Financial Risk Under Shortfall Level Uncertainty

We propose a unified approach for developing financial risk measures that are tailored to decision making in the face of two sources of uncertainty, namely, market-based and prudence-based. Common risk measures, including Value at Risk (VaR), Expected Shortfall (ES), and their extensions, are typically assessed at a fixed quantile of the risk variable’s probability distribution, given a prudence level p, thereby only considering market-based uncertainty. We advocate for risk assessment measures that incorporate both sources of uncertainty using a novel decomposition which we present as a representation theorem. Our representation theorem introduces a Mean-Covariance (MCov) decomposition of co-monotonically additive and coherent risk measures, expressed through the expected value of an investment and a covariance functional. Among these market-based uncertainty risk measures, ES has become the dominant measure used in financial decision analysis. We define a Bayesian risk measure as the expected value of ES, incorporating a variable prudence level governed by a prior probability distribution. Within a decision-theoretic framework, this Bayes Expected Shortfall (BES) serves as the optimal shortfall forecast under quadratic loss, termed the prior Bayes estimate. BES has a MCov decomposition where the covariance functional is determined by the prior distribution and is characterized by the properties given by the representation theorem. Specific prudence level distributions yield some premium principles in the existing literature. We explore both parametric and nonparametric methods for its estimation.