Modeling the Esscher Premium Principle for a System of Elliptically
Distributed Risks
Tomer Shushi
Department of Business Administration, Guilford Glazer Faculty of Business and Management,
Ben-Gurion University of the Negev, Beer-Sheva, Israel
Keywords:
Esscher Premium, Extreme Risks, Multivariate Risk Measures, Premium Principles, Tail Value at Risk,
Value-at-Risk.
Abstract:
The Esscher premium principle provides an important framework for allocating a certain loaded premium for
some claim (risk) in order to manage the risks of insurance companies. In this paper, we show how to model the
celebrated Esscher premium principle for a system of elliptically distributed dependent risks, where each risk
is greater or equal than its value-at-risk. Furthermore, we present calculations of the proposed multivariate risk
measure, investigate its properties and formulas, and show how special elliptical models can be implemented
in the theory.
1 INTRODUCTION
Recently, there is a growing interest in multivariate
risk measures. The motivation behind considering a
multivariate risk measure is that it provides more ac-
curate measurements of risks that are mutually de-
pendent on each other. There are several attempts
to obtain such multivariate measures (Jouini et al.,
2004; Molchanov and Cascos, 2016; Cousin and
Di Bernardino, 2014; Feinstein and Rudloff, 2017;
Landsman et al., 2016; Shushi, 2018). For instance,
Landsman et al. (2016) introduced the multivariate
tail conditional expectation (MTCE) with the follow-
ing form
MTCE
q
(X) = E (X|X > VaR
q
(X)).
Here X =(X
1
,X
2
,...,X
n
)
T
is n × 1 vec-
tor of risks that are mutually depend-
ing on each other, and VaR
q
(X) =
(VaR
q
1
(X
1
),VaR
q
2
(X
2
),...,VaR
q
n
(X
n
))
T
is n × 1
vector, where VaR
q
i
(X
i
) is the value at risk of X
i
under the q
i
−th quantile, q
i
∈(0,1). In this notation,
for two n-variate random vectors X and Y, X ≥Y
means that {X
i
≥
a.s.
Y
i
,i = 1, ..., n}. The multivariate
tail covariance measure was also introduced in the
literature by (Landsman et al., 2018), and obtained
for the class of elliptical distributions. The Esscher
premium principle is a widely used measure in risk
measurement and portfolio theory, which allows
to quantify insurance premiums (Kamps, 1998;
Van Heerwaarden et al., 1989; Landsman, 2004;
Shushi, 2017; Chi et al., 2017). In the theory of
risks there exist vast number of different models
to calculate insurance premiums (Goovaerts et al.,
1984; Wang and Dhaene, 1998; D
´
eniz et al., 2000)
. The Esscher premium was first introduced in the
seminal paper of Buhlmann (B
¨
uhlmann, 1980). In his
paper, Buhlmann claimed that actuaries think about
premiums as a measure of risks, which are considered
random. Unlike actuarial premiums, economical
premiums depend also on market conditions which
can be characterized by another random risk. In this
paper we focus on actuarial premiums, and thus we
are not taking into account any market conditions.
Let X be a random risk. Then, the Esscher pre-
mium of X takes the following form
π
λ
(X) =
E

Xe
λX

M
X
(λ)
, (1)
where λ > 0, M
X
(λ) = E

e
λX

is the moment gener-
ating function (MGF) of X, and E

Xe
λX

< ∞.
The Wang’s premium (Wang et al., 2002) intro-
duced as an exponential tilting of some risk, X, in-
duced by another risk, Y ,
π
λ
(X,Y ) =
E

Xe
λY

M
Y
(λ)
, (2)
and the Esscher premium is the special case of
π
λ
(X, X ) = π
λ
(X) . Furthermore, (2) has actuarial
102
Shushi, T.
Modeling the Esscher Premium Principle for a System of Elliptically Distributed Risks.
DOI: 10.5220/0007378101020110
In Proceedings of the 8th International Conference on Operations Research and Enterprise Systems (ICORES 2019), pages 102-110
ISBN: 978-989-758-352-0
Copyright
c
 2019 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

sense behind the quantitative measure. For a portfolio
consisting n ×1 risks, X, the measure π
λ
(X
i
,S) quan-
tifies the amount of risk of X
i
to the aggregate risks,
which is the sum of the risks, i.e., S = X
1
+ X
2
+ ... +
X
n
.
In Shushi (2018), a multivariate conditional ver-
sion of the Esscher premium has been introduced
which takes into account only the tail of the multivari-
ate distribution of the vector of risks. In this paper,
we generalize the Esscher premium into a conditional
framework such that each risk X
i
is greater than its
value-at-risk measure and consider the most general
case which is a portfolio that consists of n−variate
dependent risks.
The multivariate conditional Esscher premium
(MCEP) takes the following form
π
α,λ
(X) =
E

