACQ + error
Standard Error
t Stat
19.426
5.000
-4.373
11.442
P-value
0.000
0.000
Other ($/hr) vs Acquisition ($M): Summary Output
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
Intercept
Acquisition cost
0.685
0.470
0.446
21.751
25
Other[$/hr] = 45.21 + 10.12 × ACQ + error
Coefficients
Standard Error
t Stat
P-value
45.2104
10.1249
12.857
3.310
3.516
3.059
0.002
0.006
PROBABILISTIC COST
ESTIMATING
Future explanatory variable is not
always known with
certainty
Cost estimate
is a RANDOM
VARIABLE
y=a+bx+e
Intercept is
subject to
estimation
error
There always is the
model residual
error
Slope coefficient is
subject to estimation
error
PROBABILISTIC COST
ESTIMATING
What is the most appropriate
distribution function?
What is the
resulting
distribution
function?
y=a+bx+e
What is the most
appropriate
distribution function?
What is the most
appropriate
distribution
function?
What is the most
appropriate
distribution
function?
PROBABILISTIC COST
ESTIMATING
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Lognormal Distributions
Empirical Distribution Function
1.60
1.2
1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00
0.00
0.50
?
1.00
1.50
1.0
y=a+bx+e
0.8
0.6
0.4
0.2
0.0
0.0
2.0
4.0
6.0
X
2.00
2.50
x
Triangular Density
1.4
0.60
1.2
0.50
1.0
0.40
0.8
0.30
0.6
0.20
0.4
0.10
0.2
0.00
0.000
0.0
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
1.000
?
2.000
3.000
4.000
5.000
6.000
8.0
10.0
12.0
cPOL = a1 + b1×ACQ + e1
cParts = a2 + b2×ACQ + e2
cOther = a3 + b3×ACQ + e3
CO,M&S = [cPOL + cParts + cother ] × H
Annual O,M & S Cost : CASE1
0.070
0.065
0.060
0.055
$164K ≤
0.050
CO,M&S ≤ $676
0.045
0.040
0.035
0.030
0.025
0.020
0.015
0.010
0.005
900
800
700
Values in Thousands
600
500
400
300
200
100
0.000
Annual O,M & S Cost : CASE 2
0.070
0.065
0.060
0.055
$164K ≤
0.050
CO,M&S ≤ $672
0.045
0.040
0.035
0.030
0.025
0.020
0.015
0.010
0.005
900
800
700
Values in Thousands
600
500
400
300
200
100
0.000
Annual O, M & S Cost : CASE 3
0.070
0.065
0.060
$158K ≤
0.055
CO,M&S ≤ $511
0.050
0.045
0.040
0.035
0.030
0.025
0.020
0.015
0.010
0.005
900
800
700
Values in Thousands
600
500
400
300
200
100
0.000
Annual O, M & S Cost : CASE 4
0.070
$190K ≤
0.065
CO,M&S ≤ $463K
0.060
0.055
0.050
0.045
0.040
0.035
0.030
0.025
0.020
0.015
0.010
0.005
900
850
800
750
700
650
Values in Thousands
600
550
500
450
400
350
300
250
200
150
100
0.000
APPLICATION III
PROJECT
MANAGEMENT
FACILITIES FOR THE FLIGHT SIMULATOR DEPARTMENT
(ACTIVITIES, TIME ESTIMATES, AND DEPENDENCIES)
#
ACTIVITY DESCRIPTION
REQUIRED
PRECEDING
ACTIVITIES
ACTIVITY
DURATION
DAYS
A
Demolish areas 1 & 2
----
10
B
Demolish area 3
----
20
C
Dismantle Basic Simulator
----
10
D
Construct Bomber Simulator area
A
70
E
Construct Fighter Simulator area
A
40
F
Upgrade utilities
B
60
G
Construct Basic Simulator area
B
6
H
Reassemble Basic Simulator
C
48
I
Install Bomber Simulator
D
10
J
Install Fighter Simulator
E,F
27
K
Install Basic Simulator
G,H
40
New Facilities for the Flight Simulator Department
Critical Path
A
B
C
D
E
F
G
H
I
J
K
2
ACTIVITY DESCRIPTION
a
m
b
te
s
Demolish Areas 1 & 2
Demolish Areas 3
Dismantle Basic Simulator
Construct Bomber Sim Area
Construct Bomber Fighter Sim Area
Upgrade Utilities
Construct Basic Sim Area
Reassemble Basic Sim
Install Bomber Sim
Install Fighter Sim
Install Basic Sim
8
30
7
85
37
75
4
58
9
38
37
17
60
15
120
50
90
8
70
15
52
50
28
180
22
206
62
135
11
102
22
100
62
17.3
75.0
14.8
128.5
49.8
95.0
7.8
73.3
15.2
57.7
49.8
11.1
625.0
6.3
406.7
17.4
100.0
1.4
53.8
4.7
106.8
17.4
Project estimated completion time =
Variance =
Standard Deviation =
te
0.0
75.0
0.0
0.0
0.0
95.0
0.0
0.0
0.0
57.7
0.0
s
2
0.0
625.0
0.0
0.0
0.0
100.0
0.0
0.0
0.0
106.8
0.0
227.7
831.8
28.84
Shortest Time to Completion
0.08
0.07
0.06
P = 0.482
0.05
0.04
0.03
0.02
0.01
0
140
160
180
200
220
240
260
280
300
320
340
360
Shortest Time to Completion
0.08
0.07
0.06
P = 0.048
0.05
0.04
0.03
0.02
0.01
0
140
160
180
200
220
240
260
280
300
320
340
360
SUMMARY
• RISK is a factor in every decision with
significant uncertainty
• RISK is a combination of the answers to 3
questions
– what can go wrong?
– how likely is it to go wrong?
– if it does go wrong, what are the consequences?
SUMMARY
• RISK is quantified using PROBABILITY.
– use it to express the riskiness an
alternative.
– use it to find the least risky alternative.
• THINK about the RISK vs COST
tradeoff curve.
SUMMARY
• MANAGING RISK requires the information
provided by the tradeoff curve!
– THINK about where you want to be on the
curve.
– THINK about changing the tradeoff curve!
• USE THE MODEL to help find how to change
things!