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# A Review and Some Complements on Quantile Risk Measures and Their Domain

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## Abstract

In the present paper, we study quantile risk measures and their domain. Our starting point is that, for a probability measure Q on the open unit interval and a wide class L Q of random variables, we define the quantile risk measure ϱ Q as the map that integrates the quantile function of a random variable in L Q with respect to Q. The definition of L Q ensures that ϱ Q cannot attain the value + ∞ and cannot be extended beyond L Q without losing this property. The notion of a quantile risk measure is a natural generalization of that of a spectral risk measure and provides another view of the distortion risk measures generated by a distribution function on the unit interval. In this general setting, we prove several results on quantile or spectral risk measures and their domain with special consideration of the expected shortfall. We also present a particularly short proof of the subadditivity of expected shortfall.
risks
Article
A Review and Some Complements on Quantile Risk
Measures and Their Domain
Sebastian Fuchs 1*, Ruben Schlotter 2and Klaus D. Schmidt 3
1Faculty of Economics and Management, Free University of Bozen-Bolzano, 39100 Bolzano, Italy
2Fakultät für Mathematik, Technische Universität Chemnitz, 09126 Chemnitz, Germany;
ruben.schlotter@mathematik.tu-chemnitz.de
3Fachrichtung Mathematik, Technische Universität Dresden, 01062 Dresden, Germany;
klaus.d.schmidt@tu-dresden.de
*Correspondence: sebastian.fuchs@unibz.it; Tel.: +39-0471-013000
Received: 19 September 2017; Accepted: 2 November 2017 ; Published: 7 November 2017
Abstract:
In the present paper, we study quantile risk measures and their domain. Our starting
point is that, for a probability measure
Q
on the open unit interval and a wide class
LQ
of random
variables, we deﬁne the quantile risk measure
$Q as the map that integrates the quantile function of a random variable in LQ with respect to Q . The deﬁnition of LQ ensures that$Q
cannot attain the value
+
and cannot be extended beyond
LQ
without losing this property. The notion of a quantile risk
measure is a natural generalization of that of a spectral risk measure and provides another view of
the distortion risk measures generated by a distribution function on the unit interval. In this general
setting, we prove several results on quantile or spectral risk measures and their domain with special
consideration of the expected shortfall. We also present a particularly short proof of the subadditivity
of expected shortfall.
Keywords:
integrated quantile functions; quantile risk measures; spectral risk measures; subadditivity;
value at risk; expected shortfall
1. Introduction
In the present paper, we study quantile risk measures and their domain. Our starting point is
that, for a probability measure
Q
on the open unit interval and a wide class
LQ
of random variables,
we deﬁne the quantile risk measure
$Q as the map that integrates the quantile function of a random variable in LQ with respect to Q . The deﬁnition of LQ ensures that$Q
cannot attain the value
+
and
cannot be extended beyond
LQ
without losing this property. The notion of a quantile risk measure is a
natural generalization of that of a spectral risk measure and provides another view of the distortion
risk measures generated by a distribution function on the unit interval.
Quantile risk measures are thus mixtures of the values at risk at different levels and hence mixtures
of a parametric family of risk measures. Such mixtures have already been considered by Acerbi (2002),
who, however, gave little attention to the domain on which a given risk measure can be deﬁned;
he argued that in a real-world risk management application, the integral (deﬁning a risk measure) will
always be well-deﬁned and ﬁnite. Nevertheless, Acerbi (2002) proposed a maximal class of random
variables on which a given spectral risk measure is well-deﬁned and ﬁnite. In the case of a spectral
risk measure, the domain of a quantile risk measure proposed in the present paper contains the class
proposed by Acerbi (2002) and turns out to be a convex cone, which is of interest with regard to the
In this paper, we review and partly extend known results on quantile risk measures,
with particular attention to spectral risk measures and, in particular, expected shortfall, with emphasis
Risks 2017,5, 59; doi:10.3390/risks5040059 www.mdpi.com/journal/risks
Risks 2017,5, 59 2 of 16
on their maximal domain mentioned before. We deliberately adopt arguments from the literature,
with appropriate modiﬁcations if necessary, but some of our proofs and results are new.
The literature on risk measures is vast and rapidly growing. A substantial part of the theory can be
found in the monographs by Föllmer and Schied (2016), McNeil et al. (2015),
Pﬂug and Römisch (2007)
and Rüschendorf (2013) and in the references given in these books. Since the theory of risk measures is
inspired by two sources, ﬁnance and insurance, the deﬁnitions of ﬁnancial and insurance risk measures
are slightly different, and the terminology is not fully consistent; for example, the use of the term
expected shortfall is not generally agreed upon. In the present paper, we consider insurance risk
measures, which are closely related to premium principles, and to avoid more ponderous expressions,
we employ the short-term quantile risk measure for a well-deﬁned class of risk measures.
