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 (mX,pX). 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 CVaR, which typically conflate both effects. In its simplest form, (mX,pX) is obtained by mass transportation in Wasserstein metric of the law of X to a two-points {0,mX} discrete distribution with mass pX at mX. 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 usefulness of the proposed approach. Some variants, extensions and applications are also considered.