Xe
λ
T
X
|X > VaR
q
(X)

E

e
λ
T
X
|X > VaR
q
(X)

, λ
i
> 0, (3)
λ = (λ
1
,...,λ
n
)
T
,i = 1,2,...,n.
The motivation behind this multivariate risk mea-
sure is that it provides a conservative premium princi-
ple, in the sense that it quantifies the premium under
the assumption that the i−th loss is greater than its
value-at-risk, X
i
≥ VaR
q
(X
i
),i = 1,2,... . and there-
fore the MCEP measure is greater than or equal to the
Esscher premium:
π
λ
(X) ≤ π
α,λ
(X),
where π
λ
(X) = (π
λ
(X
1
),π
λ
(X
2
),...,π
λ
(X
n
))
T
.
We define the conditional analog to the Wang’s
premium, as follows:
π
α,λ
(X
i
,S) =
E

X
i
e
λS
|S >
∑
n
i=1
VaR
q
(X
i
)

E

e
λS
|S >
∑
n
i=1
VaR
q
(X
i
)

.
In the next Section, we give a concise definition
of the family of elliptical distributions, and in Section
3 we analyze the proposed MCEP measure by pro-
viding its main properties and their implications. In
Section 4 we compute the MCEP for a system of mu-
tually dependent elliptically distributed risks, and in
Section 5 we give examples. Section 6 offers a dis-
cussion to the paper.
2 THE CLASS OF ELLIPTICAL
DISTRIBUTIONS
The class of elliptical distributions consists many im-
portant distributions such as the normal, Student-t, lo-
gistic, and Laplace distributions. In fact, it is a natu-
ral generalization of the normal distribution (Camba-
nis et al., 1981). This class has attempting properties
which will be shown in the sequel.
Let X be n ×1 random vector following elliptical
distribution, X v E
n
(µ,Σ,g
n
). Then, the pdf of X is
f
X
(x) =
c
n
p
|Σ|
g
n

1
2
(x −µ)
T
Σ
−1
(x −µ)

,x ∈ R,
(4)
where c
n
is the normalizing constant, g (u) , u ≥ 0, is
called the density generator of X, µ is an n×1 location
vector, and Σ is an n ×n scale matrix.
The characteristic function of X takes the follow-
ing form
ϕ
X
(t) = exp(it
T
µ)ψ
g
n
(
1
2
t
T
Σt), (5)
some function ψ
g
n
(u) : [0,∞) → R, called the charac-
teristic generator.
The marginal distributions of the elliptical distri-
bution are also elliptical with the same characteristic
generator. For a random vector X such that
X =(X
1
,X
2
)
T
v E
n

µ
1
µ
2

,

Σ
11
Σ
12
Σ
21
Σ
22

,g
n

,
where X
1
and X
2
are m (m < n) and n −m random
vectors, the characteristic function of X, ϕ
X
(t), takes
the form
ϕ
X
(t) = exp(i

t
T
1
µ
1
+ t
T
2
µ
2

) ·ψ
1
2

t
1
t
2

T
Σ

t
1
t
2

!
,
t
1
∈ R
m
,t
2
∈ R
n−m
.
Then, for the marginal X
1
of X we take t
2
= 0 where
0 is vector of n −m zeros,
ϕ
X
((t
1
,0
2
)
T
) = exp(it
T
1
µ
1
) ·ψ

1
2
t
T
1
Σ
11
t
1

(6)
= ϕ
X
1
(t
1
).
As can be clearly seen, the above equation takes the
same form as (5) with vector of locations µ
1
, scale
matrix Σ
11
, and characteristic generator ψ(u). There-
fore, X
1
v E
m
(µ
1
,Σ
11
,g
m
).
For m ×n matrix B with rank m ≤n and m×1 vec-
tor c, the transformation BX + c is m−variate ellipti-
cal random vector, i.e., BX +c is distributed E
m
(µ
∗
=
Bµ + c,Σ
∗
= BΣB
T
,g
m
). This can be shown by the
form of elliptical characteristic function. From (5) it
follows that
ϕ
BX+c
(t) = e
it
T
c
ϕ
X
(Bt)
= exp(it
T
(Bµ + c)) ·ψ(
1
2
(Bt)
T
Σ(Bt))
= exp(it
T
µ
∗
) ·ψ(
1
2
tΣ
∗
t).
Modeling the Esscher Premium Principle for a System of Elliptically Distributed Risks
103

From this property, we immediately establish that the
marginal distribution is also elliptical as has been
shown previously.
Let matrix B be
B =

I
m×m
0
(n−m)×m
0
m×(n−m)
0
(n−m)×(n−m)

