#### Understanding Conditional VaR and Expected Shortfall

What is Conditional Value at Risk (CVaR), also called Expected Shortfall (ES)? Although the terminology can be somewhat confusing, CVaR and ES both essentially refer to the same thing: the size of the average loss when the loss exceeds the Value at Risk (VaR) metric. Please refer to our previous article for an introduction to the Value at Risk (VaR) concept, and our also here for a discussion about some of the limitations of VaR estimates. One of the most important limitations of VaR is that it is unable to tell us anything about the size of the loss once the loss is greater than the confidence threshold. Value at Risk tells us the minimum loss to expect, but it cannot quantify extreme values passed the threshold. It is for this reason that CVaR and Expected shortfall are used. Conditional VaR (CVaR) helps estimate the value of the loss when the loss exceeds the statistical threshold. Because CVaR estimates losses greater than the Value at Risk (VaR) estimated loss, it is a rule that CVaR is always greater than VaR.

The difference between VaR and CVaR can be easily seen with an example: Let us assume that the 1-Day VaR at the 95% confidence level is -6% and the 1-Day CVaR at 95% confidence level is -9%. VaR is interpreted as a 5% statistical chance of a loss of at least 6% over the following day. The key words in that intepretation are: a loss of at least. VaR cannot estimate the actual size of the loss that may occur; only the minimum loss. However, since CVaR estimates are conditional of the loss being greater than the VaR value, losses in a tail risk event are likely closer to the implied by CVaR estimate of -9%.

CVaR is likely to be a closer estimation of the actual loss once the loss exceeds Value at Risk.