# Risk Management with Tail Quasi-Linear Means

## TL;DR

This paper introduces a generalized risk measure called Tail Quasi-Linear Means, unifying several risk metrics and analyzing its properties for improved risk management strategies.

## Contribution

It extends Quasi-Linear Means to the tail of distributions, unifying Value at Risk, Tail Value at Risk, and Entropic Risk Measure in a single framework.

## Key findings

- The measure encompasses VaR, TVaR, and Entropic Risk in a unified way.
- Properties and implications of the measure in risk assessment are characterized.
- Formulas for truncated elliptical loss models are derived.

## Abstract

We generalize Quasi-Linear Means by restricting to the tail of the risk distribution and show that this can be a useful quantity in risk management since it comprises in its general form the Value at Risk, the Tail Value at Risk and the Entropic Risk Measure in a unified way. We then investigate the fundamental properties of the proposed measure and show its unique features and implications in the risk measurement process. Furthermore, we derive formulas for truncated elliptical models of losses and provide formulas for selected members of such models.