Risk measure
"TVAR" redirects here. The term may also refer to 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.
[1] Under some formulations, it is only equivalent to
expected shortfall when the underlying
distribution function is
continuous at
VaR
α
(
X
)
{\displaystyle \operatorname {VaR} _{\alpha }(X)}
, the value at risk of level
α
{\displaystyle \alpha }
.
[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.
[3] The former definition may not be a
coherent risk measure in general, however it is coherent if the underlying distribution is continuous.
[4] The latter definition is a coherent risk measure.
[3] 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
X
{\displaystyle X}
which is the payoff of a portfolio at some future time and given a parameter
0
<
α
<
1
{\displaystyle 0<\alpha <1}
then the tail value at risk is defined by
[5]
[6]
[7]
[8]
TVaR
α
(
X
)
=
E
−
X
|
X
≤
−
VaR
α
(
X
)
=
E
−
X
|
X
≤
x
α
,
{\displaystyle \operatorname {TVaR} _{\alpha }(X)=\operatorname {E} [-X|X\leq -\operatorname {VaR} _{\alpha }(X)]=\operatorname {E} [-X|X\leq x^{\alpha }],}
where
x
α
{\displaystyle x^{\alpha }}
is the upper
α
{\displaystyle \alpha }
-
quantile given by
x
α
=
inf
{
x
∈
R
:
Pr
(
X
≤
x
)
>
α
}
{\displaystyle x^{\alpha }=\inf\{x\in \mathbb {R} :\Pr(X\leq x)>\alpha \}}
. Typically the payoff random variable
X
{\displaystyle X}
is in some
Lp -space where
p
≥
1
{\displaystyle p\geq 1}
to guarantee the existence of the expectation. The typical values for
α
{\displaystyle \alpha }
are 5% and 1%.
Formulas for continuous probability distributions
Closed-form formulas exist for calculating TVaR when the payoff of a portfolio
X
{\displaystyle X}
or a corresponding loss
L
=
−
X
{\displaystyle L=-X}
follows a specific continuous distribution. If
X
{\displaystyle X}
follows some
probability distribution with the
probability density function (p.d.f.)
f
{\displaystyle f}
and the
cumulative distribution function (c.d.f.)
F
{\displaystyle F}
, the left-tail TVaR can be represented as
TVaR
α
(
X
)
=
E
−
X
|
X
≤
−
VaR
α
(
X
)
=
−
1
α
∫
0
α
VaR
γ
(
X
)
d
γ
=
−
1
α
∫
−
∞
F
−
1
(
α
)
x
f
(
x
)
d
x
.
{\displaystyle \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.}
For engineering or actuarial applications it is more common to consider the distribution of losses
L
=
−
X
{\displaystyle L=-X}
, in this case the right-tail TVaR is considered (typically for
α
{\displaystyle \alpha }
95% or 99%):
TVaR
α
right
(
L
)
=
E
L
∣
L
≥
VaR
α
(
L
)
=
1
1
−
α
∫
α
1
VaR
γ
(
L
)
d
γ
=
1
1
−
α
∫
F
−
1
(
α
)
+
∞
y
f
(
y
)
d
y
.
{\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)=E[L\mid L\geq \operatorname {VaR} _{\alpha }(L)]={\frac {1}{1-\alpha }}\int _{\alpha }^{1}\operatorname {VaR} _{\gamma }(L)d\gamma ={\frac {1}{1-\alpha }}\int _{F^{-1}(\alpha )}^{+\infty }yf(y)dy.}
Since some formulas below were derived for the left-tail case and some for the right-tail case, the following reconciliations can be useful:
TVaR
α
(
X
)
=
−
1
α
E
X
+
1
−
α
α
TVaR
α
right
(
L
)
{\displaystyle \operatorname {TVaR} _{\alpha }(X)=-{\frac {1}{\alpha }}E[X]+{\frac {1-\alpha }{\alpha }}\operatorname {TVaR} _{\alpha }^{\text{right}}(L)}
and
TVaR
α
right
(
L
)
=
1
1
−
α
E
L
+
α
1
−
α
TVaR
α
(
X
)
.
