In this paper we consider the problem of studying the gap between bounds of risk measures for sums of non-independent random variables. Owing to the choice of the context where to set the problem, namely that of distortion risk measures, we first deduce an explicit formula for the risk measure of a discrete risk by referring to its writing as sum of layers. Then, we examine the case of sums of discrete risks with identical distribution. Upper and lower bounds for risk measures of sums of risks are presented in the case of concave distortion functions. Finally, the attention is devoted to the analysis of the gap between risk measures of upper and lower bounds, with the aim of optimizing it.Distortion risk measures, discrete risks, concave risk measure, upper and lower bounds, gap between bounds

Antonella Campana

Paola Ferretti

English

Research Papers in Economics

Department of Applied Mathematics,  University of Venice  
 
 
 
WORKING PAPER SERIES 
 
 
 
 
 
 
 
 
 
 
Antonella Campana and Paola Ferretti 
 
 
 
 
On Bounds for Concave Distortion 
Risk Measures for Sums of Risks 
 
 
 
 
 
 
 
 
 
Working Paper n. 146/2006 
November 2006 
 
ISSN: 1828-6887  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
This Working Paper is published under the auspices of the Department of Applied 
Mathematics of the Ca’ Foscari University of Venice. Opinions expressed herein are 
those of the authors and not those of the Department. The Working Paper series is 
designed to divulge preliminary or incomplete work, circulated to favour discussion 
and comments. Citation of this paper should consider its provisional nature. On Bounds for Concave Distortion
Risk Measures for Sums of Risks¤
Antonella Campana Paola Ferretti
<campana@unimol.it> <ferretti@unive.it>
Dept. SEGeS Dept. of Applied Mathematics
University of Molise University of Venice and SSAV
(November 2006)
Abstract. In this paper we consider the problem of studying the gap between bounds of
risk measures of sums of non-independent random variables. Owing to the choice of the
context where to set the problem, namely that of distortion risk measures, we ﬁrst deduce
an explicit formula for the risk measure of a discrete risk by referring to its writing as sum
of layers. Then, we examine the case of sums of discrete risks with identical distribution.
Upper and lower bounds for risk measures of sums of risks are presented in the case of
concave distortion functions. Finally, the attention is devoted to the analysis of the gap
between risk measures of upper and lower bounds, with the aim of optimizing it.
Keywords: Distortion risk measures; discrete risks; concave risk measures; upper and
lower bounds; gap between bounds.
JEL Classiﬁcation Numbers: D810.
MathSci Classiﬁcation Numbers: 62H20; 60E15.
Correspondence to:
Paola Ferretti Dept. of Applied Mathematics, University of Venice
Dorsoduro 3825/e
30123 Venezia, Italy
Phone: [++39] (041)-234-6923
Fax: [++39] (041)-522-1756
E-mail: ferretti@unive.it
¤ This research was partially supported by MIUR.1 Introduction
Recently in actuarial literature, the study of the impact of dependence among risks has
become a major and ﬂourishing topic: even if in traditional risk theory individual risks have
been usually assumed to be independent, this assumption is very convenient for tractability
but it is not generally realistic. Think for example to the aggregate claim amount in which
any random variable represents the individual claim size of an insurer’s risk portfolio. When
the risk is represented by residential dwellings exposed to danger of an earthquake in a given
location or by adjoining buildings in ﬁre insurance, it is unrealistic to state that individual
risks are not correlated, because they are subject to the same claim causing mechanism.
Several notions of dependence were introduced in literature to model the fact that larger
values of one of the component of a multivariate risk tend to be associated with larger values
of the others. In ﬁnancial or actuarial situations one often encounters random variables of
the type
S =
n X
i=1
Xi
where the terms Xi are not mutually independent and the multivariate distribution function
of the random vector X = (X1;X2;:::;Xn) is not completely speciﬁed but one only knows
the marginal distribution functions of the risks. To be able to make decisions, in such cases
it may be helpful to determine approximations for the distribution of S, namely upper and
lower bounds for risk measures of the sum of risks S, in such a way that it is possible
