Risk Measures for the 21st Century (The Wiley Finance Series) [Hardcover]

Risk measures have been long been important, especially from a regulatory standpoint, but this importance has been magnified by the current `financial crisis' and the need for more robust measures of risk over and above what has been currently been in place by banks and other financial institutions. Value-At-Risk, or VAR, has been widely used in the banking industry, due in part to the Basel-II Accords and its ease in implementation. VAR of course has been criticized vociferously both by academics and practitioners alike, but alternatives to VAR, even though they seem plausible on paper, at times are difficult to implement and interpret. It remains to be seen how the relaxation of Basel requirements will affect risk management and capital requirements in the major banks of the world. One thing is clear and that is that risk management will employ even more mathematically sophisticated risk measures in the years ahead, due to the regulatory environment and hyper-technological developments. This book gives a good introduction of what to expect and what has been done in research and development in finding alternative risk measures.

Financial modelers have also been criticized recently for their use of `copula functions'. Indeed, one article in the press described copulas as a "recipe for disaster" and their use is held to be responsible for the "killing of Wall street." To counter these claims, a few articles and books have appeared in recent months, and this book contains an article that addresses the use of copulas in finance. The authors introduce copulas as a method for dealing with the aggregation of individual risks that goes beyond the Gaussian assumption.

If one begins with a vector of uniform random variables, a copula is their joint distribution, and is effectively a function that can be written as a product if the variables are independent. It also must satisfy certain properties dealing with how it increases and how it operators on the boundary of an n-dimensional hypercube. The authors believe that copulas are useful in finance in that they can quantify risk in terms of individual risk variables and the dependences between them without having to have an explicit characterization of the individual risks.

With all the press about stress testing of banks and the failure of (Gaussian) VAR models in risk management, the author detail how to use copulas in these two areas. A non-Gaussian VAR model is constructed using two different choices of copula functions and compared with the historical Gaussian VAR. The latter is show to underestimate the risk for a confidence level greater than 95%. This situation is the "tail" risk of the Gaussian assumption that has been widely discussed in the financial press in the last couple of years.

Bank stress testing, especially for European banks, is of great interest at the present time and the authors. As the name implies, stress testing deals with how resilient a bank's portfolio is to extreme shocks of the type that might be "rare" or "extreme". Regulatory requirements force the world's major banks to do this (the famous `Basel Accords'). The authors construct `extreme value' copulas to build multivariate stress scenarios. An elementary example for the bivariate case is given that deals with the DowJones and the French CAC40 risk factors. It would have been helpful if the authors would have included at least one more example in order to compare differences.

The authors also present a toy model for pricing basket (equity) derivatives that illustrates the issues in modeling the dependences in risk factors in this case. An explicit real-world example would have been helpful here, or a reference to such an example, in order that readers can see what can go wrong in a realistic scenario from an investment house or hedge fund. They do the same for credit derivatives in another section, wherein they give an interesting graph that illustrates the dependence of the loss distribution and VAR on the choice of copula function. One part of this discussion which may be new to some readers is the notion of `derivatives at risk', which the author write in terms of a conditional expectation and explain how to estimate it with Monte Carlo simulation. Readers will need to know what a `risk-neutral' probability measure to follow the discussion.