Tail value at risk
Tail value at risk, also known as tail conditional expectation or conditional tail expectation, 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. Under some formulations, it is only equivalent to expected shortfall when the underlying distribution function is continuous at, the value at risk of level. 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 in some disciplines and the right-tail 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 which is the payoff of a portfolio at some future time and given a parameter then the tail value at risk is defined by
where is the upper -quantile given by. Typically the payoff random variable is in some Lp-space where to guarantee the existence of the expectation. The typical values for are 5% and 1%.
Formulas for continuous probability distributions
Closed-form formulas exist for calculating TVaR when the payoff of a portfolio or a corresponding loss follows a specific continuous distribution. If follows some probability distribution with the probability density function and the cumulative distribution function , the left-tail TVaR can be represented asFor engineering or actuarial applications it is more common to consider the distribution of losses, in this case the right-tail TVaR is considered :
Since some formulas below were derived for the left-tail case and some for the right-tail case, the following reconciliations can be useful:
and.
Normal distribution
If the payoff of a portfolio follows normal distribution with the p.d.f. then the left-tail TVaR is equal to, where is the standard normal p.d.f., is the standard normal c.d.f., so is the standard normal quantile.If the loss of a portfolio follows normal distribution, the right-tail TVaR is equal to.
Generalized Student's t-distribution
If the payoff of a portfolio follows generalized Student's t-distribution with the p.d.f. then the left-tail TVaR is equal to, where is the standard t-distribution p.d.f., is the standard t-distribution c.d.f., so is the standard t-distribution quantile.If the loss of a portfolio follows generalized Student's t-distribution, the right-tail TVaR is equal to.
Laplace distribution
If the payoff of a portfolio follows Laplace distribution with the p.d.f. and the c.d.f. then the left-tail TVaR is equal to for.If the loss of a portfolio follows Laplace distribution, the right-tail TVaR is equal to.
Logistic distribution
If the payoff of a portfolio follows logistic distribution with the p.d.f. and the c.d.f. then the left-tail TVaR is equal to.If the loss of a portfolio follows logistic distribution, the right-tail TVaR is equal to.
Exponential distribution
If the loss of a portfolio follows exponential distribution with the p.d.f. and the c.d.f. then the right-tail TVaR is equal to.Pareto distribution
If the loss of a portfolio follows Pareto distribution with the p.d.f. and the c.d.f. then the right-tail TVaR is equal to.Generalized Pareto distribution (GPD)
If the loss of a portfolio follows GPD with the p.d.f. and the c.d.f. then the right-tail TVaR is equal to and the VaR is equal to.Weibull distribution
If the loss of a portfolio follows Weibull distribution with the p.d.f. and the c.d.f. then the right-tail TVaR is equal to, where is the upper incomplete gamma function.Generalized extreme value distribution (GEV)
If the payoff of a portfolio follows GEV with the p.d.f. and the c.d.f. then the left-tail TVaR is equal to and the VaR is equal to, where is the upper incomplete gamma function, is the logarithmic integral function.If the loss of a portfolio follows GEV, then the right-tail TVaR is equal to, where is the lower incomplete gamma function, is the Euler-Mascheroni constant.
Generalized hyperbolic secant (GHS) distribution
If the payoff of a portfolio follows GHS distribution with the p.d.f. and the c.d.f. then the left-tail TVaR is equal to, where is the Spence's function, is the imaginary unit.Johnson's SU-distribution
If the payoff of a portfolio follows Johnson's SU-distribution with the c.d.f. then the left-tail TVaR is equal to, where is the c.d.f. of the standard normal distribution.Burr type XII distribution
If the payoff of a portfolio follows the Burr type XII distribution with the p.d.f. and the c.d.f., the left-tail TVaR is equal to, where is the hypergeometric function. Alternatively,.Dagum distribution
If the payoff of a portfolio follows the Dagum distribution with the p.d.f. and the c.d.f., the left-tail TVaR is equal to, where is the hypergeometric function.Lognormal distribution
If the payoff of a portfolio follows lognormal distribution, i.e. the random variable follows normal distribution with the p.d.f., then the left-tail TVaR is equal to, where is the standard normal c.d.f., so is the standard normal quantile.Log-logistic distribution
If the payoff of a portfolio follows log-logistic distribution, i.e. the random variable follows logistic distribution with the p.d.f., then the left-tail TVaR is equal to, where 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:.
If the loss of a portfolio follows log-logistic distribution with p.d.f. and c.d.f., then the right-tail TVaR is equal to, where is the incomplete beta function.