  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 $ES_p(X)$ does not exceed a pre-specified
threshold $g(p)$ for every probability level $p\in[0,1]$. 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.
