Article from:
ARCH 2013.1 Proceedings
August 1- 4, 2012
Fan Yang
First- and Second-order Asymptotics for the Tail
Distortion Risk Measure of Extreme Risks
Fan Yang
[a]
Actuarial Science Program
College of Business and Public Administration
Drake University
Des Moines, IA 50311, USA
[b]
Applied Mathematical and Computational Sciences Program
University of Iowa
14 MacLean Hall, Iowa City, IA 52242, USA
June 21, 2012
Abstract
The tail distortion risk measure at level p was …rst introduced in Zhu and Li (2012),
where the parameter p 2 (0; 1) indicates the con…dence level. They established …rstorder asymptotics for this risk measure, as p " 1, for the Fréchet case. In this paper,
we extend their work by establishing both …rst-order and second-order asymptotics
for the Fréchet, Weibull and Gumbel cases. Numerical studies are also carried out to
examine the accuracy of both asymptotics.
Keywords: Asymptotics; Extended regular variation; Max-domain of attraction;
Regular variation; Second-order condition; Tail distortion risk measure
MSC: Primary 62P05; Secondary 60G70, 62E20
1
Introduction
Let X be a real-valued random variable, with distribution function F = 1 F on R =
( 1; 1). Let g( ) : [0; 1] ! [0; 1] be a distortion function, namely, a nondecreasing function
with g(0) = 0 and g(1) = 1. The distortion risk measure of X associated with g( ) is de…ned
as
Z
Z
1
0
Dg [X] =
g F (x)
1 dx +
g(F (x))dx:
0
1
This risk measure can be viewed as a distorted expectation of X in the sense of Choqet
integral; see Denneberg (1994). The distortion risk measure was …rst introduced by Wang
E-mail: [email protected]; Cell: 319-471-0811
1
(1996) and has been applied to deciding insurance premiums, capital requirements and capital allocations. For more details, the reader is referred to Wang (2002), Valdez and Chernih
(2003) and Tsanakas (2004), among others. Important properties of the distortion risk measure, such as coherence and second-order stochastic dominance, have been well studied; see,
for example, Hardy and Wirch (2003) and Dhaene et al. (2006).
The distortion risk measure takes into account both loss and pro…t sides. Zhu and Li
(2012) proposed the tail distortion risk measure, which focuses on the loss side only. With
a distortion function g( ), introduce gp ( ) with 0 < p < 1 as
(
0 u < 1 p;
g 1up ;
gp (u) =
1;
1 p u 1;
which again is a distortion function. Then the tail distortion risk measure at level p of a risk
variable X is de…ned as
Z 0
Z 1
Tp [X] =
gp F (x)
1 dx +
gp F (x) dx:
1
0
Note that this de…nition di¤ers from the one given by Zhu and Li (2012), but they are
identical when the risk variable X is continuous.
Clearly, if g( ) is concave, then gp ( ) is concave as well, which leads to the coherence
of the tail distortion risk measure. For the special case with g(x) = x and a continuous
risk variable X, for p > F (0) the tail distortion risk measure becomes the well-known the
expected shortfall, Tp [X] = E [XjX > F (p)], where, and throughout the paper,
F (p) = inffx : F (x)
pg
is the value at risk of X or the quantile of F with the usual convention inf ? = 1.
De…ne the function U ( ) as the quantile function of 1=F , namely,
U (t) =
Then we can rewrite Tp [X] as
Z 1
Tp [X] =
F (1
1
F
(t) = F
q)dgp (q) =
0
1
Z
0
1
t
1
U
:
1
q(1
p)
dg(q):
(1.1)
Relation (1.1) will be the starting point of our derivation of asymptotics for Tp [X].
In the de…nition of tail distortion risk measure, the parameter p clearly represents the
con…dence level. Nowadays, people are keen to measuring the tail area of a risk. The tail
area of a risk corresponds to a large loss, whose occurrence is often companied by disastrous
consequences. Therefore, in this paper we will study the asymptotic behavior of the tail
distortion risk measure as p " 1. From this point of view, the use of extreme value theory
becomes appropriate.
