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

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

=VoseSimCVARx(A1, 36.44854)