,
where I
m×m
is m×m identity matrix, 0
(n−m)×m
is (n−
m)×m matrix with zero components, 0
(n−m)×(n−m)
is
(n −m) ×(n −m) matrix with zero components, and
0
(n−m)×m
= 0
T
m×(n−m)
. Then, random vector BX is the
marginal random vector X
1
. Furthermore, in the case
that B = b is a n ×1 vector, then b
T
X, representing
weighted-sum, is distributed E
1
(b
T
µ,b
T
Σb,g
1
).
3 THE PROPERTIES OF THE
MCEP MEASURE
Let us now show some important and desirable prop-
erties of the MCEP measure for the elliptical model.
Proposition 1. Proposition 1. Let X v E
n
(µ,Σ,g
n
) be
a system of n elliptically distributed risks. Then, the
MCEP follows the properties:
1. Translation Invariance: For any random vector of
risks X and any vector of constants α ∈R
n
π
α,λ
(X + α) = π
α,λ
(X) +α. (7)
2. Independence of risks: If the vector of risks X has
independent components. Then
π
α,λ
(X) =



π
q,λ
i
(X
1
)
π
q,λ
2
(X
2
)
...
π
q,λ
n
(X
n
)



. (8)
3. Monotonicity: Suppose Y, X, are n ×1 random
vectors of risks and Y
a.s
≥ X. Then
π
α,λ
(Y −X) ≥ 0, (9)
where 0 is vector of n zeros.
4. Semi-Positive Homogeneity: For some positive
constant a > 0, The MCEP follows the following
equality
π
α,λ
(aX) = aπ
α,λ
(X). (10)
5. Semi-subadditivity of π
α,λ
(X) for elliptical dis-
tributions: Consider an (2n) ×1 elliptical ran-
dom vector X with the partition X =

X
T
1
,X
T
2

T
,
X
1
= (X
1
,...,X
n
)
T
,X
2
= (X
n+1
,...,X
2n
)
T
. Then,
the following inequality hold
π
α,λ
(X
1
+ X
2
) ≤ π
α,λ
(X)
1
+π
α,λ
(X)
2
, (11)
where π
α,λ
(X) =

π
α,λ
(X)
T
1
,π
α,λ
(X)
T
2

T
.
The motivation behind the semi-subadditivity
property can be found in Landsman et al. (Landsman
et al., 2016) . In our case (11) means that combining
risks provides less premium than separating them.
Proof.
1. The translation invariance property can be proved
after some algebraic calculations. We notice that
VaR
q
(X + α) = α+VaR
q
(X), so
π
α,λ
(X + α)
=
E

(X + α) e
λ
T
(X+α)
|X + α > V aR
q
(X + α)

E

e
λ
T
(X+α)
|X + α > V aR
q
(X + α)

=
E

(X + α) e
λ
T
X
|X > VaR
q
(X)

E

e
λ
T
X
|X > VaR
q
(X)

= α+π
α,λ
(X).
2. Since we assumed that (X
1
,X
2
,...,X
n
) are mu-
tually independent random risks the probability
density function (pdf) of X is the multiplication
for the pdf’s of the i-th component of X, so
π
α,λ
(X) =
E

Xe
λ
T
X
|X > VaR
q
(X)

E

e
λ
T
X
|X > VaR
q
(X)

=
1
n
∏
i=1
E

X
i
e
λ
i
X
i
|X
i
> VaR
q
(X
i
)

·







E

X
1
e
λ
T
X
|X
1
> VaR
q
(X
1
)

E

X
2
e
λ
T
X
|X
2
> VaR
q
(X
2
)

...
E

X
n
e
λ
T
X
|X
n
> VaR
q
(X
n
)








=



π
q,λ
i
(X
1
)
π
q,λ
2
(X
2
)
...
π
q,λ
n
(X
n
)



.
3. Notice that as Y is greater than (a.s.) X, we can
define a random vector in which its components
get only non-negative values, V = Y −X ≥ 0,
where 0 is n ×1 vector of zeros. Then, as V
is non-negative random vector VaR
q
(U)≥ 0, and
thus π
q,λ
(V) ≥ 0,
π
α,λ
(Y −X) = E (V|V > VaR
q
(V)) ≥ 0.
4. Similar to the Esscher premium, the MCEP is not
positive homogenous, but, the semi-positive ho-
ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems
104

mogeneity property holds, as follows:
π
α,λ
(aX) =
E

aXe
aλ
T
X
|aX > aVaR
q
(X)

E

e
aλ
T
X
|aX > aVaR
q
(X)

= a
E

Xe
aλ
T
X
|X > VaR
q
(X)

E

e
aλ
T
X
|X > VaR
q
(X)

= aπ
α,aλ
(X).
5. The proof is similar to the proof of the semi-
subadditivity of the MTCE for elliptical distribu-
tions shown in Landsman et al. (2016). From
(McNeil et al., 2005), Theorem 6.8, we notice
that for any matrix B with n×(2n) dimensions, in
the case of elliptical random vector X,
VaR
q
(BX) ≤ BVaR
q
(X). (12)
In our case B =

I
n×n
I
n×n

.Then, since

BX > BVaR
q
(X)

=

X > VaR
q
(X)

we have
π
α,λ
(X
1
+ X
2
)=E

BXe
Bλ
T
X
|BX > VaR
q
(BX)

≤ E

BXe
Bλ
T
X
|BX > BVaR
q
(X)

= BE

Xe
λ
T
X
|X > VaR
q
(X)

so
π
α,λ
(X
1
+ X
2
) ≤ π
α,λ
(X)
1
+π
α,λ
(X)
2
.

