Tail value at risk

Tail value at risk (TVaR), also known as tail conditional expectation (TCE), is a risk measure associated with the more general value at risk. It is equivalent to expected shortfall when the underlying distribution function is continuous at VaR_α(X).^[1] This is not a coherent risk measure in general, however it is coherent if the underlying distribution is continuous.^{[citation needed]} TVaR accounts for the severity of the failure, not only the chance of failure. The TVaR is a measure of the expectation only in the tail of the distribution.

Mathematical definition

If $X \in L^p(\mathcal{F})$ is the payoff of a portfolio at some future time and given a parameter 0 < α < 1 then we can define the tail value at risk by

$TVaR_{\alpha}(X) = E[-X|X \leq -VaR_{\alpha}(X)] = E[-X | X \leq x^{\alpha}]$^[2][3][4]

where x^α is the upper α-quantile given by $x^{\alpha} = \inf\{x \in \mathbb{R}: P(X \leq x) > \alpha\}$.

References

1. ^ "Average Value at Risk" (pdf). https://statistik.ets.kit.edu/download/doc_secure1/7_StochModels.pdf. Retrieved February 2, 2011. 
2. ^ Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc; Heath, David (1999). "Coherent Measures of Risk" (pdf). Mathematical Finance 9 (3): 203–228. http://www.math.ethz.ch/~delbaen/ftp/preprints/CoherentMF.pdf. Retrieved February 3, 2011. 
3. ^ Landsman, Zinoviy; Valdez, Emiliano (February 2004) (pdf). Tail Conditional Expectations for Exponential Dispersion Models. http://www.actuaries.org/ASTIN/Colloquia/Bergen/Landsman_Valdez.pdf. Retrieved February 3, 2011. 
4. ^ Valdez, Emiliano (May 2004) (pdf). The Iterated Tail Conditional Expectation for the Log-Elliptical Loss Process. http://www.asb.unsw.edu.au/schools/actuarialstudies/Documents/E.A.%20Valdez%20-%20The%20Iterated%20Tail%20Conditional%20Expectation%20for%20the%20Log-Elliptical%20Loss%20Process.pdf. Retrieved February 3, 2010.