VaR vs CVaR (Expected Shortfall)
VaR is a loss quantile; CVaR is the average loss beyond that quantile.
VaR at level α is the loss threshold that is only exceeded with probability 1 − α over a given horizon.
CVaR (Expected Shortfall) is the conditional average loss given that losses breach the VaR threshold.
For continuous right-tailed loss distributions, CVaR is always at least as large as VaR and is more sensitive to extreme tail behaviour.
This diagram shows a loss distribution with the VaR at level α as a red quantile and the CVaR (Expected Shortfall) as an amber line at the average loss beyond that quantile.
The red dashed line is the VaR: only a fraction 1 − α of the loss distribution lies to its right. The shaded area under the blue density shows those tail outcomes.
The amber line is the CVaR, the average loss within the shaded region. As you increase α or σ, both VaR and CVaR move to the right, but CVaR typically grows faster because it reacts to how heavy the tail is. The mental model: VaR tells you “how far the tail starts”; CVaR tells you “how bad it is, on average, once you are in the tail”.