{\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)={\frac {1}{1-\alpha }}E[L]+{\frac {\alpha }{1-\alpha }}\operatorname {TVaR} _{\alpha }(X).}
Normal distribution
If the payoff of a portfolio
X
{\displaystyle X}
follows
normal (Gaussian) distribution with the p.d.f.
f
(
x
)
=
1
2
π
σ
e
−
(
x
−
μ
)
2
2
σ
2
{\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
TVaR
α
(
X
)
=
−
μ
+
σ
ϕ
(
Φ
−
1
(
α
)
)
α
,
{\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\mu +\sigma {\frac {\phi (\Phi ^{-1}(\alpha ))}{\alpha }},}
where
ϕ
(
x
)
=
1
2
π
e
−
x
2
/
2
{\textstyle \phi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{x^{2}}/{2}}}
is the standard normal p.d.f.,
Φ
(
x
)
{\displaystyle \Phi (x)}
is the standard normal c.d.f., so
Φ
−
1
(
α
)
{\displaystyle \Phi ^{-1}(\alpha )}
is the standard normal quantile.
[9]
If the loss of a portfolio
L
{\displaystyle L}
follows normal distribution, the right-tail TVaR is equal to
[10]
TVaR
α
right
(
L
)
=
μ
+
σ
ϕ
(
Φ
−
1
(
α
)
)
1
−
α
.
{\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)=\mu +\sigma {\frac {\phi (\Phi ^{-1}(\alpha ))}{1-\alpha }}.}
Generalized Student's t-distribution
If the payoff of a portfolio
X
{\displaystyle X}
follows generalized
Student's t-distribution with the p.d.f.
f
(
x
)
=
Γ
(
ν
+
1
2
)
Γ
(
ν
2
)
π
ν
σ
(
1
+
1
ν
(
x
−
μ
σ
)
2
)
−
ν
+
1
2
{\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
TVaR
α
(
X
)
=
−
μ
+
σ
ν
+
(
T
−
1
(
α
)
)
2
ν
−
1
τ
(
T
−
1
(
α
)
)
α
,
{\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\mu +\sigma {\frac {\nu +(\mathrm {T} ^{-1}(\alpha ))^{2}}{\nu -1}}{\frac {\tau (\mathrm {T} ^{-1}(\alpha ))}{\alpha }},}
where
τ
(
x
)
=
Γ
(
ν
+
1
2
)
Γ
(
ν
2
)
π
ν
(
1
+
x
2
ν
)
−
ν
+
1
2
{\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.,
T
(
x
)
{\displaystyle \mathrm {T} (x)}
is the standard t-distribution c.d.f., so
T
−
1
(
α
)
{\displaystyle \mathrm {T} ^{-1}(\alpha )}
is the standard t-distribution quantile.
[9]
If the loss of a portfolio
L
{\displaystyle L}
follows generalized Student's t-distribution, the right-tail TVaR is equal to
[10]
TVaR
α
right
(
L
)
=
μ
+
σ
ν
+
(
T
−
1
(
α
)
)
2
ν
−
1
τ
(
T
−
1
(
α
)
)
1
−
α
.
{\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 }}.}
Laplace distribution
If the payoff of a portfolio
X
{\displaystyle X}
follows
Laplace distribution with the p.d.f.
f
(
x
)
=
1
2
b
e
−
|
x
−
μ
|
b
{\displaystyle f(x)={\frac {1}{2b}}e^{-{\frac {|x-\mu |}{b}}}}
and the c.d.f.
F
(
x
)
=
{
1
−
1
2
e
−
x
−
μ
b
if
x
≥
μ
,
1
2
e
x
−
μ
b
if
x
<
μ
.
{\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
TVaR
α
(
X
)
=
−
μ
+
b
(
1
−
ln
2
α
)
{\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\mu +b(1-\ln 2\alpha )}
for
α
≤
0.5
{\displaystyle \alpha \leq 0.5}
.
[9]
If the loss of a portfolio
L
{\displaystyle L}
follows Laplace distribution, the right-tail TVaR is equal to
[10]
TVaR
α
right
(
L
)
=
{
μ
+
b
α
1
−
α
(
1
−
ln
2
α
)
if
α
<
0.5
,
μ
+
b
1
−
ln
(
2
(
1
−
α
)
)
if
α
≥
0.5.
{\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}}}
Logistic distribution
If the payoff of a portfolio
X
{\displaystyle X}
follows
logistic distribution with the p.d.f.
f
(
x
)
=
1
s
e
−
x
−
μ
s
(
1
+
e
−
x
−
μ
s
)
−
2
{\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.