2
For a risk variable X with distribution function F , we denote by x^ its upper endpoint
F (1). From (1.1), we see that Tp [X] " x^ if and only if p " 1. We will assume that Pr(X =
x^) = 0 because otherwise Tp [X] = x^ when p is close to 1. We will consider risk variables with
distributions from the max-domain of attraction of an extreme value distribution. For the
Fréchet and Gumbel cases with x^ = 1, we derive …rst-order asymptotics for Tp [X] diverging
to 1 as p " 1. For the Weibull and Gumbel cases with x^ < 1, we derive …rst-order
asymptotics for x^ Tp [X] converging to 0 as p " 1. Furthermore, we derive second-order
asymptotics for the three cases. In this paper, we follow the methodology of Mao and Hu
(2012), who studied the second-order properties of Haezendonck–Goovaerts risk measure.
The rest of this paper consists of four sections. In section 2, we introduce the extreme
value theory. In section 3, we derive …rst-order asymptotics for Tp [X] for the Fréchet, Weibull
and Gumbel cases separately. In section 4, we derive second-order asymptotics for Tp [X]
for all three cases. In section 5, we numerically examine the accuracy of …rst-order and
second-order asymptotics.
2
Extreme Value Theory
A distribution function F is said to belong to the max-domain of attraction of a distribution
function G, denoted by F 2 MDA(G), if for a sequence of independent and identically
distributed random variables, fXn ; n = 1; 2; : : :g, with common distribution function F , the
normalized block maximum Mn = max1 i n Xi , has a distribution weakly converging to G.
The classical Fisher-Tippett theorem (see Fisher and Tippett (1928)) states that G has to
be one of the Fréchet, Weibull and Gumbel distributions whose standard forms are given by,
respectively,
(x) = exp f x g for x > 0,
(x) = exp f jxj g for x 0, and
(x) = exp f e x g for x 2 R.
It is usually convenient to characterize the three max-domains of attraction in terms of
regular variation; see, for example, Embrechts et al. (1997). A positive measurable function
f ( ) is said to be regularly varying at 1, with a regularity index 2 R, denoted by f ( ) 2 R ,
if
f (tx)
=x ;
x > 0:
(2.1)
lim
t!1 f (x)
For the extreme case of (2.1) with = 1, the function f ( ) is said to be rapidly varying
at 1, denoted by f ( ) 2 R 1 .
Using the concept of extended regular variation leads to a uni…ed description for the
three max-domains of attraction. By de…nition, a positive measurable function f ( ) is said
to be extended regularly varying with index 2 R, denoted by f ( ) 2 ERV , if there exists
3
an auxiliary function a( ) > 0 such that, for all x > 0,
f (tx) f (t)
x
=
t!1
a(t)
lim
1
;
(2.2)
where the right-hand side is interpreted as log x when = 0. We will use this usual convention and throughout the paper. The auxiliary function a( ) appearing in (2.2) is often
chosen to be
8
> 0;
< f (t);
(f (1) f (t));
< 0;
a0 (t) =
(2.3)
R
:
1 t
f (t) t
f (s)ds;
= 0:
0
Note that for = 0, if f (1) = 1 then a0 (t) = o(f (t)) as t ! 1 while if f (1) < 1 then
a0 (t) = o(f (1) f (t)) as t ! 1. Theorem 1.1.6 of de Haan and Ferreira (2006) essentially
proves that F 2 MDA(G ) if and only if U 2 ERV , where
8
> 0;
< 1= ;
;
< 0;
G =
1=
:
;
= 0;
and U is the quantile function of 1=F as de…ned above.
Very often we need not only to derive …rst-order asymptotics such as (2.1) and (2.2) but
also to know their convergence speed. For the latter purpose, we introduce the concepts of
second-order regular variation and second-order extended regular variation.
