# Measures of risk - Conditional Value-at-Risk (CVaR)

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.

ModelRisk’s CVaR functions use the convention that the underlying variable
**is a distribution
of loss**. It then calculates the mean loss *conditional*
on the loss exceeding
the threshold value *T*,
or the losses in the top fraction *P* of the
distribution, as appropriate. If *f(l)* and *F(l)*
are the probability density the cumulative distribution functions for
the loss distribution *L*
at value *l*,
then at the target value *T*
(where ),
we have:

**Note**

If your model is simulating the distribution of profit, simply enter
negative this distribution as an input to the VoseCVARx or VoseCVARp functions.

**Example**

Assume losses as following a Normal(20,10) distribution, being simulated
in cell A1 using the formula “=VoseNormal(20,10)”. Also assume the *P* value of interest to be
5%, i.e. that we are interested in the highest 5% of possible losses.

**The VaR (value at risk)** is just the 95^{th}
percentile (1-5%) of the normal distribution. This could be calculated
directly as VoseNormal(20,10,1-5%) if the entire model was just the one
distribution. More commonly, however, cell A1 would represent the simulated
loss of a more complex model, in which case we can get the VaR value using:

=VoseSimPercentile(A1,1-5%)

Again, if the entire model was just the one distribution, *the CVaR* could be calculated
directly as:

=VoseMean(VoseNormalObject(20,10,VosePBounds(1-5%,))

However, more normally cell A1 would represent the simulated loss of
a more complex model, in which case we can get the VaR value using:

=VoseSimCVARp(A1,5%)

The 95^{th}
percentile of a Normal distribution equals 36.44854.. We can define the
CVaR using this threshold value too:

=VoseSimCVARx(A1,
36.44854)