This paper is organized as follows: We ﬁrst ﬁx some notation, recall some basic properties of
the quantile function and present a couple of examples of distortion functions (Section 2). We then
introduce quantile risk measures and provide several alternative representations of quantile risk
measures and their domain, as well as conditions under which certain quantile risk measures can be
compared (Section 3). In the next step, we consider spectral risk measures and characterize spectral
risk measures within the class of all quantile risk measures (Section 4). We then present a particularly
short proof of the subadditivity of expected shortfall and use this result to show that a quantile risk
measure is subadditive if and only if it is spectral (Section 5). As a major issue of this paper, we proceed
with a detailed comparison of the domain of a quantile risk measure with the classes of random
variables proposed by Acerbi (2002) and Pichler (2013) in the spectral case (Section 6). Finally, and as a
complement, we brieﬂy discuss related integrated quantile functions occurring in the measurement of
economic inequality (Section 7).
2. Preliminaries
We use the terms positive and increasing in the weak sense which admits equality in the
inequalities deﬁning these terms. For
BR
, we denote by
χB
the indicator function of
B
(such
that χB(x) = 1 if xBand χB(x) = 0 if x/B). Furthermore, we denote:
by B(R)the σ-ﬁeld of all Borel sets of R,
by B((0, 1)) the σ-ﬁeld of all Borel sets of (0, 1)and
by λthe Lebesgue measure on B(R)or its restriction to B((0, 1)).
By the correspondence theorem, there exists a bijection between the distribution functions on
R
and the
probability measures on
B(R)
such that the probability measure
QG
corresponding to the distribution
function
G
satisﬁes
QG[(x
,
y]] = G(y)G(x)
for all
x
,
yR
such that
xy
. Correspondingly, there
exists a bijection between the distribution functions on
(
0, 1
)
and the probability measures on
B((
0, 1
))
.
Throughout this paper, we consider a ﬁxed probability space
(
,
F
,
P)
and random variables
(,F)(R,B(R)), and we denote:
by L0the vector lattice of all random variables,
by L1the vector lattice of all integrable random variables,
by L2the vector lattice of all square integrable random variables and
by Lthe vector lattice of all almost surely bounded random variables.
Then, we have
L⊆ L2⊆ L1⊆ L0
. For a random variable
X∈ L0
, we denote by
FX
its distribution
function R[0, 1]given by:
FX(x):=P[{Xx}]
and by F
Xits (lower) quantile function (0, 1)Rgiven by:
F
X(u):=infnxR
FX(x)uo
Risks 2017,5, 59 3 of 16
For
u(
0, 1
)
and
xR
, the quantile function satisﬁes
F
X(u)x
if and only if
uFX(x)
. Moreover,
the quantile function is increasing and has the following properties:
Lemma 1. Consider X,Y∈ L0. Then:
(1) If X Y, then F
XF
Y.
(2) If a R+, then F
aX =a F
X.
(3) If c R, then F
X+c=F
X+c.
(4) If X and Y are comonotone, then F
X+Y=F
X+F
Y.
(5) F
X+= (F
X)+.
A function
D:[
0, 1
][
0, 1
]
is said to be a distortion function if it is increasing and continuous
from the right and satisﬁes
D(
0
) =
0 and
supu(0,1)D(u) =
1 (and hence,
D(
1
) =
1). The restriction
of a distortion function
D
to
(
0, 1
)
is a distribution function on
(
0, 1
)
, and for simplicity, the probability
measure corresponding to the restriction of
D
to
(
0, 1
)
will be referred to as the probability measure
corresponding to D.
Example 1.
The terms attached to the following examples are the names of the risk measures resulting from the
respective distortion functions.
(1) Expectation: The function DE:[0, 1][0, 1]given by:
DE(u):=u
is a distortion function.
(2) Value at risk: For α(0, 1), the function DVaRα:[0, 1][0, 1]given by:
DVaRα(u):=χ[α,1](u)
is a distortion function.
(3) Expected shortfall: For α[0, 1), the function DESα:[0, 1][0, 1]given by:
DESα(u):=uα
1αχ[α,1](u)
is a distortion function; in particular, DES0=DE.
(4)
Expected shortfall of higher degree: For
nN
and
α[
0, 1
)
, the function
DESn;α(u):[
0, 1
][
0, 1
]
given by:
DESn;α(u):=uα
1αn
χ[α,1](u)
is a distortion function; in particular, DES1;α=DESα.
(5) Range value at risk: For α[0, 1)and β(0, α), the function DESα,β:[0, 1][0, 1]given by:
DESα,β(u):=uα+β
1αχ[αβ,1β)(u) + χ[1β,1](u)
is a distortion function; in particular, limβ0DESα,β(u) = DESα(u).
The distortion functions
DESn;α
, and in particular
DESα
and
DE
, are convex, whereas
DVaRα
and
DESα,β
are not
convex. Further distortion functions may be found e.g., in Hardy (2006).
Risks 2017,5, 59 4 of 16
Throughout this paper, we consider pairs
(D
,
Q)
consisting of a distortion function
D:[
0, 1
]
[
0, 1
]
and the probability measure
Q:B((
0, 1
)) [
0, 1
]
corresponding to
D
, and we use identical sub-
or super-scripts for both, Dand Q, in the case of a particular choice of Dor Q.