to consider a riskiest portfolio and a safest portfolio, where riskiness and safety are both
evaluated in terms of risk measures.
With the aim of studying the gap between the riskiest and the safest portfolio, the
present contribution addresses the analysis to a particular class of risk measures, namely
that of distortion risk measures introduced by Wang [8]. In this class the risk measure of a
non-negative real valued random variable X is written in the following way
Wg(X) =
Z 1
0
g(HX(x))dx
where the distortion function g is deﬁned as a non-decreasing function g : [0;1] ! [0;1]
such that g(0) = 0 and g(1) = 1.
Given the choice of this context, it is possible to write an explicit formula for the risk
measure of a discrete risk by referring to its writing as sum of layers (Campana and Ferretti
[1]). Starting from this result, the attention is therefore devoted to the study of bounds of
sums of risks in the case of discrete identically distributed random variables. Now the key
role is played by the choice of the framework where to set the study: by referring to concave
distortion risk measures, in fact, it is possible to characterize the riskiest portfolio where
the multivariate distribution refers to mutually comonotonic risks and the safest portfolio
where the multivariate distribution is that of mutually exclusive risks. Again, starting from
the representation of risks as sums of layers, it is possible to derive explicit formulas for
risk measures of upper and lower bounds of sums of risks. The attention is then devoted
to the study of the diﬀerence between risk measures of upper and lower bounds, with the
1aim of obtaining some information on random variables for which the gap is maximum or
minimum.
The paper is organized as follows. In Section 2 we ﬁrst review some basic settings for
describing the problem of measuring a risk and we remind some deﬁnitions and preliminary
results in the ﬁeld of distortion risk measures; then we propose the study of the case of a
discrete risk with ﬁnitely many mass points in such a way that it is possible to give an explicit
formula for its distortion risk measure. Section 3 is devoted to the problem of detecting
upper and lower bounds for sums of not mutually independent risks: we present the study
of the case of sums of discrete and identically distributed risks in order to obtain upper and
lower bounds for concave distortion measures of aggregate claims of the portfolio. Then in
Section 4 the attention is focused on the problem of characterizing risks for which the gap
between bounds of risk measures is maximum or minimum. Some concluding remarks in
Section 5 end the paper.
2 The class of distortion risk measures
As it is well-known, an insurance risk is deﬁned as a non-negative real-valued random
variable X deﬁned on some probability space.
Here we consider a set Γ of risks with bounded support [0;c]. For each risk X 2 Γ we
denote by HX its tail function, i.e. HX(x) = Pr[X > x], for all x ¸ 0.
A risk measure is deﬁned as a mapping from the set of random variables, namely losses
or payments, to the set of real numbers. In actuarial science common risk measures are
premium principles; other risk measures are used for determining provisions and capital
requirements of an insurer in order to avoid insolvency (see e.g. Dhaene et al. [5]).
In this paper we consider the distortion risk measure introduced by Wang [8]:
Wg(X) =
Z 1
0
g(HX(x))dx (1)
where the distortion function g is deﬁned as a non-decreasing function g : [0;1] ! [0;1]
such that g(0) = 0 and g(1) = 1. As it is well-known, the quantile risk measure and the
Tail Value-at-Riskare examples of risk measures belonging to this class. In the particular
case of a power g function, i.e. g(x) = x1=½, ½ ¸ 1, the corresponding risk measure is the
PH-transform risk measure proposed by Wang [7].
Distortion risk measures satisfy the following properties (see Wang [8] and Dhaene et al. [5]):
P1. Additivity for comonotonic risks
Wg(Sc) =
n X
i=1
Wg(Xi) (2)
where Sc is the sum of the components of the random vector Xc with the same marginal
distributions of X and with the comonotonic dependence structure.
P2. Positive homogeneity
Wg(aX) = aWg(X) for any non-negative costant a; (3)
2P3. Translation invariance
Wg(X + b) = Wg(X) + b for any costant b; (4)
P4. Monotonicity
Wg(X) · Wg(Y ) (5)
for any two random variables X and Y where X · Y with probability 1.
2.1 Discrete risks with ﬁnitely many mass points
In the particular case of a discrete risk X 2 Γ with ﬁnitely many mass points it is possible
