Risk measures based on target risk profiles

As obvious for Value-at-Risk (VaR), even Expected Shortfall (ES) may not detect tail risk adequately. The current literature proposes the adjusted ES as a possible solution. It is defined as the supremum of ES values over different confidence levels, adjusted with a deterministic function, the so-called target risk profile. By using a family of general monetary risk measures instead of a family of ESs we unify this concept. This leads to a new class of risk measures, called adjusted risk measures. As a main finding we present equivalent assumptions for an adjusted risk measure to be positively homogeneous, subadditive, convex and consistent with second order stochastic dominance. Furthermore, we show that these conditions hold for several adjusted risk measures beyond the adjusted ES and we derive their dual representations. Finally, a case study based on the S&P 500 demonstrates similarities and differences between the adjusted ES and several new adjusted risk measures. Numerical aspects for the calculation of these risk measures are discussed.