3. Quantile Risk Measures
Deﬁne:
LQ:=X∈ L0
Z(0,1)(F
X(u))+dQ(u)<
Then, we have L⊆ LQ, and the map $Q:LQ[,)given by:$Q[X]:=Z(0,1)F
X(u)dQ(u)
is said to be a quantile risk measure.
For every
X∈ L0
, we have
X∈ LQ
if and only if
X+∈ LQ
, by Lemma 1. This implies that, for
every
Z∈ L0
satisfying
ZX
for some
X∈ LQ
, we have
Z∈ LQ
. Lemma 1also yields the following
properties of a quantile risk measure:
Lemma 2. Consider X,Y∈ LQ. Then:
(1) If X Y, then $Q[X]$Q[Y].
(2) If a R+, then aX ∈ LQand $Q[aX] = a$Q[X].
(3) If c R, then X +c∈ LQand $Q[X+c] =$Q[X] + c.
(4) If X and Y are comonotone, then X +Y∈ LQand $Q[X+Y] =$Q[X] + $Q[Y]. The quantile risk measure$Q
is said to be subadditive if
$Q[X+Y]$Q[X] + $Q[Y] holds for all X , Y∈ LQ such that X+Y∈ LQ . We shall show that$Q
is subadditive if and only if
D
is convex
and that, in this case, LQis a convex cone; see Theorem 4below.
To obtain alternative representations of a quantile risk measure and its domain, we need the
following Lemma:
Lemma 3. The identities:
Z(0,1)(F
X(u))+dQ(u) = ZRx+dQDFX(x) = Z(0,)1(DFX)(x)dλ(x)
and: Z(0,1)(F
X(u))dQ(u) = ZRxdQDFX(x) = Z(,0)(DFX)(x)dλ(x)
hold for every X ∈ L0.
The following result is immediate from Lemma 3:
Theorem 1. The domain of $Qsatisﬁes: LQ=X∈ L0 ZRx+dQDFX(x)<=X∈ L0 Z(0,)1(DFX)(x)dλ(x)< and the identities:$Q[X] = ZRx dQDFX(x) = Z(0,)1(DFX)(x)dλ(x)Z(,0)(DFX)(x)dλ(x)
hold for every X ∈ LQ.
Risks 2017,5, 59 5 of 16
Because of the previous result, the quantile risk measure generated by the probability measure
Q
corresponds to the distortion risk measure generated by the distortion function
D
; the latter is also
Example 2.
(1) Expectation: The distortion function DEsatisﬁes DEFX=FX. Because of Theorem 1, this yields:
LQE=nX∈ L0
E[X+]<o
and:
$QE[X] = E[X] for every X ∈ LQE. (2) Value at risk: For α( 0, 1 ) , the probability measure QVaRα corresponding to DVaRα is the Dirac measure at α. This yields: LQVaRα=L0 and:$QVaRα[X] = F
X(α)
for every
X∈ LQVaRα
; in particular,
$QVaRα is ﬁnite. The quantile risk measure$QVaRα
is called value at
risk at level αand is usually denoted by VaRα.
(3) Expected shortfall: For α[0, 1), the probability measure QESαcorresponding to DESαsatisﬁes:
QESα=Z1
1αχ(α,1)(u)dλ(u)
Since F
Xis increasing and F
X(α)is ﬁnite for α(0, 1), this yields, because of (1),
LQESα=X∈ L0
Z(α,1)(F
X(u))+dλ(u)<
=X∈ L0
Z(0,1)(F
X(u))+dλ(u)<
=nX∈ L0
E[X+]<o
=LQE
and:
$QESα[X] = Z(0,1)F X(u)1 1αχ(α,1)(u)dλ(u) for every X∈ LQESα . In particular,$QES0=$QE , and$QESα
is ﬁnite for every
α(
0, 1
)
. The quantile
risk measure $QESαis called expected shortfall at level αand is usually denoted by ESα. (4) Expected shortfall of higher degree: For nN and α[ 0, 1 ) , the probability measure QESn;α corresponding to DESn;αsatisﬁes: QESn;α=Zn 1αuα 1αn1 χ(α,1)(u)dλ(u) This yields: LQESn;α=LQE and:$QESn;α[X] = Z(0,1)F
X(u)n
1αuα
1αn1
χ(α,1)(u)dλ(u)
Risks 2017,5, 59 6 of 16
for every
X∈ LQESn;α
. In particular,
$QES1;α=$QESα
, and
$QESn;α is ﬁnite for every nN and α( 0, 1 ) . The quantile risk measure$QESn;αis called expected shortfall of degree n at level α.
