Tail value at risk

Tail value at risk (TVaR), also known as tail conditional expectation (TCE) or conditional tail expectation (CTE), is a risk measure associated with the more general value at risk. It quantifies the expected value of the loss given that an event outside a given probability level has occurred.

Background

There are a number of related, but subtly different, formulations for TVaR in the literature. A common case in literature is to define TVaR and average value at risk as the same measure.^[1] Under some formulations, it is only equivalent to expected shortfall when the underlying distribution function is continuous at ${\displaystyle \operatorname {VaR} _{\alpha }(X)}$, the value at risk of level $\alpha$.^[2] Under some other settings, TVaR is the conditional expectation of loss above a given value, whereas the expected shortfall is the product of this value with the probability of it occurring.^[3] The former definition may not be a coherent risk measure in general, however it is coherent if the underlying distribution is continuous.^[4] The latter definition is a coherent risk measure.^[3] 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

The canonical tail value at risk is the left-tail (large negative values) in some disciplines and the right-tail (large positive values) in other, such as actuarial science. This is usually due to the differing conventions of treating losses as large negative or positive values. Using the negative value convention, Artzner and others define the tail value at risk as:

Given a random variable $X$ which is the payoff of a portfolio at some future time and given a parameter $0<\alpha <1$ then the tail value at risk is defined by^[5][6][7][8]

${\displaystyle \operatorname {TVaR} _{\alpha }(X)=\operatorname {E} [-X|X\leq -\operatorname {VaR} _{\alpha }(X)]=\operatorname {E} [-X|X\leq x^{\alpha }],}$

where ${\displaystyle x^{\alpha }}$ is the upper $\alpha$-quantile given by ${\displaystyle x^{\alpha }=\inf\{x\in \mathbb {R} :\Pr(X\leq x)>\alpha \}}$. Typically the payoff random variable $X$ is in some L^p-space where $p\geq 1$ to guarantee the existence of the expectation.

References

1. Bargès; Cossette, Marceau (2009). "TVaR-based capital allocation with copulas". Insurance: Mathematics and Economics. 45: 348–361. doi:10.1016/j.insmatheco.2009.08.002. Retrieved 20 July 2012.
2. "Average Value at Risk" (pdf). Retrieved February 2, 2011.
3. Sweeting, Paul (2011). "15.4 Risk Measures". Financial Enterprise Risk Management. International Series on Actuarial Science. Cambridge University Press. pp. 397–401. ISBN 978-0-521-11164-5. LCCN 2011025050.
4. Acerbi, Carlo; Tasche, Dirk (2002). "On the coherence of Expected Shortfall" (pdf). Journal of Banking and Finance. 26 (7): 1487–1503. doi:10.1016/s0378-4266(02)00283-2. Retrieved April 25, 2012.
5. Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc; Heath, David (1999). "Coherent Measures of Risk" (pdf). Mathematical Finance. 9 (3): 203–228. doi:10.1111/1467-9965.00068. Retrieved February 3, 2011.
6. Landsman, Zinoviy; Valdez, Emiliano (February 2004). "Tail Conditional Expectations for Exponential Dispersion Models" (pdf). Retrieved February 3, 2011.
7. Landsman, Zinoviy; Makov, Udi; Shushi, Tomer (July 2013). "Tail Conditional Expectations for Generalized Skew - Elliptical distributions" (pdf). Retrieved June 30, 2013.
8. Valdez, Emiliano (May 2004). "The Iterated Tail Conditional Expectation for the Log-Elliptical Loss Process" (pdf). Retrieved February 3, 2010.