Modeling Dependence Between Loss Triangles With Hierarchical Archimedean Copulas

Abstract

One of the most critical problems in property/casualty insurance is to determine an appropriate reserve for incurred but unpaid losses. These provisions generally comprise most of the liabilities of a non-life insurance company. The global provisions are often determined under an assumption of independence between the lines of business. Recently, Shi and Frees (2011) proposed to put dependence between lines of business with a copula that captures dependence between two cells of two different runoff triangles. In this paper, we propose to generalize this model in two steps. First, by using an idea proposed by Barnett and Zehnwirth (1998), we will suppose a dependence between all the observations that belong to the same calendar year for each line of business. Thereafter, we will then suppose another dependence structure that links the calendar years of different lines of business. This model is done by using hierarchical Archimedean copulas. We show that the model provides more flexibility than existing models, and offers a better, more realistic and more intuitive interpretation of the dependence between the lines of business. For illustration, the model is first applied to a dataset from a major US property-casualty insurer, and then to two lines of business from a large Canadian insurer.

Sarmanov Family of Bivariate Distributions for Multivariate Loss Reserving Analysis

“…However, for the gamma distribution, as noted by Abdallah et al (2015) we use the exponential link instead of the canonical inverse link to ensure positive means, with µ…”

“…In the same view of Abdallah et al (2015), we propose a model that allows a dependence relation between all the observations that belong to the same calendar year for each line of business using random effects instead of multivariate Archimedean copulas. Additionally, we use another dependence structure that links the losses of calendar years of different lines of business with a Sarmanov family of bivariate distributions instead of hierarchical copula.…”

“…Kirschner et al (2008) use a synchronized bootstrap and Taylor and McGuire (2007) extend this result to a generalized linear model context. More recently, Abdallah et al (2015) use Hierarchical Archimedean copulas to accommodate correlation within and between runoff triangles.…”

On Unbalanced Data and Common Shock Models in Stochastic Loss Reserving

“…Outstanding claims reserves are typically some of the most critical components in the financial statement of a non-life insurer (Abdallah, Boucher and Cossette, 2015;Heberle and Thomas, 2016;Saluz and Gisler, 2014). When estimating reserves, the insurer often has to provide the central estimate as well as a risk margin to accommodate for the stochastic nature of outstanding claims.…”

“…the dependence between cells that are in the same position) across loss triangles. Alternatively, parametric approaches utilising distributional assumptions can be used, see for example, Shi and Frees (2011); Zhang and Dukic (2013); De Jong (2012); Abdallah, Boucher and Cossette (2015); Shi (2014).…”