In financial mathematics, 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.
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.
[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.
[4] The latter definition is a coherent risk measure.
The TVaR is a measure of the expectation only in the tail of the distribution.
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
which is the payoff of a portfolio at some future time and given a parameter
then the tail value at risk is defined by[5][6][7][8]
Typically the payoff random variable
Closed-form formulas exist for calculating TVaR when the payoff of a portfolio
For engineering or actuarial applications it is more common to consider the distribution of losses
, in this case the right-tail TVaR is considered (typically for
follows normal distribution, the right-tail TVaR is equal to[10]
follows generalized Student's t-distribution, the right-tail TVaR is equal to[10]
follows Laplace distribution, the right-tail TVaR is equal to[10]
follows logistic distribution, the right-tail TVaR is equal to[10]
is the upper incomplete gamma function.
is the upper incomplete gamma function,
follows GEV, then the right-tail TVaR is equal to
is the lower incomplete gamma function,
follows the Burr type XII distribution with the p.d.f.
follows lognormal distribution, i.e. the random variable
follows log-logistic distribution, i.e. the random variable
is the regularized incomplete beta function,
As the incomplete beta function is defined only for positive arguments, for a more generic case the left-tail TVaR can be expressed with the hypergeometric function:[13]
follows log-Laplace distribution, i.e. the random variable
follows log-GHS distribution, i.e. the random variable