With reference to risk adjusted premium principle, in this paper we study excess of loss reinsurance with reinstatements in the case in which the aggregate claims are generated by a discrete distribution. In particular, we focus our study on conditions ensuring feasibility of the initial premium, for example with reference to the limit on the payment of each claim. Comonotonic exchangeability shows the way forward to a more general definition of the initial premium: some properties characterizing the proposed premium are presented.Excess of loss reinsurance; reinstatements; distortion risk measures; initial premium; exchangeability.

Antonella Campana

Paola Ferretti

English

Research Papers in Economics

Department of Applied Mathematics,  University of Venice  
 
 
 
WORKING PAPER SERIES 
 
 
 
 
 
 
 
Antonella Campana  and  Paola Ferretti 
 
 
 
Initial premium, aggregate claims  
and distortion risk measures in X L 
reinsurance with reinstatements 
 
 
 
 
 
 
 
 
 
 
Working Paper n. 203/2010 
October 2010 
 
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. Initial premium, aggregate claims and distortion risk
measures in XL reinsurance with reinstatements∗
Antonella Campana Paola Ferretti
<campana@unimol.it> <ferretti@unive.it>
Dept. SEGeS Dept. of Applied Mathematics and
University of Molise Advanced School of Economics
University of Venice
(October 2010)
Abstract. With reference to risk adjusted premium principle, in this paper we study
excess of loss reinsurance with reinstatements in the case in which the aggregate claims are
generated by a discrete distribution. In particular, we focus our study on conditions ensuring
feasibility of the initial premium, for example with reference to the limit on the payment
of each claim. Comonotonic exchangeability shows the way forward to a more general
deﬁnition of the initial premium: some properties characterizing the proposed premium are
presented.
Keywords: Excess of loss reinsurance; reinstatements; distortion risk measures; initial
premium; exchangeability.
JEL Classiﬁcation Numbers: G22.
MathSci Classiﬁcation Numbers: 62P05.
Correspondence to:
Paola Ferretti Dept. of Applied Mathematics, University of Venice
San Giobbe, Cannaregio 387
30121 Venezia, Italy
Phone: [++39] (041)-234-6923
Fax: [++39] (041)-234-7444
E-mail: ferretti@unive.it
∗ We are grateful to some MAF2010 conference members for suggestions and valuable comments on an
earlier version of the paper. This research was partially supported by MIUR.1 Introduction
The excess of loss reinsurance model we study in this paper is related to the model that
has been originally proposed and analyzed by Sundt [9] and that it has been subsequently
generalized (see [1] and [3]). Given the initial premium P, the limit on the payment of each
claim m, the number of reinstatements K (such that the aggregate limit M satisﬁes M =
(K +1)m), the aggregate deductible D, the percentages of reinstatement ci, (i = 1,...,K),
the total premium income results to be a random variable, say δ(P), which is so deﬁned
δ(P) = P
 
