Distorted Copula-Based Probability Distribution of a Counting Hierarchical Variable: A Credit Risk Application

In this paper, we propose a novel approach for the computation of the probability distribution of a counting variable linked to a multivariate hierarchical Archimedean copula function. The hierarchy has a twofold impact: it acts on the aggregation step but also it determines the arrival policy of the random event. The novelty of this work is to introduce this policy, formalized as an arrival matrix, i.e., a random matrix of dependent 0–1 random variables, into the model. This arrival matrix represents the set of distorted (by the policy itself) combinatorial distributions of the event, i.e., of the most probable scenarios. To this distorted version of the CHC approach [see Ref. 7 and Ref. 27], we are now able to apply a pure hierarchical Archimedean dependence structure among variables. As an empirical application, we study the problem of evaluating the probability distribution of losses related to the default of various type of counterparts in a structured portfolio exposed to the credit risk of a selected set among the major banks of European area and to the correlations among these risks.