We introduce and study the main properties of a class of convex risk measures that refine Expected Shortfall by simultaneously controlling the expected losses associated with different portions of the tail distribution. The corresponding adjusted Expected Shortfalls quantify risk as the minimum amount of capital that has to be raised and injected into a financial position X to ensure that Expected Shortfall ESp(X) does not exceed a pre-specified threshold g(p) for every probability level p∈[0,1]. Through the choice of the benchmark risk profile g one can tailor the risk assessment to the specific application of interest. We devote special attention to the study of risk profiles defined by the Expected Shortfall of a benchmark random loss, in which case our risk measures are intimately linked to second-order stochastic dominance.

We introduce and explore Gini-type measures of risk and variability, and develop the corresponding economic capital allocation rules. The new measures are coherent, additive for co-monotonic risks, convenient computationally, and require only finiteness of the mean. To elucidate our theoretical considerations, we derive closed-form expressions for several parametric families of distributions that are of interest in insurance and finance, and further apply our findings to a risk portfolio of a bancassurance company.