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 ${\displaystyle \phi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}}$ 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 ${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)=\mu +\sigma {\frac {\phi (\Phi ^{-1}(\alpha ))}{1-\alpha }}}$.[10]

If the payoff of a portfolio ${\displaystyle X}$ follows generalized Student's t-distribution with the p.d.f. ${\displaystyle f(x)={\frac {\Gamma {\bigl (}{\frac {\nu +1}{2}}{\bigr )}}{\Gamma {\bigl (}{\frac {\nu }{2}}{\bigr )}{\sqrt {\pi \nu }}\sigma }}{\Bigl (}1+{\frac {1}{\nu }}{\bigl (}{\frac {x-\mu }{\sigma }}{\bigr )}^{2}{\Bigr )}^{-{\frac {\nu +1}{2}}}}$ 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 ${\displaystyle \tau (x)={\frac {\Gamma {\bigl (}{\frac {\nu +1}{2}}{\bigr )}}{\Gamma {\bigl (}{\frac {\nu }{2}}{\bigr )}{\sqrt {\pi \nu }}}}{\Bigl (}1+{\frac {x^{2}}{\nu }}{\Bigr )}^{-{\frac {\nu +1}{2}}}}$ 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 ${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)=\mu +\sigma {\frac {\nu +(\mathrm {T} ^{-1}(\alpha ))^{2}}{\nu -1}}{\frac {\tau (\mathrm {T} ^{-1}(\alpha ))}{1-\alpha }}}$.[10]

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 ${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)={\begin{cases}\mu +b{\frac {\alpha }{1-\alpha }}(1-\ln 2\alpha )&{\text{if }}\alpha <0.5,\\\mu +b[1-\ln(2(1-\alpha ))]&{\text{if }}\alpha \geq 0.5.\end{cases}}}$.[10]

If the payoff of a portfolio ${\displaystyle X}$ follows logistic distribution with the p.d.f. ${\displaystyle f(x)={\frac {1}{s}}e^{-{\frac {x-\mu }{s}}}{\Bigl (}1+e^{-{\frac {x-\mu }{s}}}{\Bigr )}^{-2}}$ and the c.d.f. ${\displaystyle F(x)={\Bigl (}1+e^{-{\frac {x-\mu }{s}}}{\Bigr )}^{-1}}$ then the left-tail TVaR is equal to ${\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\mu +s\ln {\frac {(1-\alpha )^{1-{\frac {1}{\alpha }}}}{\alpha }}}$.[9]

If the loss of a portfolio ${\displaystyle L}$ follows logistic distribution, the right-tail TVaR is equal to ${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)=\mu +s{\frac {-\alpha \ln \alpha -(1-\alpha )\ln(1-\alpha )}{1-\alpha }}}$.[10]

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 ${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)={\frac {-\ln(1-\alpha )+1}{\lambda }}}$.[10]

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 ${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)={\frac {x_{m}a}{(1-\alpha )^{1/\alpha }(a-1)}}}$.[10]

If the loss of a portfolio ${\displaystyle L}$ follows GPD with the p.d.f. ${\displaystyle f(x)={\frac {1}{s}}{\Bigl (}1+{\frac {\xi (x-\mu )}{s}}{\Bigr )}^{{\bigl (}-{\frac {1}{\xi }}-1{\bigr )}}}$ and the c.d.f. ${\displaystyle F(x)={\begin{cases}1-{\Big (}1+{\frac {\xi (x-\mu )}{s}}{\Big )}^{-{\frac {1}{\xi }}}&{\text{if }}\xi \neq 0,\\1-\exp {\bigl (}-{\frac {x-\mu }{s}}{\bigr )}&{\text{if }}\xi =0.\end{cases}}}$ then the right-tail TVaR is equal to ${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)={\begin{cases}\mu +s{\Bigl [}{\frac {(1-\alpha )^{-\xi }}{1-\xi }}+{\frac {(1-\alpha )^{-\xi }-1}{\xi }}{\Bigr ]}&{\text{if }}\xi \neq 0,\\\mu +s[1-\ln(1-\alpha )]&{\text{if }}\xi =0.\end{cases}}}$ and the VaR is equal to ${\displaystyle VaR_{\alpha }(L)={\begin{cases}\mu +s{\frac {(1-\alpha )^{-\xi }-1}{\xi }}&{\text{if }}\xi \neq 0,\\\mu -s\ln(1-\alpha )&{\text{if }}\xi =0.\end{cases}}}$.[10]