to deduce an explicit formula of the distortion risk measure Wg(X) of X. The key-point
relies on the fact that each risk X 2 Γ can be written as sum of layers that are pairwise
mutually comonotonic risks.
Let X 2 Γ be a discrete risk with ﬁnitely many mass points: then, there exist a positive
integer m, a ﬁnite sequence fxjg, (j = 0;¢¢¢;m), 0 ´ x0 < x1 < ::: < xm ´ c and a ﬁnite
sequence fpjg, (j = 0;¢¢¢;m ¡ 1), 1 ¸ p0 > p1 > p2 > ::: > pm¡1 > 0 such that the tail
function HX of X is so deﬁned
HX(x) =
m¡1 X
j=0
pj I(xj·x<xj+1); x ¸ 0; (6)
where I(xj·x<xj+1) is the indicator function of the set fx : xj · x < xj+1g. Then
X =
m¡1 X
j=0
L(xj;xj+1) (7)
where a layer at (xj;xj+1) of X is deﬁned as the loss from an excess-of-loss cover, namely
L(xj;xj+1) =
8
<
:
0 0 · X · xj
X ¡ xj xj < X < xj+1
xj+1 ¡ xj X ¸ xj+1
(8)
and the tail function of the layer L(xj;xj+1) is given by
HL(xj;xj+1)(x) =
½
pj 0 · x < xj+1 ¡ xj
0 x ¸ xj+1 ¡ xj
: (9)
If we consider a Bernoulli random variable Bpj such that Pr[Bpj = 1] = pj = 1 ¡
Pr[Bpj = 0] then L(xj;xj+1) is a two-points distributed random variable which satisﬁes the
equality in distribution L(xj;xj+1)
d = (xj ¡ xj+1)Bpj:
Additivity for comonotonic risks and positive homogeneity of distorted risk measures
Wg ensure that
3Wg(X) =
m¡1 X
j=0
Wg(L(xj;xj+1)) =
m¡1 X
j=0
(xj+1 ¡ xj)g(pj):
In this way for any discrete risk X 2 Γ for which representation (6) holds and any
distortion function g we can assert that
Wg(X) =
m¡1 X
j=0
(xj+1 ¡ xj)g(pj): (10)
3 The class of concave distortion risk measures
In the particular case of a concave distortion measure, the related distortion risk measure
satisfying properties P1-P4 is also sub-additive and it preserves stop-loss order. As it is
well-known, examples of concave distortion risk measures are the Tail Value-at-Risk and the
PH-transform risk measure, whereas quantile risk measure is not a concave risk measure.
In the previous section we deduced an explicit formula for the distortion risk measure
Wg(X) when a discrete risk X 2 Γ with ﬁnitely many mass points is considered. This
result may be used to obtain upper and lower bounds for sums of discrete and identically
distributed risks with common tail function given by (6) when we consider the following
framework where to set the study: the Fr´ echet space consisting of all n-dimensional random
vectors X possessing (HX1;HX2;:::;HXn) as marginal tail functions, for which the condi-
tion
Pn
i=1 HXi(0) · 1 is fulﬁlled and the distortion function g is assumed to be concave.
3.1 Upper bound for sums of discrete and identically distributed risks
Let X be a random vector with discrete and identically distributed risks Xi 2 Γ. The least
attractive random vector with given marginal distribution functions has the comonotonic
joint distribution (see e.g. Dhaene et. al. [3] and Kaas et al. [6]), namely
Wg(S) · Wg(Sc):
Now we want to give an explicit formula for Wg(Sc). Let the common tail function of Xi
be written as
HXi(x) =
m¡1 X
j=0
pj I(xj·x<xj+1); x ¸ 0 (11)
where m is a positive integer and 1 ¸ p0 > p1 > p2 > ::: > pm¡1 > 0, 0 ´ x0 < x1 < ::: <
xm ´ c. Then
Sc d = nX1;
and by subadditivity of the concave risk measure Wg it follows that
4Wg(S) · Wg(Sc) =
m¡1 X
j=0
(xj+1 ¡ xj)ng(pj):
Namely, under representation (11) the riskiest portfolio Sc exhibits the following risk mea-
sure
m¡1 X
j=0
(xj+1 ¡ xj)ng(pj): (12)
3.2 Lower bound for sums of discrete and identically distributed risks
As in the previous subsection, let X be a random vector with discrete and identically
distributed risks Xi 2 Γ. In the Fr´ echet space consisting of all n-dimensional random vec-
tors X possessing (HX1;HX2;:::;HXn) as marginal tail functions, for which the condition Pn
i=1 HXi(0) · 1 is fulﬁlled, the safest random vector is given by (see Dhaene et Denuit,
[2]) the vector Xe = (Xe
1;Xe
2;:::;Xe
n) where the components are said to be mutually ex-
clusive because Pr[Xe
i > 0;Xe
j > 0] = 0 for all i 6= j. Let Se denote the sum of mutually
exclusive risks Xe
1;Xe
2;:::;Xe
n. In order to have an explicit formula for Wg(Se), note that
its tail function is given by
HSe(x) =
n X
i=1
HXi(x); for all x ¸ 0: (13)
Owing to the fact that the common tail function of Xi is written as (11) where np0 · 1,
the tail function of the sum Se of mutually exclusive risks becomes
HSe(x) = n
m¡1 X
j=0
pj I(xj·x<xj+1); for all x ¸ 0: (14)
Note that Se can be written as a sum of layers
Se =
m¡1 X
j=0
˜ L(xj;xj+1)
where ˜ L(xj;xj+1) is a two-points distribution with ˜ L(xj;xj+1)
d = Bnpj. By considering a
concave distortion risk measure, it is
Wg(S) ¸ Wg(Se) =
m¡1 X
