{{Afd-merge to|Expected shortfall|discussion=Tail value at risk|date=18 May 2026}} {{Short description|Measure giving the average loss beyond a specified Value-at-Risk level}} {{Redirect-synonym|TVAR|Time variance}}
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.
==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.<ref name=Bar/> Under some formulations, it is only equivalent to expected shortfall when the underlying distribution function is continuous at <math>\operatorname{VaR}_{\alpha}(X)</math>, the value at risk of level <math>\alpha</math>.<ref name=web1/> 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.<ref name = "Sweeting"/> The former definition may not be a coherent risk measure in general, however it is coherent if the underlying distribution is continuous.<ref name=Acerbi/> The latter definition is a coherent risk measure.<ref name = "Sweeting" /> 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 <math>X</math> which is the payoff of a portfolio at some future time and given a parameter <math>0 < \alpha < 1</math> then the tail value at risk is defined by<ref name=Artzner/><ref name=Landsman/><ref name=Landsman2/><ref name=Valdez/> <math display="block">\operatorname{TVaR}_{\alpha}(X) = \operatorname{E} [-X|X \leq -\operatorname{VaR}_{\alpha}(X)] = \operatorname{E} [-X | X \leq x^{\alpha}] ,</math> where <math>x^{\alpha}</math> is the upper <math>\alpha</math>-quantile given by <math>x^{\alpha} = \inf\{x \in \mathbb{R}: \Pr(X \leq x) > \alpha\}</math>. Typically the payoff random variable <math>X</math> is in some L<sup>p</sup>-space where <math>p \geq 1</math> to guarantee the existence of the expectation. The typical values for <math>\alpha</math> are 5% and 1%.
== Formulas for continuous probability distributions ==
Closed-form formulas exist for calculating TVaR when the payoff of a portfolio <math>X</math> or a corresponding loss <math>L = -X</math> follows a specific continuous distribution. If <math>X</math> follows some probability distribution with the probability density function (p.d.f.) <math>f</math> and the cumulative distribution function (c.d.f.) <math>F</math>, the left-tail TVaR can be represented as
<math display="block">\operatorname{TVaR}_{\alpha}(X) = \operatorname{E} [-X|X \leq -\operatorname{VaR}_{\alpha}(X)] = \frac{1}{\alpha} \int_0^\alpha \operatorname{VaR}_\gamma(X)d\gamma = -\frac{1}{\alpha}\int_{-\infty}^{F^{-1}(\alpha)}xf(x)dx.</math>
For engineering or actuarial applications it is more common to consider the distribution of losses <math>L=-X</math>, in this case the right-tail TVaR is considered (typically for <math>\alpha</math> 95% or 99%):
<math display="block">\operatorname{TVaR}^\text{right}_\alpha(L) = E[L\mid L \geq \operatorname{VaR}_{\alpha}(L)] = \frac{1}{1-\alpha} \int^1_\alpha \operatorname{VaR}_\gamma(L)d\gamma = \frac{1}{1-\alpha}\int^{+\infty}_{F^{-1}(\alpha)}yf(y)dy.</math>
Since some formulas below were derived for the left-tail case and some for the right-tail case, the following reconciliations can be useful:
<math display="block">\operatorname{TVaR}_{\alpha}(X) = -\frac{1}{\alpha}E[X]+\frac{1-\alpha}{\alpha}\operatorname{TVaR}^\text{right}_\alpha(L)</math> and <math display="block">\operatorname{TVaR}^\text{right}_\alpha(L) = \frac{1}{1-\alpha}E[L]+\frac{\alpha}{1-\alpha}\operatorname{TVaR}_{\alpha}(X).</math>
