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Coherent Stress Testing: A Bayesian Approach to the Analysis of Financial Stress


In Coherent Stress Testing: A Bayesian Approach to the Analysis of Financial Stress, industry expert Riccardo Rebonato presents an all new approach to his important but often undervalued part of the risk management toolkit.

Based on the author’s extensive work, research and presentation in the area, the book fills a gap in quantitative risk management by introducing a new and very intuitively appealing approach to stress testing based on expert judgment and Bayesian networks. It constitutes a radical departure from the traditional statistical methodologies based on Economic Capital or Extreme-Value-Theory approaches.

The book is split into four parts. Part I looks at stress testing and its role in modern risk management. It discusses the distinctions between risk and uncertainty, the different types of probability that are used in risk management today and for which tasks they are best used. Stress testing is positioned as a bridge between the statistical areas where VaR can be effective and the domain of total Keynesian uncertainty. Part II lays down the quantitative foundations for the concepts described in the rest of the book. Part III takes readers through the applications of the tools discussed in part II, and introduces two different systematic approaches to obtaining a coherent stress testing output that can satisfy the ends of industry users and regulators. In part IV the author addresses more practical questions such as embedding the suggestions of the book into a viable governance structure.

“Riccardo Renato examines the deficiencies of current financial modelling practice and the limitations of the purely statistical approaches to risk quantification that underpin VaR methodology. Taking his cue from Knightian uncertainty – and Rumsfeldian unknown unknowns – the author argues that a program of stress testing carried out within a Bayesian paradigm can offer risk managers a route to redemption after the crisis. Written in his usual lucid and engaging style, Coherent Stress Testing is a thought provoking text on a vitally important issue, and a serious proposal of a workable solution.”
Alexander J. McNeil, Maxwell Professor, Heriot-Watt University

“Riccardo Rebonato’s book shows how managerial judgments can be combined with analysis to improve the way stress testing is done. The book is well written and very timely. In the aftermath of the 2007&ndash2009 financial crisis risk management groups at all financial institutions are looking for ways they can make stress testing more effective.”
John Hull, Maple Financial Professor of Derivatives and Risk Management, Joseph L. Rotman School of Management, University of Toronto

“Rebonato’s interesting book provides a refreshingly different and thought-provoking perspective of stress-testing and quantitative risk management – exactly what the field needs in these troubled times.”
Rüdiger Frey, Professor of Financial Mathematics and Optimization, Universität Leipzig, Co-author of Quantitative Risk Management: Concepts, Techniques, Tools

DR. RICCARDO REBONATO (London, UK) is Head of Front Office Risk Management and Head of the Clients Analytics team at BGM RBS. He is visiting lecturer at Oxford University (Mathematical Finance) and adjunct professor at Imperial College (Tanaka Business School). He sits on the Board of Directors of ISDA and on the Board of Trustees for GARP. He is an editor for the International Journal of Theoretical and Applied Finance, Applied Mathematical Finance, Journal of Risk, and the Journal of Risk Management in Financial Institutions. He holds doctorates in Nuclear Engineering and in Science of Material/Solid State Phsyics. He was a research fellow in Physics at Corpus Christi College, Oxford, UK.

1 Introduction.

1.1 Why We Need Stress Testing.

1.2 Plan of the Book.

1.3 Suggestions for Further Reading.

I Data, Models and Reality.

2 Risk and Uncertainty – or, Why Stress Testing is Not Enough.

2.1 The Limits of Quantitative Risk Analysis.

2.2 Risk or Uncertainty?

2.3 Suggested Reading.

3 The Role of Models in Risk Management and Stress Testing.

3.1 How Did We Get Here?

3.2 Statement of the Two Theses of this Chapter.

3.3 Defence of the First Thesis (Centrality of Models).

3.3.1 Models as Indispensable Interpretative Tools.

3.3.2 The Plurality-of-Models View.

3.4 Defence of the Second Thesis (Coordination).

3.4.1 Traders as Agents.

3.4.2 Agency Brings About Coordination.

3.4.3 From Coordination to Positive Feedback.

3.5 The Role of Stress and Scenario Analysis.

3.6 Suggestions for Further Reading.

4 What Kind of Probability Do We Need in Risk Management?

4.1 Frequentist versus Subjective Probability.

4.2 Tail Co-dependence.

4.3 From Structural Models to Co-dependence.

4.4 Association or Causation?

4.5 Suggestions for Further Reading.

II The Probabilistic Tools and Concepts.

5 Probability with Boolean Variables I: Marginal and Conditional Probabilities.

5.1 The Set-up and What We are Trying to Achieve.

5.2 (Marginal) Probabilities.

