On the Measurement of Economic Tail Risk

This paper attempts to provide a decision-theoretic foundation for the measurement of economic tail risk, which is not only closely related to utility theory but also relevant to statistical model uncertainty. The main result is that the only risk measures that satisfy a set of economic axioms for the Choquet expected utility and the statistical property of general elicitability (i.e., there exists an objective function such that minimizing the expected objective function yields the risk measure) are the mean functional and value-at-risk (VaR), in particular the median shortfall, which is the median of tail loss distribution and is also the VaR at a higher confidence level. We also discuss various approaches of backtesting and their relations to elicitability and co-elicitability; in particular, we show that the co-elicitability of VaR and expected shortfall does not lead to a reliable backtesting method for expected shortfall and there have been only indirect backtesting methods for expected shortfall. Furthermore, we extend the result to address model uncertainty by incorporating multiple scenarios. As an application, we argue that median shortfall is a better alternative than expected shortfall for setting capital requirements in Basel Accords.