=== Normal distribution === If the payoff of a portfolio <math>X</math> follows normal (Gaussian) distribution with the p.d.f. <math display="block">f(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}</math> then the left-tail TVaR is equal to <math display="block">\operatorname{TVaR}_{\alpha}(X) = -\mu+\sigma\frac{\phi(\Phi^{-1}(\alpha))}{\alpha},</math> where <math display="inline">\phi(x) = \frac{1}{\sqrt{2\pi}}e^{-{x^2}/{2}}</math> is the standard normal p.d.f., <math>\Phi(x)</math> is the standard normal c.d.f., so <math>\Phi^{-1}(\alpha)</math> is the standard normal quantile.<ref name=":0">{{Cite journal|last=Khokhlov|first=Valentyn|date=2016|title=Conditional Value-at-Risk for Elliptical Distributions|journal=Evropský časopis Ekonomiky a Managementu|volume=2|issue=6|pages=70–79}}</ref>
If the loss of a portfolio <math>L</math> follows normal distribution, the right-tail TVaR is equal to<ref name=":1">{{cite arXiv|last1=Norton|first1=Matthew|last2=Khokhlov|first2=Valentyn|last3=Uryasev|first3=Stan|date=2018-11-27|title=Calculating CVaR and bPOE for Common Probability Distributions With Application to Portfolio Optimization and Density Estimation|eprint=1811.11301|class=q-fin.RM}}</ref> <math display="block">\operatorname{TVaR}^\text{right}_\alpha(L) = \mu+\sigma\frac{\phi(\Phi^{-1}(\alpha))}{1-\alpha}.</math>
=== Generalized Student's t-distribution === If the payoff of a portfolio <math>X</math> follows generalized Student's t-distribution with the p.d.f. <math display="block">f(x) = \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)\sqrt{\pi\nu}\sigma}\left(1+\frac{1}{\nu}\left(\frac{x-\mu}{\sigma}\right)^2\right)^{-\frac{\nu+1}{2}}</math> then the left-tail TVaR is equal to <math display="block">\operatorname{TVaR}_{\alpha}(X) = -\mu+\sigma\frac{\nu+(\Tau^{-1}(\alpha))^2}{\nu-1}\frac{\tau(\Tau^{-1}(\alpha))}{\alpha},</math> where <math display="block">\tau(x)=\frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)\sqrt{\pi\nu}}\left(1+\frac{x^2}{\nu}\right)^{-\frac{\nu+1}{2}}</math> is the standard t-distribution p.d.f., <math>\Tau(x)</math> is the standard t-distribution c.d.f., so <math>\Tau^{-1}(\alpha)</math> is the standard t-distribution quantile.<ref name=":0" />
If the loss of a portfolio <math>L</math> follows generalized Student's t-distribution, the right-tail TVaR is equal to<ref name=":1" /> <math display="block">\operatorname{TVaR}^\text{right}_\alpha(L) = \mu+\sigma\frac{\nu+(\Tau^{-1}(\alpha))^2}{\nu-1}\frac{\tau(\Tau^{-1}(\alpha))}{1-\alpha}.</math>
=== Laplace distribution === If the payoff of a portfolio <math>X</math> follows Laplace distribution with the p.d.f. <math display="block">f(x) = \frac{1}{2b}e^{-\frac{|x-\mu|}{b}}</math> and the c.d.f. <math display="block">F(x) = \begin{cases}1 - \frac{1}{2} e^{-\frac{x-\mu}{b}} & \text{if }x \geq \mu,\\ \frac{1}{2} e^\frac{x-\mu}{b} & \text{if }x < \mu.\end{cases}</math> then the left-tail TVaR is equal to <math>\operatorname{TVaR}_{\alpha}(X) = -\mu+b(1-\ln2\alpha)</math> for <math>\alpha \le 0.5</math>.<ref name=":0" />
If the loss of a portfolio <math>L</math> follows Laplace distribution, the right-tail TVaR is equal to<ref name=":1" /> <math display="block">\operatorname{TVaR}^\text{right}_\alpha(L) = \begin{cases} \mu + b \frac{\alpha}{1-\alpha} (1-\ln2\alpha) & \text{if }\alpha < 0.5,\\[1ex] \mu + b[1 - \ln(2(1-\alpha))] & \text{if }\alpha \ge 0.5. \end{cases}</math>
