Risk Quantization by Magnitude and Propensity

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 high 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 mX and the propensity pX 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.