Risk Measures on $\mathcal{P}(\mathbb{R})$ and Value At Risk with Probability/Loss function

TL;DR

This paper introduces a generalized class of law invariant risk measures that extend Value at Risk by incorporating both loss probability and magnitude, with a dual representation on probability measures.

Contribution

It proposes a new framework for risk measures that generalizes V@R, capturing the balance between loss probability and size, and provides their dual representation.

Findings

Generalized risk measures encompass V@R as a special case.

Dual representation of risk measures on probability measures established.

Framework accounts for probability and loss magnitude in risk assessment.

Abstract

We propose a generalization of the classical notion of the $V@R_{\lambda}$ that takes into account not only the probability of the losses, but the balance between such probability and the amount of the loss. This is obtained by defining a new class of law invariant risk measures based on an appropriate family of acceptance sets. The $V@R_{\lambda}$ and other known law invariant risk measures turn out to be special cases of our proposal. We further prove the dual representation of Risk Measures on $\mathcal{P}(% \mathbb{R}).$