=== Logistic distribution === If the payoff of a portfolio <math>X</math> follows logistic distribution with the p.d.f. <math display="block">f(x) = \frac{1}{s}e^{-\frac{x-\mu}{s}}\left(1+e^{-\frac{x-\mu}{s}}\right)^{-2}</math> and the c.d.f. <math display="block">F(x) = \left(1+e^{-\frac{x-\mu}{s}}\right)^{-1}</math> then the left-tail TVaR is equal to<ref name=":0" /> <math display="block">\operatorname{TVaR}_{\alpha}(X) = -\mu+s\ln\frac{(1-\alpha)^{1-\frac{1}{\alpha}}}{\alpha}.</math>
If the loss of a portfolio <math>L</math> follows logistic distribution, the right-tail TVaR is equal to<ref name=":1" /> <math display="block">\operatorname{TVaR}^\text{right}_\alpha(L) = \mu + s\frac{-\alpha\ln\alpha-(1-\alpha)\ln(1-\alpha)}{1-\alpha}.</math>
=== Exponential distribution === If the loss of a portfolio <math>L</math> follows exponential distribution with the p.d.f. <math display="block">f(x) = \begin{cases}\lambda e^{-\lambda x} & \text{if }x \geq 0,\\ 0 & \text{if }x < 0.\end{cases}</math> and the c.d.f. <math display="block">F(x) = \begin{cases}1 - e^{-\lambda x} & \text{if }x \geq 0,\\ 0 & \text{if }x < 0.\end{cases}</math> then the right-tail TVaR is equal to<ref name=":1" /> <math display="block">\operatorname{TVaR}^\text{right}_\alpha(L) = \frac{-\ln(1-\alpha)+1}{\lambda}.</math>
=== Pareto distribution === If the loss of a portfolio <math>L</math> follows Pareto distribution with the p.d.f. <math display="block">f(x) = \begin{cases}\frac{a x_m^a}{x^{a+1}} & \text{if }x \geq x_m,\\ 0 & \text{if }x < x_m.\end{cases}</math> and the c.d.f. <math display="block">F(x) = \begin{cases}1 - (x_m/x)^a & \text{if }x \geq x_m,\\ 0 & \text{if }x < x_m.\end{cases}</math> then the right-tail TVaR is equal to<ref name=":1" /> <math display="block">\operatorname{TVaR}^\text{right}_\alpha(L) = \frac{x_m a}{(1-\alpha)^{1/a}(a-1)}.</math>
=== Generalized Pareto distribution (GPD) === If the loss of a portfolio <math>L</math> follows GPD with the p.d.f. <math display="block">f(x) = \frac{1}{s} \left( 1+\frac{\xi (x-\mu)}{s} \right)^{\left(-\frac{1}{\xi}-1\right)}</math> and the c.d.f. <math display="block">F(x) = \begin{cases}1 - \left(1+\frac{\xi(x-\mu)}{s}\right)^{-\frac{1}{\xi}} & \text{if }\xi \ne 0,\\ 1-\exp \left( -\frac{x-\mu}{s} \right) & \text{if }\xi = 0.\end{cases}</math> then the right-tail TVaR is equal to <math display="block">\operatorname{TVaR}^\text{right}_\alpha(L) = \begin{cases}\mu + s \left[ \frac{(1-\alpha)^{-\xi}}{1-\xi}+\frac{(1-\alpha)^{-\xi}-1}{\xi} \right] & \text{if }\xi \ne 0,\\ \mu + s[1 - \ln(1-\alpha)] & \text{if }\xi = 0.\end{cases}</math> and the VaR is equal to<ref name=":1" /> <math display="block">\mathrm{VaR}_\alpha(L) = \begin{cases} \mu + s \frac{(1-\alpha)^{-\xi}-1}{\xi} & \text{if }\xi \ne 0,\\ \mu - s \ln(1-\alpha) & \text{if }\xi = 0. \end{cases}</math>
=== Weibull distribution === If the loss of a portfolio <math>L</math> follows Weibull distribution with the p.d.f. <math display="block">f(x) = \begin{cases}\frac{k}{\lambda} \left(\frac{x}{\lambda}\right)^{k-1} e^{-(x/\lambda)^k} & \text{if }x \geq 0,\\ 0 & \text{if }x < 0.\end{cases}</math> and the c.d.f. <math display="block">F(x) = \begin{cases}1 - e^{-(x/\lambda)^k} & \text{if }x \geq 0,\\ 0 & \text{if }x < 0.\end{cases}</math> then the right-tail TVaR is equal to <math display="block">\operatorname{TVaR}^\text{right}_\alpha(L) = \frac{\lambda}{1-\alpha} \Gamma\left(1+\frac{1}{k},-\ln(1-\alpha)\right),</math> where <math>\Gamma(s,x)</math> is the upper incomplete gamma function.<ref name=":1" />
