By Werner Hürlimann, Switzerland
Abstract.
To provide incentive for active risk management, it is argued that a sound coherent distortion risk measure should preserve some higher degree stop-loss orders, at least the degree three convex order. Such risk measures are called free of tail risk or simply tail-free risk measures.
It is shown that under some common axioms and other plausible conditions, a tail-free coherent distortion risk measure identifies necessarily with the Wang right-tail measure or the expected value measure. This main result is applied to derive an optimal economic capital formula.
Key words : coherent risk measure, distortion risk measure, tail-free risk measure, higher degree convex order, Bernoulli distribution, Pareto distribution, economic capital
1.
Introduction.
The axiomatic approach to risk measures is an important and very active subject, which applies to different topics of actuarial and financial interest like premium calculation and capital requirements. Besides the coherent risk measures by Arztner et al.(1997/99), one is interested in the distortion risk measures by Denneberg(1990/94), Wang(1995/96) and Wang et al.(1997). Under certain circumstances, distortion risk measures are coherent risk measures
(e.g. Wang et al.(1997), Theorem 3). For this reason, they can be used to determine the capital requirements of a risky business, as suggested by several authors including Wirch and
Hardy(1999), Goovaerts et al.(2002) and Wang(2002).
However, despite of being coherent, a lot of distortion risk measures, like conditional value-at-risk (identical to expected shortfall) or the very recent Wang transform risk measure, do not always provide incentive for risk management because they lack of giving a capital relief in some simple two scenarios situations of reduced risk (Examples 3.1 and 3.2). To prevent the existence of such pathological counterexamples, one is interested in distortion risk measures that preserve the higher degree stop-loss orders (e.g. Hürlimann(2000b), Yoshiba and Yamai(2001)). More precisely, it is known that a coherent distortion risk measure preserves the usual stochastic order and the usual stop-loss order (e.g.
Hürlimann(1998a)).This is a desirable property because increased risk should be penalized with an increased risk measure. With equal means and variances, a stop-loss order relation between different random variables cannot exist. In this situation, increased risk can be modeled by the degree three stop-loss order or more specifically, by equal mean and variance, the degree three convex order. Thus, one is interested in distortion risk measures that preserve this higher degree convex orders. Such measures are called free of tail risk or simply tail-free distortion risk measures. The present contribution shows that under some plausible conditions, a non-trivial degree two tail-free coherent distortion risk measure must necessarily coincide with the Wang right-tail risk measure proposed by Wang(1998). A more detailed account of the paper follows.

2
Section 2 recalls the notions of coherent risk measure and distortion risk measure and identifies the distortion risk measures that induce coherent risk measures. Section 3 defines the notion of a higher degree tail-free risk measure by requiring the preservation property under a higher degree convex order. It is shown through counterexamples that the conditional value-at-risk or expected shortfall and the new Wang transform risk measure are not degree two tail-free coherent risk measures. In Proposition 4.1, we identify a necessary and sufficient condition for a coherent distortion risk measure to be a degree two tail-free risk measure on the subset of biatomic losses. Then, in Proposition 4.2, we show that the coherent proportional hazard or PH-distortion risk measure g ( t )
 t

,
 

0 , 1

, is degree two tail-free if, and only if, one has
 

0 , 1
2

or
 
1 . In Section 5, it is shown that the special case
 
1
2
, that is the Wang right-tail measure, remains finite and degree two tail-free on the subset of all positive affine transformed Pareto losses with finite means and variances. These simple results are the essential ingredients of our main Theorem 6.3. Suppose the set of losses contain all Bernoulli losses, all Pareto losses, and the positive affine transforms thereof. If the risk measure remains finite for losses with finite means and variances, and if it satisfies some plausible axioms, then the risk measure identifies necessarily with the Wang right-tail risk measure or the trivial expected value measure. This main result is applied to the determination of an optimal economic capital formula in Section 7. As a practical illustration, we recover the quite old “ 
2.
Coherent distortion risk measures.
Let


, A , P

be a probability space such that

is the space of outcomes or states of the world, A is the

-algebra of events and P is the probability measure. For a measurable real-valued random variable X on this probability space, that is a map X :
 
R , the probability distribution of X is defined and denoted by F
X
( x )

P ( X
 x ) .
 
