Modelling Dependence with Copulas and Applications to Risk Management

by Paul Embrechts of the Department of Mathematics ETHZ,
Filip Lindskog of the Department of Mathematics ETHZ, and
Alexander McNeil of the Department of Mathematics ETHZ

September 10, 2001

Introduction: Integrated Risk Management (IRM) is concerned with the quantitative description of risks to a financial business. Whereas the qualitative aspects of IRM are extremely important, in the present contribution we only concentrate on the quantitative ones. Since the emergence of Value-at-Risk (VaR) in the early nineties and its various generalisations and refinements more recently, regulators and banking and insurance professionals have build up a huge system aimed at making the global financial system safer. Whereas the steps taken no doubt have been very important towards increasing the overall risk awareness, continuously questions have been asked concerning the quality of the safeguards as constructed.

All quantitative models are based on assumptions vis-a-vis the markets on which they are to be applied. Standard hedging techniques require a high level of liquidity of the underlying instruments, prices quoted for many financial products are often based on "normal" conditions. The latter may be interpreted in a more economic sense, or more specifically referring to the distributional (i.e. normal, Gaussian) behaviour of some underlying data. Especially for IRM, deviations from the "normal" would constitute a prime source of investigation. Hence the classical literature is full of deviations from the so-called random walk (Brownian motion) model and heavy tails appear prominently. The latter has for instance resulted in the firm establishment of Extreme Value Theory (EVT) as a standard tool within IRM. Within market risk management, the so-called stylised facts of econometrics summarise this situation: market data returns tend to be uncorrelated, but dependent, they are heavy tailed, extremes appear in clusters and volatility is random.

Our contribution aims at providing tools for going one step further: what would be the stylised facts of dependence in financial data? Is there a way of understanding so-called normal (i.e.Gaussian) dependence and how can we construct models which allow to go beyond normal dependence? Other problems we would like to understand better are spillover, the behaviour of correlations under extreme market movements, the pros and contras of linear correlation as a measure of dependence, the construction of risk measures for functions of dependent risks. One example concerning the latter is the following: suppose we have two VaR numbers corresponding to two different lines of business. In order to cover the joint position, can we just add the VaR? Under which conditions is this always the upper bound? What can go wrong if these conditions are not fulfilled? A further type of risk where dependence play a crucial role is credit risk: how to define, stress test and model default correlation. The present paper is not solving the above problem, it presents however tools which are crucial towards the construction of solutions.

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