What Is a Good Risk Measure: Bridging the Gaps between Data, Coherent Risk Measures, and Insurance Risk Measures

Two main axiomatically based risk measures are the coherent risk measure, which assumes subadditivity for random variables, and the insurance risk measure, which assumes additivity for comonotonic random variables. We propose a new, data based, risk measure, called natural risk statistic, that is characterized by a new set of axioms. The new axioms only require subadditivity for comonotonic random variables, which is consistent with the prospect theory in psychology. Comparing to two previous measures, the natural risk statistic includes the tail conditional median which is more robust than the tail conditional expectation suggested by the coherent risk measure; and, unlike insurance risk measures, the natural risk statistics can also incorporate scenario analysis. The natural risk statistic includes the VaR as a special case and therefore shows that VaR, though simple, is not irrational.