By de…nition, a positive measurable function f ( ) is said to be second-order regularly
varying with …rst-order index 2 R and second-order index
0, denoted by f ( ) 2 2RV ; ,
if there exists an auxiliary function A( ), which does not change sign eventually and converges
to 0, such that, for all x > 0,
lim
t!1
f (tx)
f (t)
x
A(t)
=x
x
1
:
More generally, a positive measurable function f ( ) is said to be second-order extended
regularly varying with …rst-order index 2 R and second-order index
0, denoted by
f ( ) 2 2ERV ; , if there exist a …rst-order auxiliary function a( ) and a second-order auxiliary
function A( ), where A( ) does not change sign eventually and converges to 0, such that, for
all x > 0,
f (tx) f (t)
x 1
1 x +
1 x
1
a(t)
lim
=
;
x > 0:
(2.4)
t!1
A(t)
+
Some immediate explanations of (2.4) follow. The right-hand side of (2.4), written as H
is equal to
8
1 x + 1
x 1
>
;
6= 0;
>
+
<
1
H ; (x) =
x log x x 1 ;
= 0; 6= 0;
>
>
: 1
(log x)2 ;
= = 0:
2
4
;
(x),
In (2.4), a( ) is itself second-order regularly varying, a( ) 2 2RV ; with auxiliary function
A( ) 2 R . The two auxiliary functions a( ) and A( ) are unique up to asymptotic equivalence. Moreover, according to Corollary 2.3.5 of de Haan and Ferreira (2006), if a( ) and
A( ) are chosen appropriately, then H ; (x) is reduced to ; (x) as
8 +
x
1
>
;
< 0;
<
+
1
(2.5)
x log x;
6= = 0;
; (x) =
>
: 1 (log x)2 ;
= = 0:
2
3
First-order Asymptotics
Firstly, we derive …rst-order asymptotics for the tail distortion risk measure of a general risk
variable X.
Theorem 3.1 Let X be a random variable with Pr(X = x^) = 0. Assume that U 2 ERV
with
2 R and the …rst-order auxiliary function a0 ( ). In case
0, assume that
R1
1=( + )
g x
dx < 1 for some > 0. Then when x^ = 1
1
Tp [X]
when x^ < 1
x^
F (p) + a0
Tp [X]
x^
F (p)
Z
1
1
p
a0
1
q
1
dg(q);
(3.1)
0
1
1
p
Z
1
q
1
dg(q):
(3.2)
0
Proof. By Theorem B.2.18 in de Haan and Ferreira (2006), there exists some 0 < p0 =
p0 ( ) < 1 such that for p0 p < 1 and 0 < q < 1,
1
q(1 p)
U
a0
Therefore, if the inequalities
Z 1
U
1
1 p
q
and
q
1
:
1 p
dg(q) < 1
q
0
1
Z
1
q
1
0
for some
(3.3)
> 0;
dg(q) < 1
(3.4)
hold, then applying the dominated convergence theorem we obtain the following: for x^ = 1,
lim
p"1
Z
0
1
U
1
q(1 p)
a0
U
1
1 p
1
dg(q) =
Z
0
1 p
5
1
q
1
dg(q);
which gives the desired result (3.1); similarly, for x^ < 1,
lim
p"1
Z
1
x^
U
1
q(1 p)
a0
0
x^
U
1
1 p
1
dg(q) =
Z
1
q
1
dg(q);
0
1 p
which gives the desired result (3.2).
It remains to verify inequalities (3.3) and (3.4). We consider the three cases, < 0, = 0
and > 0, respectively. For < 0, inequality (3.3) with =
=2 and inequality (3.4) hold
R1
obviously. For = 0, inequality (3.3) is veri…ed by using the condition 1 g(x 1= )dx < 1
and integration by parts, while inequality (3.4) can be veri…ed as
Z 1
Z 1
1
log q dg(q)
q dg(q) < 1;
0
0
where the last step is due to (3.3). For > 0, inequality (3.3) is veri…ed the same as before,
while inequality (3.4) is veri…ed as
Z
Z
Z 1
1 1
1 1
q
1
1=
g(x
)dx
g(x 1=( + ) )dx < 1:
dg(q) =
0
1
1
This ends the proof.
