# Extreme-value based estimation of the conditional tail moment

Activity: Talks and presentationsGuest lectures, external teaching and course activities at other universities

## Description

Quantifying the risk related to extreme events is of crucial importance in insurance and finance. Occasionally insurance companies are faced with extreme claims which can jeopardise the solvency of a portfolio, or even a substantial part of a company, e.g., claims due to flooding, storms, industrial fires or earthquakes. The quantification of the risk is done by risk measures, e.g. value-at-risk and conditional tail expectation.

In this presentation, I will talk about the estimation of the conditional tail moment, defined for a non-negative random variable $X$ as $\theta_{p,\beta}=\mathbb E(X^\beta | X> U(1/p))$, $\beta > 0$, $p \in (0,1)$, provided $\mathbb E(X^\beta)< \infty$, where $U$ denotes the tail quantile function given by $U(x) = \inf \{ y: F(y) \ge 1-1/x \}$, $x>1$. The focus will be on situations where $p$ is small, i.e. smaller than $1/n$, where $n$ is the number of observations on $X$ that is available for estimation. This situation corresponds to extrapolation outside the data range, and requires extreme value arguments to construct an appropriate estimator. The asymptotic properties of the estimator, properly normalised, are established under suitable conditions.
Period 8. Apr 2022 Aarhus University, Denmark National