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See also: Insurance and finance risk analysis modeling introduction, Measures of risk - Value at Risk
Conditional Value-at-Risk (CVaR) is also known as Expected Shortfall (ES),
Average Value-at-Risk (AVaR) ad Expected Tail Loss (ETL).
CVaR is superior to VaR
because it satisfies all the requirements for a coherent risk measure
(Artzner
et al, 1997) including subadditivity.
It is simply negative the mean of the loss returns in the distribution
with a greater loss than specified by the confidence interval. For example,
the 99% CVaR is negative the mean of the returns under the remaining 1%
of the distribution. The 100% CVaR is just negative the mean of the revenue
distribution.
ModelRisk’s CVaR functions use the convention that the underlying variable
is a distribution
of loss. It then calculates the mean loss conditional on it exceeding
the threshold value x, or the losses in the top fraction P of the distribution,
as appropriate. If 
and 
are the probability density the cumulative distribution
functions for the loss distribution L, then at the target value x (where
), we have:
