RISK MEASURES BASED ON TARGET RISK PROFILES

As obvious for Value-at-Risk (VaR), even Expected Shortfall (ES) may not detect tail risk adequately. A possible solution proposed in the literature is the adjusted ES. 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[Formula: see text]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.