j=0
(xj+1 ¡ xj)g(npj): (15)
In other words, under hypothesis (11) where np0 · 1, the safest portfolio Se exhibits
the following risk measure
5m¡1 X
j=0
(xj+1 ¡ xj)g(npj): (16)
4 Optimal gap between bounds of risk measures
In the previous section lower and upper bounds for sums of discrete and identically dis-
tributed risks Xi 2 Γ have been obtained: the attention now is devoted to the study of
the diﬀerence between risk measures of upper and lower bounds, with the aim of obtaining
some information on random variables for which the gap is maximum or minimum. Starting
from formulations (16) and (12) exhibiting bounds for aggregate claims S of the portfolio
X, we face the problem of studying the diﬀerence between bounds in order to minimize and
maximize it. The problem in pj (j = 1;¢¢¢;m ¡ 1) and xj (j = 1;¢¢¢;m) is then
max=min
m¡1 X
j=0
(xj+1 ¡ xj)[ng(pj) ¡ g(npj)] (17)
where
x0 ´ 0;xj < xj+1;xm ´ c;
0 < pm¡1 < ¢ ¢ ¢ < p0 · 1
n;n > 1;
g : [0;1] ! [0;1]; g(0) = 0; g(1) = 1;g is non-decreasing and concave:
Note that the optimization problem is related to the maximization/minimization of
the gap between upper and lower bounds for risk measures, namely to the maximiza-
tion/minimization of the diﬀerence Wg(Sc) ¡ Wg(Se).
Let Án(pj) be equal to ng(pj) ¡ g(npj): since g is concave and g(0) = 0, Án is non-
negative; in particular Án(0) = 0. Moreover, non-negativity and concavity of g imply that
Án is non-decreasing. After setting ∆j = xj+1 ¡ xj the problem becomes
max=min
m¡1 X
j=0
∆j Án(pj) (18)
where
∆j > 0;
Pm¡1
j=0 ∆j = c;
0 < pm¡1 < ::: < p0 · 1
n; n > 1;
Án : [0; 1
n] ! R; Án(0) = 0; Án( 1
n) = ng( 1
n) ¡ 1 ¸ 0; Án is non-decreasing:
Note that the feasible set is not closed, so at any ﬁrst step a relaxed problem with closed
constraints will be faced.
64.1 The problem of minimizing the gap
The problem of minimizing the diﬀerence Wg(Sc) ¡ Wg(Se) may be faced both in terms of
pj both in terms of ∆j.
a) Solution with respect to the pjs.
At a ﬁrst step the hypothesis that the constraints for the pjs admit equality is assumed
(namely we consider the closure of the feasible set); by monotonicity of Án it follows
that pj = 0; moreover, if there exists 0 < ² < 1
n such that Án(²) = 0 then all the pjs
could be set in the interval (0;²] and a solution would exist also in the original open
set.
b) Solution with respect to the ∆js.
Starting from the case that the constraints for the ∆js admit equality (the closure
of the feasible set is assumed), by non-decreasing monotonicity of Án it follows that
the minimum is when ∆m¡1 = c and all the other ∆j are set equal to 0 (that is
x0 = x1 = ::: = xm¡1 = 0 and xm = c). In the particular case of a constant function
Án in the interval [pm¡1;p0] the problem would admit interior minima given by any
feasible choice of the ∆js.
4.2 The problem of maximizing the gap
The problem of maximizing the diﬀerence Wg(Sc)¡Wg(Se) exhibits the following solutions
in terms of pj and in terms of ∆j.
a) Solution with respect to the pjs.
By referring to the case of closed feasible set (that is the constraints for the pjs admit
equality) the optimal solution is given by pj = 1
n; if moreover there exists 0 < ² < 1
n
such that Án is constant in the interval [ 1
n ¡²; 1
n], then all the pjs could be set in that
interval and a solution would exist also in the original open set.
b) Solution with respect to the ∆js. Under the relaxed hypothesis of equality constraints
on ∆js, the maximum is when ∆0 = c and all the other ∆js are set equal to 0 (that
is x0 = 0 and x1 = ::: = xm = c). Note that if Án were constant in the interval
[pm¡1;p0] the problem would admit interior maxima: any feasible choice of the ∆js
is solution.
5 Concluding remarks
In this paper we face the problem of studying the gap between bounds for risk measures
of sums of discrete and identically distributed risks. Starting from the representation of
risks as sums of layers, explicit formulas for risk measures of upper and lower bounds of
sums of risks are obtained in the particular case of concave distortion risk measures. A
maximization(minimization) problem related to the maximization (minimization) of the
7gap between risk measures of upper and lower bounds is solved with respect to information
characterizing the random vector X.
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On Bounds for Concave Distortion Risk Measures for Sums of Risks

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On Bounds for Concave Distortion Risk Measures for Sums of Risks

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