In the present paper, the random variable X represents a financial loss such that for

the real number for a loss and X (

)

0
X (

) is the realization of a loss and profit function with X (

)

0
for a profit. A set of financial losses is denoted by

. A risk measure is a functional from the set of losses to the extended non-negative real numbers described by a map R :
 
 
. A coherent risk measure is a risk measure, which satisfies the following desirable properties (e.g. Arztner et al.(1997/99)) :
(M) (monotonicity) If
F
X
( x )

F
Y
( x )
X , Y
 
are ordered in stochastic dominance of first order, that is
for all x, written X
 st
Y , then R
   
(P) (positive homogeneity) If a

0 is a positive constant and
(S) (subadditivity) If X , Y , X

Y
 
then R

X
(T) (translation invariance) If c is a constant and

X
Y

R X

X
R

Y

    

then R

X
then
 c
R
 
 aR
 
  
 c
Definitions 2.1.
A continuous increasing function g ( 1 )

1 g :
   
is called distortion function. The dual transform

( x )

1
such that
 g ( 1
 x ) g ( 0 )

0 and
of a distortion function is called dual distortion function. For X
 
with probability distribution F
X
( x ) , the transform F
X g
( x ) :
 g ( F
X
( x )) defines a distribution function, which is called distorted distribution function. The dual distortion function defines a transformed distribution function
F
X

( x ) :
 
( F
X
( x )) , which is called dual distorted distribution function.

3
Taking the mean value with respect to the distorted distribution of a loss X
 
with probability distribution F
X
( x ) , one obtains the distortion (risk) measure
R g



0
1


F
X g
( x ))
 dx

0

 
F
X g
( x ) dx . (2.1)
Similarly, the dual distorted distribution defines the dual distortion (risk) measure
R




0
1


F
X

( x ))
 dx

0

 
F
X

( x ) dx (2.2)
One notes that the dual transform

( x )

1
 g ( 1
 x ) implies the following alternative dual representations of the distortion measures (2.1) and (2.2) in terms of the distorted survival function F
X g
( x ) :
 g ( F
X
( x ))

1

F
X

( x ) and the dual distorted survival function
F
X

( x ) :
 
( F
X
( x ))

1

F
X g
( x ) associated to the survival function F
X
( x )

1

F
X
( x ) :
R g
R





0
F
X g
( x ) dx


0
F
X

( x ) dx





0
1




0
1



F
X g
F
X

( x ))
 dx

R

 
( x ))
 dx

R g
 
(2.3)
(2.4)
Wang et al.(1997), Theorem 3, implies that the risk measures (2.3) and (2.4) are coherent risk measures provided that g ( x ) is a concave (

( x ) is a convex) function. This implies that
(2.1) and (2.2) are coherent provided that g ( x ) is a convex (

( x ) is a concave) function.
For completeness, let us also mention a further duality between losses and gains, the latter being defined as negative losses. With this result, it suffices to study risk measures of either losses or gains.
Lemma 2.1.
Let

( x )

1
 g ( 1
 x )
X
 
be a loss random variable, g ( x ) a distortion function, and
the dual distortion function. Then one has the relationships
R g
 
 
R

 
, R g
 
 
R

 
. (2.5)
Proof.
Using that F
(

X )
( x )

1

F
X
(
 x ) and making the substitution x
  t one obtains
R g

  
1
0
 



0


( F
X
( t )) dt g ( 1

F
X
(
 x ))
 dx

 
0


1
 
( F
X
( t ))
 dt



0
 g ( 1

F
X
(
 x )) dx

R

 
.
The second relation in (2.5) is shown similarly.


4
3.
Tail-free distortion risk measures.
Besides monotonicity, that is preservation of stochastic dominance of first order, it is known that a distortion measure R g
 
with concave distortion function preserves the stoploss order or increasing convex order (e.g. Hürlimann(1998a)).This is a desirable property because increased risk should be penalized with an increased measure. With equal means and variances, a stop-loss order relation between different random variables cannot exist. In this situation, increased risk can be modeled by the degree three stop-loss order or equivalently, by equal mean and variance, the degree three convex order. Thus, one is interested in distortion measures, which preserve this higher degree convex orders. As suggested by
Yoshiba and Yamai(2001), such measures should be called free of tail risk or simply tail-free distortion measures. Some more formal definitions and properties are required.
For any real random variable X with distribution partial moments
 n
X
( x )

