Common Shock Models for Claim Arrays

Abstract

The paper is concerned with multiple claim arrays. 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.

In recognition of the extensive use by practitioners of large correlation matrices for the estimation of diversification benefits in capital modelling, substantial effort is applied to practical interpretation of such matrices generated by the models constructed here. Indeed, the literature does not document any methodology by which practitioners, who often parametrise those correlations by means of informed guesswork, may do so in a disciplined and parsimonious manner. In fact, this motivated the work presented in this paper.

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. Finally, the generated correlation matrix is then combined with heuristic estimates of tail dependency to arrive at a t-copula which might be used to construct capital margins dealing with the extreme right tail.

Keywords: Cell-wise dependency, Claim array, Claim dependency, Claim triangle, Column-wise dependency, Common shock, Diagonal-wise dependency, Heuristic estimation, Row-wise dependency

JEL Classification: G22, C52, C55