Correlation of powers of H\"usler-Reiss vectors and Brown-Resnick fields, and application to insured wind losses

TL;DR

This paper derives analytical formulas for the correlation of powers of H"usler-Reiss vectors and Brown-Resnick fields, explores their properties, and applies these findings to model insured wind losses in Germany.

Contribution

It provides new analytical formulas for correlations of powered components of H"usler-Reiss vectors and Brown-Resnick fields, with applications to actuarial risk assessment.

Findings

Correlation formulas enable better modeling of wind extremes.

Power transforms improve dependence measurement for weather-related damages.

Case study offers insights for insurance risk management.

Abstract

H\"usler-Reiss vectors and Brown-Resnick fields are popular models in multivariate and spatial extreme-value theory, respectively, and are widely used in applications. We provide analytical formulas for the correlation between powers of the components of the bivariate H\"usler-Reiss vector, extend these to the case of the Brown-Resnick field, and thoroughly study the properties of the resulting dependence measure. The use of correlation is justified by spatial risk theory, while power transforms are insightful when taking correlation as dependence measure, and are moreover very suited damage functions for weather events such as wind extremes or floods. This makes our theoretical results worthwhile for, e.g., actuarial applications. We finally perform a case study involving insured losses from extreme wind speeds in Germany, and obtain valuable conclusions for the insurance industry.