Subadditivity and Parameter Uncertainty of VaR and Solvency Capital Requirement (SCR) in Tail Region of a Non-life Insurance Portfolio

Authors

1 Professor, KU Leuven

2 PhD Candidate of Actuarial Science, Maastricht University

Abstract

Although VaR is important due to its widespread usage to obtain overall Solvency Capital Requirement (SCR) in the standard model of Solvency II directives, it is not subadditive. Without subadditivity, the summation of SCRs of different lines of business, which is usually used by risk managers, may underestimate overall SCR for an insurance company. This research examines the subadditivity property of VaR for fat-tailed insurance losses in a dependent structure. The foundation of the paper is based on Danielson et al (2013), a study on subadditivity of VaR in the tail region of asset return data. We applied the same idea by using Generalized Pareto Distribution (GPD) to model the fat-tailed insurance losses and capture their dependence structure by the Gumbel-Houggard copula through the tail of the joint distribution. Using these instruments, we proposed a simulation method to examine subadditivity of VaR and SCR. By empirical methods, we found that, similar to the fat-tailed asset returns, insurance losses are also more subadditive in tail region. We found that only going deep into the tail, will not guarantee monotonically more subadditivity, where “Variation of dependence” and “shape parameter” through the tail of the distribution are other important factors that Danielson et al didn’t take into account. More specially, when the correlation measure in different thresholds is changed, subadditivity of VaR deviates to increase monotonically in the tail. Furthermore, we observed that the uncertainty of VaR estimation is not always monotonically increasing through the tail; it may increase in the first thresholds of the right tail, it decreases in higher thresholds.

Keywords