What is the expected relationship between Conditional VaR and VaR?

The expected relationship between Conditional Value-at-Risk (Conditional VaR, or CVaR) and Value-at-Risk (VaR) is that Conditional VaR is greater than VaR. This is because CVaR measures the expected loss in the tail of the distribution beyond the VaR level.

VaR provides a threshold value such that the probability of a loss exceeding this threshold over a specified period is at a certain confidence level. For example, if the VaR at a 95% confidence level is $1 million, this means that there is a 5% chance of a loss exceeding $1 million.

On the other hand, Conditional VaR not only considers the fact that losses can exceed the VaR threshold, but it also gives an average loss that might be expected in those extreme tail events. Therefore, since CVaR averages out the losses that occur in that tail region, it will always be equal to or greater than the VaR. Thus, it is conceptually and mathematically expected that CVaR will be greater than VaR for any given loss distribution.

Therefore, the assertion that Conditional VaR is greater than VaR is consistent with their definitions and the roles they play in risk assessment.