F
(
x
)
=
(
1
+
e
−
x
−
μ
s
)
−
1
{\displaystyle F(x)=\left(1+e^{-{\frac {x-\mu }{s}}}\right)^{-1}}
then the left-tail TVaR is equal to
[9]
TVaR
α
(
X
)
=
−
μ
+
s
ln
(
1
−
α
)
1
−
1
α
α
.
{\displaystyle \operatorname {TVaR} _{\alpha }(X)=-\mu +s\ln {\frac {(1-\alpha )^{1-{\frac {1}{\alpha }}}}{\alpha }}.}
If the loss of a portfolio
L
{\displaystyle L}
follows
logistic distribution , the right-tail TVaR is equal to
[10]
TVaR
α
right
(
L
)
=
μ
+
s
−
α
ln
α
−
(
1
−
α
)
ln
(
1
−
α
)
1
−
α
.
{\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)=\mu +s{\frac {-\alpha \ln \alpha -(1-\alpha )\ln(1-\alpha )}{1-\alpha }}.}
Exponential distribution
If the loss of a portfolio
L
{\displaystyle L}
follows
exponential distribution with the p.d.f.
f
(
x
)
=
{
λ
e
−
λ
x
if
x
≥
0
,
0
if
x
<
0.
{\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.
F
(
x
)
=
{
1
−
e
−
λ
x
if
x
≥
0
,
0
if
x
<
0.
{\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]
TVaR
α
right
(
L
)
=
−
ln
(
1
−
α
)
+
1
λ
.
{\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)={\frac {-\ln(1-\alpha )+1}{\lambda }}.}
Pareto distribution
If the loss of a portfolio
L
{\displaystyle L}
follows
Pareto distribution with the p.d.f.
f
(
x
)
=
{
a
x
m
a
x
a
+
1
if
x
≥
x
m
,
0
if
x
<
x
m
.
{\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.
F
(
x
)
=
{
1
−
(
x
m
/
x
)
a
if
x
≥
x
m
,
0
if
x
<
x
m
.
{\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]
TVaR
α
right
(
L
)
=
x
m
a
(
1
−
α
)
1
/
a
(
a
−
1
)
.
{\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)={\frac {x_{m}a}{(1-\alpha )^{1/a}(a-1)}}.}
Generalized Pareto distribution (GPD)
If the loss of a portfolio
L
{\displaystyle L}
follows
GPD with the p.d.f.
f
(
x
)
=
1
s
(
1
+
ξ
(
x
−
μ
)
s
)
(
−
1
ξ
−
1
)
{\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.
F
(
x
)
=
{
1
−
(
1
+
ξ
(
x
−
μ
)
s
)
−
1
ξ
if
ξ
≠
0
,
1
−
exp
(
−
x
−
μ
s
)
if
ξ
=
0.
{\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
TVaR
α
right
(
L
)
=
{
μ
+
s
(
1
−
α
)
−
ξ
1
−
ξ
+
(
1
−
α
)
−
ξ
−
1
ξ
if
ξ
≠
0
,
μ
+
s
1
−
ln
(
1
−
α
)
if
ξ
=
0.
{\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]
V
a
R
α
(
L
)
=
{
μ
+
s
(
1
−
α
)
−
ξ
−
1
ξ
if
ξ
≠
0
,
μ
−
s
ln
(
1
−
α
)
if
ξ
=
0.
{\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}}}
Weibull distribution
If the loss of a portfolio
L
{\displaystyle L}
follows
Weibull distribution with the p.d.f.
f
(
x
)
=
{
k
λ
(
x
λ
)
k
−
1
e
−
(
x
/
λ
)
k
if
x
≥
0
,
0
if
x
<
0.
{\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.
F
(
x
)
=
{
1
−
e
−
(
x
/
λ
)
k
if
x
≥
0
,
0
if
x
<
0.
{\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
TVaR
α
right
(
L
)
=
λ
1
−
α
Γ
(
1
+
1
k
,
−
ln
(
1
−
α
)
)
,
{\displaystyle \operatorname {TVaR} _{\alpha }^{\text{right}}(L)={\frac {\lambda }{1-\alpha }}\Gamma \left(1+{\frac {1}{k}},-\ln(1-\alpha )\right),}
where
Γ
(
s
,
x
)
{\displaystyle \Gamma (s,x)}
is the
upper incomplete gamma function .
[10]
Generalized extreme value distribution (GEV)
If the payoff of a portfolio
X
{\displaystyle X}
follows
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 }}\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.
