Tail value at risk

If the payoff of a portfolio ${\displaystyle X}$ follows normal (Gaussian) distribution with the p.d.f.

$${\displaystyle f(x)={\frac {1}{{\sqrt {2\pi }}\sigma }}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}$$

then the left-tail TVaR is equal to

$${\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\mu +\sigma {\frac {\phi (\Phi ^{-1}(\alpha ))}{\alpha }},}$$

where ${\textstyle \phi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{x^{2}}/{2}}}$ is the standard normal p.d.f., ${\displaystyle \Phi (x)}$ is the standard normal c.d.f., so ${\displaystyle \Phi ^{-1}(\alpha )}$ is the standard normal quantile.[9] If the loss of a portfolio ${\displaystyle L}$ follows normal distribution, the right-tail TVaR is equal to[10]

$${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)=\mu +\sigma {\frac {\phi (\Phi ^{-1}(\alpha ))}{1-\alpha }}.}$$

If the payoff of a portfolio ${\displaystyle X}$ follows generalized Student's t-distribution with the p.d.f.

$${\displaystyle f(x)={\frac {\Gamma \left({\frac {\nu +1}{2}}\right)}{\Gamma \left({\frac {\nu }{2}}\right){\sqrt {\pi \nu }}\sigma }}\left(1+{\frac {1}{\nu }}\left({\frac {x-\mu }{\sigma }}\right)^{2}\right)^{-{\frac {\nu +1}{2}}}}$$

then the left-tail TVaR is equal to

$${\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\mu +\sigma {\frac {\nu +(\mathrm {T} ^{-1}(\alpha ))^{2}}{\nu -1}}{\frac {\tau (\mathrm {T} ^{-1}(\alpha ))}{\alpha }},}$$

where

$${\displaystyle \tau (x)={\frac {\Gamma \left({\frac {\nu +1}{2}}\right)}{\Gamma \left({\frac {\nu }{2}}\right){\sqrt {\pi \nu }}}}\left(1+{\frac {x^{2}}{\nu }}\right)^{-{\frac {\nu +1}{2}}}}$$

is the standard t-distribution p.d.f., ${\displaystyle \mathrm {T} (x)}$ is the standard t-distribution c.d.f., so ${\displaystyle \mathrm {T} ^{-1}(\alpha )}$ is the standard t-distribution quantile.[9] If the loss of a portfolio ${\displaystyle L}$ follows generalized Student's t-distribution, the right-tail TVaR is equal to[10]

$${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)=\mu +\sigma {\frac {\nu +(\mathrm {T} ^{-1}(\alpha ))^{2}}{\nu -1}}{\frac {\tau (\mathrm {T} ^{-1}(\alpha ))}{1-\alpha }}.}$$

If the payoff of a portfolio ${\displaystyle X}$ follows Laplace distribution with the p.d.f.

$${\displaystyle f(x)={\frac {1}{2b}}e^{-{\frac {|x-\mu |}{b}}}}$$

and the c.d.f.

$${\displaystyle F(x)={\begin{cases}1-{\frac {1}{2}}e^{-{\frac {x-\mu }{b}}}&{\text{if }}x\geq \mu ,\\{\frac {1}{2}}e^{\frac {x-\mu }{b}}&{\text{if }}x<\mu .\end{cases}}}$$

then the left-tail TVaR is equal to ${\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\mu +b(1-\ln 2\alpha )}$ for ${\displaystyle \alpha \leq 0.5}$.[9] If the loss of a portfolio ${\displaystyle L}$ follows Laplace distribution, the right-tail TVaR is equal to[10]

$${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)={\begin{cases}\mu +b{\frac {\alpha }{1-\alpha }}(1-\ln 2\alpha )&{\text{if }}\alpha <0.5,\\[1ex]\mu +b[1-\ln(2(1-\alpha ))]&{\text{if }}\alpha \geq 0.5.\end{cases}}}$$

If the payoff of a portfolio ${\displaystyle X}$ follows logistic distribution with the p.d.f.

$${\displaystyle f(x)={\frac {1}{s}}e^{-{\frac {x-\mu }{s}}}\left(1+e^{-{\frac {x-\mu }{s}}}\right)^{-2}}$$

and the c.d.f.