(5)
Range value at risk: For
α[
0, 1
)
and
β(
0,
α)
, the probability measure
QESα,β
corresponding to
DESα,β
satisﬁes:
QESα,β=Z1
1αχ(αβ,1β)(u)dλ(u)
This yields:
LQESα,β=L0
and:
$QESα,β[X] = Z(0,1)F X(u)1 1αχ(αβ,1β)(u)dλ(u) for every X∈ LQESα,β . In particular,$QESα,β
is finite for every
α(
0, 1
)
and
β(
0,
α)
. The quantile risk
measure
$QESα,β is called range value at risk at levels α and β ; see Cont et al. (2010) and Embrechts et al. (2017). The examples show that the domains of different quantile risk measures may be distinct. Lemma 3and Theorem 1have several applications. For example, they provide a condition on D under which$Qis ﬁnite:
Corollary 1. Assume that there exists some δ(0, 1)such that D(u) = 0holds for every u (0, δ). Then:
LQ=X∈ L0
Z(0,1)|F
X(u)|dQ(u)<
=X∈ L0
ZR|x|dQDFX(x)<
=X∈ L0
Z(0,)1(DFX)(x)dλ(x) + Z(,0)(DFX)(x)dλ(x)<
and $Qis ﬁnite. Proof. For every X∈ L0, the assumption yields: Z(0,1)(F X(u))dQ(u) = Z(,0)(DFX)(x)dλ(x) =Z(,0)(DFX)(x)χ[δ,1)(FX(x)) dλ(x) =Z(,0)(DFX)(x)χ[F X(δ),0)(x)dλ(x) (DFX)(0)Z(,0)χ[F X(δ),0)(x)dλ(x) Since F X(δ)is ﬁnite, this proves the assertion. Theorem 1also provides a condition for the comparison of the domains of quantile risk measures: Corollary 2. Assume that there exists some δ( 0, 1 ) such that D1(u)D2(u) holds for every u[δ , 1 ) . Then, LQ1⊆ LQ2. Risks 2017,5, 59 7 of 16 Proof. For every X∈ L0, we have: Z(0,)1(D2FX)(x)dλ(x) =Z(0,)1(D2FX)(x)χ(0,F X(δ))(x)dλ(x) + Z(0,)1(D2FX)(x)χ[F X(δ),)(x)dλ(x) Z(0,)χ(0,F X(δ))(x)dλ(x) + Z(0,)1(D1FX)(x)dλ(x) Since F X(δ)is ﬁnite, Theorem 1yields LQ1⊆ LQ2. Corollary 3. Assume that there exist some n Nand α,δ(0, 1)such that: DESn,α(u)D(u)DE(u) holds for every u [δ, 1). Then, LQ=LQE. Proof. Because of Corollary 2, we have LQESn,α⊆ LQ⊆ LQE . Now, the assertion follows from LQESn,α=LQE. Combining Corollaries 1and 3yields a condition under which LQ=LQE and$Q
is ﬁnite.
Corollary 2also yields some further results on the comparison of quantile risk measures and their
domains:
Corollary 4.
(1) D1D2if and only if $Q2[X]$Q1[X]holds for every X ∈ LQ1 LQ2,and in this case, LQ1⊆ LQ2.
(2) DDEif and only if E[X]$Q[X]holds for every X ∈ LQ LQE,and in this case, LQ⊆ LQE. (3) If D is convex, then LQ⊆ LQEand E[X]$Q[X]holds for every X ∈ LQ.
(4) Consider α,β[0, 1). Then, αβif and only if $QES α[X]$QESβ[X]holds for every X ∈ LQE.
(5) The identity E[X] = infα(0,1)$QESα[X]holds for every X ∈ LQE. Proof. Assume ﬁrst that D1D2 . Then Corollary 2yields LQ1⊆ LQ2 and Theorem 1yields$Q2[X]$Q1[X] for every X∈ LQ1∩ LQ2=LQ1 . Assume now that$Q2[X]$Q1[X] holds for every X∈ LQ1∩ LQ2 and consider u( 0, 1 ) . Then, for any choice of a , bR such that a<b and for every random variable X satisfying P[{X=a}] = u= 1 P[{X=b}] , we have X∈ L⊆ LQ1∩ LQ2 . Straightforward computation yields$Di[X] = b(ba)Di(u)
for
all
i∈ {
1, 2
}
, and hence,
D1(u)D2(u)
. Since
u(
0, 1
)
was arbitrary, it follows that
D1D2
. This
proves (1). Assertions (2)–(4) are immediate from (1), and Assertion(5) follows from the dominated
convergence theorem.
Assertion (1) of Corollary 4extends a result of Wang et al. (2015), who considered risk measures
that are deﬁned on a common convex cone containing L.
4. Spectral Risk Measures
A map
s:(
0, 1
)R+
is said to be a spectral function if it is increasing and satisﬁes
R(0,1)s(u)dλ(u) = 1.
The quantile risk measure
$Q is said to be a spectral risk measure if there exists a spectral function ssuch that: Q=Zs(u)dλ(u) Risks 2017,5, 59 8 of 16 Thus, if$Qis a spectral risk measure with spectral function s, then the domain of $Qsatisﬁes: LQ=X∈ L0 Z(0,1)(F X(u))+s(u)dλ(u)< and the identity:$Q[X] = Z(0,1)F
X(u)s(u)dλ(u)
holds for every
X∈ LQ
. Note that the spectral function of a spectral risk measure is unique almost
Example 3.
(1) Expectation: Since DE(u) = u, we have:
QE=λ
and the function sE:(0, 1)R+given by:
sE(u):=1
is a spectral function. Therefore, $QEis a spectral risk measure. (2) Value at risk: For every α( 0, 1 ) , QVaRα is the Dirac measure at α and hence does not have a density with respect to λ. Therefore,$QVaRαis not a spectral risk measure.
