Risk Quantization by Magnitude and Propensity

TL;DR

This paper introduces a new bivariate risk measure, combining magnitude and propensity, to better assess the severity and likelihood of risks, contrasting with traditional univariate measures like VaR.

Contribution

It proposes a novel magnitude-propensity risk measure using Wasserstein mass transportation and optimal quantization, providing a dual perspective on risk assessment.

Findings

The measure effectively distinguishes risk severity and likelihood.

Illustrative examples demonstrate its practical application.

The approach offers a new way to compare risks on two scales.

Abstract

We propose a novel approach in the assessment of a random risk variable $X$ by introducing magnitude-propensity risk measures $(m_X,p_X)$. This bivariate measure intends to account for the dual aspect of risk, where the magnitudes $x$ of $X$ tell how hign are the losses incurred, whereas the probabilities $P(X=x)$ reveal how often one has to expect to suffer such losses. The basic idea is to simultaneously quantify both the severity $m_X$ and the propensity $p_X$ of the real-valued risk $X$. This is to be contrasted with traditional univariate risk measures, like VaR or Expected shortfall, which typically conflate both effects. In its simplest form, $(m_X,p_X)$ is obtained by mass transportation in Wasserstein metric of the law $P^X$ of $X$ to a two-points $\{0, m_X\}$ discrete distribution with mass $p_X$ at $m_X$. The approach can also be formulated as a constrained optimal quantization problem. This allows for an informative comparison of risks on both the magnitude and propensity scales. Several examples illustrate the proposed approach.