5.3 Deterministic Causal Relationship.

5.4 Conditional Probabilities.

5.5 Time Ordering and Causation.

5.6 An Important Consequence: Bayes’ Theorem.

5.7 Independence.

5.8 Two Worked-Out Examples.

5.8.1 Dangerous Running.

5.8.2 Rare and Even More Dangerous Diseases.

5.9 Marginal and Conditional Probabilities: A Very Important Link.

5.10 Interpreting and Generalizing the Factors x k/i.

5.11 Conditional Probability Maps.

6 Probability with Boolean Variables II: Joint Probabilities.

6.1 Conditioning on More Than One Event.

6.2 Joint Probabilities.

6.3 A Remark on Notation.

6.4 From the Joint to the Marginal and the Conditional Probabilities.

6.5 From the Joint Distribution to Event Correlation.

6.6 From the Conditional and Marginal to the Joint Probabilities?

6.7 Putting Independence to Work.

6.8 Conditional Independence.

6.9 Obtaining Joint Probabilities with Conditional Independence.

6.10 At a Glance.

6.11 Summary.

6.12 Suggestions for Further Reading.

7 Creating Probability Bounds.

7.1 The Lay of the Land.

7.2 Bounds on Joint Probabilities.

7.3 How Tight are these Bounds in Practice?

8 Bayesian Nets I: An Introduction.

8.1 Bayesian Nets: An Informal Definition.

8.2 Defining the Structure of Bayesian Nets.

8.3 More About Conditional Independence.

8.4 What Goes in the Conditional Probability Tables?

8.5 Useful Relationships.

8.6 A Worked-Out Example.

8.7 A Systematic Approach.

8.8 What Can We Do with Bayesian Nets?

8.8.1 Unravelling the Causal Structure.

8.8.2 Estimating the Joint Probabilities.

8.9 Suggestions for Further Reading.

9 Bayesian Nets II: Constructing Probability Tables.

9.1 Statement of the Problem.

9.2 Marginal Probabilities – First Approach.

9.2.1 Starting from a Fixed Probability.

9.2.2 Starting from a Fixed Magnitude of the Move.

9.3 Marginal Probabilities – Second Approach.

9.4 Handling Events of Different Probability.

9.5 Conditional Probabilities: A Reasonable Starting Point.

9.6 Conditional Probabilities: Checks and Constraints.

9.6.1 Necessary Conditions.

9.6.2 Triplet Conditions.

9.6.3 Independence.

9.6.4 Deterministic Causation.

9.6.5 Incompatibility of Events.

9.7 Internal Compatibility of Conditional Probabilities: The Need for a Systematic Approach.

III Applications.

10 Obtaining a Coherent Solution I: Linear Programming.

10.1 Plan of the Work Ahead.

10.2 Coherent Solution with Conditional Probabilities Only.

10.3 The Methodology in Practice: First Pass.

10.4 The CPU Cost of the Approach.

10.5 Illustration of the Linear Programming Technique.

10.6 What Can We Do with this Information?

10.6.1 Extracting Information with Conditional Probabilities Only.

10.6.2 Extracting Information with Conditional and Marginal Probabilities.

11 Obtaining a Coherent Solution II: Bayesian Nets.

11.1 Solution with Marginal and n-conditioned Probabilities.

11.1.1 Generalizing the Results.

11.2 An ‘Automatic’ Prescription to Build Joint Probabilities.

11.3 What Can We Do with this Information?

11.3.1 Risk-Adjusting Returns.

IV Making It Work In Practice.

12 Overcoming Our Cognitive Biases.

12.1 Cognitive Shortcomings and Bounded Rationality.

12.1.1 How Pervasive are Cognitive Shortcomings?

12.1.2 The Social Context.

12.1.3 Adaptiveness.

12.2 Representativeness.

12.3 Quantification of the Representativeness Bias.

12.4 Causal/Diagnostic and Positive/Negative Biases.

12.5 Conclusions.

12.6 Suggestions for Further Reading.

13 Selecting and Combining Stress Scenarios.

13.1 Bottom Up or Top Down?

13.2 Relative Strengths and Weaknesses of the Two Approaches.

13.3 Possible Approaches to a Top-Down Analysis.

13.4 Sanity Checks.

13.5 How to Combine Stresses – Handling the Dimensionality Curse.

13.6 Combining the Macro and Bottom-Up Approaches.

14 Governance.

14.1 The Institutional Aspects of Stress Testing.

14.1.1 Transparency and Ease of Use.

14.1.2 Challenge by Non-specialists.

14.1.3 Checks for Completeness.

14.1.4 Interactions among Different Specialists.

14.1.5 Auditability of the Process and of the Results.

14.2 Lines of Criticism.

14.2.1 The Role of Subjective Inputs.

14.2.2 The Complexity of the Stress-testing Process.

Appendix A Simple Introduction to Linear Programming.

A.1 Plan of the Appendix.

A.2 Linear Programming – A Refresher.

A.3 The Simplex Method.