If the loss of a portfolio ${\displaystyle L}$ follows Weibull distribution with the p.d.f. ${\displaystyle f(x)={\begin{cases}{\frac {k}{\lambda }}{\Big (}{\frac {x}{\lambda }}{\Big )}^{k-1}e^{-(x/\lambda )^{k}}&{\text{if }}x\geq 0,\\0&{\text{if }}x<0.\end{cases}}}$ 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 ${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)={\frac {\lambda }{1-\alpha }}\Gamma {\Big (}1+{\frac {1}{k}},-\ln(1-\alpha ){\Big )}}$, 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. ${\displaystyle f(x)={\begin{cases}{\frac {1}{\sigma }}{\Bigl (}1+\xi {\frac {x-\mu }{\sigma }}{\Bigr )}^{-{\frac {1}{\xi }}-1}\exp {\Bigl [}-{\Bigl (}1+\xi {\frac {x-\mu }{\sigma }}{\Bigr )}^{-{\frac {1}{\xi }}}{\Bigr ]}&{\text{if }}\xi \neq 0,\\{\frac {1}{\sigma }}e^{-{\frac {x-\mu }{\sigma }}}e^{-e^{-{\frac {x-\mu }{\sigma }}}}&{\text{if }}\xi =0.\end{cases}}}$ and the c.d.f. ${\displaystyle F(x)={\begin{cases}\exp {\Big (}-{\big (}1+\xi {\frac {x-\mu }{\sigma }}{\big )}^{-{\frac {1}{\xi }}}{\Big )}&{\text{if }}\xi \neq 0,\\\exp {\Big (}-e^{-{\frac {x-\mu }{\sigma }}}{\Big )}&{\text{if }}\xi =0.\end{cases}}}$ then the left-tail TVaR is equal to ${\displaystyle \operatorname {TVaR} _{\alpha }(X)={\begin{cases}-\mu -{\frac {\sigma }{\alpha \xi }}{\big [}\Gamma (1-\xi ,-\ln \alpha )-\alpha {\big ]}&{\text{if }}\xi \neq 0,\\-\mu -{\frac {\sigma }{\alpha }}{\big [}{\text{li}}(\alpha )-\alpha \ln(-\ln \alpha ){\big ]}&{\text{if }}\xi =0.\end{cases}}}$ and the VaR is equal to ${\displaystyle VaR_{\alpha }(X)={\begin{cases}-\mu -{\frac {\sigma }{\xi }}{\big [}(-\ln \alpha )^{-\xi }-1{\big ]}&{\text{if }}\xi \neq 0,\\-\mu +\sigma \ln(-\ln \alpha )&{\text{if }}\xi =0.\end{cases}}}$, 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 ${\displaystyle \operatorname {TVaR} _{\alpha }(X)={\begin{cases}\mu +{\frac {\sigma }{(1-\alpha )\xi }}{\big [}\gamma (1-\xi ,-\ln \alpha )-(1-\alpha ){\big ]}&{\text{if }}\xi \neq 0,\\\mu +{\frac {\sigma }{1-\alpha }}{\big [}y-{\text{li}}(\alpha )+\alpha \ln(-\ln \alpha ){\big ]}&{\text{if }}\xi =0.\end{cases}}}$, 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. ${\displaystyle f(x)={\frac {1}{2\sigma }}{\text{sech}}({\frac {\pi }{2}}{\frac {x-\mu }{\sigma }})}$and the c.d.f. ${\displaystyle F(x)={\frac {2}{\pi }}\arctan {\Big [}\exp {\Big (}{\frac {\pi }{2}}{\frac {x-\mu }{\sigma }}{\Big )}{\Big ]}}$ then the left-tail TVaR is equal to ${\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\mu -{\frac {2\sigma }{\pi }}\ln {\Big (}\tan {\frac {\pi \alpha }{2}}{\Big )}-{\frac {2\sigma }{\pi ^{2}\alpha }}i{\Big [}{\text{Li}}_{2}{\Big (}-i\tan {\frac {\pi \alpha }{2}}{\Big )}-{\text{Li}}_{2}{\Big (}i\tan {\frac {\pi \alpha }{2}}{\Big )}{\Big ]}}$, 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. ${\displaystyle F(x)=\Phi {\Big [}\gamma +\delta \sinh ^{-1}{\Big (}{\frac {x-\xi }{\lambda }}{\Big )}{\Big ]}}$ then the left-tail TVaR is equal to ${\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\xi -{\frac {\lambda }{2\alpha }}{\Big [}exp{\Big (}{\frac {1-2\gamma \delta }{2\delta ^{2}}}{\Big )}\Phi {\Big (}\Phi ^{-1}(\alpha )-{\frac {1}{\delta }}{\Big )}-exp{\Big (}{\frac {1+2\gamma \delta }{2\delta ^{2}}}{\Big )}\Phi {\Big (}\Phi ^{-1}(\alpha )+{\frac {1}{\delta }}{\Big )}{\Big ]}}$, 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. ${\displaystyle f(x)={\frac {ck}{\beta }}{\Big (}{\frac {x-\gamma }{\beta }}{\Big )}^{c-1}{\Big [}1+{\Big (}{\frac {x-\gamma }{\beta }}{\Big )}^{c}{\Big ]}^{-k-1}}$ and the c.d.f. ${\displaystyle F(x)=1-{\Big [}1+{\Big (}{\frac {x-\gamma }{\beta }}{\Big )}^{c}{\Big ]}^{-k}}$, the left-tail TVaR is equal to ${\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\gamma -{\frac {\beta }{\alpha }}{\Big (}(1-\alpha )^{-1/k}-1{\Big )}^{1/c}{\Big [}\alpha -1+{_{2}F_{1}}{\Big (}{\frac {1}{c}},k;1+{\frac {1}{c}};1-(1-\alpha )^{-1/k}{\Big )}{\Big ]}}$, where ${\displaystyle _{2}F_{1}}$ is the hypergeometric function. Alternatively, ${\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\gamma -{\frac {\beta }{\alpha }}{\frac {ck}{c+1}}{\Big (}(1-\alpha )^{-1/k}-1{\Big )}^{1+{\frac {1}{c}}}{_{2}F_{1}}{\Big (}1+{\frac {1}{c}},k+1;2+{\frac {1}{c}};1-(1-\alpha )^{-1/k}{\Big )}}$.[11]