Plugging the expression for a0 ( ) given in (2.3) into Theorem 3.1, we immediately obtain
…rst-order asymptotics for the three max-domains of attraction.
Corollary 3.1 Under the conditions of Theorem 3.1, we have the following:
(a) the Fréchet case: if
> 0 then Tp [X]
(b) the Weibull case: if
< 0 then x^
(c) the Gumbel case: if
then x^ Tp [X] x^
= 0 and x^ = 1 then Tp [X]
F (p).
F (p) 1 +
Tp [X]
(^
x
R1
1
g x
F (p)) 1
1=
dx ;
R1
g x
0
F (p), while if
1=
dx ;
= 0 and x^ < 1
The result for the Fréchet case coincides with the one earlier obtained by Zhu and Li
(2012).
4
Second-order Asymptotics
To derive second-order asymptotics, similar as in the proof of Theorem 3.1, we need a uniform
inequality for the second-order extended regularly varying function, which is a restatement
of Theorem B.3.10 of de Haan and Ferreira (2006).
6
Lemma 4.1 Suppose f ( ) 2 2ERV ; with 2 R,
0, …rst-order auxiliary function a( )
and second-order auxiliary function A( ). Then there exist functions a0 ( ) and A0 ( ) such
that for every "; > 0, some t0 = t0 ("; ) and all t; tx t0 ,
f (tx) f (t)
a0 (t)
x
1
;
A0 (t)
where
;
(x)
+ +
" max x
;x
+
;
( ) is de…ned in (2.5).
The next lemma establishes the relation between 2RV and 2ERV:
Lemma 4.2 Let X be a random variable and let
0.
(a) When > 0, if U 2 ERV with auxiliary function a0 ( ) and U 2 2RV ; with auxiliary
function A( ), then U 2 2ERV ; ;
(b) When < 0, if U 2 ERV with auxiliary function a0 ( ) and x^ U 2 2RV ; with
auxiliary function A( ), then x^ U 2 2ERV ; .
Proof. (a) In this case, a0 (t) = U (t). We have
U (tx) U (t)
a0 (t)
A(t)
x
1
=
=
!
U (t)
U (tx)
U (t)
1
x
1
U (t)
A(t)
U (tx)
U (t)
x
A(t)
x x
1
;
as t ! 1:
Thus, U 2 2ERV ; by de…nition.
(b) In this case, a0 (t) =
(^
x U (t)). Similarly as above, we can prove that x^
2ERV ; with the same limit function.
U 2
We conclude that, under the conditions of Lemma 4.2 and for 6= 0, the limit function
H ; (x) in (2.4) is H ; (x) = x x 1 . Now we derive second-order asymptotics for the tail
distortion risk measure of a general variable X.
