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An accurate assessment of the risk of extreme environmental events is ofgreat importance for populations, authorities and thebanking/insurance/reinsurance industry. Koch (2017) introduced a notion ofspatial risk measure and a corresponding set of axioms which are well suited toanalyze the risk due to events having a spatial extent, precisely such asenvironmental phenomena. The axiom of asymptotic spatial homogeneity is ofparticular interest since it allows one to quantify the rate of spatialdiversification when the region under consideration becomes large. In thispaper, we first investigate the general concepts of spatial risk measures andcorresponding axioms further and thoroughly explain the usefulness of thistheory for both actuarial science and practice. Second, in the case of ageneral cost field, we give sufficient conditions such that spatial riskmeasures associated with expectation, variance, Value-at-Risk as well asexpected shortfall and induced by this cost field satisfy the axioms ofasymptotic spatial homogeneity of order $0$, $-2$, $-1$ and $-1$, respectively.Last but not least, in the case where the cost field is a function of amax-stable random field, we provide conditions on both the function and themax-stable field ensuring the latter properties. Max-stable random fields arerelevant when assessing the risk of extreme events since they appear as anatural extension of multivariate extreme-value theory to the level of randomfields. Overall, this paper improves our understanding of spatial risk measuresas well as of their properties with respect to the space variable andgeneralizes many results obtained in Koch (2017).

Spatial risk measures and rate of spatial diversification