4 DERIVATION OF MCEP FOR
ELLIPTICAL MODELS
In risk measurement, the family of elliptical distri-
butions is important since this family has desirable
properties which were shown in the previous Section.
This class is used to model loss distributions of some
random risks associated with this family (Landsman,
2004; Valdez and Chernih, 2003; Xiao and Valdez,
2015). Therefore, it is natural to derive the condi-
tional Esscher premium for the family of elliptical dis-
tributions.
Before we derive the MCEP measure for ellipti-
cal models, we define a cumulative generator
G
n
(u),
(Landsman and Valdez, 2003), which takes the fol-
lowing form
G
n
(u) =
∞
Z
u
g
n
(q)dq. (13)
Furthermore, let us define a shifted cumulative gener-
ator G
∗
n−1
(u) (Landsman et al., 2016)
G
∗
n−1
(u) =
∞
Z
u
g
n
(q + a)dq,a ≥ 0, n > 1,
under the condition that G
∗
n−1
(0) < ∞. For the
sequel, let us define the random vector of risks
X v E
n
(µ,Σ,g
n
), and a standard random vector
Z = Σ
−1/2
(X − µ) v E
n
(0,I,g
n
). Furthermore, de-
fine ζ
q
= Σ
−1/2
(VaR
q
(X) −µ) , x
q
= VaR
q
(X), and
ζ
q,−i
=

ζ
q
1
,1
,...,ζ
q
i−1
,i−1
,ζ
q
i+1
,i+1
,ζ
q
n
,n

T
, and we
introduce the tail function of (n −1)−variate random
vector Y
i
,
F
Y
i
(y),
F
Y
i
(y)
=
Z
∞
y
f
Y
i
(u)du, u,y ∈ R
n−1
, du = du
1
du
2
...du
n
,
where f
Y
i
(u) is the elliptical pdf
f
Y
i
(y)
= c
∗
n−1,i
G
∗
n−1,i

1
2
y
T
y

= c
∗
n−1,i
G
n

1
2
y
T
y+
1
2
ζ
2
q,i

,
i = 1,2,...,n.
Lemma 1. Lemma 1. If M
X,q
(λ) < ∞, the conditional
moment generating function of X is given by
M
X,q
(λ) = e
λ
T
µ
ψ
g
n

1
2
λ
T
Σλ

F
θ
(ζ
q
)
F
X
(x
q
)
, (14)
where F
θ
is the tail function of a random vector θ with
the pdf
f
θ
(t) = ψ
g
n

1
2
t
T
Σt

−1
e
λ
T
Σ
1/2
t
·c
n
g
n

1
2
t
T
t

.
(15)
Proof. From the definition of M
X,q
(λ), we have
M
X,q
(λ)
=
c
n
∞
R
VaR
q
(X)
e
λ
T
x
·
|
Σ
|
−1/2
g
n

1
2
(x −µ)
T
Σ
−1
(x −µ)

dx
F
X
(x
q
)
after the transformation z =Σ
−1/2
(x −µ), we have
M
X,q
(λ) =
c
n
∞
R
ζ
q
e
λ
T
(µ+Σ
1/2
z)
·
|
Σ
|
−1/2
g
n

1
2
z
T
z

|
Σ
|
1/2
dz
F
X
(x
q
)
=
c
n
e
λ
T
µ
∞
R
ζ
q
e
λ
T
Σ
1/2
z
·g
n

1
2
z
T
z

dz
F
X
(x
q
)
.
Taking into account (15), we conclude that
c
n
∞
Z
ζ
q
e
λ
T
Σ
1/2
z
·g
n

1
2
z
T
z

dz=ψ
g
n

1
2
λ
T
Σλ

F
θ
(ζ
q
),
Modeling the Esscher Premium Principle for a System of Elliptically Distributed Risks
105

and finally,
M
X,q
(λ) = c
n
e
λ
T
µ
ψ
g
n

1
2
λ
T
Σλ

F
θ
(ζ
q
)
F
X
(x
q
)
.
We note that the proof of Lemma 1 is based on the
same method introduced in (Landsman et al., 2013)
which derived the TCE and TV of the elliptical distri-
butions, respectively.
Theorem 1. Theorem 1. Suppose that the conditional
moment generating function of X, M
X,q
(λ), exist,
and that E

