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Spatial Risk Measures and Rate of Spatial Diversification

EPFL, Chair of Statistics STAT, EPFL-SB-MATH-STAT, MA B1 433 (Bâtiment MA), Station 8, 1015 Lausanne, Switzerland
Risks 2019, 7(2), 52; https://doi.org/10.3390/risks7020052
Received: 21 December 2018 / Revised: 25 March 2019 / Accepted: 4 April 2019 / Published: 2 May 2019
(This article belongs to the Special Issue Risk, Ruin and Survival: Decision Making in Insurance and Finance)
An accurate assessment of the risk of extreme environmental events is of great importance for populations, authorities and the banking/insurance/reinsurance industry. Koch (2017) introduced a notion of spatial risk measure and a corresponding set of axioms which are well suited to analyze the risk due to events having a spatial extent, precisely such as environmental phenomena. The axiom of asymptotic spatial homogeneity is of particular interest since it allows one to quantify the rate of spatial diversification when the region under consideration becomes large. In this paper, we first investigate the general concepts of spatial risk measures and corresponding axioms further and thoroughly explain the usefulness of this theory for both actuarial science and practice. Second, in the case of a general cost field, we give sufficient conditions such that spatial risk measures associated with expectation, variance, value-at-risk as well as expected shortfall and induced by this cost field satisfy the axioms of asymptotic 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 a max-stable random field, we provide conditions on both the function and the max-stable field ensuring the latter properties. Max-stable random fields are relevant when assessing the risk of extreme events since they appear as a natural extension of multivariate extreme-value theory to the level of random fields. Overall, this paper improves our understanding of spatial risk measures as well as of their properties with respect to the space variable and generalizes many results obtained in Koch (2017). View Full-Text
Keywords: central limit theorem; insurance; max-stable random fields; rate of spatial diversification; reinsurance; risk management; risk theory; spatial dependence; spatial risk measures and corresponding axiomatic approach central limit theorem; insurance; max-stable random fields; rate of spatial diversification; reinsurance; risk management; risk theory; spatial dependence; spatial risk measures and corresponding axiomatic approach
MDPI and ACS Style

Koch, E. Spatial Risk Measures and Rate of Spatial Diversification. Risks 2019, 7, 52.

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