Theorem 4.1 Let X be a random variable with Pr(X = x^) = 0. Assume that U 2 2ERV ;
with 6= 0,
0, the …rst-order auxiliary function a0 ( ) and the second-order auxiliary
R1
function A( ). In case > 0, assume that 1 g x 1=( + ) dx < 1 for some > 0. Then
when x^ = 1,
Z 1
q
1
1
Tp [X] = F (p) + a0
dg(q)
1 p
0
Z 1
1
1
+a0
A
H ; q 1 dg(q) + o(1) ;
(4.1)
1 p
1 p
0
7
when x^ < 1,
x^
Tp [X] = x^
F (p)
a0
1
1
p
1
1 p
= c2 1
U( 1 1p )
)
p)
1
A
p
x x
Proof. Note that H ; (x) =
c2 = = ( + ). Thus
U ( q(11
a0
1
Z
1
0
U ( q(11
a0 ( 1
=
1
A0
A
1
1
1 p
0
+ @
H
q
B
B
@
U ( q(11
1
q
;
dg(q) + o(1) :
c1 x
1
(q 1 )
;
1
1 p
A
1
p)
)
U( 1 1p )
H
(q 1 )
;
1 p
q
1
1
1 p
A0
1
1
1 p
( + + )
1A
c2
+ Cq
;
q
;
( + )
1
c1 q
c2
+
1
q
1
1
C
C
A
:
In the last step, we used Lemma 4.1 and the following facts: for every
there exists some positive constant C such that, for all 0 < q < 1,
q
( + )
1
( + + )
Cq
+
;
q
1
Cq
( + )
;
log q
6= 0 and
1
lim
p"1
1
U ( q(11
p)
U( 1 1p )
)
a0 ( 1
1
A
0
q
1
p)
dg(q) =
1
Z
1
H
;
(q 1 )dg(q);
0
1 p
which gives (4.1); similarly, for x^ < 1,
lim
p"1
Z
0
1
x
^ U ( q(11
) (x^ U ( 1 1 p ))
+
a0 ( 1 1 p )
p)
A
q
1
dg(q) =
1
Z
0
1 p
8
1
H
;
0,
<q ;
R1
and A0 (t)
A(t). Since one easily checks that 0 q
dg(q) < 1 hold for some
applying the dominated convergence theorem we obtain the following: for x^ = 1,
Z
(4.2)
, where c1 = = ( + ) and
a0 ( 1 1 p )
1 p
A
H
A0
p
1
(1 + ") q
1
1
A0 1
0
A0
1
1 p
p)
1 p
dg(q)
1
1
U( 1 1p )
)
p)
Z
1
(x)
a0 ( 1 1 p )
A
q
0
;
q
1
(q 1 )dg(q);
> 0,
which gives (4.2). This ends the proof.
Next we develop second-order asymptotics for the three max-domains of attraction.
Corollary 4.1 Let X be a random variable with Pr(X = x^) = 0. In case
R1
that 1 g x 1=( + ) dx < 1 for some > 0.
0, assume
(a) The Fréchet case: If for some > 0 and
0, U 2 ERV with auxiliary function
a0 ( ) and U 2 2RV ; with auxiliary function A( ), then we have
Z 1
1
g x 1= dx + F (p)A
Tp [X] = F (p) 1 +
(I ; + o(1)) ;
1 p
1
where
I
;
=
(
1
R1
R
1 1
1
0
1=( + )
g x
g x
dx +
1=( + )
g x
R1
1
1=
1=
g x
j j;
dx ;
> j j:
dx;
(b) The Weibull case: If for some < 0 and
0, U 2 ERV with auxiliary function
a0 ( ) and x^ U 2 2RV ; with auxiliary function A( ), then we have
x^
Tp [X] = (^
x
Z
F (p)) 1
1
g x
1=
0
+ (^
x
F (p))A
1
(c) The Gumbel case: Assume = 0,
De…ne I0; by
(
I0; =
1
p
Z
1
g x
1=
g x
1=( + )
dx + o(1) :
0
0 and U 2 ERV0 with auxiliary function a0 ( ).