E (

X
 x ) n


, n

0 , 1 , 2 ,...
F
X
( x ) , consider the higher order
, called degree n stop-loss transforms . For n=0 the convention is made that ( x
 d )
0

coincides with the indicator function 1  x
 d
 , hence
 0
X
( x )

F
X
( x )

1
 stop-loss transform
F
X

( x ) is simply the survival function of X . For n=1 this is the usual
( x ) , written without upper index. It is not difficult to establish the
X recursion (e.g. Hürlimann(2000a), Theorem 2.1)

X n
( x )
 n
  x

 n

1
( t ) dt ,
X n

1 , 2 ,...
. (3.1)
It will be useful to consider the following variants of the higher degree stop-loss orders (see
Kaas et al.(1994), Hürlimann(2000a) among others).
Definitions 3.1.
For n=0,1,2,..., a random variable X precedes Y in degree n stop-loss transform order , written X
 ( n ) slt
Y , if for all x one has
 n
X
( x )
  n
Y
( x ) . A random variable
X precedes Y in degree n stop-loss order , written inequalities
E
   
, k

E
   
0 ,..., j ,
, k

1 ,...,
for some
X
 ( sl n )
Y , if X
 ( n slt
)
Y and the moment n j


1 ,

0 are satisfied. With equal moments
,..., n

, the relation is written X
 ( n sl ,
) j
Y . In particular, the one extreme case
 ( n ) sl , 0
 ( n ) sl
defines a general degree n stop-loss order and the other one
 ( n ) sl , n

( n

1 )
 cx
defines the so-called (n+1)-convex order recently studied by
Denuit et al.(1998). Note that the special case
 ( 0 ) sl
is identical with the usual stochastic order or stochastic dominance of first order, also denoted coincides with the usual stop-loss order
 sl

. For n=1 the stochastic order
 st
or equivalently increasing convex order
 icx
. sl
For fixed n , the above stop-loss order variants satisfy the following hierarchical relationship

( n

1 )
 cx
 ( n ) sl , n
  ( n ) sl , n

1

...
  ( n ) sl , 1
  ( n ) sl , 0
 ( n ) sl
  ( n ) slt
. (3.2)
Moreover, the higher degree stop-loss orders build a hierarchical class of partial orders (Kaas et al.(1994), Theorem 2.2), that is one has
 ( n ) sl , j
  ( n

1 ) sl , j
, j


0 ,..., n

. (3.3)

5
Definition 3.2.
A risk measure R :
 
0 , it is preserved under the (n+1)convex order, that is if
R
   
.
is called a degree n tail-free risk measure if
X , Y
 
satisfy X

( n

1 )
 cx
Y then
As mentioned above, it is known that a distortion measure distortion function preserves

( n

1 )
 cx
for
R g
with concave n

0 , 1 , and is thus a tail-free risk measure of degree zero and one. In the present paper, we are interested in specific concave distortion functions g ( x ) such that R g
 
is a degree two tail-free risk measure. For motivation, it is very important to emphasize the practical relevance of tail-free distortion measures. Our field of application is here risk management.
Example 3.1 : Conditional value-at-risk versus Wang right-tail measure
Consider the coherent distortion measure (2.3) defined by the increasing concave distortion function g

( x )
 min
 
, where

is a small probability of loss, say
 
0 .
05 . By definition, the measure associated to X
 
is denoted R g 
. It is known that this risk measure coincides with several other known risk measures like the conditional value-at-risk measure and the expected shortfall measure (e.g. Hürlimann(2001a)). In now standard notation, conditional value-at-risk at the confidence level
 
1
 
, written CVaR

 
, coincides with R g 
. For comparison, consider the distortion function g ( x )
 x . The coherent distortion measure (2.3), called Wang right-tail measure and denoted by
WRT
 
:

R g
 
, has been proposed by Wang(1998) as a measure of right-tail risk. For illustration, let now Y be a loss consisting of two scenarios with loss amounts 20$, 2100$ such that P ( Y

20 )

1

P ( Y

2100 )

25
26
. Through active risk management, assume that the lower amount can be eliminated and that the higher loss amount can be reduced to 1700$.
By equal mean and variance, this results in a loss X such that
P ( X

0 )

1

P ( X

1700 )

16
17
. Suppose a risk manager is weighing the cost of risk management against the benefit of capital relief. Then CVaR does not promote risk management because CVaR

 

1700

CVaR


20

2080

 
26

1620 , which shows that there is a capital penalty instead of a capital relief for either removing or reducing the initial loss amounts. However, the Wang right-tail measure offers a capital relief because
WRT
 