=== Generalized extreme value distribution (GEV) === If the payoff of a portfolio <math>X</math> follows GEV with the p.d.f. <math display="block">f(x) = \begin{cases} \frac{1}{\sigma} \left( 1+\xi \frac{ x-\mu}{\sigma} \right)^{-\frac{1}{\xi}-1} \exp\left[-\left( 1+\xi \frac{x-\mu}{\sigma} \right)^{-\frac{1}{\xi}}\right] & \text{if } \xi \ne 0,\\ \frac{1}{\sigma}e^{-\frac{x-\mu}{\sigma}}e^{-e^{-\frac{x-\mu}{\sigma}}} & \text{if } \xi = 0. \end{cases}</math> and the c.d.f. <math display="block">F(x) = \begin{cases} \exp\left(-\left(1+\xi\frac{x-\mu}{\sigma}\right)^{-\frac{1}{\xi}}\right) & \text{if } \xi \ne 0,\\ \exp\left(-e^{-\frac{x-\mu}{\sigma}}\right) & \text{if }\xi = 0. \end{cases}</math> then the left-tail TVaR is equal to <math display="block">\operatorname{TVaR}_{\alpha}(X) = \begin{cases}-\mu - \frac{\sigma}{\alpha \xi} \left[ \Gamma(1-\xi,-\ln\alpha)-\alpha \right] & \text{if }\xi \ne 0,\\ -\mu - \frac{\sigma}{\alpha} \left[ \text{li}(\alpha) - \alpha \ln(-\ln \alpha) \right] & \text{if }\xi = 0.\end{cases}</math> and the VaR is equal to <math display="block">\mathrm{VaR}_\alpha(X) = \begin{cases}-\mu - \frac{\sigma}{\xi} \left[(-\ln \alpha)^{-\xi}-1 \right] & \text{if }\xi \ne 0,\\ -\mu + \sigma \ln(-\ln\alpha) & \text{if }\xi = 0.\end{cases}</math> where <math>\Gamma(s,x)</math> is the upper incomplete gamma function, <math>\text{li}(x)=\int \frac{dx}{\ln x}</math> is the logarithmic integral function.<ref name=":3">{{Cite journal|ssrn=3200629|title=Conditional Value-at-Risk for Uncommon Distributions|last=Khokhlov|first=Valentyn|date=2018-06-21|website=SSRN}}</ref>
If the loss of a portfolio <math>L</math> follows GEV, then the right-tail TVaR is equal to <math display="block">\operatorname{TVaR}_{\alpha}(X) = \begin{cases}\mu + \frac{\sigma}{(1-\alpha) \xi} \left[ \gamma(1-\xi,-\ln\alpha)-(1-\alpha) \right] & \text{if }\xi \ne 0,\\ \mu + \frac{\sigma}{1-\alpha} \left[y - \text{li}(\alpha) + \alpha \ln(-\ln \alpha) \right] & \text{if }\xi = 0.\end{cases}</math> where <math>\gamma(s,x)</math> is the lower incomplete gamma function, <math>y</math> is the Euler-Mascheroni constant.<ref name=":1" />
=== Generalized hyperbolic secant (GHS) distribution === If the payoff of a portfolio <math>X</math> follows GHS distribution with the p.d.f. <math display="block">f(x) = \frac{1}{2 \sigma} \operatorname{sech}\left(\frac{\pi}{2}\frac{x-\mu}{\sigma}\right)</math>and the c.d.f. <math display="block">F(x) = \frac{2}{\pi}\arctan\left[\exp\left(\frac{\pi}{2}\frac{x-\mu}{\sigma}\right)\right]</math> then the left-tail TVaR is equal to <math display="block">\operatorname{TVaR}_{\alpha}(X) = -\mu - \frac{2\sigma}{\pi} \ln\left( \tan \frac{\pi\alpha}{2} \right) - \frac{2\sigma}{\pi^2\alpha}i\left[\text{Li}_2\left(-i\tan\frac{\pi\alpha}{2}\right)-\text{Li}_2\left(i\tan\frac{\pi\alpha}{2}\right)\right],</math> where <math>\text{Li}_2</math> is the dilogarithm and <math>i=\sqrt{-1}</math> is the imaginary unit.<ref name=":3" />
=== Johnson's SU-distribution === If the payoff of a portfolio <math>X</math> follows Johnson's SU-distribution with the c.d.f. <math display="block">F(x) = \Phi\left[\gamma+\delta\sinh^{-1}\left(\frac{x-\xi}{\lambda}\right)\right]</math> then the left-tail TVaR is equal to <math display="block">\operatorname{TVaR}_{\alpha}(X) = -\xi - \frac{\lambda}{2\alpha} \left[ \exp\left(\frac{1-2\gamma\delta}{2\delta^2}\right) \Phi\left(\Phi^{-1}(\alpha)-\frac{1}{\delta}\right) - \exp\left(\frac{1+2\gamma\delta}{2\delta^2}\right)\Phi\left(\Phi^{-1}(\alpha)+\frac{1}{\delta}\right) \right],</math> where <math>\Phi</math> is the c.d.f. of the standard normal distribution.<ref>{{Cite journal|ssrn=1855986|title=Moment-Based CVaR Estimation: Quasi-Closed Formulas|last=Stucchi|first=Patrizia|date=2011-05-31|website=SSRN}}</ref>