$${\displaystyle F(x)=\left(1+e^{-{\frac {x-\mu }{s}}}\right)^{-1}}$$

then the left-tail TVaR is equal to[9]

$${\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\mu +s\ln {\frac {(1-\alpha )^{1-{\frac {1}{\alpha }}}}{\alpha }}.}$$

If the loss of a portfolio ${\displaystyle L}$ follows logistic distribution, the right-tail TVaR is equal to[10]

$${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)=\mu +s{\frac {-\alpha \ln \alpha -(1-\alpha )\ln(1-\alpha )}{1-\alpha }}.}$$

If the loss of a portfolio ${\displaystyle L}$ follows exponential distribution with the p.d.f.

$${\displaystyle f(x)={\begin{cases}\lambda e^{-\lambda x}&{\text{if }}x\geq 0,\\0&{\text{if }}x<0.\end{cases}}}$$

and the c.d.f.

$${\displaystyle F(x)={\begin{cases}1-e^{-\lambda x}&{\text{if }}x\geq 0,\\0&{\text{if }}x<0.\end{cases}}}$$

then the right-tail TVaR is equal to[10]

$${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)={\frac {-\ln(1-\alpha )+1}{\lambda }}.}$$

If the loss of a portfolio ${\displaystyle L}$ follows Pareto distribution with the p.d.f.

$${\displaystyle f(x)={\begin{cases}{\frac {ax_{m}^{a}}{x^{a+1}}}&{\text{if }}x\geq x_{m},\\0&{\text{if }}x<x_{m}.\end{cases}}}$$

and the c.d.f.

$${\displaystyle F(x)={\begin{cases}1-(x_{m}/x)^{a}&{\text{if }}x\geq x_{m},\\0&{\text{if }}x<x_{m}.\end{cases}}}$$

then the right-tail TVaR is equal to[10]

$${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)={\frac {x_{m}a}{(1-\alpha )^{1/a}(a-1)}}.}$$

If the loss of a portfolio ${\displaystyle L}$ follows GPD with the p.d.f.

$${\displaystyle f(x)={\frac {1}{s}}\left(1+{\frac {\xi (x-\mu )}{s}}\right)^{\left(-{\frac {1}{\xi }}-1\right)}}$$

and the c.d.f.

$${\displaystyle F(x)={\begin{cases}1-\left(1+{\frac {\xi (x-\mu )}{s}}\right)^{-{\frac {1}{\xi }}}&{\text{if }}\xi \neq 0,\\1-\exp \left(-{\frac {x-\mu }{s}}\right)&{\text{if }}\xi =0.\end{cases}}}$$

then the right-tail TVaR is equal to

$${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)={\begin{cases}\mu +s\left[{\frac {(1-\alpha )^{-\xi }}{1-\xi }}+{\frac {(1-\alpha )^{-\xi }-1}{\xi }}\right]&{\text{if }}\xi \neq 0,\\\mu +s[1-\ln(1-\alpha )]&{\text{if }}\xi =0.\end{cases}}}$$

and the VaR is equal to[10]

$${\displaystyle \mathrm {VaR} _{\alpha }(L)={\begin{cases}\mu +s{\frac {(1-\alpha )^{-\xi }-1}{\xi }}&{\text{if }}\xi \neq 0,\\\mu -s\ln(1-\alpha )&{\text{if }}\xi =0.\end{cases}}}$$

If the loss of a portfolio ${\displaystyle L}$ follows Weibull distribution with the p.d.f.

$${\displaystyle f(x)={\begin{cases}{\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}e^{-(x/\lambda )^{k}}&{\text{if }}x\geq 0,\\0&{\text{if }}x<0.\end{cases}}}$$

and the c.d.f.

$${\displaystyle F(x)={\begin{cases}1-e^{-(x/\lambda )^{k}}&{\text{if }}x\geq 0,\\0&{\text{if }}x<0.\end{cases}}}$$

then the right-tail TVaR is equal to

$${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)={\frac {\lambda }{1-\alpha }}\Gamma \left(1+{\frac {1}{k}},-\ln(1-\alpha )\right),}$$

where ${\displaystyle \Gamma (s,x)}$ is the upper incomplete gamma function.[10]