1700

1
17

412 .
3

WRT

20

2080

1
26

427 .
9 . Since Y is evidently a higher loss than X , the CVaR measure fails to recognize this feature. Even more, in this simple example X precedes Y in the degree three convex order. This shows through a meaningful counterexample that CVaR is not a degree two tail-free coherent risk measure.
Being aware of the fact that CVaR ignores useful information in a large part of the loss distribution, Wang(2002) has proposed a new coherent distortion measure, which should adjust more properly extreme low frequency and high severity losses. However, as the following counterexample shows, Wang’s most recent proposal does not generate a degree two tail-free coherent risk measure.

6
Example 3.2 : Wang transform measure versus Wang right-tail measure
Consider the distortion function normal distribution and g

( x )
 

 
1
( x )
  
1
(

)

, where

( x )

is a small probability of loss, say
 
0 .
05
is the standard
. This interesting choice finds further motivation in Wang(2000/2001) and defines the Wang transform measure
WT

 
:

R g
 
, where
 
1
 
. Similarly to Example 3.1, consider a biatomic loss Y such that P ( Y

90 )

1

P ( Y mean and variance such that

P (
100 ' 100 )
X

0 )


10 ' 000
10 ' 001
1

P (
. Let
X

X be a biatomic loss with the same
10 ' 010 )

100
101
. Obviously Y is a higher loss than X , but the Wang measure does not provide incentive for risk management because
WT

 

10 ' 010
 g

 
101

2468 .
5

WT


90

100 ' 100
 g

 
10 ' 001

1993 .
3 . However, the
Wang
WRT
  right-tail

10 ' 010

1
101 measure

996 .
0

WRT offers a

90

100 ' 100
 capital
1
10 ' 001 relief because

1090 .
1 . Since X precedes
Y in the degree three convex order, the Wang transform measure is not a degree two tail-free coherent risk measure.
4.
Tail-free distortion risk measures for biatomic losses.
To prevent counterexamples of the type presented in the Examples 3.1 and 3.2, we derive a condition on the distortion function, which guarantees that the measure defined in (2.3) is a degree two tail-free distortion measure when restricted to the subset of biatomic losses with equal mean and variance. For this one must show that the distortion measure is preserved under the degree three convex order.
Let D
2
denote the subset of all real-valued standard biatomic random variables with mean zero and variance one. Recall that an element X

D
2
is uniquely determined by its support
 
with x

0 , x
  x

1
, and probabilities P ( X
 x )

1

P ( X
 x )
It is convenient to identify X with its support and use the short-hand notation

( 1
X

 x x
2
,
)

1
.
(e.g. Hürlimann(1998b), Theorem I.5.1).
Lemma 4.1.
Let only if, one has x
X

 
and
 y

0
 x
 y .
Y

 
belong to D . Then one has
2
X

3
 cx
Y if, and
Proof.
If x
 y

0
 difference affirmation x

F
X
X
X
(
 y x
3

)
3

. Conversely, if the latter inequalities hold, one has

 cx cx
F
Y
Y
Y
(
then one has necessarily x )
E
 x
 x

E
E
follows from Denuit et al.(1998), Theorem 4.3.

X
 
3 

E
in distributions has two sign changes in the order y
 y , hence

Y

3
,

and the
,


. The
An arbitrary biatomic loss with mean

and standard deviation transform
   
X with X

 

D
2

is a positive affine
. The distortion measure (2.3) satisfies the relationship R g

   
X

    
R g
with
R g

1
 x
2 x
 g
1

1 x
2

1 x
. (4.1)

7
Applying Lemma 4.1 it follows that the distortion measure R g
is tail-free of degree two for the subset of biatomic losses if, and only if, the function of one variable in (4.1) is monotone increasing.
Proposition 4.1.
Let g ( x ) be a continuous and differentiable increasing concave distortion function. The coherent distortion risk measure R g
 
is a degree two tail-free risk measure for the subset of biatomic losses if, and only if, the following condition holds : t
1
 t

1
 g ( t )

 g ( t )

2 tg ' ( t )

0 , t

. (4.2)
Proof.
The function h ( x )

1
 x x
2
 g
1

1 x
2

1 x
is monotone increasing if, and only if, one has
1 x
2 variables

 1
 t
 g (
( 1

1
 x
1
2
) x
2

1

) 
 g (
1

1 x
2
)