=== Burr type XII distribution === If the payoff of a portfolio <math>X</math> follows the Burr type XII distribution with the p.d.f. <math display="block">f(x) = \frac{ck}{\beta}\left(\frac{x-\gamma}{\beta}\right)^{c-1}\left[1+\left(\frac{x-\gamma}{\beta}\right)^c\right]^{-k-1}</math> and the c.d.f. <math display="block">F(x) = 1-\left[1+\left(\frac{x-\gamma}{\beta}\right)^c\right]^{-k},</math> the left-tail TVaR is equal to <math display="block">\operatorname{TVaR}_{\alpha}(X) = -\gamma -\frac{\beta}{\alpha}\left( (1-\alpha)^{-1/k}-1 \right)^{1/c} \left[ \alpha -1+{_2F_1}\left(\frac{1}{c},k;1+\frac{1}{c};1-(1-\alpha)^{-1/k}\right) \right],</math> where <math>_2F_1</math> is the hypergeometric function. Alternatively,<ref name=":3" /> <math display="block">\operatorname{TVaR}_{\alpha}(X) = -\gamma -\frac{\beta}{\alpha}\frac{ck}{c+1}\left( (1-\alpha)^{-1/k}-1 \right)^{1+\frac{1}{c}} {_2F_1}\left(1+\frac{1}{c},k+1;2+\frac{1}{c};1-(1-\alpha)^{-1/k}\right). </math>
=== Dagum distribution === If the payoff of a portfolio <math>X</math> follows the Dagum distribution with the p.d.f. <math display="block">f(x) = \frac{ck}{\beta}\left(\frac{x-\gamma}{\beta}\right)^{ck-1}\left[1+\left(\frac{x-\gamma}{\beta}\right)^c\right]^{-k-1}</math> and the c.d.f. <math display="block">F(x) = \left[1+\left(\frac{x-\gamma}{\beta}\right)^{-c}\right]^{-k},</math> the left-tail TVaR is equal to <math display="block">\operatorname{TVaR}_{\alpha}(X) = -\gamma -\frac{\beta}{\alpha}\frac{ck}{ck+1}\left( \alpha^{-1/k}-1 \right)^{-k-\frac{1}{c}} {_2F_1}\left(k+1,k+\frac{1}{c};k+1+\frac{1}{c};-\frac{1}{\alpha^{-1/k}-1}\right), </math> where <math>_2F_1</math> is the hypergeometric function.<ref name=":3" />
=== Lognormal distribution === If the payoff of a portfolio <math>X</math> follows lognormal distribution, i.e. the random variable <math>\ln(1+X)</math> follows normal distribution with the p.d.f. <math display="block">f(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}},</math> then the left-tail TVaR is equal to <math display="block">\operatorname{TVaR}_{\alpha}(X) = 1-\exp\left(\mu+\frac{\sigma^2}{2}\right) \frac{\Phi(\Phi^{-1}(\alpha)-\sigma)}{\alpha},</math> where <math>\Phi(x)</math> is the standard normal c.d.f., so <math>\Phi^{-1}(\alpha)</math> is the standard normal quantile.<ref name=":2">{{Cite journal|ssrn=3197929|title=Conditional Value-at-Risk for Log-Distributions|last=Khokhlov|first=Valentyn|date=2018-06-17|website=SSRN}}</ref>
=== Log-logistic distribution === If the payoff of a portfolio <math>X</math> follows log-logistic distribution, i.e. the random variable <math>\ln(1+X)</math> follows logistic distribution with the p.d.f. <math display="block">f(x) = \frac{1}{s}e^{-\frac{x-\mu}{s}}\left(1+e^{-\frac{x-\mu}{s}}\right)^{-2},</math> then the left-tail TVaR is equal to <math display="block">\operatorname{TVaR}_{\alpha}(X) = 1-\frac{e^\mu}{\alpha}I_\alpha(1+s,1-s)\frac{\pi s}{\sin\pi s},</math> where <math>I_\alpha</math> is the regularized incomplete beta function, <math>I_\alpha(a,b)=\frac{\Beta_\alpha(a,b)}{\Beta(a,b)}</math>.