1 +
1
m
K−1 X
i=0
ci+1LX(D + im,D + (i + 1)m)
!
(1)
where LX(a,a + b) = min{(X − a)+,b} and (X − a)+ = X − a if X > a, otherwise
(X − a)+ = 0 (a,b ∈ R+). From the point of view of the reinsurer, the aggregate claims S
paid by the reinsurer for this XL reinsurance treaty, namely
S = LX(D,D + (K + 1)m) (2)
satisfy the relation
S =
K X
i=0
LX(D + im,D + (i + 1)m). (3)
This reinsurance cover is called an XL reinsurance for the layer m xs d with aggregate
deductible D and K reinstatements and provides total cover for the amount S. The initial
premium P is supposed to cover the original layer
LX(D,D + m) = min{(X − D)+,m}. (4)
The condition that the reinstatement is paid pro rata means that the premium for the i-th
reinstatement is a random variable given by
ciP
m
LX(D + (i − 1)m,D + im) (5)
where 0 ≤ ci ≤ 1 is the i-th percentage of reinstatement.
As it is well-known, excess of loss reinsurance with reinstatement has been essentially
studied in Actuarial Literature in the framework of collective model of risk theory and this
choice requires the knowledge of the claim size distribution in the classical evaluation of
pure premiums. Conversely, as a rule, only few characteristics of aggregate claims can be
explained and the interest for general properties characterizing the involved premiums is
still ﬂourishing. In order to contribute to this research, we set our analysis in the framework
of risk adjusted premiums (a more general choice than that of Walhin and Paris [10]) and
we concentrate our attention on properties exhibited by initial premiums, both with respect
to equilibrium condition (section 2), both with respect to feasibility with reference to the
limit on the payment of each claim (section 3). Then the attention is moved to the recent
framework of generalized risk adjusted premiums (Wu and Zhou [12]) where the main role is
played by comonotonic exchangeability assumption: we propose the deﬁnition of the related
generalized initial premium and we state some regularity properties displayed by it (section
4). Finally section 5 ends the paper with some concluding remarks.
12 Excess of loss reinsurance with reinstatements: some pre-
liminary results
Recently (see [3]) we studied the initial risk adjusted premium P as solution of the equilib-
rium condition expressing the fact that the value of the total premium income δ(P) equals
the distorted expected value of the aggregate claims S, that is
Wg1(δ(P)) = Wg2(S) (6)
where
Wgj(X) =
Z ∞
0
gj(SX(x))dx (7)
and the functions gj (j = 1,2) are distortion function, i.e. non-decreasing functions gj :
[0,1] → [0,1] such that gj(0) = 0 and gj(1) = 1; SX(x) = P(X > x) is the decumulative
distribution function of X. This initial premium P is well-deﬁned and it is given by
P =
PK
i=0 Wg2(LX(im,(i + 1)m))
1 + 1
m
PK−1
i=0 ci+1 Wg1(LX(im,(i + 1)m))
. (8)
We see that (6) gives a sort of global equilibrium: the distortion risk measure associated with
the aggregate claims to the layer S must be equal to the distortion risk measure associated
with the total premium income δ(P). If we apply the same scheme to each layer, we derive
that the premium Pi, for all i ∈ {0,1,...,K}, must satisfy the following equation
Wg1(Pi) = Wg2(LX(im,(i + 1)m)). (9)
Given that distortion risk measures obey positive homogeneity, the initial premium P0 is
easily deﬁned by the previous equation; in fact
Wg1(P0) = P0 = Wg2(LX(0,m)). (10)
The premium for the i-th reinstatement is given by
Pi =
ciP0
m
LX(D + (i − 1)m,D + im), (11)
then we have
ciP0
m
Wg1(LX((i − 1)m,im)) = Wg2(LX(im,(i + 1)m)). (12)
Hitherto we have implicitly assumed the reinstatements percentages ci to be known. If the
reinstatement percentages ci are not ﬁxed in advance, we obtain
ci =
mWg2(LX(im,(i + 1)m))
P0Wg1(LX((i − 1)m,im))
(13)
where the initial premium P0 given by (10). We note that both the initial premium P0
both the percentages ci do not depend on the number of reinstatements K. The values
2given by (13) are useful even when the ci are to be determined in advance. They will give
us a point of reference in order to obtain local equilibrium for each reinstatement. If the
reinstatements percentages ci are determined by formula (13), one can easily verify that the
following equality holds
P = P0
both in the case of same distortion functions g1 = g2 (see [1]) both in the case of not
necessarily equal distortion functions gj (j = 1,2).
In fact, by (8) and (13) it is
P =
PK
i=0 Wg2(LX(im,(i + 1)m))
1 + 1
P0
PK−1
i=0 Wg2(LX((i + 1)m,(i + 2)m))
, (14)
that is,
P =
PK
i=0 Wg2(LX(im,(i + 1)m))
1 +
PK−1
i=0
Wg2(LX((i+1)m,(i+2)m))
Wg2(LX(0,m))
(15)
and, ﬁnally,
P = Wg2(LX(0,m)) = P0. (16)
Therefore local equilibrium condition for each reinstatement ensures global equilibrium as
deﬁned in (6).
3 Initial premiums and limit on the payment of each claim
Generally, reinstatement percentages are ﬁxed in advance: this assumption moves our at-
tention to the study of the initial premium deﬁned by local equilibrium condition (9). From
now on we make the assumption that the premium for the i-th reinstatement (5) is a two-
point random variable distributed as ci P Bpi and Bpi denotes a Bernoulli random variable
such that Pr[Bpi = 1] = pi = 1 − Pr[Bpi = 0]. Closely related to equation (12) is the deﬁ-
nition of the initial premium P0i (i = 1,...,K) for which the local equilibrium condition is
satisﬁed in each layer, that is
P0i =
mWg2(LX(im,(i + 1)m))
ciWg1(LX((i − 1)m,im))
=
mg2(pi+1)
cig1(pi)
. (17)
Clearly acceptability of the treaty in the market requires that P0i ≤ m for each i = 1,...,K.
In order to determine conditions ensuring the validity of this condition which is not generally
true (see Campana [2]), we direct our attention to the analysis of the relation between two
consecutive values of the initial premium, that is to the comparison of P0i to P0i+1.
Let us consider the following real-valued function H deﬁned on [0,pi+1]:
H(x) = ci+1g1(pi+1)g2(pi+1) − cig1(pi)g2(x).
Note that H(0) = ci+1g1(pi+1)g2(pi+1) ≥ 0. Given that
H(pi+1) = g2(pi+1)[ci+1g1(pi+1) − cig1(pi)] ≤ g2(pi+1)ci+1[g1(pi+1) − g1(pi)]
3if the percentages of reinstatement are decreasing, then increasing monotonicity of the
distortion function g ensures both non-positivity of H(pi+1) both decreasing monotonicity
of H. If the distortion function g2 is continuous, intermediate value theorem ensures that
there exists at least one point x in [0,pi+1] such that
H(x) = ci+1g1(pi+1)g2(pi+1) − cig1(pi)g2(x) = 0.
As a consequence, we have that
H(x)