F
(
x
)
=
{
exp
(
−
(
1
+
ξ
x
−
μ
σ
)
−
1
ξ
)
if
ξ
≠
0
,
exp
(
−
e
−
x
−
μ
σ
)
if
ξ
=
0.
{\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
TVaR
α
(
X
)
=
{
−
μ
−
σ
α
ξ
Γ
(
1
−
ξ
,
−
ln
α
)
−
α
if
ξ
≠
0
,
−
μ
−
σ
α
li
(
α
)
−
α
ln
(
−
ln
α
)
if
ξ
=
0.
{\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
V
a
R
α
(
X
)
=
{
−
μ
−
σ
ξ
(
−
ln
α
)
−
ξ
−
1
if
ξ
≠
0
,
−
μ
+
σ
ln
(
−
ln
α
)
if
ξ
=
0.
{\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
Γ
(
s
,
x
)
{\displaystyle \Gamma (s,x)}
is the
upper incomplete gamma function ,
li
(
x
)
=
∫
d
x
ln
x
{\displaystyle {\text{li}}(x)=\int {\frac {dx}{\ln x}}}
is the
logarithmic integral function .
[11]
If the loss of a portfolio
L
{\displaystyle L}
follows
GEV , then the right-tail TVaR is equal to
TVaR
α
(
X
)
=
{
μ
+
σ
(
1
−
α
)
ξ
γ
(
1
−
ξ
,
−
ln
α
)
−
(
1
−
α
)
if
ξ
≠
0
,
μ
+
σ
1
−
α
y
−
li
(
α
)
+
α
ln
(
−
ln
α
)
if
ξ
=
0.
{\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
γ
(
s
,
x
)
{\displaystyle \gamma (s,x)}
is the
lower incomplete gamma function ,
y
{\displaystyle y}
is the
Euler-Mascheroni constant .
[10]
Generalized hyperbolic secant (GHS) distribution
If the payoff of a portfolio
X
{\displaystyle X}
follows
GHS distribution with the p.d.f.
f
(
x
)
=
1
2
σ
sech
(
π
2
x
−
μ
σ
)
{\displaystyle f(x)={\frac {1}{2\sigma }}\operatorname {sech} \left({\frac {\pi }{2}}{\frac {x-\mu }{\sigma }}\right)}
and the c.d.f.
F
(
x
)
=
2
π
arctan
exp
(
π
2
x
−
μ
σ
)
{\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
TVaR
α
(
X
)
=
−
μ
−
2
σ
π
ln
(
tan
π
α
2
)
−
2
σ
π
2
α
i
Li
2
(
−
i
tan
π
α
2
)
−
Li
2
(
i
tan
π
α
2
)
,
{\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
Li
2
{\displaystyle {\text{Li}}_{2}}
is the
dilogarithm and
i
=
−
1
{\displaystyle i={\sqrt {-1}}}
is the imaginary unit.
[11]
Johnson's SU-distribution
If the payoff of a portfolio
X
{\displaystyle X}
follows
Johnson's SU-distribution with the c.d.f.
F
(
x
)
=
Φ
γ
+
δ
sinh
−
1
(
x
−
ξ
λ
)
{\displaystyle F(x)=\Phi \left[\gamma +\delta \sinh ^{-1}\left({\frac {x-\xi }{\lambda }}\right)\right]}
then the left-tail TVaR is equal to
TVaR
α
(
X
)
=
−
ξ
−
λ
2
α
exp
(
1
−
2
γ
δ
2
δ
2
)
Φ
(
Φ
−
1
(
α
)
−
1
δ
)
−
exp
(
1
+
2
γ
δ
2
δ
2
)
Φ
(
Φ
−
1
(
α
)
+
1
δ
)
,
{\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
Φ
{\displaystyle \Phi }
is the c.d.f. of the standard normal distribution.
[12]
Burr type XII distribution
If the payoff of a portfolio
X
{\displaystyle X}
follows the
Burr type XII distribution with the p.d.f.
f
(
x
)
=
c
k
β
(
x
−
γ
β
)
c
−
1
1
+
(
x
−
γ
β
)
c
−
k
−
1
{\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.