1

2 x
2 g ' (
1

1 x
2
)

0 ,
this condition identifies with (4.2).
 x

0 . Making the change of
Among the many concave distortion functions known in the literature, only a few turn out to generate degree two tail-free coherent risk measures for the subset of biatomic losses. For illustration, we present two attractive possibilities.
Example 4.1 : Proportional hazard or PH-distortion risk measure
The PH-transform g ( t

)
 t

,
 

0 , 1

, is due to Wang(1995) and has been justified on an axiomatic basis in Wang et al.(1997). If g ( t )

2 tg ' ( t )

Therefore, for

1

2
 
 

0 , 1
2
 t
 
0 , t

. Since g ( t )

 
 

0 , 1
2
 one has
the condition (4.2) is satisfied.
, the PH-distortion measure is a degree two tail-free coherent risk measure for the subset of biatomic losses.
Example 4.2 : Lookback distortion risk measure
The lookback transform g ( t )
 t
 

1
   ln( t )

,
 

0 , 1

, is an interesting alternative to the
PH-transform. For
 
1 the lookback transform has been motivated in Hürlimann(1998c).
We show that for
 

0 , 1
2

the lookback distortion risk measure is a degree two tail-free coherent risk measure for the subset of biatomic losses. Denote by H ( t ) condition (4.2). Using that
 
1
2
, a calculation shows the inequality
the left-hand side in
H ( t )
 t


1
 t
 
1 

1

2 t

1
2 t
 ln( t )
 
.
It suffices to show that the curly bracket is non-negative. Setting t

1
 u this is equivalent to the inequality
G ( u )


1
 u
  
1 

1

2 u

1
2

1
 u

 ln(
1
1
 u
)


1 , u

.

8
Using the series expansion
 ln( 1
 u )
 k



1
1 k u k
, one obtains that
G ( u )


1
 u
  
1 



1
 u

1
2

 u
 k



1

1 k
 k
1

1
 u k

1 



 which shows the desired inequality.


1
 u
  
1 ,
The following precise characterization will be crucial in Section 6.
Proposition 4.2.
Let g ( t )
 t

,
 

0 , 1

, be the PH-transform. Then the PH-distortion risk measure is degree two tail-free for the subset of biatomic losses if, and only if, one has
 

0 , 1
2

or
 
1 .
Proof.
With the result of Example 4.1, it remains to prove that the PH-distortion measure is not degree two tail-free in case
 
 
2
. Let Y


2 x , 1
2 x

, X

 
be biatomic losses in D such that
2
Y

3
 cx
X . Using (4.1) we show that there exists x

0 such that
R g
 

R g
 

1
2 x


2

2

1
 x
2

1
 

4


4
 x
2

1
 


0 . (4.3)
With the substitution t

( 1
 x
2
)

1
it suffices to show that there exists t

0 such that h ( t )

2
2
 
1

1

3 t

1
  
1
 t
1
  
0 . (4.4)
Clearly, for on
 
 
 
2
, 1
0 , 1 there exists t
0
0 , 1 h ( 0 )
 2
one has

 
2
such that
 
1 
1
 h ( t
0
)

0 and h ( 1 )

0 . Since h ( t ) is continuous
0 .

Remarks 4.1.
The lookback distortion risk measure is degree two tail-free for biatomic losses if, and only if, one has
 

0 , 1
2

(Hürlimann(2002), Proposition 5.2). Though the Wang transform in
Example 3.2 is not degree two tail-free, numerical calculation suggests that the modified
Wang transform g ( t )
 


 
1
( t )
 
1
(

)

 (4.5) is degree two tail-free for biatomic losses. However, a simple proof of this conjectural statement is at present not available.
5.
A tail-free distortion risk measure for Pareto losses.
Though discrete scenarios (the two-state scenarios in Section 4 being special cases) are more and more important in practice, a great deal of risk management methodology is based

9 on continuous distributions for the random losses. A prominent example is the three parameter Pareto random variable denoted X

P ( a
X
, c
X
,

X
) with survival function
F
X
( x )



1
 x
 a
X c
X


 
X
, x
 a x
, c
X

0 ,

X

0 , (5.1) and which plays an essential role in the numerous applications of Extreme Value Theory
(EVT) (see Embrechts et al.(1997) for background). Note that c is a scale parameter, and
X variance

