# Correlation of powers of Hüsler–Reiss vectors and Brown–Resnick fields, and application to insured wind losses

## TL;DR

This paper studies the correlation of extreme weather events using statistical models and applies the findings to insurance risk assessment.

## Contribution

The paper provides new analytical formulas for correlation in Hüsler–Reiss and Brown–Resnick models with power transforms.

## Key findings

- Analytical formulas for correlation between powers of Hüsler–Reiss vector components are derived.
- The results are extended to Brown–Resnick fields and their dependence properties are studied.
- The findings are applied to assess insured wind losses in Germany, offering insights for the insurance industry.

## Abstract

Hüsler–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 Hüsler–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.