If the payoff of a portfolio ${\displaystyle X}$ follows GEV with the p.d.f.

$${\displaystyle f(x)={\begin{cases}{\frac {1}{\sigma }}\left(1+\xi {\frac {x-\mu }{\sigma }}\right)^{-{\frac {1}{\xi }}-1}\exp \left[-\left(1+\xi {\frac {x-\mu }{\sigma }}\right)^{-{\frac {1}{\xi }}}\right]&{\text{if }}\xi \neq 0,\\{\frac {1}{\sigma }}e^{-{\frac {x-\mu }{\sigma }}}e^{-e^{-{\frac {x-\mu }{\sigma }}}}&{\text{if }}\xi =0.\end{cases}}}$$

and the c.d.f.

$${\displaystyle F(x)={\begin{cases}\exp \left(-\left(1+\xi {\frac {x-\mu }{\sigma }}\right)^{-{\frac {1}{\xi }}}\right)&{\text{if }}\xi \neq 0,\\\exp \left(-e^{-{\frac {x-\mu }{\sigma }}}\right)&{\text{if }}\xi =0.\end{cases}}}$$

then the left-tail TVaR is equal to

$${\displaystyle \operatorname {TVaR} _{\alpha }(X)={\begin{cases}-\mu -{\frac {\sigma }{\alpha \xi }}\left[\Gamma (1-\xi ,-\ln \alpha )-\alpha \right]&{\text{if }}\xi \neq 0,\\-\mu -{\frac {\sigma }{\alpha }}\left[{\text{li}}(\alpha )-\alpha \ln(-\ln \alpha )\right]&{\text{if }}\xi =0.\end{cases}}}$$

and the VaR is equal to

$${\displaystyle \mathrm {VaR} _{\alpha }(X)={\begin{cases}-\mu -{\frac {\sigma }{\xi }}\left[(-\ln \alpha )^{-\xi }-1\right]&{\text{if }}\xi \neq 0,\\-\mu +\sigma \ln(-\ln \alpha )&{\text{if }}\xi =0.\end{cases}}}$$

where ${\displaystyle \Gamma (s,x)}$ is the upper incomplete gamma function, ${\displaystyle {\text{li}}(x)=\int {\frac {dx}{\ln x}}}$ is the logarithmic integral function.[11] If the loss of a portfolio ${\displaystyle L}$ follows GEV, then the right-tail TVaR is equal to

$${\displaystyle \operatorname {TVaR} _{\alpha }(X)={\begin{cases}\mu +{\frac {\sigma }{(1-\alpha )\xi }}\left[\gamma (1-\xi ,-\ln \alpha )-(1-\alpha )\right]&{\text{if }}\xi \neq 0,\\\mu +{\frac {\sigma }{1-\alpha }}\left[y-{\text{li}}(\alpha )+\alpha \ln(-\ln \alpha )\right]&{\text{if }}\xi =0.\end{cases}}}$$

where ${\displaystyle \gamma (s,x)}$ is the lower incomplete gamma function, ${\displaystyle y}$ is the Euler-Mascheroni constant.[10]

If the payoff of a portfolio ${\displaystyle X}$ follows GHS distribution with the p.d.f.

$${\displaystyle f(x)={\frac {1}{2\sigma }}\operatorname {sech} \left({\frac {\pi }{2}}{\frac {x-\mu }{\sigma }}\right)}$$

and the c.d.f.

$${\displaystyle F(x)={\frac {2}{\pi }}\arctan \left[\exp \left({\frac {\pi }{2}}{\frac {x-\mu }{\sigma }}\right)\right]}$$

then the left-tail TVaR is equal to

$${\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\mu -{\frac {2\sigma }{\pi }}\ln \left(\tan {\frac {\pi \alpha }{2}}\right)-{\frac {2\sigma }{\pi ^{2}\alpha }}i\left[{\text{Li}}_{2}\left(-i\tan {\frac {\pi \alpha }{2}}\right)-{\text{Li}}_{2}\left(i\tan {\frac {\pi \alpha }{2}}\right)\right],}$$

where ${\displaystyle {\text{Li}}_{2}}$ is the Spence's function, ${\displaystyle i={\sqrt {-1}}}$ is the imaginary unit.[11]

