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Modeling dependence between loss triangles with hierarchical Archimedean copulas. (English) Zbl 1390.91154

Summary: 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. In [“Dependent loss reserving using copulas, ASTIN Bull. 41, No. 2, 449–486 (2011; doi:10.2143/AST.41.2.2136985)], P. Shi and E. W. Frees 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 G. Barnett and B. Zehnwirth [“Best estimates for reserves”, in: Proceedings of the Casuality Actuarial Society, Vol. 87. 245–321 (2000), http://www.casact.org/pubs/proceed/proceed00/00245.pdf]. , we will suppose a dependence between all the observations that belong to the same calendar year (CY) for each line of business. Thereafter, we will then suppose another dependence structure that links the CYs 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 applied to a dataset from a major US property-casualty insurer, where a bootstrap method is proposed to estimate the distribution of the reserve.

MSC:

91B30 Risk theory, insurance (MSC2010)
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62P05 Applications of statistics to actuarial sciences and financial mathematics

Software:

nacopula
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References:

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