Common Shock Models for Claim Arrays

Abstract

The paper is concerned with multiple claim arrays. In recognition of the extensive use by practitioners of large correlation matrices for the estimation of diversification benefits in capital modelling, we develop a methodology for the construction of such correlation structures (to any dimension). Indeed, the literature does not document any methodology by which practitioners, who often parameterise those correlations by means of informed guesswork, may do so in a disciplined and parsimonious manner.We construct a broad and flexible family of models, where dependency is induced by common shock components. Models incorporate dependencies between observations both within arrays and between arrays. Arrays are of general shape (possibly with holes), but include the usual cases of claim triangles and trapezia that appear in the literature. General forms of dependency are considered with cell-, row-, column-, diagonal-wise, and other forms of dependency as special cases. Substantial effort is applied to practical interpretation of such matrices generated by the models constructed here.Reasonably realistic examples are examined, in which an expression is obtained for the general entry in the correlation matrix in terms of a limited set of parameters, each of which has a straightforward intuitive meaning to the practitioner. This will maximise chance of obtaining a reliable matrix. This construction is illustrated by a numerical example.

On Unbalanced Data and Common Shock Models in Stochastic Loss Reserving

“…In this section, we examine the unbalanced feature of loss reserving data in detail. The general common shock framework developed in Avanzi et al (2018) is then described. Challenges that arise in applying common shock models to reserving data due to its unbalanced feature are then discussed.…”

“…Many multivariate models with different types of dependence can be generalised by the common shock framework in Avanzi, Taylor and Wong (2018) with…”

“…i,j , control the extent to which the set-wide common shock contributes to individual members of the set. In this section, we have wished to preserve the link to the general notation of Avanzi et al (2018) through the use of the notation π (i,j) . This notation will be simplified in Section 2.3 for specific examples.…”

“…It is hence essential to develop an accurate approach to the modelling of outstanding losses while allowing for dependencies (Côté, Genest and Abdallah, 2016;Shi, Basu and Meyers, 2012). This not only allows the insurer to accurately assess their performance, but also to hold an appropriate amount of reserves and capital to optimise its internal use while satisfying regulatory requirements (Ajne, 1994;Avanzi, Taylor and Wong, 2018).…”

“…Correlation matrices are tools extensively used by practitioners to specify dependence in the aggregation of outstanding claims liabilities or risk-based capital. Explicit dependence structures captured using common shock approaches allow correlation matrices to be specified in a more disciplined and parsimonious manner (see e.g., Avanzi, Taylor and Wong, 2018).…”