If the payoff of a portfolio ${\displaystyle X}$ follows the Dagum distribution with the p.d.f. ${\displaystyle f(x)={\frac {ck}{\beta }}{\Big (}{\frac {x-\gamma }{\beta }}{\Big )}^{ck-1}{\Big [}1+{\Big (}{\frac {x-\gamma }{\beta }}{\Big )}^{c}{\Big ]}^{-k-1}}$ and the c.d.f. ${\displaystyle F(x)={\Big [}1+{\Big (}{\frac {x-\gamma }{\beta }}{\Big )}^{-c}{\Big ]}^{-k}}$, the left-tail TVaR is equal to ${\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\gamma -{\frac {\beta }{\alpha }}{\frac {ck}{ck+1}}{\Big (}\alpha ^{-1/k}-1{\Big )}^{-k-{\frac {1}{c}}}{_{2}F_{1}}{\Big (}k+1,k+{\frac {1}{c}};k+1+{\frac {1}{c}};-{\frac {1}{\alpha ^{-1/k}-1}}{\Big )}}$, 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 ${\displaystyle \operatorname {TVaR} _{\alpha }(X)=1-\exp {\Bigl (}\mu +{\frac {\sigma ^{2}}{2}}{\Bigr )}{\frac {\Phi (\Phi ^{-1}(\alpha )-\sigma )}{\alpha }}}$, 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. ${\displaystyle f(x)={\frac {1}{s}}e^{-{\frac {x-\mu }{s}}}{\Bigl (}1+e^{-{\frac {x-\mu }{s}}}{\Bigr )}^{-2}}$, 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: ${\displaystyle \operatorname {TVaR} _{\alpha }(X)=1-{\frac {e^{\mu }\alpha ^{s}}{s+1}}{_{2}F_{1}}(s,s+1;s+2;\alpha )}$.[13]

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 ${\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)={\frac {a}{1-\alpha }}{\Bigl [}{\frac {\pi }{b}}\csc {\Bigl (}{\frac {\pi }{b}}{\Bigr )}-\mathrm {B} _{\alpha }{\Bigl (}{\frac {1}{b}}+1,1-{\frac {1}{b}}{\Bigr )}{\Bigr ]}}$, 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 ${\displaystyle \operatorname {TVaR} _{\alpha }(X)={\begin{cases}1-{\frac {e^{\mu }(2\alpha )^{b}}{b+1}}&{\text{if }}\alpha \leq 0.5,\\1-{\frac {e^{\mu }2^{-b}}{\alpha (b-1)}}{\big [}(1-\alpha )^{(1-b)}-1{\big ]}&{\text{if }}\alpha >0.5.\end{cases}}}$.[13]

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. ${\displaystyle f(x)={\frac {1}{2\sigma }}{\text{sech}}({\frac {\pi }{2}}{\frac {x-\mu }{\sigma }})}$, then the left-tail TVaR is equal to ${\displaystyle \operatorname {TVaR} _{\alpha }(X)=1-{\frac {1}{\alpha (\sigma +{\pi /2})}}{\Big (}\tan {\frac {\pi \alpha }{2}}\exp {\frac {\pi \mu }{2\sigma }}{\Big )}^{2\sigma /\pi }\tan {\frac {\pi \alpha }{2}}{_{2}F_{1}}{\Big (}1,{\frac {1}{2}}+{\frac {\sigma }{\pi }};{\frac {3}{2}}+{\frac {\sigma }{\pi }};-\tan {\big (}{\frac {\pi \alpha }{2}}{\big )}^{2}{\Big )}}$, where ${\displaystyle _{2}F_{1}}$ is the hypergeometric function.[13]