2
and skewness
X

X
is a shape parameter called Pareto index . The mean

X a is a location parameter,
X

X
,
exist only under restriction of the Pareto index and are given by

X
 a
X


X c
X

1
,

X

1 ,

2
X

X

(

X

2


X
 a
X
)




X

X

2


,

X
2





X
X


1
3


,

X

X

2 ,

3 .
(5.2)
(5.3)
(5.4)
To check the degree two tail-free property of distortion risk measures for the set of Pareto losses, one needs a characterization of degree three convex ordering within the class with equal finite mean and variance. Under the assumption

X

2 , we reparameterize the Pareto
P ( a
X
, c
X
,

X
) to P ( a
X
, m
X
,

X
) with the substitution m
X


X c
X

1
,

X
2 


X
X

2
. (5.5)
Lemma 5.1.
Let X

P ( a
X
, m
X
,

X
) and equal mean and variance. Then one has X
Y

3
 cx

Y
P ( a
Y
, m
Y
,

Y
) be two Pareto losses with
if, and only if, the following conditions hold : a m
Y
Y

 a
X m
X
,
 a
X

Y
 m
X m
Y
 
X
.
 a
Y
,
(5.6)
(5.7)
(5.8)
Proof.
With (5.5) the condition of equal mean is (5.7) and the condition of equal variance is
(5.8). If

X
 
Y
X

3
 cx
, hence
Y

Y
then one has necessarily
 
X
, m
X
 m
Y
and a
Y
F
X
 a
(
X x )

F
Y
( x ) as x
 
, which implies that
. Conversely, if (5.6)-(5.8) hold, consider the logarithmic derivatives of the density functions

X
( t )
 d dt ln f
X
( t ), t
 a
X
,

Y
( t )
 d dt ln f
Y
( t ), t
 a
Y
. Their difference on the common support is

( t )
 
Y
( t )
 
X
( t )
 c
X
1
 
X
 a
X
 t
 c
Y
1

  a
Y
Y
 t
, t
 a
Y
.

10
One has t lim
 

( t )

0 and

( t )

0 as t
 
. Next, let us show that

( a
Y
)

( 1
 
X
) c
Y

( c
X
( 1

 a
X

Y

)( c a
Y
X

) c
Y a
Y
 a
X
)

0 .
Indeed, one has
( 1



Y
( 1

)( c
X

Y

)( m
X a
Y

X

 a
X m
X
)

( 1

 a
Y


X a
X
) c
Y
)

( 1
 
X
)( m
Y

Y
 m
Y
).
With (5.7) this equals
( 1
 
Y
)

X m
X

( 1
 
X
)

Y m
Y
 m
Y
(

X
 
Y
).
But m
X
 m
Y
by (5.6) and (5.7), which implies that the displayed expression is always greater or equal to 2 (

Moreover, one sees that
X

 
( t )
Y
'

) m
Y
0
, hence non-negative. This shows that
has a unique stationary solution t
0

( a
Y
)

0 .
in the interval
( a
Y
,

) satisfying the equation c
X
 a
X
 t
0

1
1




X
Y
( c
Y
 a
Y
 t
0
).
Since

'' ( t
0 affirmation
)
X


0 the function
3
 cx
Y

( t ) has one sign change in ( a
Y
,

) from
follows from Denuit et al.(1998), Theorem 4.6.


to +. The
Our goal is to show that the coherent Wang right-tail measure WRT
 

R g
 
with g ( x )
 x is degree two tail-free for the subset of Pareto losses with finite means and variances. A straightforward calculation shows the formula
R g




X

,


( 2
2 ( 1

 
1 )


)

2
X
2
X

1
 m
X
,

1
2
 
2
X

2
X

1


.
 
1
2
 
2
X

2
X

1


,
(5.9)
To remain finite in practical applications for all choices of
 
 
2
. This fact will be crucial in Section 6.

X

1 requires necessarily that
Proposition 5.1.
Let g ( x )
 coherent distortion risk measure x
R g
be the Wang right-tail distortion function. Then the
 
is finite and degree two tail-free for the subset of all
Pareto losses with finite means and variances.

11
Proof.
It remains to show that if X

3
 cx
Y then R g
 

R g
 
. By (5.9) the latter holds if the inequality

Y
2 m
Y
 
2
X m
X


Y
2 m
Y
 
2
X m
X
is fulfilled. Using (5.8) and the proof of Lemma 5.1 one gets

2
X m
X

 m
X
 m
Y m
Y



0 , which shows the desired result.