F
(
x
)
=
1
−
1
+
(
x
−
γ
β
)
c
−
k
,
{\displaystyle F(x)=1-\left[1+\left({\frac {x-\gamma }{\beta }}\right)^{c}\right]^{-k},}
the left-tail TVaR is equal to
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 }}\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
2
F
1
{\displaystyle _{2}F_{1}}
is the
hypergeometric function . Alternatively,
[11]
TVaR
α
(
X
)
=
−
γ
−
β
α
c
k
c
+
1
(
(
1
−
α
)
−
1
/
k
−
1
)
1
+
1
c
2
F
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}}\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).}
Dagum distribution
If the payoff of a portfolio
X
{\displaystyle X}
follows the
Dagum distribution with the p.d.f.
f
(
x
)
=
c
k
β
(
x
−
γ
β
)
c
k
−
1
1
+
(
x
−
γ
β
)
c
−
k
−
1
{\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.
F
(
x
)
=
1
+
(
x
−
γ
β
)
−
c
−
k
,
{\displaystyle F(x)=\left[1+\left({\frac {x-\gamma }{\beta }}\right)^{-c}\right]^{-k},}
the left-tail TVaR is equal to
TVaR
α
(
X
)
=
−
γ
−
β
α
c
k
c
k
+
1
(
α
−
1
/
k
−
1
)
−
k
−
1
c
2
F
1
(
k
+
1
,
k
+
1
c
;
k
+
1
+
1
c
;
−
1
α
−
1
/
k
−
1
)
,
{\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
2
F
1
{\displaystyle _{2}F_{1}}
is the
hypergeometric function .
[11]
Lognormal distribution
If the payoff of a portfolio
X
{\displaystyle X}
follows
lognormal distribution , i.e. the random variable
ln
(
1
+
X
)
{\displaystyle \ln(1+X)}
follows normal distribution with the p.d.f.
f
(
x
)
=
1
2
π
σ
e
−
(
x
−
μ
)
2
2
σ
2
,
{\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
TVaR
α
(
X
)
=
1
−
exp
(
μ
+
σ
2
2
)
Φ
(
Φ
−
1
(
α
)
−
σ
)
α
,
{\displaystyle \operatorname {TVaR} _{\alpha }(X)=1-\exp \left(\mu +{\frac {\sigma ^{2}}{2}}\right){\frac {\Phi (\Phi ^{-1}(\alpha )-\sigma )}{\alpha }},}
where
Φ
(
x
)
{\displaystyle \Phi (x)}
is the standard normal c.d.f., so
Φ
−
1
(
α
)
{\displaystyle \Phi ^{-1}(\alpha )}
is the standard normal quantile.
[13]
Log-logistic distribution
If the payoff of a portfolio
X
{\displaystyle X}
follows
log-logistic distribution , i.e. the random variable
ln
(
1
+
X
)
{\displaystyle \ln(1+X)}
follows logistic distribution with the p.d.f.
f
(
x
)
=
1
s
e
−
x
−
μ
s
(
1
+
e
−
x
−
μ
s
)
−
2
,
{\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
TVaR
α
(
X
)
=
1
−
e
μ
α
I
α
(
1
+
s
,
1
−
s
)
π
s
sin
π
s
,
{\displaystyle \operatorname {TVaR} _{\alpha }(X)=1-{\frac {e^{\mu }}{\alpha }}I_{\alpha }(1+s,1-s){\frac {\pi s}{\sin \pi s}},}
where
I
α
{\displaystyle I_{\alpha }}
is the
regularized incomplete beta function ,
I
α
(
a
,
b
)
=
B
α
(
a
,
b
)
B
(
a
,
b
)
{\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]
TVaR
α
(
X
)
=
1
−
e
μ
α
s
s
+
1
2
F
1
(
s
,
s
+
1
;
s
+
2
;
α
)
.
{\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
L
{\displaystyle L}
follows log-logistic distribution with p.d.f.
f
(
x
)
=
b
a
(
x
/
a
)
b
−
1
(
1
+
(
x
/
a
)
b
)
2
{\displaystyle f(x)={\frac {{\frac {b}{a}}(x/a)^{b-1}}{(1+(x/a)^{b})^{2}}}}
and c.d.f.
F
(
x
)
=
1
1
+
(
x
/
a
)
−
b
,
{\displaystyle F(x)={\frac {1}{1+(x/a)^{-b}}},}
then the right-tail TVaR is equal to
TVaR
α
right
(
L
)
=
a
1
−
α
π
b
csc
(
π
b
)
−
B
α
(
1
b
+
1
,
1
−
1
b
)
,
{\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
B
α
{\displaystyle B_{\alpha }}
is the
incomplete beta function .