If the payoff of a portfolio ${\displaystyle X}$ follows Johnson's SU-distribution with the c.d.f.

$${\displaystyle F(x)=\Phi \left[\gamma +\delta \sinh ^{-1}\left({\frac {x-\xi }{\lambda }}\right)\right]}$$

then the left-tail TVaR is equal to

$${\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\xi -{\frac {\lambda }{2\alpha }}\left[\exp \left({\frac {1-2\gamma \delta }{2\delta ^{2}}}\right)\Phi \left(\Phi ^{-1}(\alpha )-{\frac {1}{\delta }}\right)-\exp \left({\frac {1+2\gamma \delta }{2\delta ^{2}}}\right)\Phi \left(\Phi ^{-1}(\alpha )+{\frac {1}{\delta }}\right)\right],}$$

where ${\displaystyle \Phi }$ is the c.d.f. of the standard normal distribution.[12]

If the payoff of a portfolio ${\displaystyle X}$ follows the Burr type XII distribution with the p.d.f.

$${\displaystyle f(x)={\frac {ck}{\beta }}\left({\frac {x-\gamma }{\beta }}\right)^{c-1}\left[1+\left({\frac {x-\gamma }{\beta }}\right)^{c}\right]^{-k-1}}$$

and the c.d.f.

$${\displaystyle F(x)=1-\left[1+\left({\frac {x-\gamma }{\beta }}\right)^{c}\right]^{-k},}$$

the left-tail TVaR is equal to

$${\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\gamma -{\frac {\beta }{\alpha }}\left((1-\alpha )^{-1/k}-1\right)^{1/c}\left[\alpha -1+{_{2}F_{1}}\left({\frac {1}{c}},k;1+{\frac {1}{c}};1-(1-\alpha )^{-1/k}\right)\right],}$$

where ${\displaystyle _{2}F_{1}}$ is the hypergeometric function. Alternatively,[11]

$${\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\gamma -{\frac {\beta }{\alpha }}{\frac {ck}{c+1}}\left((1-\alpha )^{-1/k}-1\right)^{1+{\frac {1}{c}}}{_{2}F_{1}}\left(1+{\frac {1}{c}},k+1;2+{\frac {1}{c}};1-(1-\alpha )^{-1/k}\right).}$$

If the payoff of a portfolio ${\displaystyle X}$ follows the Dagum distribution with the p.d.f.

$${\displaystyle f(x)={\frac {ck}{\beta }}\left({\frac {x-\gamma }{\beta }}\right)^{ck-1}\left[1+\left({\frac {x-\gamma }{\beta }}\right)^{c}\right]^{-k-1}}$$

and the c.d.f.

$${\displaystyle F(x)=\left[1+\left({\frac {x-\gamma }{\beta }}\right)^{-c}\right]^{-k},}$$

the left-tail TVaR is equal to

$${\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\gamma -{\frac {\beta }{\alpha }}{\frac {ck}{ck+1}}\left(\alpha ^{-1/k}-1\right)^{-k-{\frac {1}{c}}}{_{2}F_{1}}\left(k+1,k+{\frac {1}{c}};k+1+{\frac {1}{c}};-{\frac {1}{\alpha ^{-1/k}-1}}\right),}$$

where ${\displaystyle _{2}F_{1}}$ is the hypergeometric function.[11]

If the payoff of a portfolio ${\displaystyle X}$ follows lognormal distribution, i.e. the random variable ${\displaystyle \ln(1+X)}$ follows normal distribution with the p.d.f.

$${\displaystyle f(x)={\frac {1}{{\sqrt {2\pi }}\sigma }}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}},}$$

then the left-tail TVaR is equal to

$${\displaystyle \operatorname {TVaR} _{\alpha }(X)=1-\exp \left(\mu +{\frac {\sigma ^{2}}{2}}\right){\frac {\Phi (\Phi ^{-1}(\alpha )-\sigma )}{\alpha }},}$$

where ${\displaystyle \Phi (x)}$ is the standard normal c.d.f., so ${\displaystyle \Phi ^{-1}(\alpha )}$ is the standard normal quantile.[13]

