Spatial risk measures and rate of spatial diversification

TL;DR

This paper advances the theory of spatial risk measures, analyzing their properties and asymptotic behavior, especially for extreme environmental events modeled by max-stable fields, with implications for actuarial science.

Contribution

It generalizes previous results by providing conditions under which spatial risk measures satisfy axioms of asymptotic homogeneity, especially for max-stable random fields.

Findings

Spatial risk measures satisfy asymptotic homogeneity of order 0, -2, -1, and -1 for expectation, variance, VaR, and expected shortfall.

Conditions are established for cost fields derived from max-stable random fields to meet these axioms.

The theory enhances understanding of risk diversification over large regions for environmental extremes.

Abstract

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).