(3) Expected shortfall: For every α[0, 1), we have:
QESα=Z1
1αχ(α,1)(u)dλ(u)
and the function sESα:(0, 1)R+given by:
sESα(u):=1
1αχ(α,1)(u)
is a spectral function. Therefore, $QESαis a spectral risk measure. (4) Expected shortfall of higher degree: For every n Nand α[0, 1), we have: QESn;α=Zn 1αuα 1αn1 χ(α,1)(u)dλ(u) and the function sESn;α:(0, 1)R+given by: sESn;α(u):=n 1αuα 1αn1 χ(α,1)(u) is a spectral function. Therefore,$QESn;αis a spectral risk measure.
(5) Range value at risk: For every α[0, 1)and β(0, α), we have:
QESα,β=Z1
1αχ(αβ,1β)(u)dλ(u)
and the function sESα,β:(0, 1)R+given by:
sESα,β=1
1αχ(αβ,1β)(u)
Risks 2017,5, 59 9 of 16
fails to be increasing and hence fails to be a spectral function. Therefore,
$QESα,β is not a spectral risk measure. Our aim is to characterize the spectral risk measures within the class of all quantile risk measures. The following result is inspired by Gzyl and Mayoral (2008), who considered distortion risk measures on the positive cone of L2: Theorem 2. The following are equivalent: (a) D is convex. (b) There exists a spectral function s such that Q =Rs(u)dλ(u). (c)$Qis a spectral risk measure.
In this case, every spectral function s representing Q satisfies s =D0almost everywhere (with respect to λ).
Proof.
Since
limu0D(u) =
0
=D(
0
)
and
limu1D(u) =
1
=D(
1
)
,
D
is convex if and only if
D
is
convex on (0, 1).
Assume ﬁrst that (a) holds. The following arguments are taken from Aliprantis and
Burkinshaw (1990, chp. 29). Since
D
is increasing,
D
is differentiable almost everywhere, and since
D
is convex, its derivative
D0
is increasing. Consider now an arbitrary interval
[u
,
v](
0, 1
)
. Since
D
is convex, the restriction of
D
to
[u
,
v]
is Lipschitz continuous, hence absolutely continuous and,
thus, continuous and of bounded variation. Therefore, the restriction of
Q
to the
σ
-ﬁeld of all Borel
sets in
[u
,
v]
is absolutely continuous with respect to the restriction of
λ
derivative agrees with
D0
. Since
[u
,
v](
0, 1
)
was arbitrary, it follows that
Q
is absolutely continuous
with respect to
λ
, and since the Radon–Nikodym derivative
s:(
0, 1
)R+
of
Q
with respect to
λ
is
unique almost everywhere, it follows that
s=D0
almost everywhere. This yields the existence of an
increasing function s:(0, 1)R+satisfying Q=Rs(u)dλ(u). Therefore, (a) implies (b).
Assume now that (b) holds. Since
s
is increasing, we have, for any
u
,
v
,
w(
0, 1
)
such that
u<v<w,
D(v)D(u)
vu=1
vuZ(u,v]s(t)dλ(t)s(v)1
wvZ(v,w]s(t)dλ(t) = D(w)D(v)
wv
which implies that Dis convex. Therefore, (b) implies (a).
The following result is inspired by Kusuoka (2001), who studied risk measures on L:
Theorem 3. If D is convex, then there exists a measure ν:B([0, 1)) [0, ]such that:
$Q[X] = Z[0,1)(1α)$QESα[X]dν(α)
holds for every X ∈ LQ.
Proof.
Without loss of generality, we may and do assume that
s
is continuous from the right.
Deﬁne
s(
0
):=infu(0,1)s(u)
. Then, there exists a unique
σ
-ﬁnite measure
ν:B([
0, 1
)) [
0,
]
satisfying
ν[[
0,
u]] = s(u)
for all
u(
0, 1
)
. Since the map
(
0, 1
)×[
0, 1
)R:(u
,
α)F
X(u)χ[0,u](α)
is measurable and its positive part is integrable with respect to the product measure
νλ
, Fubini’s
theorem yields:
$Q[X] = Z(0,1)F X(u)s(u)dλ(u) =Z(0,1)F X(u)Z[0,1)χ[0,u](α)dν(α)dλ(u) Risks 2017,5, 59 10 of 16 =Z[0,1)Z(0,1)F X(u)χ(α,1)(u)dλ(u)dν(α) =Z[0,1)(1α)$QESα[X]dν(α)
This proves the assertion.
5. Subadditivity of Spectral Risk Measures
In the present section, we show that a quantile risk measure is subadditive if and only if its
distortion function is convex. To prove that the convexity of the distortion function is sufﬁcient for
subadditivity of the quantile risk measure, we use Theorem 3. Since the expectation is additive and
hence subadditive, it remains to show that the expected shortfall at any level is subadditive.
