How to Deal with Uncertainty Concerning Dependence?

Abstract

Which copula should we use in situations where we have no, or only limited, historical data? In other words, a classical estimation is not feasible and we are exposed to uncertainty concerning the dependence structure. This is not at all an academic question: consider a financial institution that wants to assess its overall risk across different pillars such as market risk, operational risk, credit risk, and so on. If these risk figures are measured on a yearly basis only, then classical estimation strategies are hopeless due to insufficient observations. So what is done in such a situation? In most cases, it is a matter of experience to understand the nature and stylized facts of the risks one faces and a good financial engineer can try to translate those considerations into a suitable parametric family of copulas. But still, the parameters of this family need to be specified. The next step would typically be to stress the parameters of the chosen model to the extremes, which reflects the hope of also understanding the extremes of the application one has in mind. In principle, this is a good idea. The weak point, however, is that the chosen family of copulas need not contain the worst-case scenario across all possible copulas and the extremes within the constructed model might be significantly smaller than the extremes across all copulas. A typical fallacy, which we encounter below, is the mis-belief that the comonotonicity copula implies the biggest risk in many cases. In the following we discuss two important applications where in both cases the worst-case dependence structure is not the comonotonicity copula.