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

## Author

Listed:
• Enrico Bernardi

(University of Bologna, Department of Statistics, Via Belle Arti 41, Bologna 40126, Italy)

• Silvia Romagnoli

(University of Bologna, Department of Statistics, Via Belle Arti 41, Bologna 40126, Italy)

## Abstract

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.

## Suggested Citation

• Enrico Bernardi & Silvia Romagnoli, 2016. "Distorted Copula-Based Probability Distribution of a Counting Hierarchical Variable: A Credit Risk Application," International Journal of Information Technology & Decision Making (IJITDM), World Scientific Publishing Co. Pte. Ltd., vol. 15(02), pages 285-310, March.
• Handle: RePEc:wsi:ijitdm:v:15:y:2016:i:02:n:s021962201650005x
DOI: 10.1142/S021962201650005X
as

File URL: http://www.worldscientific.com/doi/abs/10.1142/S021962201650005X

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## References listed on IDEAS

as
1. Cornelia Savu & Mark Trede, 2010. "Hierarchies of Archimedean copulas," Quantitative Finance, Taylor & Francis Journals, vol. 10(3), pages 295-304.
2. Valdez, Emiliano A., 2009. "On the Distortion of a Copula and its Margins," MPRA Paper 20524, University Library of Munich, Germany.
3. Okhrin, Ostap & Okhrin, Yarema & Schmid, Wolfgang, 2013. "On the structure and estimation of hierarchical Archimedean copulas," Journal of Econometrics, Elsevier, vol. 173(2), pages 189-204.
4. repec:wsi:ijitdm:v:11:y:2012:i:06:n:s0219622012500320 is not listed on IDEAS
5. Umberto Cherubini & Silvia Romagnoli, 2009. "Computing the Volume of n-Dimensional Copulas," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(4), pages 307-314.
6. Choroś-Tomczyk, Barbara & Härdle, Wolfgang Karl & Okhrin, Ostap, 2013. "Valuation of collateralized debt obligations with hierarchical Archimedean copulae," Journal of Empirical Finance, Elsevier, vol. 24(C), pages 42-62.
7. Hofert, Marius & Maechler, Martin, 2011. "Nested Archimedean Copulas Meet R: The nacopula Package," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 39(i09).
8. Hofert, Marius, 2008. "Sampling Archimedean copulas," Computational Statistics & Data Analysis, Elsevier, vol. 52(12), pages 5163-5174, August.
9. repec:wsi:ijtafx:v:05:y:2002:i:08:n:s0219024902001742 is not listed on IDEAS
Full references (including those not matched with items on IDEAS)

### Keywords

Distorted copula; random matrix; copula volume; hierarchical Archimedean copula; probability distribution of counting variable;

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