6.
Axiomatic justification of the Wang right-tail risk measure.
We rely on the axiomatic characterization of the PH-distortion measure in Wang et al.(1997), which is briefly reviewed . Adding some extra conditions, we show that the unique non-trivial choice is necessarily the Wang right-tail measure defined in Example 3.1. The following axioms have been proposed.
(A1) ( conditional state independence ) The risk measure only on its distribution function F
X
( x )

P ( X
 x ) .
(A2) ( monotonicity ) If X , Y
 
satisfy X
 st
Y , then R
R
X
 

of a loss X
 
depends
(A3) ( comonotone additivity ) If X , Y
  increasing real functions f and g such that
R

X

Y
    
.
are comonotone, that is there exists Z
 
X
 f ( Z ), Y
 g ( Z )
and
, then
(A4) ( continuity
Furthermore, if
) For X
 
one has d lim
 
X

0 and d

0 one has
R
 max( d lim

0

R

(
X
X
, d )
 d )


, d lim
 
R
 min( X , d )
.

.
Special types of losses, which play a crucial role in the sequel are Bernoulli random losses
B p
, p

 
, with probability function p

P ( B p

1 )

1

P ( B p

0 ) , and Pareto losses
P

,
 
1 , with survival function P ( P

 x )

( 1
 x )
 
. We will also use positive affine transforms thereof, that is random variables of the type cB p
 a and which define the biatomic losses and the three parameter Pareto loss cP


P ( a , a c ,

, with c

0 ,
) studied in
Sections 4 and 5. For coherent risk measures the subadditive property (S) of Section 2 is also assumed. One has the following remarkable result.
Theorem 6.1.
( Characterization of coherent distortion risk measures ) Assume

contains all Bernoulli losses B p
, p

 
. The risk measure R :
 
 
satisfies the axioms (A1)-
(A4), the property R
 

1 and subadditivity if, and only if, there exists a continuous increasing concave distortion function g ( x ) such that R
 

R g
 
for all X
 
.
Proof.
This follows from Theorem 3 and the Appendix A in Wang et al.(1997).

Next, consider compound Bernoulli losses of the type X

B p
Y , p

, where Y represents the loss severity given that a loss has occurred, that is Y

( X X

0 ) , and Y is independent of B p
. The following axiom has been shown desirable.

12
(A5) ( reduction of compound Bernoulli losses ) If then R
    p p

R
 
.
X

B p
Y is a compound Bernoulli loss
Theorem 6.2.
( Characterization of the coherent PH-distortion risk measure ) Assume
 contains all Bernoulli losses B p
, p

 
. The risk measure R :
 
0 ,

satisfies the axioms (A1)-(A5) and subadditivity if, and only if, one has g ( t )
 t

,
 

0 , 1

, for all X
 
.
 
R
 

R g
 
with
Proof.
This follows from Theorem 4 and the Appendix A in Wang et al.(1997).

As motivated in Section 3, a sound risk measure, which provides incentive for risk management, should satisfy the following axiom.
(A6) ( degree two tail-free condition ) If relation X

3
 cx
Y , then R
   
.
X , Y
 
satisfy a degree three convex order
The following result derives the Wang right-tail measure in an axiomatic way.
Theorem 6.3.
Assume

contains all Bernoulli losses B p
, p

P

,
 
1 , and positive affine transforms thereof. If the risk measure R :

, all Pareto losses

 
is finite for losses with finite means and variances, satisfies the axioms (A1)-(A6) and subadditivity, then one has necessarily R
 

R g
 
with g ( t )
 t or g ( t )
 t , for all X
 
.
Proof.
By Theorem 6.2 one has that R g
    g ( t )
 t

,
 

0 , 1

. It is clear that the choice
satisfies all the required conditions. Therefore, let
 
0 , g ( t )

  t such
. As shown by formula (5.9), to remain finite for all three parameter Pareto losses with finite means and variances, the measure must satisfy the condition
 

1
2
, 1

. On the other hand, by Proposition
4.2, the tail-free condition (A6) is fulfilled for the set of biatomic losses if, and only if, one has
 

0 , 1
2

. Together this shows that
 
1
2
.

