Impact of an Aggregation Tree Over the Skewness of the Final Risk Distribution

Abstract

Following the implementation of Solvency 2 in Europe and other parts of the world, many (re)insurance companies decided to put in place Internal Models. The central part of an internal model is its aggregation tree and the calibration of the aggregation tree has been the subject of many articles. Among the different articles, “Estimating copula for insurance from scarce observations, expert opinion and prior information: a Bayesian approach”, ASTIN Bulletin (2012) by Arbenz et al. sets a general framework for estimating dependencies.

In practice, most of the calibration exercises focus on elements which are similar in nature to “correlation” as this is the easiest part of the dependence structure: Practitioners (including underwriters, claims managers, actuaries and finance managers) usually understand and have a slight feeling for correlations if relevant explanations are provided.

But the other impacts of these calibration choices are usually forgotten as practitioners do not have a view on these elements. In particular, the number of layers of the aggregation tree as well as the order of the aggregation play a role on both the overall volatility and the overall skewness of the final risk distribution. But, usually, these elements are not discussed in detail. As volatility and skewness are the main determinants of the risk measure (Value at Risk or Expected Shortfall), understanding the impact of the calibration choices on these elements is certainly important and should also be in the focus of the calibration exercise.

As a consequence, this article aims at estimating the impact of the different choices of calibration of the aggregation tree on both volatility and skewness of the overall risk distribution. Examples of calibration are provided to demonstrate the impact of the different (forgotten) elements of choice in a calibration exercise.

Keywords: Coefficient of variation, diversification, value at risk, skewness, kurtosis, aggregation tree, copula, Cornish-fisher expansion, Kendall Tau

JEL Classification: G22, M41