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Coherent Stress Testing:A Bayesian Approach to the Analysis of Financial Stress by Riccardo Rebonato
This edition first published 2010
© 2010 Riccardo Rebonato
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Contents
Acknowledgements xi
1 Introduction 1
1.1 Why We Need Stress Testing 1
1.2 Plan of the Book 5
1.3 Suggestions for Further Reading 6
I Data, Models and Reality 7
2 Risk and Uncertainty – or, Why Stress Testing is Not Enough 9
2.1 The Limits of Quantitative Risk Analysis 9
2.2 Risk or Uncertainty? 10
2.3 Suggested Reading 13
3 The Role of Models in Risk Management and Stress Testing 15
3.1 How Did We Get Here? 16
3.2 Statement of the Two Theses of this Chapter 17
3.3 Defence of the First Thesis (Centrality of Models) 18
3.3.1 Models as Indispensable Interpretative Tools 18
3.3.2 The Plurality-of-Models View 21
3.4 Defence of the Second Thesis (Coordination) 23
3.4.1 Traders as Agents 23
3.4.2 Agency Brings About Coordination 25
3.4.3 From Coordination to Positive Feedback 26
3.5 The Role of Stress and Scenario Analysis 27
3.6 Suggestions for Further Reading 29
4 What Kind of Probability Do We Need in Risk Management? 31
4.1 Frequentist versus Subjective Probability 31
4.2 Tail Co-dependence 36
4.3 From Structural Models to Co-dependence 38
4.4 Association or Causation? 39
4.5 Suggestions for Further Reading 42
II The Probabilistic Tools and Concepts 45
5 Probability with Boolean Variables I: Marginal and Conditional
Probabilities 47
5.1 The Set-up and What We are Trying to Achieve 47
5.2 (Marginal) Probabilities 50
5.3 Deterministic Causal Relationship 53
5.4 Conditional Probabilities 55
5.5 Time Ordering and Causation 56
5.6 An Important Consequence: Bayes’ Theorem 57
5.7 Independence 58
5.8 Two Worked-Out Examples 59
5.8.1 Dangerous Running 59
5.8.2 Rare and Even More Dangerous Diseases 61
5.9 Marginal and Conditional Probabilities: A Very Important Link 62
5.10 Interpreting and Generalizing the Factors xki 65
5.11 Conditional Probability Maps 67
6 Probability with Boolean Variables II: Joint Probabilities 71
6.1 Conditioning on More Than One Event 71
6.2 Joint Probabilities 73
6.3 A Remark on Notation 75
6.4 From the Joint to the Marginal and the Conditional Probabilities 76
6.5 From the Joint Distribution to Event Correlation 77
6.6 From the Conditional and Marginal to the Joint Probabilities? 83
6.7 Putting Independence to Work 84
6.8 Conditional Independence 86
6.9 Obtaining Joint Probabilities with Conditional Independence 88
6.10 At a Glance 89
6.11 Summary 90
6.12 Suggestions for Further Reading 90
7 Creating Probability Bounds 93
7.1 The Lay of the Land 93
7.2 Bounds on Joint Probabilities 93
7.3 How Tight are these Bounds in Practice? 96
8 Bayesian Nets I: An Introduction 99
8.1 Bayesian Nets: An Informal Definition 99
8.2 Defining the Structure of Bayesian Nets 101
8.3 More About Conditional Independence 104
8.4 What Goes in the Conditional Probability Tables? 106
8.5 Useful Relationships 107
8.6 A Worked-Out Example 109
8.7 A Systematic Approach 111
8.8 What Can We Do with Bayesian Nets? 113
8.8.1 Unravelling the Causal Structure 113
8.8.2 Estimating the Joint Probabilities 114
8.9 Suggestions for Further Reading 115
9 Bayesian Nets II: Constructing Probability Tables 117
9.1 Statement of the Problem 117
9.2 Marginal Probabilities – First Approach 118
9.2.1 Starting from a Fixed Probability 119
9.2.2 Starting from a Fixed Magnitude of the Move 120
9.3 Marginal Probabilities – Second Approach 120
9.4 Handling Events of Different Probability 122
9.5 Conditional Probabilities: A Reasonable Starting Point 123
9.6 Conditional Probabilities: Checks and Constraints 125
9.6.1 Necessary Conditions 125
9.6.2 Triplet Conditions 126
9.6.3 Independence 127
9.6.4 Deterministic Causation 127
9.6.5 Incompatibility of Events 128
9.7 Internal Compatibility of Conditional Probabilities: The Need for
a Systematic Approach 129
III Applications 131
10 Obtaining a Coherent Solution I: Linear Programming 133
10.1 Plan of the Work Ahead 133
10.2 Coherent Solution with Conditional Probabilities Only 135
10.3 The Methodology in Practice: First Pass 141
10.4 The CPU Cost of the Approach 144
10.5 Illustration of the Linear Programming Technique 144
10.6 What Can We Do with this Information? 149
10.6.1 Extracting Information with Conditional Probabilities Only 149
10.6.2 Extracting Information with Conditional
and Marginal Probabilities 151
11 Obtaining a Coherent Solution II: Bayesian Nets 155
11.1 Solution with Marginal and n-conditioned Probabilities 156
11.1.1 Generalizing the Results 164
11.2 An ‘Automatic’ Prescription to Build Joint Probabilities 165
11.3 What Can We Do with this Information? 167
11.3.1 Risk-Adjusting Returns 168
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