R
1 1
2
1
dx
0
1
R1
0
g e
gp x
x
1=
dx;
< 0;
= 0:
dx;
(c1) When x^ = 1 further assume U 2 2ERV0; with auxiliary functions a0 ( ) and
A0 ( ). Then we have
Z 1
1
1
1
g e x dx+a0
A0
(I0; + o(1)):
Tp [X] = F (p)+a0
1 p 0
1 p
1 p
(c2) When x^ < 1, further assume x^
A0 ( ). Then we have
x^
Tp [X] = x^
U 2 2ERV0; with auxiliary functions a0 ( ) and
F (p)
a0
a0
1
1
9
p
A0
Z
1
1
p
1
1
1
g e
x
dx
0
p
(I0; + o(1)) :
Proof. (a) By Lemma 4.2 and Theorem 4.1, we obtain
Z 1
1
Tp [X] = F (p) 1 +
q dg(q) + F (p)A
1 p
0
Write I
I
;
;
=
=
R1
1
0
Z
q
q
1
dg(q). If
q
1
dg(q) =
q
q
Z
1
g x
R1
Notice that if =
then 0 g x 1=( +
theorem. If > j j > 0 then
Z 1
q
1
q
dg(q) =
0
)
1
dg(q) + o(1) :
0
1=( + )
dx +
Z
1
g x
1=
dx :
1
0
0
When
1
j j, we continue to derive
1
q
Z
dx = 0, where we used the dominated convergence
1
Z
1
g x
1=( + )
g x
1=
dx:
1
= 0, by the dominated convergence theorem we obtain
Z 1
Z 1
q
1
1
q log q dg(q) = lim
q
dg(q)
!0 0
0
Z 1
1
= lim
g x 1=( + )
g x
!0
1=
dx:
1
R1
dx.
g x 1=
Thus for > j j 0, we have I ; = 1 1 g x 1=( + )
(b) By Lemma 4.2 and Theorem 4.1, we obtain
Z 1
x^ Tp [X] = (^
x F (p)) 1
g x 1= dx
0
Z 1
1
q
1
q
dg(q) + o(1) :
+ (^
x F (p)) A
1 p
0
Similar as in (a), it holds for all
0 that
Z 1
Z
q
1
1
dg(q) =
q
1
g x
1=
g x
1=( + )
dx:
0
0
Then we obtain the desired result.
(c) Notice that U 2 2ERV0; with auxiliary functions a0 ( ) and A0 ( ) such that
lim
t!1
U (tx) U (t)
a0 (t)
A0 (t)
log x
x
=
1
2
1
;
(log x)2 ;
< 0;
= 0:
Then the asymptotics for the Gumbel case can be derived similarly by using Lemma 4.1.
10
5
Numerical Examples
In this section, we use R to numerically examine the accuracy of …rst-order and second-order
asymptotics derived in section 3 and 4.
Example 5.1 (The Fréchet case) Assume that F is a Pareto distribution given by
F (x) = 1
;
x+
x; ; > 0:
Thus F 2 MDA( ) and U (t) = (t 1). Easily one can check U 2 ERV with = 1= and
U 2 2RV ; with the second-order auxiliary function A(t) = t . We choose the distortion
p
function g(x) = x and set = 2:1 and = 1. In Graph 5.1, we compare the exact value,
…rst-order and second-order asymptotic values for Tp [X] on the left and show the ratios of
exact value to both asymptotic values on the right. We …nd that both ratios converge to 1
as p " 1 and second-order asymptotics is more accurate than …rst-order asymptotics.
Graph 5.1 about here.
Example 5.2 (The Weibull case) Assume that F is a beta distribution with probability
density function given by
f (x) =
xa 1 (1 x)b
B(a; b)
1
;
0 < x < 1; a; b > 0:
Thus, F 2 MDA( b ). One can check U 2 ERV with
the second-order auxiliary function
A(t) =
a 1
b(b + 1)
t
bB(a; b)
1=b and 1
=
U 2 2RV
;
with
1
b
;
p
see, for example, Mao and Hu (2012). We choose the distortion function g(x) = x and
set a = 2 and b = 6. Similarly as in Graph 5.1, we compare the exact value for 1 Tp [x]
and its asymptotic values of beta distribution in Graph 5.2. Again we can see that both
ratios converge to 1 as p " 1 and second-order asymptotics is more accurate than …rst-order
asymptotics.
Graph 5.2 about here.