To establish subadditivity of the expected shortfall, we need the following lemma, which provides
another representation of the values of the expected shortfall:
Lemma 4. For every α(0, 1),the identity:
$QESα[X] = F X(α) + 1 1αEhXF X(α)+i=inf cRc+1 1αE[(Xc)+] holds for every X ∈ LQESα. Lemma 4is well-known and is frequently used to establish the subadditivity of expected shortfall on L ; see, e.g., Embrechts and Wang (2015), who used a general extension procedure to extend this result beyond L . Here, we use Lemma 4to establish the subadditivity of expected shortfall on its (maximal) domain LQESαin a single step: Lemma 5. For every α[0, 1),LQESαis a convex cone and$QESαis subadditive.
Proof.
Since
LQESα=LQE
, we see that
LQESα
is a convex cone. Furthermore, since
QES0=QE
, we see
that
$QES0 is subadditive. Consider now α( 0, 1 ) and X , Y∈ LQESα . Then, we have X+Y∈ LQESα and, for any x,yR, Lemma 4yields:$QESα[X+Y](x+y) + 1
1αEh(X+Y)(x+y)+i
=x+y+1
1αEh(Xx) + (Yy)+i
x+1
1αE[(Xx)+]+y+1
1αE[(Yy)+]
Now, minimization over
x
,
yR
and using Lemma 4again yields:
$QESα[X+Y]$QESα[X] + $QESα[Y] . Therefore,$QESαis subadditive for every α(0, 1).
The previous result provides the key for proving the main implication of the following theorem;
see also Wang and Dhaene (1998), who considered distortion risk measures on the positive cone of
L1
and used a proof based on comonotonicity.
Theorem 4. The following are equivalent:
(a) D is convex.
(b) $Qis subadditive. (c) LQis a convex cone, and$Qis subadditive.
Risks 2017,5, 59 11 of 16
Proof.
Assume ﬁrst that (a) holds, and consider a spectral function
s
representing
Q
and the measure
ν
constructed in the proof of Theorem 3. Consider
X
,
Y∈ LQ
and
aR+
. Then, we have
aX ∈ LQ
.
Moreover, since
D
is convex, Corollary 4yields
X
,
Y∈ LQE
. For every
α[
0, 1
)
, this yields
X,Y∈ LQESα
; hence,
X+Y∈ LQESα
, by Lemma 5; and thus,
X+
,
Y+
,
(X+Y)+∈ LQESα
. Proceeding
as in the proof of Theorem 3and using Lemma 5again, we obtain:
Z(0,1)F
(X+Y)+(u)s(u)dλ(u) = Z[0,1)(1α)$QESα[(X+Y)+]dν(α) Z[0,1)(1α)$QESα[X+] + $QESα[Y+]dν(α) =Z[0,1)(1α)$QESα[X+]dν(α) + Z[0,1)(1α)$QESα[Y+]dν(α) =$Q[X+] + $Q[Y+] < This yields (X+Y)+∈ LQ , and hence, X+Y∈ LQ . Thus, LQ is a convex cone, and Theorem 3 together with Lemma 5implies that$Q
is subadditive. Therefore, (a) implies (c). Obviously, (c) implies
(b), and it follows from Example 4below that (b) implies (a).
For the discussion of the subsequent Example 4, we need the following lemma:
Lemma 6. The following are equivalent:
(a) D is convex.
(b) The inequality:
D(u)1
2D(uε) + D(u+ε)
holds for all u (0, 1)and ε(0, min{u, 1u}).
Proof. Assume that (b) holds. Then, the inequality:
Du+v
21
2D(u) + D(v)
holds for all
u
,
v(
0, 1
)
, and this implies that
D
is continuous on
(
0, 1
)
. Since
D
is a distortion function,
it follows that
D
is continuous on
[
0, 1
]
, and now, the previous inequality implies that
D
is convex.
Therefore, (b) implies (a). The converse implication is obvious.
The bivariate distribution discussed in the following example was proposed by Wirch and Hardy (2002).
Example 4.
Assume that
D
is not convex. Then, Lemma 6yields the existence of some
u(
0, 1
)
and
ε(0, min{u, 1u})such that:
2D(u)>D(uε) + D(u+ε)
Consider random variables
X
,
Y∈ L
whose joint distribution is given by the following table with
a(
0,
)
:
xyP[{X=x}] P[{Xx}]
(a+ε)(a+ε/2)0
(a+ε)uε0εu u
0 0 ε1uε1u1
P[{Y=y}]uε ε 1u
P[{Yy}]uεu1
Risks 2017,5, 59 12 of 16
Then, the distribution of the sum X +Y is given by the table:
z2(a+ε)(a+ε)(a+ε/2)0
P[{X+Y=z}]uε ε ε 1uε
P[{X+Yz}]uεu u +ε1
Because of Theorem 1, this yields:
$Q[X] = (a+ε)D(u)$Q[Y] = (ε/2)D(uε)(a+ε/2)D(u)
$Q[X+Y] = (a+ε)D(uε)(ε/2)D(u)(a+ε/2)D(u+ε) and hence:$Q[X+Y] = $Q[X] +$Q[Y]+(a+ε/2)2D(u)D(uε)D(u+ε)
>$Q[X] +$Q[Y]
Therefore, $Qfails to be subadditive. 6. On the Domain of a Quantile Risk Measure In this section, we compare the domain: LQ=X∈ L0 Z(0,1)(F X(u))+dQ(u)< of the quantile risk measure$Qwith two other classes of random variables. Deﬁne:
LAcerbi
Q:=X∈ L0
Z(0,1)|F
X(u)|dQ(u)<
and:
LPichler
Q:=X∈ L0
Z(0,1)F
|X|(u)dQ(u)<
In the case where
Q
is represented by a spectral function, these classes were introduced by
Acerbi (2002)
and Pichler (2013), respectively. We have
LAcerbi
Q⊆ LQ
, and Corollary 1provides a sufﬁcient condition
for
LAcerbi
Q=LQ
. Moreover, since
X+≤ |X|
, we also have
LPichler
Q⊆ LQ
. Below, we shall show that
LPichler
Q⊆ LAcerbi
Qwhenever Dis convex. To this end, we need the following lemma:
Lemma 7.
