Expected shortfall( ES) : a new tool for risk measure and risk management

Based on loss variables describing corresponding financial portfolios' risk and α-quantile( upper tail) of loss distribution,the definitions of expected shortfall( ES) and conditional value at risk( CVaR) were set up. Under general loss distributions,it has been proved by some direct calculations that the definition of ES was independent on the choice of α-quantile for any loss variable; also by some direct calculations the equivalence between ES and CVaR has been checked; and furthermore by constructing a set of probability measures with which ES could be represented for any loss variable,the coherence of ES as a risk measure has been discovered. More over, there were some important remarks in this paper for some correlated topics,e. g.,α-quantile,coherent risk measure, tail conditional expectation( TCE),etc.