If the payoff of a portfolio ${\displaystyle X}$ follows log-logistic distribution, i.e. the random variable ${\displaystyle \ln(1+X)}$ follows logistic distribution with the p.d.f.

$${\displaystyle f(x)={\frac {1}{s}}e^{-{\frac {x-\mu }{s}}}\left(1+e^{-{\frac {x-\mu }{s}}}\right)^{-2},}$$

then the left-tail TVaR is equal to

$${\displaystyle \operatorname {TVaR} _{\alpha }(X)=1-{\frac {e^{\mu }}{\alpha }}I_{\alpha }(1+s,1-s){\frac {\pi s}{\sin \pi s}},}$$

where ${\displaystyle I_{\alpha }}$ is the regularized incomplete beta function, ${\displaystyle I_{\alpha }(a,b)={\frac {\mathrm {B} _{\alpha }(a,b)}{\mathrm {B} (a,b)}}}$. 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]

$${\displaystyle \operatorname {TVaR} _{\alpha }(X)=1-{\frac {e^{\mu }\alpha ^{s}}{s+1}}{_{2}F_{1}}(s,s+1;s+2;\alpha ).}$$

If the loss of a portfolio ${\displaystyle L}$ follows log-logistic distribution with p.d.f.

$${\displaystyle f(x)={\frac {{\frac {b}{a}}(x/a)^{b-1}}{(1+(x/a)^{b})^{2}}}}$$

and c.d.f.

$${\displaystyle F(x)={\frac {1}{1+(x/a)^{-b}}},}$$

then the right-tail TVaR is equal to

$${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)={\frac {a}{1-\alpha }}\left[{\frac {\pi }{b}}\csc \left({\frac {\pi }{b}}\right)-\mathrm {B} _{\alpha }\left({\frac {1}{b}}+1,1-{\frac {1}{b}}\right)\right],}$$

where ${\displaystyle B_{\alpha }}$ is the incomplete beta function.[10]

If the payoff of a portfolio ${\displaystyle X}$ follows log-Laplace distribution, i.e. the random variable ${\displaystyle \ln(1+X)}$ follows Laplace distribution the p.d.f.

$${\displaystyle f(x)={\frac {1}{2b}}e^{-{\frac {|x-\mu |}{b}}},}$$

then the left-tail TVaR is equal to[13]

$${\displaystyle \operatorname {TVaR} _{\alpha }(X)={\begin{cases}1-{\frac {e^{\mu }(2\alpha )^{b}}{b+1}}&{\text{if }}\alpha \leq 0.5,\\1-{\frac {e^{\mu }2^{-b}}{\alpha (b-1)}}\left[(1-\alpha )^{(1-b)}-1\right]&{\text{if }}\alpha >0.5.\end{cases}}}$$

If the payoff of a portfolio ${\displaystyle X}$ follows log-GHS distribution, i.e. the random variable ${\displaystyle \ln(1+X)}$ follows GHS distribution with the p.d.f.

$${\displaystyle f(x)={\frac {1}{2\sigma }}\operatorname {sech} \left({\frac {\pi }{2}}{\frac {x-\mu }{\sigma }}\right),}$$

then the left-tail TVaR is equal to

$${\displaystyle \operatorname {TVaR} _{\alpha }(X)=1-{\frac {1}{\alpha (\sigma +{\pi /2})}}\left(\tan {\frac {\pi \alpha }{2}}\exp {\frac {\pi \mu }{2\sigma }}\right)^{2\sigma /\pi }\tan {\frac {\pi \alpha }{2}}{_{2}F_{1}}\left(1,{\frac {1}{2}}+{\frac {\sigma }{\pi }};{\frac {3}{2}}+{\frac {\sigma }{\pi }};-\tan \left({\frac {\pi \alpha }{2}}\right)^{2}\right),}$$

where ${\displaystyle _{2}F_{1}}$ is the hypergeometric function.[13]