Modelling parameter uncertainty for risk capital calculation

Abstract

For risk capital calculation within the framework of Solvency II the possible loss of basic own funds over the next business year of an insurance undertaking is usually interpreted as a random variable X. If we assume that the parametric distribution family XðhÞjh 2 I R d È É is known, but the parameter h is unknown and has to be estimated from the available historical data, the undertaking faces parameter uncertainty. To assess methods to model parameter uncertainty for risk capital calculations we apply a criterion going back to the theory of predictive inference which has already been used in the context of Solvency II. In particular, we show that the bootstrapping approach is not appropriate to model parameter uncertainty from the undertaking's perspective. Based on ideas closely related to the concept of fiducial inference we introduce a new approach to model parameter uncertainty. For a wide class of distributions and for common estimators including the maximum likelihood method we prove that this approach is appropriate to model parameter uncertainty according to the criterion mentioned above. Several examples demonstrate that our method can easily be applied in practice.

Parameter uncertainty and reserve risk under Solvency II

“…The "inversion method" introduced in Fröhlich and Weng (2015) expresses the parameter estimateθ in terms of the true parameter θ and to invert this relation to obtain an expression of θ in terms ofθ. Since we assume that all payments are settled after n development years, we set f sim n ≡ 1 and σ sim n ≡ 0 and apply the idea of the inversion method to the parameter vector θ = (f 1 , σ 2 1 , f 2 , σ 2 , .…”

“…Considering the probability of solvency already introduced in Gerrard and Tsanakas (2011); Fröhlich and Weng (2015) we assessed several methods to model parameter uncertainty for risk capital calculations considering a very simple model -the normal model. In particular, we demonstrate that the popular bootstrapping approach does not guarantee the required probability of solvency.…”

“…Barndorff-Nielsen and Cox (1996); Young and Smith (2005); Severini et al (2002), and has been investigated in the context of Solvency II for several distributional assumptions for the random variable X in various articles (see e.g. Gerrard and Tsanakas (2011); Fröhlich and Weng (2015); Bignozzi and Tsanakas (2016a,b)). Furthermore, Question 1.1 corresponds to an unbiased estimate of the value-at-risk in the sense of Pitera and Schmidt (2016).…”

“…Wüthrich and Merz (2008) Chapter 4, Verrall (1990) or Peters et al (2017)). However, in order to find a solution to Question 1.1 for the reserve risk we follow the approach introduced in Fröhlich and Weng (2015) (in the sequel this method is called "inversion method") based on Fisher's idea of fiducial inference Fisher (1930). While there was a lot of criticism of Fisher's original argument (see Zabell (1992) or Hannig et al (2016), p.2), in the last decades many authors reinvestigated Fisher's idea and showed that fiducial inference, properly generalized, yields solutions to many important inference problems (see e.g.…”

“…Recall the difference between modelled and true quantities from Remark 2.2. In particular, it is important to understand that we do not assume the true unknown parameter to be random.2 For an introduction to fiducial inference and for a discussion of the history as well as the strengthens and weaknesses of the original fiducial approach we refer toFröhlich and Weng (2015),Zabell (1992). The fiducial approach has properly been generalized in the last decades by many researches (see e.g Hannig et al (2016).…”