X
i
e
λ
T
X
|X > VaR
q
(X)

< ∞ ∀i = 1,2...
. Then, the MCEP for the multivariate elliptical dis-
tribution, n > 1, takes the form
π
α,λ
(X) = µ + Σ
1/2
χ
q,λ
. (16)
Here χ
q,λ
is n ×1 vector of that depends on the q −th
percentile
χ
q,λ
=

χ
1,q
χ
2,q
... χ
n,q

T
, (17)
where each component of (17) is
χ
i,q
=
ψ
G
∗
n−1,i

1
2

λ
T
Σ
1/2

T
−i

λ
T
Σ
1/2

−i

ψ
g
n

1
2
λ
T
Σλ

F
θ
∗
i

ζ
q,−i

c
n
F
θ

ζ
q

c
∗
n−1,i
+

λ
T
Σ
1/2

i
ψ
G
n

1
2
λ
T
Σλ

ψ
g
n

1
2
λ
T
Σλ

F
θ
∗∗

ζ
q

c
n
F
θ

ζ
q

c
∗
n
,
with the pdf’s of θ
∗∗
∈R
n
and θ
∗
i
∈R
n−1
, i = 1,2,...,n,
f
θ
∗∗
(t) = ψ
G
n

1
2
t
T
Σt

−1
e
λ
T
Σ
1/2
t
·c
∗
n
G
n

1
2
t
T
t

,t ∈ R
n
,
(18)
and
f
θ
∗
i
(u) = ψ
G
∗
n−1,i

1
2
u
T
u

−1
e
(
λ
T
Σ
1/2
)
−i
u
(19)
c
∗
n−1,i
G
∗
n−1,i

1
2
u
T
u

,u ∈ R
n−1
,
with the cumulative generator
G
∗
n−1,i
(u) =
∞
Z
u
g
n
(q +
1
2
z
2
q,i
)dq.
where c
∗
n
and c
∗
n−1,i
are the normalizing constants of
(18) and (19), respectively.
Proof. From the definition of π
α,λ
(X), we have
π
α,λ
(X)
=
1
M
X,q
(λ)
·
c
n
∞
R
VaR
q
(X)
xe
λ
T
x
·
|
Σ
|
−1/2
g
n

1
2
(x −µ)
T
Σ
−1
(x −µ)

dx
F
X
(x
q
)
.
Now, substituting z =Σ
−1/2
(x −µ) , we obtain
π
q,λ
(X)
=
c
n
∞
R
ζ
q
(µ + Σ
1/2
z)e
λ
T
(µ+Σ
1/2
z)
·
|
Σ
|
−1/2
g
n

1
2
z
T
z

|
Σ
|
1/2
dz
e
λ
T
µ
ψ
g
n

1
2
t
T
Σt

F
θ

ζ
q

= µ +
c
n
Σ
1/2
∞
R
ζ
q
ze
λ
T
Σ
1/2
z
g
n

1
2
z
T
z

dz
ψ
g
n

1
2
t
T
Σt

F
θ

ζ
q

= µ + Σ
1/2
χ
q,λ
,
with χ
q,λ
an n ×1 vector of the form
χ
q,λ
=
c
n
ψ
g
n

1
2
t
T
Σt

F
θ

ζ
q


α
1,q
α
2,q
... α
n,q

T
,
(20)
where
α
i,q
=
∞
Z
ζ
q
z
i
e
λ
T
Σ
1/2
z
g
n

1
2
z
T
z

dz.
From (13), and after some algebraic calculations, we
have
α
i,q
= −
∞
Z
ζ
q,−i
dz
n−1,−i
e
(
λ
T
Σ
1/2
)
−i
z
n−1,i
∞
Z
ζ
q,i
e
(
λ
T
Σ
1/2
)
i
z
i
·d
i
G
n

1
2
z
T
n−1,−i
z
n−1,−i
+
1
2
z
2
i

.
where z
n
= (z
1
,...,z
n
)
T
, z
n−1,−i
=
(z
1
,z
2
,...,z
i−1
,z
i+1
,...,z
n
)
T
, and ζ
q,i
is the i − th
ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems
106

element of vector ζ
q
.
α
i,q
=
∞
Z
ζ
q,−i
dz
n−1,−i
e
(
λ
T
Σ
1/2
)
−i
z
n−1,i
[e
(
λ
T
Σ
1/2
)
i
z
i,q
G
n

