VaR is (usually) not a consistent measure of risk, but TVaR is. VaR is arguably more shareholder oriented and TVaR more regulatory/customer-oriented, see VaR versus TVaR mentalities. A relatively easy-to-use formula for the extreme risk value for the distribution of the mixture will be presented in the next article. If the payment of a wallet X {displaystyle X} GEV with the p.d.f. f ( x ) = { 1 σ ( 1 + ξ x − μ σ ) − 1 ξ − 1 exp [ − ( 1 + ξ x − μ σ ) − 1 ξ ] if ξ ≠ 0 , 1 σ e − x − μ σ e − e − x − μ σ if ξ = 0. {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. F ( x ) = { exp ( − ( 1 + ξ x − μ σ ) − 1 ξ ) Wenn ξ 0 ≠, exp ( − e − x − μ σ ) wenn ξ = 0. {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}}} dann ist der linke Schwanz TVaR gleich TVaR α ( X ) = { − μ − σ α ξ [ Γ ( 1 − ξ , − ln α ) − α ] si ξ ≠ 0 , − μ − σ α [ li ( α ) − α ln ( − ln α ) ] wenn ξ = 0. {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}}} und der VaR gleich V a R α ( X ) = { − μ − σ ξ [ ( − ln α ) − ξ − 1 ] if ξ ≠ 0 , − μ + σ ln ( − ln α ) wenn ξ = 0. {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 Γ ( s , x ) {displaystyle gamma (s,x)} is the incomplete upper gamma function, li ( x ) = ∫ d x ln x {displaystyle {text{li}}(x)=int {frac {dx}{ln x}}} is the logarithmic integral function. [11] For a random variable X {displaystyle X}, which represents the payment of a wallet at a later date, and a parameter 0 < α < 1 {displaystyle 0<alpha <1}, the risk value is defined by[5][6][7][8] If the payment of a wallet X {displaystyle X} of the Burr Distribution Type XII with the p.d.f. f f ( x ) = c k β ( x − γ β ) c − 1 [ 1 + ( x − γ β ) c ] − k − 1 {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.
F ( x ) = 1 − [ 1 + ( x − γ β ) c ] − k {displaystyle F(x)=1-{Big [}1+{Big (}{frac {x-gamma }{beta }}{Big )}^{c}{Big ]}^{-k}} , der linke Schwanz TVaR ist gleich TVaR α ( X ) = − γ − β α ( ( 1 − α ) − 1 / k − 1 ) 1 / c [ α − 1 + 2 F 1 ( 1 c , k ; 1 + 1 c ; 1 − ( 1 − α ) − 1 / k ) ] {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 ]}} , wobei 2 F 1 {displaystyle _{2}F_{1}} die hypergeometrische Funktion ist. Alternativ kann TVaR α ( X ) = − γ − β α c k c + 1 ( ( 1 − α ) − 1 / k − 1 ) 1 + 1 c 2F 1 ( 1 + 1 c , k + 1 ; 2 + 1 c ; 1 − ( 1 − α ) − �� 1 / k ) {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] When several different exposures contribute to the overall RVTV, it is often important to determine the contribution of each person in total. This can be done with the marginal value of the tail at risk (or marginal TVaR). For technical or actuarial applications, it is more common to consider the loss distribution L = − X {displaystyle L=-X}, in which case the right tail TVaR is taken into account (typically for α {displaystyle alpha } 95% or 99%): Value at Risk does not evaluate the kurtosis of the loss distribution.