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:<ref name=":2" /> <math display="block">\operatorname{TVaR}_{\alpha}(X) = 1-\frac{e^\mu \alpha^s}{s+1} {_2F_1}(s,s+1;s+2;\alpha).</math>
If the loss of a portfolio <math>L</math> follows log-logistic distribution with p.d.f. <math display="block">f(x) = \frac{\frac{b}{a}(x/a)^{b-1}}{(1+(x/a)^b)^2}</math> and c.d.f. <math display="block">F(x) = \frac{1}{1+(x/a)^{-b}},</math> then the right-tail TVaR is equal to <math display="block">\operatorname{TVaR}^\text{right}_\alpha(L) = \frac{a}{1-\alpha}\left[\frac{\pi}{b}\csc \left(\frac{\pi}{b}\right)-\Beta_\alpha\left(\frac{1}{b}+1,1-\frac{1}{b}\right)\right],</math> where <math>B_\alpha</math> is the incomplete beta function.<ref name=":1" />
=== Log-Laplace distribution === If the payoff of a portfolio <math>X</math> follows log-Laplace distribution, i.e. the random variable <math>\ln(1+X)</math> follows Laplace distribution the p.d.f. <math display="block">f(x) = \frac{1}{2b}e^{-\frac{|x-\mu|}{b}},</math> then the left-tail TVaR is equal to<ref name=":2" /> <math display="block">\operatorname{TVaR}_{\alpha}(X) = \begin{cases}1 - \frac{e^\mu (2\alpha)^b}{b+1} & \text{if }\alpha \le 0.5,\\ 1 - \frac{e^\mu 2^{-b}}{\alpha(b-1)}\left[(1-\alpha)^{(1-b)}-1\right] & \text{if }\alpha > 0.5.\end{cases}</math>
=== Log-generalized hyperbolic secant (log-GHS) distribution === If the payoff of a portfolio <math>X</math> follows log-GHS distribution, i.e. the random variable <math>\ln(1+X)</math> follows GHS distribution with the p.d.f. <math display="block">f(x) = \frac{1}{2 \sigma} \operatorname{sech}\left(\frac{\pi}{2}\frac{x-\mu}{\sigma}\right),</math> then the left-tail TVaR is equal to <math display="block">\operatorname{TVaR}_{\alpha}(X) = 1-\frac{1}{\alpha(\sigma+{\pi/2})} \left(\tan\frac{\pi \alpha}{2}\exp\frac{\pi \mu}{2\sigma}\right)^{2\sigma/\pi} \tan\frac{\pi \alpha}{2} {_2F_1}\left(1,\frac{1}{2}+\frac{\sigma}{\pi};\frac{3}{2}+\frac{\sigma}{\pi};-\tan\left(\frac{\pi \alpha}{2}\right)^2\right),</math> where <math>_2F_1</math> is the hypergeometric function.<ref name=":2" />
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{{DEFAULTSORT:Tail Value At Risk}} Category:Actuarial science Category:Financial risk modeling Category:Monte Carlo methods in finance