≥ 0, 0 ≤ x ≤ x;
≤ 0, x ≤ x ≤ pi+1.
This means that, given any choice of probabilities pi+1 and pi (pi+1 ≤ pi), it is possible to
state that there exists al least one probability x ∈ [0,pi+1] such that P0i+1 = P0i. Moreover,
we can conclude that it is possible to anticipate the relation underlying two consecutive
initial premiums P0i and P0i+1 for any choice of a probability pi+2 such as follows
P0i+1

≤ P0i, 0 ≤ pi+2 ≤ x;
≥ P0i, x ≤ pi+2 ≤ pi+1.
(18)
In this way, it is possible to deﬁne a ﬁnite sequence of initial premiums P0i (i = 1,...,K)
such that P0i is increasing (decreasing): it is suﬃcient to choose a ﬁnite sequence of prob-
abilities p0i (i = 2,...,K + 1) where at any step, x ≤ pi ≤ pi−1, (respectively, 0 ≤ pi ≤ x).
In this way we obtain the following results.
Proposition 1 Given an XL reinsurance with K reinstatements and distortion functions
g1 and g2, where g2 is continuous and the percentages of reinstatement are decreasing, then
there exists a ﬁnite sequence of probabilities pi (i = 1,...,K+1) such that the ﬁnite sequence
of initial risk adjusted premiums P0i (i = 1,...,K) is monotone (increasing or decreasing).
Proposition 2 Given an XL reinsurance with K reinstatements and distortion functions
g1 and g2, where g2 is continuous and the percentages of reinstatement are decreasing, there
exists a ﬁnite sequence of probabilities pi (i = 1,...,K + 1), with g2(pK+1) ≤ g1(pK) or
g2(p2) ≤ g1(p1), such that P0i ≤ m, for all i = 1,...,K.
4 Generalized initial premiums and exchangeability
As it is well-known, insurance premium principles and risk measures can be characterized
with reference to a set of axioms, that is to a set of desirable properties that they should
have for actuarial practice. Among them, recently Goovaerts et al. ([5], [6]), Wu and
Zhou (see [12]) propose a generalization of Yaari’s risk measure by relaxing his proposal.
Comonotonic additivity has been generalized to comonotonic exchangeability in [5], while
the link between comonotonic additivity and independent additivity has been addressed in
[6]. Additivity for a ﬁnite number of comonotonic risks is substituted by countable additivity
and countable exchangeability in [12] in order to characterize generalized distortion premium
4principles. In this context, we refer to a non-empty collection of risks L on (Ω,F,P) such
that min{X,a}−min{X,b}, aX and X +a are all in L for any X ∈ L and for any a,b ∈ R.
In studying diﬀerent characterizations of insurance premium principles and risk measures,
Goovaerts et al. [5] refer to some realistic situations owing to speciﬁc sets of axioms: they
refer to the notion of exchangeability, that with reference to a principle H states
A1. Exchangeability
H(Xc + Yc) = H(X∗c + Yc) provided that H(X) = H(X∗) (19)
where the random vectors Xc and Yc admit the same marginal distributions of X and Y,
respectively, with the comonotonic dependence structure. The following theorem states a
representation result of the so called generalized distortion principle
Theorem 1 A premium principle H : L → R+ admits for all X ∈ L the representation
H(X) = h
Z ∞
0
g(SX(x))dx

(20)
where h is a continuous non-decreasing function on R+, if and only if H satisﬁes exchange-
ability, monotonicity in stochastic order and continuity.
As comonotonic additivity is the most essential property of distortion integral (and Choquet
integral), comonotonic exchangeability by generalizing comonotonic additivity suggests a
new deﬁnition of distortion premium principle (and Choquet price principle), that is
Wh
g (X) = h
Z ∞
0
g(SX(x))dx