1
2
z
T
n−1,−i
z
n−1,−i
+
1
2
z
2
i,q

+

λ
T
Σ
1/2

i
∞
Z
ζ
q,i
e
(
λ
T
Σ
1/2
)
i
z
i
·G
n

1
2
z
T
n−1,−i
z
n−1,−i
+
1
2
z
2
i

dz
i
]
= e
(
λ
T
Σ
1/2
)
i
z
i,q
∞
Z
ζ
q,−i
e
(
λ
T
Σ
1/2
)
−i
z
n−1,i
G
n

1
2
z
T
n−1,−i
z
n−1,−i
+
1
2
z
2
i,q

dz
n−1,−i
+

λ
T
Σ
1/2

i
∞
Z
ζ
q
e
λ
T
Σ
1/2
z
n
G
n

1
2
z
T
n
z
n

dz
n
.
Finally, from the random vectors θ
∗
i
and θ
∗∗
, we ob-
tain
α
i,q
=
1
c
∗
n−1,i
ψ
G
∗
n−1,i

1
2

λ
T
Σ
1/2

T
−i

λ
T
Σ
1/2

−i

·F
θ
∗
i
(ζ
q,−i
)
+

λ
T
Σ
1/2

i
c
∗
n
ψ
G
n

1
2
λ
T
Σλ

F
θ
∗∗
(ζ
q
).
Remark 1. Remark. The calculation of the compo-
nents χ
i,q
can be computed explicitly for special mem-
bers of the elliptical distributions (e.g., the normal,
logistic and Laplace distributions), in the same way,
that was obtained in (Dhaene et al., 2008).
Corollary 1. Corollary 1. Suppose that
X =

X
T
1
,X
T
2

T
v E
2n
(µ =

µ
X
1
µ
X
2

,Σ,g
2n
)
where

X
T
1
,X
T
2

T
, X
1
,X
2
∈ R
n
, has uncorrelated
components (i.e. Σ is a diagonal matrix). Then, the
MCEP takes the form
π
α,λ
(X) = µ + σχ
q,λ
. (21)
Here σ = diag(
√
σ
11
,...,
√
σ
nn
)
T
where σ
ii
is the
variance of the i-th random variable of X, and χ
i,q
is expressed as follows:
χ
i,q
=
ψ
G
∗
n−1,i

1
2
(λσ)
T
−i
(σλ)
−i

ψ
g
n

1
2
λ
T
Σλ

F
θ
∗
1
(z
q
1
2n−1
)c
n
F
θ
(z
q
1
2n
)c
∗
n−1,i
+ λ
i
√
σ
ii
ψ
G
n

1
2
λ
T
Σλ

ψ
g
n

1
2
λ
T
Σλ

F
θ
∗∗
(z
q
1
2n
)c
n
F
θ
(z
q
1
2n
)c
∗
n
,
where 1
k
= (1,1,...,1) is vector of k ones, and z
q
=
VaR
q
(Z), Z v E
1
(0,1,g
1
).
Proof. As X
1
and X
2
are uncorrelated random vec-
tors, Σ is a diagonal matrix, so
ζ
q
= Σ
−1/2
(VaR
q
(X) −µ) = z
q
1
2n
.
This gives us the following expression of π
q,λ
(X)
π
q,λ
(X) = µ + χ
q,λ
σ.
Lemma 2. Lemma 2. For the random vector
(X
1
,X
2
)
T
(X
1
,X
2
)
T
v E
2

µ
1
µ
2

,

σ
11
σ
12
σ
12
σ
22

,g
2

.
Then, the conditional Wang Esscher premium
π
q,λ
(X
1
,X
2
) is
π
q,λ
(X
1
,X
2
) = µ
1
+
σ
1
c
2
c
∗
1
ψ
G
1

1
2
λ
2
2
σ
22

F
θ
∗∗
(ζ
2,q
)
ψ
g
1

1
2
λ
2
σ
22

F
θ
(ζ
2,q
)
(22)
where σ
i
=
√
σ
ii
.
Proof. From the definition of π
α,λ
(X
1
,X
2
), we have
π
α,λ
(X
1
,X
2
)
= E(X
1
e
λX
2
|X
2
> VaR
q
(X
2
))
=
1
M
X
2
,q
(λ)
·
c
2
∞
R
−∞
∞
R
VaR
q
(X
2
)
x
1
e
λx
2
·
|
Σ
|
−1/2
g
2

1
2
(x −µ)
T
Σ
−1
(x −µ)

dx
F
X
2
(x
2,q
)
=
c
2
∞
R
−∞
∞
R
ζ
2,q
(µ
1
+ z
1
σ
1
)e
λ(µ
2
+z
2
σ
2
)
·g
2

1
2
z
T
z

dz
e
λµ
2
ψ
g
1

1
2
λ
2
σ
22

F
θ

ζ
2,q

= µ
1
+ σ
1
c
2
∞
R
−∞
∞
R
ζ
2,q
e
λσ
2
z
2
·G
2

1
2
z
T
z

dz
ψ
g
1

1
2
λ
2
σ
22

F
θ

ζ
2,q

.
Then, after some calculations, and by using the
marginality property of the elliptical distributions,
Modeling the Esscher Premium Principle for a System of Elliptically Distributed Risks
107