Assume that
D
is convex and consider
X∈ L0
. If
X+∈ LAcerbi
Q
and
X∈ LAcerbi
Q
,then
X∈ LAcerbi
Q
.
Proof. From (F
X)+=F
X+and X+∈ LAcerbi
Q, we obtain:
Z(0,1)(F
X(u))+dQ(u)<
To prove that the integral
R(0,1)(F
X(u))dQ(u)
is ﬁnite, as well, we need the upper quantile function
F
X:(0, 1)Rgiven by:
F
X(u):=supnxR
FX(x)uo
Risks 2017,5, 59 13 of 16
The lower and upper quantile functions satisfy F
XF
X, and we have:
(F
X(u))=F
X(u)χ(0,FX(0)](u)
and:
F
X(1u) = F
X(u)χ(0,FX(0))(u)
almost everywhere with respect to
λ
. Since
D
is convex and hence continuous,
Q
is absolutely
continuous with respect to λ. This yields:
0Z(0,1)F
X(u)F
X(u)dQ(u)
=Z(0,1)ZRχ[F
X(u),F
X(u))(x)dλ(x)dQ(u)
ZRZ(0,1)χ{FX(x)}(u)dQ(u)dλ(x)
=0
and hence.
F
X=F
X
almost everywhere with respect to
Q
. Consider now a spectral function
s
representing Q. Since sis positive and increasing, we obtain:
Z(0,1)(F
X(u))dQ(u) = Z(0,1)(F
X(u)) χ(0,FX(0)](u)dQ(u)
=Z(0,1)(F
X(u)) χ(0,FX(0))(u)dQ(u)
=Z(0,1)(F
X(u)) χ(0,FX(0))(u)s(u)dλ(u)
=Z(0,1)F
X(1u)s(u)dλ(u)
=Z(0,1)F
X(u)s(1u)dλ(u)
Z(0,1/2)F
X(1/2)s(1u)dλ(u) + Z(1/2,1)F
X(u)s(u)dλ(u)
F
X(1/2) + Z(0,1)F
X(u)dQ(u)
Since X∈ LAcerbi
Q, the last expression is ﬁnite, and this yields:
Z(0,1)(F
X(u))dQ(u)<
Therefore, we have X∈ LAcerbi
Q.
Theorem 5. If D is convex, then LPichler
Q⊆ LAcerbi
Q.
Proof.
Consider
X∈ LPichler
Q
. Then, we have
|X|∈LPichler
Q
, hence
X+
,
X∈ LPichler
Q
, and thus,
X+,X∈ LAcerbi
Q. Now, Lemma 7yields X∈ LAcerbi
Q.
The following examples provide some further insight into the relationships between these three
classes of random variables:
Example 5.
(1) If D =DVaRα, then LPichler
Q=LAcerbi
Q=LQ=L0.
Risks 2017,5, 59 14 of 16
(2) If D =DE, then LPichler
Q=LAcerbi
Q=L16=LQ.
(3) If D =DESαfor some α(0, 1), then LPichler
Q6=LQ=LAcerbi
Q.
(4) Assume that there exists some δ(0, 1)such that D satisﬁes:
D(u) = uχ[0,δ)(u) + χ[δ,1](u)
(and hence, fails to be convex). Then, every X ∈ L0satisﬁes:
Z(0,1)F
|X|(u)dQ(u)<and Z(0,1)(F
X(u))+dQ(u)<
This yields LPichler
Q=L0=LQ, as well as:
LAcerbi
Q=X∈ L0
Z(0,1)(F
X(u))dQ(u)<
=X∈ L0
Z(0,δ)(F
X(u))dλ(u)<
=X∈ L0
Z(0,1)(F
X(u))dλ(u)<
=nX∈ L0
E[X]<o
such that LPichler
Q6=LAcerbi
Qand LAcerbi
Q6=LQ.