(21)
which is called generalized risk adjusted premium principle; the function g is a distortion
function and h is a continuous non-decreasing function on R+.
Now we turn our attention to the analysis of local equilibrium condition (9) and in order
to consider the case of generalized risk adjusted premium principle we deﬁne and study the
following generalized local equilibrium condition
Wh1
g1 (Pi) = Wh2
g2 (LX(im,(i + 1)m)) (22)
where the functions gj are distortion functions and hj are continuous non-decreasing func-
tions on R+ (j = 1,2). Accordingly, the deﬁnition of generalized initial premium naturally
follows: it is solution, write P0i (i = 1,...,K), of the generalized local equilibrium condi-
tion (22). Clearly, positivity of P0i is assumed. In the following we focus our attention on
properties exhibited by the generalized initial premium P0i, with reference to some assump-
tions made on the functions involved, namely the distortion functions gj and the composing
functions hj, (j = 1,2). The analysis of equation (22) when aggregate claims are generated
by a discrete distribution may be conducted by introducing the following related function
Fh1h2
g1g2 (x,pi,pi+1) = h1
hcix
m
g1(pi)
i
− h2 [g2(pi+1)]. (23)
5In this way, the generalized initial premium P0i is such that
Fh1h2
g1g2 (P0i,pi,pi+1) = 0.
Then if gj (j = 1,2) are continuously diﬀerentiable distortion functions and hj are con-
tinuously diﬀerentiable and non-decreasing functions on R+ (j = 1,2), where g1(pi) 6= 0
and h0
1
cix
m g1(pi)

6= 0, P0i results to be a continuously diﬀerentiable function of (pi,pi+1).
Moreover
∂P0i
∂pi
(pi,pi+1) = −
"
∂Fh1h2
g1g2 (x,pi,pi+1)
∂pi
#
.
"
∂Fh1h2
g1g2 (x,pi,pi+1)
∂x
#
= −x
g0
1(pi)
g1(pi)
∂P0i
∂pi+1
(pi,pi+1) = −
"
∂Fh1h2
g1g2 (x,pi,pi+1)
∂pi+1
#."
∂Fh1h2
g1g2 (x,pi,pi+1)
∂x
#
=
m
ci
h0
2[g2(pi+1)]
h0
1[cix
m g1(pi)]
g0
2(pi+1)
g1(pi)
where x = P0i. Then we can state the following result.
Proposition 3 Let gj be continuously diﬀerentiable distortion functions and let hj be con-
tinuously diﬀerentiable and non-decreasing functions on R+ (j = 1,2), where g1(pi) 6= 0 and
h0
1
cix
m g1(pi)

6= 0. Then the generalized initial premium P0i (i = 1,...,K) is a continuously
diﬀerentiable function of (pi,pi+1) with
∇P0i(pi,pi+1) =

−x
g0
1(pi)
g1(pi)
,
m
ci
h0
2[g2(pi+1)]
h0
1[cix
m g1(pi)]
g0
2(pi+1)
g1(pi)

where x = P0i.
Note that the generalized initial premium P0i is monotone function of the probabilities:
as a function of the probability pi it is decreasing, while as a function of the probability
pi+1 it is increasing.
65 Concluding remarks
In the considered XL model of reinsurance, when the reinstatements are paid the total
premium income becomes a random variable which is correlated to aggregate claims. In-
complete information on aggregate claims distribution ampliﬁes the interest for general
properties characterizing the involved premiums. In this paper our main goal is to study
properties exhibited by initial premium: our analysis refers to two settings, that of risk
adjusted premiums and that of generalized risk adjusted premiums. We assume that the
premium for each reinstatement is a two-point random variable, a particularly interesting
hypothesis given that the reinsurance companies often assess treaties under the conjecture
that there are only total losses. This happens, for example, when they use the rate on line
method to price catastrophe reinsurance. As a matter of fact, conditions ensuring feasibility
of the initial premium with respect to comparison with the limit on the payment of each
claim and regularity properties displayed by the generalized initial premium are presented.
In particular, the regularity results of the generalized initial premium hint that further
research may be addressed to the analysis of optimal initial premiums.
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8

A comonotonic image of independence for additive risk measures.

A new characterization of distortion premiums via countable additivity for comonotonic risks.

Excess of loss reinsurance with reinstatements: premium calculation and ruin probability of the cedent.

Premium calculation by transforming the layer premium density.

Pricing excess of loss reinsurance with reinstatements.

Reinstatement premiums for an excess of loss reinsurance contract. Giornale Istituto Italiano degli Attuari,

Risk adjusted premiums for excess of loss reinsurance with reinstatements.

Risk measures and comonotonicity: a review.

Some new classes of consistent risk measures.

W.: Excess of loss reinsurance with reinstatements revisited.

What do distortion risk measures tell us on excess of loss reinsurance with reinstatements?

Initial premium, aggregate claims and distortion risk measures in XL reinsurance with reinstatements

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Initial premium, aggregate claims and distortion risk measures in XL reinsurance with reinstatements

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