At this stage, some comments are in order. By Proposition 5.1, the converse of Theorem
6.3 holds provided the set

is restricted to the families of Bernoulli losses, or Pareto losses, and positive affine transforms thereof. It is not known for which largest set

the Wang right-tail measure is still tail-free of degree two, and thus satisfies (A6). However, numerical evidence shows that, for many of the common two-parameter continuous distributions used to model insurance claims data, the axiom (A6) is fulfilled for the Wang right-tail measure (see
Hürlimann(2000b)). It would be worthwhile to construct some simple counterexample if possible. Let us also mention that the Wang right-tail measure has been used to derive a
“golden pricing” rule in Hürlimann(2001b), Section 4. A further application to economic capital is proposed in the next Section.

13
7.
Application to economic capital.
The use of distortion risk measures to determine capital requirements of a risky insurance business can be found in many papers (e.g. Wirch and Hardy(1999), Goovaerts et al.(2002),
Wang(2002)). We follow the note by Dhaene and Goovaerts(2002).
Let X
 
represent the insurance loss (e.g. claims minus premiums) at the end of a oneyear period. To avoid the insolvency risk, an insurer borrows at the beginning of the period and at the interest rate i some economic capital C

EC
 
, which is assumed to depend only on the loss. The insurance company invests this capital at the risk-free interest rate r
 i . The resulting (net) interest on capital ( i
 r ) C should be as small as possible. On the other hand, insolvency occurs if X

C ( 1
 r ) , hence C should be as large as possible.
Therefore, an “optimal” compromise solution must be found. Theoretically, to eliminate the insolvency risk, the insurer could buy on the insurance market (if available) a stop-loss contract with payoff

X

C ( 1
 r )


. If the price of such a contract is set using a risk measure, then the cost of solvability equals R
 
X

C ( 1
 r )



. The aggregate cost of solvability and interest on capital determines the cost of capital function f ( C )

R
 
X

C ( 1
 r )




( i
 r ) C , which should be minimized. Assume insurance market prices are set using a coherent distortion measure such that R
 

R g
 
for all X
 
.
Using the distorted survival function F
X g
( x )
 g ( F
X
( x )) associated to the survival function of X , the cost of capital function can be rewritten as f ( C )

E g
 
X

C ( 1
 r )




( i
 r ) C , (7.1) where E g
 
denotes expectation of X under the distorted survival function. Under the assumption of continuous distributions, the optimal economic capital , which minimizes (7.1), and the corresponding minimum cost of capital are determined as follows (formulas (5) and
(7) in Dhaene and Goovaerts(2002)) : f
EC
 

1
( EC
 
)
1
 r
 i

1

 
X

1 r r
E g

X i

1

X r r

1
1
 r
F
X

1

 1
 g

1

( 1
 r ) EC
   i

1
 r r

 (7.2)
(7.3)
The formula (7.2) identifies the end of the period value of the optimal economic capital, that is
( 1
( 1
 r ) EC
 
 r ) EC
 

, as the value-at-risk of X at the confidence level
VaR

 
1
 g

1 i

1
 r r
, that is
in the usual notation. Similarly, the end of the period value of the minimum cost of capital identifies with the interest at the net rate ( i
 r ) on the distorted conditional value-at-risk of X at the same confidence level evaluated with respect to the distorted survival function, that is ( i
 r ) E g

X X

VaR

  

( i
 r ) CVaR g

 
, where the latter notation remembers the usual notation of conditional value-at-risk. In this setting, the considerable amount of recent research on related matters remains applicable.
Furthermore, if insurance market prices are set using the axioms (A1)-(A6), if
 contains Bernoulli losses, Pareto losses, and positive affine transforms thereof, and if the

14 distortion function is non-trivial, then Theorem 6.3 implies the unique choice g ( t )
 follows that the optimal confidence level for capital requirement should be equal to
 
1
 i

1
 r r

2
. t .
It
(7.4)
Let us conclude this study with a significant illustration.
Example 7.1.
Suppose the normal approximation can be applied, that is the standard normal distribution,

is the mean, and
F
X
( x )
 
( x



) , where

( x ) is

is the standard deviation. For the special choice i

7 .
5 %, r

3 .
75 % , one obtains from (7.2) with g ( t )
 t the optimal economic capital formula
EC
 

1
1
 r
  
3 .
01
 
, (7.5) which is reminiscent of the handy and quite old “
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Werner Hürlimann
Schönholzweg 24
CH-8409 Winterthur
Switzerland e-mail : whurlimann@bluewin.ch
Homepage : www.geocities.com/hurlimann53