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 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.