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. Under some formulations, it is only equivalent to expected shortfall when the underlying distribution  function is continuous at formula_1, the value at risk of level formula_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. 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 formula_3 which is the payoff of a portfolio at some future time and given a parameter formula_4 then the tail value at risk is defined by where formula_6 is the upper  formula_2-quantile given by formula_8. Typically the payoff random variable formula_3 is in some L-space where formula_10 to guarantee the existence of the expectation. The typical values for formula_2 are 5% and 1%. Formulas for continuous probability distributions. Closed-form formulas exist for calculating TVaR when the payoff of a portfolio formula_3 or a corresponding loss formula_13 follows a specific continuous distribution. If formula_3 follows some probability distribution with the probability density function (p.d.f.) formula_15 and the cumulative distribution function (c.d.f.) formula_16, the left-tail TVaR can be represented as formula_17 For engineering or actuarial applications it is more  common to consider the distribution of losses formula_18, in this case the right-tail TVaR is considered (typically for formula_2 95% or 99%): formula_20. Since some formulas below were derived for the left-tail case and some for the right-tail case, the following reconciliations can be useful: formula_21 and formula_22. Normal distribution. If the payoff of a portfolio formula_3 follows normal (Gaussian) distribution with the p.d.f. formula_24 then the left-tail TVaR is equal to formula_25, where formula_26 is the standard normal p.d.f., formula_27 is the standard normal c.d.f., so formula_28 is the standard normal quantile. If the loss  of a portfolio formula_29 follows normal distribution, the right-tail TVaR is equal to formula_30. Generalized Student's t-distribution. If the payoff of a portfolio formula_3 follows generalized Student's t-distribution with the p.d.f. formula_32 then the left-tail TVaR is equal to formula_33, where formula_34 is the standard t-distribution p.d.f., formula_35 is the standard t-distribution c.d.f., so formula_36 is the standard t-distribution quantile. If the loss of a portfolio formula_29 follows generalized Student's t-distribution, the right-tail TVaR is equal to formula_38. Laplace distribution. If the payoff of a portfolio formula_3 follows  Laplace distribution with the p.d.f. formula_40 and the c.d.f. formula_41 then the left-tail TVaR is equal to formula_42 for formula_43. If the loss of a portfolio formula_29 follows Laplace distribution, the right-tail TVaR is equal to formula_45. Logistic distribution. If the payoff of a portfolio formula_3 follows logistic distribution with the p.d.f. formula_47 and the c.d.f. formula_48 then the left-tail TVaR is equal to formula_49. If the loss of a portfolio formula_29 follows logistic distribution, the right-tail TVaR is equal to formula_51. Exponential distribution. If the loss of a portfolio formula_29 follows exponential distribution with the  p.d.f. formula_53 and the c.d.f. formula_54 then the right-tail TVaR is equal to formula_55. Pareto distribution. If the loss of a portfolio formula_29 follows Pareto distribution with the p.d.f. formula_57 and the c.d.f. formula_58 then the right-tail TVaR is equal to formula_59. Generalized Pareto distribution (GPD). If the loss of a portfolio formula_29 follows GPD with the p.d.f. formula_61 and the c.d.f. formula_62 then the right-tail TVaR is equal to formula_63 and the VaR is equal to formula_64. Weibull distribution. If the loss of a portfolio formula_29 follows Weibull distribution with the p.d.f. formula_66 and the c.d.f . formula_67 then the right-tail TVaR is equal to formula_68, where formula_69 is the upper incomplete gamma function. Generalized extreme value distribution (GEV). If the payoff of a portfolio formula_3 follows GEV with the p.d.f. formula_71 and the c.d.f. formula_72 then the left-tail TVaR is equal to formula_73 and the VaR is equal to formula_74, where formula_69 is the upper incomplete gamma function, formula_76 is the logarithmic integral function. If the loss of a portfolio formula_29 follows GEV, then the right-tail TVaR is equal to formula_78, where formula_79 is the lower incomplete gamma function, formula_80 is the Euler -Mascheroni constant. Generalized hyperbolic secant (GHS) distribution. If the payoff of a portfolio formula_3 follows GHS distribution with the p.d.f. formula_82and the c.d.f. formula_83 then the left-tail TVaR is equal to formula_84, where formula_85 is the Spence's function, formula_86 is the imaginary unit. Johnson's SU-distribution. If the payoff of a portfolio formula_3 follows Johnson's SU-distribution with the c.d.f. formula_88 then the left-tail TVaR is equal to formula_89, where formula_90 is the c.d.f. of the standard normal distribution. Burr type XII distribution. If the payoff of a portfolio formula_3 follows the Burr type XII  distribution with the p.d.f. formula_92 and the c.d.f. formula_93, the left-tail TVaR is equal to formula_94, where formula_95 is the hypergeometric function. Alternatively, formula_96. Dagum distribution. If the payoff of a portfolio formula_3 follows the Dagum distribution with the p.d.f. formula_98 and the c.d.f. formula_99, the left-tail TVaR is equal to formula_100, where formula_95 is the hypergeometric function. Lognormal distribution. If the payoff of a portfolio formula_3 follows lognormal distribution, i.e. the random variable formula_103 follows normal distribution with the p.d.f. formula_24, then the left-tail TVaR is equal to formula_105, where formula_27 is the standard normal c.d.f ., so formula_28 is the standard normal quantile. Log-logistic distribution. If the payoff of a portfolio formula_3 follows log-logistic distribution, i.e. the random variable formula_103 follows logistic distribution with the p.d.f. formula_47, then the left-tail TVaR is equal to formula_111, where formula_112 is the regularized incomplete beta function, formula_113. 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: formula_114. If the loss of a portfolio formula_29 follows log-logistic distribution with p.d.f. formula_116 and c.d.f. formula_117, then the right-tail  TVaR is equal to formula_118, where formula_119 is the incomplete beta function. Log-Laplace distribution. If the payoff of a portfolio formula_3 follows log-Laplace distribution, i.e. the random variable formula_103 follows Laplace distribution the p.d.f. formula_40, then the left-tail TVaR is equal to formula_123. Log-generalized hyperbolic secant (log-GHS) distribution. If the payoff of a portfolio formula_3 follows log-GHS distribution, i.e. the random variable formula_103 follows GHS distribution with the p.d.f. formula_82, then the left-tail TVaR is equal to formula_127, where formula_95 is the hypergeometric function.