Risk Quantization by Magnitude and Propensity
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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.
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