Copulas: Tales and Facts

When I started writing the paper [19] in 2003 a Google search of the word “copula” gave 10,000 responses. In September 2005 the same search gives 650,000 responses. There is an explosion of activity. What is going on? Many of the web-sites found in the Google search are related to mathematical finance, statistics, extreme value theory, and risk management. Everybody who opens any journal on stochastic processes, probability theory, statistics, econometrics, risk management, finance, insurance, etc., observes that there is a fast growing industry on copulas. The commercial statistics software Splus provides the module FinMetrics that includes copula fitting which is written by R. Carmona, see also Carmona [4]. One can also get copula modules in other major software packages (R, Mathematica, MatLab, etc.). The International Actuarial Association [14] in its hefty paper on Solvency II recommends using copulas for modeling dependence in insurance portfolios. Moody’s uses Gaussian copulas for modeling credit risk and provides software for it which is used in many financial institutions. Since Basle II copulas are now standard tools in credit risk management. The main purpose of this paper is to ask some näıve questions about the fast ascent of copula technology that has become so fashionable. My main concern is that this very simple concept might be something like the emperor’s new clothes because it promises to solve all problems of stochastic dependence but it falls short in achieving the goal. Although I do appreciate that practitioners, in contrast to academic researchers, have to come up with solutions to their risk problems within deadlines and that “quick and dirty methods” cannot always be avoided. Yet one may of course ask how much safety the banking and insurance industry (and maybe the rest of the world) really gains by using the copula concept. I have been watching my colleagues for some years and have been wondering why more and more of them became immersed in copulas without discussing the pros and cons of the concept. I suspect that some include the word “copula” in the title of their papers not because they contribute to the theory on copulas, but because they believe that one can publish easier. Some have adopted the language of copulas which has led them to publish papers with complicated technical assumptions, whereas the results are not new when considered in the usual language of distributions. I also