Asymptotic Analysis of Conditional Tail Expectations

The conditional tail expectation used in continuous risk analysis describes the expected amount of risk that could be experienced given that a potential risk exceeds a threshold value, and provides a coherent risk measure that is preferable than the value-at-risk, a risk measure that is widely used but fails to satisfy the coherency principle. In this paper, we study the asymptotic relations between the conditional tail expectation and the value-at-risk, and show that for a large class of continuous heavy-tailed risks, the conditional tail expectation is asymptotically proportional to the value-at-risk. The asymptotic results for the conditional tail expectations of multivariate risks are also derived.