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

Riccardo Rebonato

Wiley 2010
A book review by Danny Yee © 2010
In Plight of the Fortune Tellers Rebonato explained some of the limitations of data-driven approaches to managing extreme financial risks, based on VaR or similar approaches, but didn't really present an alternative. In Coherent Stress Testing he explains how a risk manager might use conditional probabilities, coherency constraints, and causal networks in stress testing, incorporating basic macroeconomic knowledge into their understanding and handling of tail risks. A theoretical framework is illustrated with worked examples and accompanied by detailed advice on practical implementation.

Rebonato begins by arguing that models are indispensable and that we should accept a plurality of models. Model diversity helps explain coordination between market participants, which in turn contributes to "fat tails" in asset price movements. He also argues for a Bayesian view of probabilities, introduces the problems of tail co-dependence, and looks at what is involved in moving from associative to causal models.

Stress testing traditionally examines what might happen to a bank following extreme events, say "Australian house prices fall by 40%". But even if one can give crude, order of magnitude, estimates for the probability of such events, it is difficult to determine the relationships between them — what is the probability of an Australian house price crash and a 50% fall in the S&P 500, for example, or the probability of either of these given the other (or a Chinese recession)? Even simple attempts to estimate such conditional probabilities can easily produce incoherent — logically incompatible — results.

Rebonato envisages a risk manager making order of magnitude estimates of marginal and conditional probabilities for maybe a dozen binary events. The goal then is to find the complete set of joint probabilities, which allow calculation of such useful things as total expected losses and risk-adjusted returns. It turns out that having only some marginal probabilities puts significant bounds on the joint probabilities, and having some singly conditioned probabilities makes those bounds tighter.

Further constraints can be obtained using "Bayesian nets", where the risk manager delineates the causal relationships between events, effectively marking some as independent or conditionally independent. And if the probabilities assigned by the risk manager prove incoherent, Rebonato provides an algorithm for modifying them. (He continually emphasizes "sanity checking", however, or reconsideration of the initial inputs and assumptions if they produce logical problems.)

Rebonato concludes with some practical concerns. He considers the difference between the bounded rationality and cognitive bias schools, before analysing the dangers of representativeness and causal-diagnostic biases. Human intuitions work better in causal than diagnostic mode, making some conditional probabilities easier to estimate than others.

Perhaps the most difficult part of stress testing is the selection of stress scenarios. Here Rebonato suggests some ways of combining "top down" approaches driven by macroeconomic structural risks with "bottom up" approaches driven by trading book risks. (Rebonato makes no attempt to use macroeconomic theory here, perhaps unsurprisingly given how primitive that is.) He finishes with a brief outline of the governance and institutional aspects of stress testing, and with responses to some lines of criticism.

The mathematics at the core of this looks intimidating at first glance, but is actually trivial — Venn diagrams, elementary probability, simple formalism — with some linear programming consigned to an appendix. Rebonato also works through examples in more detail than the pure mathematicians may like. But it is a great strength of his approach that the relationship of the theoretical framework and its mathematics to the underlying modelling is always visible, and that the sensitivity of the conclusions to the assumptions and inputs is transparent.

Rebonato's repeated emphasis on Bayesianism seems like a distraction, however. There is nothing specifically Bayesian about conditional probabilities or, despite the name, Bayes' Theorem, so that part of his analysis works regardless of one's interpretation of probabilities. And when it comes to causality Rebonato uses Judea Pearl's work, claiming that as "deeply Bayesian in nature", but Pearl himself states that "the debate between Bayesians and frequentists ... is orthogonal to the distinct problems confronted by causal analysis" ("Causal Inference in Statistics", Statistics Surveys Vol 3, 2009).

This is an entirely peripheral concern, however. A bigger limitation for most readers is that, while many of its ideas are more general, much of Coherent Stress Testing is directed at professional risk managers. This doesn't make it inaccessible — occasional technical terms such as "short gamma position" and "PV01" are either explained or not critical and easily looked up — but those whose interests are largely outside finance might prefer other books.

I certainly hope the risk managers at my banks understand Rebonato's ideas. And I really enjoyed Coherent Stress Testing myself. Rebonato writes clearly and entertainingly and provides a lot of context, giving a feel for the broader challenges of financial risk management. He ranges widely intellectually — his footnotes are always worth reading — and offers useful suggestions for further reading at the end of each chapter.

October 2010

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%T Coherent Stress Testing
%S A Bayesian Approach to the Analysis of Financial Stress
%A Rebonato, Riccardo
%I Wiley
%D 2010
%O hardcover, references, index
%G ISBN-13 9780470666012
%P 227pp