  Expected Shortfall (ES) in several variants has been proposed as remedy for
the defi-ciencies of Value-at-Risk (VaR) which in general is not a coherent
risk measure. In fact, most definitions of ES lead to the same results when
applied to continuous loss distributions. Differences may appear when the
underlying loss distributions have discontinuities. In this case even the
coherence property of ES can get lost unless one took care of the details in
its definition. We compare some of the definitions of Expected Shortfall,
pointing out that there is one which is robust in the sense of yielding a
coherent risk measure regardless of the underlying distributions. Moreover,
this Expected Shortfall can be estimated effectively even in cases where the
usual estimators for VaR fail.
  Key words: Expected Shortfall; Risk measure; worst conditional expectation;
tail con-ditional expectation; value-at-risk (VaR); conditional value-at-risk
(CVaR); tail mean; co-herence; quantile; sub-additivity.
