Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk

CHAPTERSIX

Families of Closed-Form Single Risk LDA Models

6.1 Motivating the Consideration of Closed-Form Models in LDA Frameworks

In Cruz et al. (2015) and Chapter 4, we have discussed in detail the idea of developing severity models which admit flexible skewness, kurtosis and heavy-tail features for OpRisk modeling. These models represent an important aspect of OpRisk modeling but they comprise only a single component of loss distribution approach (LDA) modeling, and when combined into a compound process setting to obtain the annual loss distribution, several questions naturally arise. The first question to consider involves

Which of these families of models admit a closed-form density and distribution function for the compound process?

In answering this question, one could just enumerate some models with special properties. However, perhaps a more constructive approach would be to consider the question

Under what conditions on the frequency and severity models do closed-form LDA models arise and how best can one characterize such families of OpRisk models?

In characterizing these conditions, one may explain a general LDA model framework that admits closed-form representations for important quantities in OpRisk modeling. The answer to these questions will be addressed in the first part of this chapter.

The second part of this chapter is associated with work with such LDA models in practical settings. It is recognized that even in the case of the single risk LDA model in OpRisk, ...