Tail value at risk

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. Under some formulations, it is only equivalent to expected shortfall when the underlying distribution function is continuous at VaR α ⁡ ( X ) , the value at risk of level α . 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. The former definition may not be a coherent risk measure in general, however it is coherent if the underlying distribution is continuous. The latter definition is a coherent risk measure. 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 < α < 1 then the tail value at risk is defined by

where x α is the upper α -quantile given by x α = inf { x ∈ R : Pr ( X ≤ x ) > α } . Typically the payoff random variable X is in some L^p-space where p ≥ 1 to guarantee the existence of the expectation.