Example 5.3 (The Gumbel case) Assume that F is a Weibull distribution given by
a
F (x) = 1
x
e (b) ;
x > 0; 0 < a < 1; b > 0:
Thus, F 2 MDA( ) and U (t) = b (log t)1=a . If we set a = 1=2 and b = 1, then U 2 2ERV0;0
with …rst-order auxiliary function a(t) = 2 log t and second-order auxiliary function A(t) =
11
1= log t. We choose the distortion function g(x) = x2 . Similarly as in Graph 5.1, we compare
the exact value and asymptotic values for Tp [x] of Weibull distribution in Graph 5.3. Again
we …nd that both ratios converge to 1 as p " 1 and second-order asymptotics is more accurate
than …rst-order asymptotics.
Graph 5.3 about here.
Acknowledgments. The author would like to thank Professor Qihe Tang for his useful
comments on this paper. The project was supported by a Centers of Actuarial Excellence
(CAE) Research Grant from the Society of Actuaries.
References
[1] Denneberg, D. Non-additive Measure and Integral. Kluwer Academic Publishers Group,
Dordrecht, 1994.
[2] Dhaene, J.; Vandu¤el, S.; Goovaerts, M. J.; Kaas, R.; Tang, Q.; Vyncke, D. Risk
measures and comonotonicity: a review. Stoch. Models 22 (2006), no. 4, 573–606.
[3] Embrechts, P.; Klüppelberg, C.; Mikosch, T. Modelling Extremal Events for Insurance
and Finance. Springer-Verlag, Berlin, 1997.
[4] Fisher, R. A.; Tippett, L. H. C. Limiting forms of the frequency distribution of the
largest or smallest member of a sample. Math. Proc. Cambridge Philos. Soc. 24 (1928),
no. 2, 180–190.
[5] Hardy, M.; Wirch, J. Ordering of risk measures for capital adequacy, in Proceedings of
the AFIR Colloquium, Tromsoe, 2003.
[6] Mao, T.; Hu, T. Second-order properties of Haezendonck–Goovaerts risk measure for
extreme risks. Insurance Math. Econom. (2012) to appear.
[7] Tsanakas, A. Dynamic capital allocation with distortion risk measures. Insurance Math.
Econom. 35 (2004), no. 2, 223–243.
[8] Valdez, E.; Chernih, A. Wang’s capital allocation formula for elliptically contoured
distributions. Insurance Math. Econom. 33 (2003), no. 3, 517–532.
[9] Wang, S. Premium calculation by transforming the layer premium density. Astin Bull.
26 (1996), no. 1, 71–92.
[10] Wang, S. A universal framework for pricing …nancial and insurance risks. Astin Bull.
32 (2002), no. 2, 213–234.
[11] Zhu, L.; Li, H. Tail distortion risk and its asymptotic analysis. Insurance Math. Econom.
51 (2012), no. 1, 115–121.
12
1.30
600
1.20
1.15
1.05
1.10
Ratio of exact value to asymptotic estimate
500
400
300
200
1.00
100
0
Tail distortion risk measure
Exact / 1st−order
Exact / 2nd−order
1.25
Exact
1st−order
2nd−order
0.95
0.96
0.97
0.98
0.99
1.00
0.95
p
0.96
0.97
0.98
p
Graph 5.1 Frechet case: Pareto distribution
0.99
1.00
0.995
0.975
0.980
Ratio of exact value to asymptotic estimate
0.985
0.990
0.35
0.30
0.25
0.95
0.96
Exact / 1st−order
Exact / 2nd−order
0.970
Tail distortion risk measure
0.20
0.15
0.10
Exact
1st−order
2nd−order
0.97
0.98
0.99
1.00
0.95
p
0.96
0.97
0.98
p
Graph 5.2 Weibull case: Beta distribution
0.99
1.00
1.4
100
Exact / 1st−order
Exact / 2nd−order
1.3
1.2
1.1
Ratio of exact value to asymptotic estimate
60
40
1.0
20
0
Tail distortion risk measure
80
Exact
1st−order
2nd−order
0.95
0.96
0.97
0.98
0.99
1.00
0.95
p
0.96
0.97
0.98
p
Graph 5.3 Gumbel case: Weibull distribution
0.99
1.00