π
α,λ
(X
1
,X
2
) = µ
1
+
σ
1
c
2
c
∗
1
c
∗
2
∞
R
−∞
∞
R
ζ
2,q
e
λσ
2
z
2
·G
2

1
2
z
T
z

dz
ψ
g
1

1
2
λ
2
σ
22

F
θ

ζ
2,q

= µ
1
+
σ
1
c
2
c
∗
1
c
∗
1
∞
R
ζ
2,q
e
λσ
2
z
2
·G
1

1
2
z
2
2

dz
2
ψ
g
1

1
2
λ
2
σ
22

F
θ

ζ
2,q

,
and from the characteristic function of the elliptical
distributions
π
α,λ
(X
1
,X
2
) = µ
1
+
σ
1
c
2
c
∗
1
ψ
G
1

1
2
λ
2
2
σ
22

F
θ
∗∗
(ζ
2,q
)
ψ
g
1

1
2
λ
2
σ
22

F
θ
(ζ
2,q
)
Theorem 2. Theorem 2. Let X v E
n
(µ,Σ,g
n
) and let
S = X
1
+ X
2
+ ...+ X
n
, so
(X
i
,S)
T
(23)
v E
2







µ
i
n
∑
j=1
µ
j


,





σ
ii
n
∑
j=1
σ
i j
n
∑
j=1
σ
i j
σ
SS





,g
2





,
where σ
SS
=
n
∑
i, j=1
σ
i j
. Then
π
q,λ
(X
i
,S) = µ
i
+
σ
i
c
2
c
∗
1
ψ
G
1

1
2
λ
2
σ
SS

F
θ
∗∗
(ζ
S,q
)
ψ
g
1

1
2
λ
2
σ
SS

F
θ
(ζ
S,q
)
,
where ζ
S,q
= VaR
q
(S).
Proof. From the marginal properties of the elliptical
distributions, we know that the distribution of (X
i
,S)
T
is (23). Then, from Lemma 2, we immediately have
π
α,λ
(X
i
,S) = µ
i
+
σ
i
c
2
c
∗
1
ψ
G
1

1
2
λ
2
σ
SS

F
θ
∗∗
(ζ
S,q
)
ψ
g
1

1
2
λ
2
σ
SS

F
θ
(ζ
S,q
)
.
5 EXAMPLES
In this Section, we show several special members of
the elliptical family where the MCEP can be com-
puted. For computing the MCEP we need to compute
χ
q,λ
, (17).
5.1 Normal Distribution
Suppose that X v N
n
(µ,Σ). Then g
n
(u) = e
−u
,
so c
n
g
n
(
1
2
x
T
x) = φ
n
(x) = (2π)
−n/2
exp(−
1
2
x
T
x)
and Φ
n
(x) is the n − th multivariate standard
normal pdf and cdf, respectively. In this case
c
n
= (2π)
−n/2
, G
n
(u) = e
−u
= g
n
(u). Thus
G
∗
n−1,i

1
2
y
T
y

= exp(−
1
2

y
T
y+ζ
2
q,i

) so
f
θ
∗∗
(t) = f
θ
(t) ∝ exp

1
2
t
T
Σt + λ
T
Σ
1/2
t −
1
2
t
T
t

,t ∈ R
n
,
and
f
θ
∗
i
(u) ∝ exp

1
2
u
T
u+

λ
T
Σ
1/2

−i
u −
1
2
u
T
u

,u ∈ R
n−1
.
5.2 Logistic Distribution
Suppose that X has a logistic distribution. Then its
pdf is
f
X
(x) =
c
n
p
|Σ|
exp

−
1
2
(x −µ)
T
Σ
−1
(x −µ)


1 + exp

−
1
2
(x −µ)
T
Σ
−1
(x −µ)

2
,
and we write X v Lo
n
(µ,Σ) (Gupta et al., 2013). In
this case the density generator is
g
n
(u) =
exp(−u)
[1 + exp(−u)]
2
,
and c
n
is
c
n
= (2π)
−n/2
"
∞
∑
j=0
(−1)
j−1
j
1−n/2
#
−1
,
see (Landsman and Valdez, 2003), and the cumulative
generator G
n
(u) is
G
n
(u) =
∞
Z
u
e
−x
[1 + e
−x
]
2
dx =
e
−u
1 + e
−u
.
Then, f
θ
(t), f
θ
∗∗
(t), and f
θ
∗
i
(u) are, respectively,
f
θ
(t) ∝ ψ
g
n