(5) Assume that D satisﬁes:
D(u) = 1
2uχ[0,1/4)(u) + uχ[1/4,1](u)
Then, Corollary 3yields LQ=LQE. Moreover, straightforward calculation yields:
Z(0,1)F
|X|(u)dQ(u)λ[(0, F
|X|(1/4))] + Z[F
|X|(1/4),)1(DF|X|)(x)dλ(x)
and: Z(0,1)F
|X|(u)dλ(u)λ[(0, F
|X|(1/4))] + Z[F
|X|(1/4),)1F|X|(x)dλ(x)
Since: Z[F
|X|(1/4),)1(DF|X|)(x)dλ(x) = Z[F
|X|(1/4),)1F|X|(x)dλ(x)
we see that LPichler
Q=L16=LQE=LQ. Consider, ﬁnally, a random variable X satisfying:
FX(x) = β
x2
χ(,β)(x) + χ[β,)(x)
for some
β(
0,
)
. Then,
X
has a Pareto distribution with ﬁnite expectation. This yields
X∈ L1=
LPichler
Q⊆ LQ. Since D(u)(1/2)uχ[0,1/4)(u), we obtain:
Z(0,1)|F
X(u)|dQ(u)Z(0,1)(F
X(u))dQ(u)
=Z(,0)(DFX)(x)dλ(x)
Z(,0)
1
2qFX(x)χ[0,1/4)(FX(x)) dλ(x)
Risks 2017,5, 59 15 of 16
=Z(,0)
1
2β
xχ(,β)(x) + χ[β,)(x)χ(,2β)(x)dλ(x)
=Z(,2β)
β
2xdλ(x)
=β
2Z(2β,)
1
zdλ(z)
and hence, X /∈ LAcerbi
Q. Therefore, any two of the three classes LQ,LAcerbi
Qand LPichler
Qare distinct.
7. Related Integrated Quantile Functions
Integrated quantile functions also occur in the measurement of economic inequality. To brieﬂy
give an idea of this topic, consider the class:
LLorenz :=nX∈ L0
X0 and E[X] = 1o
and the map L:LLorenz ×[0, 1)Rgiven by:
L(X,t):=Z(0,1)F
X(u)χ(0,t](u)dλ(u)
Then, for any X∈ LLorenz, the function LX:(0, 1)[0, 1]given by:
LX(t):=Z(0,1)F
X(u)χ(0,t](u)dλ(u)
is called the Lorenz curve of
X
. If the distribution of
X
is interpreted as the normalized income
distribution of a population, then the value
LX(t)
represents the proportion of the poorest 100
t
percent
of the population; see Rüschendorf (2013). On the other hand, for any t(0, 1)and with:
QLorenz,t:=Zχ(0,t](u)dλ(u)
the map $QLorenz,t:LLorenz [0, 1]given by:$QLorenz,t[X]:=Z(0,1)F
X(u)dQLorenz,t(u)
can be used to compare the proportions of the poorest 100
t
percent of different populations. Moreover,
the map $QGini :LLorenz [0, 1]given by:$QGini [X]:=2Z(0,1)(tLX(t)) dλ(t)
is called the Gini index of
X
and can be used to measure the inequality of the incomes within a given
population; see Bennett and Zitikis (2015) and Greselin and Zitikis (2015). Letting:
QGini :=Z(2u1)dλ(u)
we obtain:
$QGini [X] = Z(0,1)F X(u)dQGini (u) Formally, each of the maps$QLorenz,t
and
$QGini looks like a quantile risk measure, but it should be noted that the integrating measures QLorenz,t fail to be probability measures and that QGini is only a signed measure. Risks 2017,5, 59 16 of 16 Because of these examples, it appears to be reasonable to extend the notion of a quantile risk measure$Q
to the case of an arbitrary integrating measure or even an integrating signed measure
Q:B((0, 1)) R, although in the latter case, Property (1) of Lemma 2, would be lost.
Acknowledgments:
The authors are most grateful to the referees whose comments led to a substantial
improvement of this paper. The ﬁrst and the last author gratefully acknowledge substantial discussions on
risk measures with Désiré Dörner, André Neumann and Sarah Santo. The ﬁrst author also acknowledges
the support of the Faculty of Economics and Management, Free University of Bozen-Bolzano, via the project
NEW-DEMO.
Author Contributions: These authors contributed equally to this work.
Conﬂicts of Interest: The authors declare no conﬂict of interest.
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The potential for superadditivity that a risk measure displays, is directly linked to the potential for penalizing portfolio diversification. In this paper, we study the extent to which any risk measure can lead to superadditive risk assessments. For this purpose we introduce the notion of extreme-aggregation risk measures. The extreme-aggregation measure characterizes the most superadditive behavior of a risk measure, by yielding the worst-possible diversification ratio across dependence structures. One of the main novelties in our contribution is demonstrating, for a wide range of risk measures, including distortion and convex risk measures, that the extreme-aggregation measure corresponds to the smallest dominating coherent risk measure. In our main result, it is shown that the extreme-aggregation measure induced by a general distortion risk measure is a coherent distortion risk measure. In the case of convex risk measures, a general robust representation of coherent extreme-aggregation measures is provided. Specifically, the extreme-aggregation measure induced by a convex shortfall risk measure is a coherent expectile. These results show that, in the presence of dependence uncertainty, quantification of a coherent risk measure is often necessary, an observation that lends further support to the use of coherent risk measures in portfolio risk management.