[10]
Log-Laplace distribution
If the payoff of a portfolio
X
{\displaystyle X}
follows
log-Laplace distribution , i.e. the random variable
ln
(
1
+
X
)
{\displaystyle \ln(1+X)}
follows Laplace distribution the p.d.f.
f
(
x
)
=
1
2
b
e
−
|
x
−
μ
|
b
,
{\displaystyle f(x)={\frac {1}{2b}}e^{-{\frac {|x-\mu |}{b}}},}
then the left-tail TVaR is equal to
[13]
TVaR
α
(
X
)
=
{
1
−
e
μ
(
2
α
)
b
b
+
1
if
α
≤
0.5
,
1
−
e
μ
2
−
b
α
(
b
−
1
)
(
1
−
α
)
(
1
−
b
)
−
1
if
α
>
0.5.
{\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}}}
Log-generalized hyperbolic secant (log-GHS) distribution
If the payoff of a portfolio
X
{\displaystyle X}
follows log-GHS distribution, i.e. the random variable
ln
(
1
+
X
)
{\displaystyle \ln(1+X)}
follows
GHS distribution with the p.d.f.
f
(
x
)
=
1
2
σ
sech
(
π
2
x
−
μ
σ
)
,
{\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
TVaR
α
(
X
)
=
1
−
1
α
(
σ
+
π
/
2
)
(
tan
π
α
2
exp
π
μ
2
σ
)
2
σ
/
π
tan
π
α
2
2
F
1
(
1
,
1
2
+
σ
π
;
3
2
+
σ
π
;
−
tan
(
π
α
2
)
2
)
,
{\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
2
F
1
{\displaystyle _{2}F_{1}}
is the
hypergeometric function .
[13]
References
^ Bargès; Cossette, Marceau (2009). "TVaR-based capital allocation with copulas". Insurance: Mathematics and Economics . 45 (3): 348–361.
CiteSeerX
10.1.1.366.9837 .
doi :
10.1016/j.insmatheco.2009.08.002 .
^
"Average Value at Risk" (PDF) . Archived from
the original (PDF) on July 19, 2011. Retrieved February 2, 2011 .
^
a
b Sweeting, Paul (2011). "15.4 Risk Measures". Financial Enterprise Risk Management . International Series on Actuarial Science.
Cambridge University Press . pp. 397–401.
ISBN
978-0-521-11164-5 .
LCCN
2011025050 .
^ Acerbi, Carlo; Tasche, Dirk (2002). "On the coherence of Expected Shortfall". Journal of Banking and Finance . 26 (7): 1487–1503.
arXiv :
cond-mat/0104295 .
doi :
10.1016/s0378-4266(02)00283-2 .
S2CID
511156 .
^ Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc; Heath, David (1999).
"Coherent Measures of Risk" (PDF) . Mathematical Finance . 9 (3): 203–228.
doi :
10.1111/1467-9965.00068 .
S2CID
6770585 . Retrieved February 3, 2011 .
^ Landsman, Zinoviy; Valdez, Emiliano (February 2004).
"Tail Conditional Expectations for Exponential Dispersion Models" (PDF) . Retrieved February 3, 2011 .
^ Landsman, Zinoviy; Makov, Udi; Shushi, Tomer (July 2013). "Tail Conditional Expectations for Generalized Skew - Elliptical distributions".
doi :
10.2139/ssrn.2298265 .
S2CID
117342853 .
SSRN
2298265 .
^ Valdez, Emiliano (May 2004).
"The Iterated Tail Conditional Expectation for the Log-Elliptical Loss Process" (PDF) . Retrieved February 3, 2010 .
^
a
b
c
d Khokhlov, Valentyn (2016). "Conditional Value-at-Risk for Elliptical Distributions". Evropský časopis Ekonomiky a Managementu . 2 (6): 70–79.
^
a
b
c
d
e
f
g
h
i
j Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2018-11-27). "Calculating CVaR and bPOE for Common Probability Distributions With Application to Portfolio Optimization and Density Estimation".
arXiv :
1811.11301 [
q-fin.RM ].
^
a
b
c
d Khokhlov, Valentyn (2018-06-21). "Conditional Value-at-Risk for Uncommon Distributions". SSRN .
SSRN
3200629 .
^ Stucchi, Patrizia (2011-05-31). "Moment-Based CVaR Estimation: Quasi-Closed Formulas". SSRN .
SSRN
1855986 .
^
a
b
c
d Khokhlov, Valentyn (2018-06-17). "Conditional Value-at-Risk for Log-Distributions". SSRN .
SSRN
3197929 .