1
2
t
T
Σt

−1
e
λ
T
Σ
1/2
t
exp

−
1
2
t
T
t


1 + exp

−
1
2
t
T
t

2
.
f
θ
∗∗
(t) ∝ ψ
G
n

1
2
t
T
Σt

−1
e
λ
T
Σ
1/2
t
exp

−
1
2
t
T
t

1 + exp

−
1
2
t
T
t

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems
108

and
f
θ
∗
i
(u) ∝ ψ
G
∗
n−1,i

1
2
u
T
u

−1
e
(
λ
T
Σ
1/2
)
−i
u
·
exp

−
1
2
u
T
u+
1
2
z
2
q,i

1 + exp

−
1
2
u
T
u+
1
2
z
2
q,i

,u ∈ R
n−1
.
We note that while ψ
G
n
and ψ
G
∗
n−1,i
can be difficult
to calculate, these characteristic functions are reduced
when applying them in the MCEP π
q,λ
(X). In fact,
for the i −th component of χ
q,λ
χ
i,q
=
c
n
F
θ
(ζ
q
)
[e
(
λ
T
Σ
1/2
)
i
z
i,q
+
1
2
z
2
i,q
∞
Z
ζ
q,−i
exp

−
1
2
u
T
u +

λ
T
Σ
1/2

−i
u

1 + exp(−
1
2
u
T
u +
1
2
z
2
i,q
)
du
+
∞
Z
ζ
q
exp

−
1
2
z
T
z + λ
T
Σ
1/2
z

1 + exp

−
1
2
z
T
z

dz].
5.3 Laplace Distribution
We say that X is multivariate Laplace random vector
if its pdf has the form (Fang, 2017)
f
X
(x) =
Γ(n/2)
2π
n/2
Γ(n)
exp

−

(x −µ)
T
Σ
−1
(x −µ)

1/2

and we write X v La
n
(µ,Σ). Then, the density gener-
ator and the characteristic generator are, respectively,
g
n
(u) = e
−
√
2u
and
ψ
g
n
(u) =
1
1 + u
.
In this case G
n
(u) is
G
n
(u) =
∞
Z
u
e
−
√
2x
dx =

1 +
√
2u

e
−
√
2u
,
Then, f
θ
(t), f
θ
∗∗
(t), and f
θ
∗
i
(u) are, respectively,
f
θ
(t) ∝

1 +
1
2
t
T
Σt

e
λ
T
Σ
1/2
t−
√
t
T
t
.
f
θ
∗∗
(t)
∝ ψ
G
n

1
2
t
T
Σt

−1
e
λ
T
Σ
1/2
t−
√
t
T
t

1 +
√
t
T
t

,t ∈ R
n
,
and
f
θ
∗
i
(u) ∝ ψ
G
∗
n−1,i

1
2
u
T
u

−1
e
(
λ
T
Σ
1/2
)
−i
u
· 1 +
r
1
2
u
T
u+
1
2
z
2
q,i
!
exp −
r
1
2
u
T
u+
1
2
z
2
q,i
!
,
u ∈ R
n−1
,
Notice that although ψ
G
n
and ψ
G
∗
n−1,i
can be difficult
to calculate they can be reduced when applying them
in the MCEP π
q,λ
(X). For the i −th component of
χ
q,λ
χ
i,q
=
c
n
F
θ
(ζ
q
)
[e
(
λ
T
Σ
1/2
)
i
z
i,q
·
∞
Z
ζ
q,−i
exp


λ
T
Σ
1/2

−i
u −
q
u
T
u + z
2
i,q

du
+
∞
Z
ζ
q
exp

−
√
z
T
z + λ
T
Σ
1/2
z

dz].
6 DISCUSSION
In this paper, we have shown how to model the Ess-
cher premium principle for a system of mutually de-
pendent risks with the underlying elliptical model,
which is common in the world of risk measurement
and actuarial science. Furthermore, we derived the
conditional moment generating function for the fam-
ily of multivariate elliptical distribution, in which the
MTCE measure is a special case,
MTCE
q
(X) =
∂
∂λ
M
X,q
(λ)|
λ=0
.
The MCEP measure quantifies the premium of a
vector of dependent risks under the condition that an
event outside a given probability level has occurred.
We derived a general formula of the MCEP for the
elliptical distributions
π
α,λ
(X) = µ + Σ
1/2
χ
q,λ
.
We then derived the MCEP for aggregate risks, based
on the Wang’s premium with exponential tilting,
π
α,λ
(X
i
,S) = µ
i
+
σ
1
c
∗
1
ψ
G
1

1
2
λ
2
σ
SS

F
θ
∗
(ζ
S,q
)
ψ
g
1

1
2
λ
2
σ
SS

F
θ
∗∗
(ζ
S,q
)
.
Modeling the Esscher Premium Principle for a System of Elliptically Distributed Risks
109

ACKNOWLEDGMENT
The author is grateful to anonymous referees for their
careful reading of the paper and useful comments.
This research was supported by the Israel Science
Foundation (grant No. 1686/17).
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