In this paper we focused our attention to the study of an excess of loss reinsurance with reinstatements, a problem previously studied by Sundt [5] and, more recently, by Mata [4] and HÄurlimann [3]. As it is well-known, the evaluation of pure premiums requires the knowledge of the claim size distribution of the insurance risk: in order to face this question, different approaches have been followed in the actuarial literature. In a situation of incomplete information in which only some characteristics of the involved elements are known, it appears to be particularly interesting to set this problem in the framework of risk adjusted premiums. It is shown that if risk adjusted premiums satisfy a generalized expected value equation, then the initial premium exhibits some regularity properties as a function of the percentages of reinstatement.Excess of loss reinsurance, reinstatements, distortion risk measures, expected value equation

Antonella Campana

Paola Ferretti

English

Research Papers in Economics

Department of Applied Mathematics,  University of Venice  
WORKING PAPER SERIES 
Antonella Campana  and  Paola Ferretti 
What do distortion risk measures tell us 
on excess of loss reinsurance with 
reinstatements ? 
Working Paper n. 175/2008 
November 2008 
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. What do distortion risk measures tell us
on excess of loss reinsurance with reinstatements?∗
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 2008)
Abstract. In this paper we focused our attention to the study of an excess of loss reinsur-
ance with reinstatements, a problem previously studied by Sundt [5] and, more recently, by
Mata [4] and H¨ urlimann [3]. As it is well-known, the evaluation of pure premiums requires
the knowledge of the claim size distribution of the insurance risk: in order to face this
question, diﬀerent approaches have been followed in the actuarial literature. In a situation
of incomplete information in which only some characteristics of the involved elements are
known, it appears to be particularly interesting to set this problem in the framework of
risk adjusted premiums. It is shown that if risk adjusted premiums satisfy a generalized
expected value equation, then the initial premium exhibits some regularity properties as a
function of the percentages of reinstatement.
Keywords: Excess of loss reinsurance; reinstatements; distortion risk measures; expected
value equation.
JEL Classiﬁcation Numbers: G22.
MathSci Classiﬁcation Numbers: 62P05.
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
In recent years the study of excess of loss reinsurance with reinstatements has become a
major topic, in particular with reference to the classical evaluation of pure premiums which
is based on the collective model of risk theory.
The problem, previously studied by Sundt [5] and, more recently, by Mata [4] and
H¨ urlimann [3], requires the evaluation of pure premiums given the knowledge of the claim
size distribution of the insurance risk: in order to face this question, diﬀerent approaches
have been followed in the actuarial literature. Sundt [5] based the computation on the Panjer
recursion numerical method and H¨ urlimann [3] provided distribution-free approximations
to pure premiums.
In a situation of incomplete information in which only some characteristics of the in-
volved elements are known, it appears to be particularly interesting to set this problem in
the framework of risk adjusted premiums.
We start from the methodology developed by Sundt [5] to price excess of loss reinsurance
with reinstatements for pure premiums and, with the aim of relaxing the basic hypothesis
made by Walhin and Paris [6] who calculated the initial premium P under the Proportional
Hazard transform premium principle, we address our analysis to the study of the role played
by risk adjusted premium principles. The particular choice in the proposal of Walhin and
Paris of the PH-transform risk measure strengthens our interest in the study of risk adjusted
premiums which belong to the class of distortion risk measures deﬁned by Wang [7].
In the mathematical model we studied (for more details see Campana [1]), when the
reinstatements are paid (0 ≤ ci ≤ 1 is the i-th percentage of reinstatement) the total
premium income δ(P) becomes a random variable which is correlated to the aggregate claims
S. Since risk measures satisfy the properties of linearity and additivity for comonotonic risks
(see [2]) and layers are comonotonic risks, we can deﬁne the function
F(P,c1,c2,...,cK) = P
 
1 + 1
m
 K−1
i=0 ci+1 Wg1(LX(im,(i + 1)m))
 
−
−
 K
i=0 Wg2(LX(im,(i + 1)m))
(1)
where g1 and g2 are distortion functions. This function gives a measure of the distance
between the total premium income δ(P) and the distortion risk measures of the aggregate
claims S. It is shown that if risk adjusted premiums satisfy the expected value equation,
that is the previous distance is null, then the initial premium exhibits some regularity
properties as a function of the percentages of reinstatement.
The paper is organized as follows. In Section 2 we ﬁrst review some basic settings
for describing the excess of loss reinsurance model and we remind some deﬁnitions and
preliminary results in the ﬁeld of non-proportional reinsurance covers. Section 3 is devoted
to the problem of detecting the total initial premium: we present the study of the case
in which the reinstatements are paid in order to consider the total premium income as a
random variable which is correlated to the aggregate claims. The analysis is set in the
framework of distortion risk measures: some basic deﬁnitions and results in this ﬁeld are
recalled. Section 4 presents the main results related to the problem of measuring the total
initial premium as a function of the percentages of reinstatement, dependence that it is
generally neglected in the literature. Some concluding remarks in Section 5 end the paper.
12 Excess of loss reinsurance with reinstatements: problem
setting
The excess of loss reinsurance model we study in this paper is related to the model that has
been proposed and analyzed by Sundt [5]. Some notations, abbreviations and conventions
used throughout the paper are the following.
An excess of loss reinsurance for the layer m in excess of d, written m xs d, is a reinsur-
ance which covers the part of each claim that exceeds the deductible d but with a limit on
the payment of each claim, which is set equal to m; in other words, the reinsurer covers for
each claim of size Y the amount
LY (d,d + m) = min{(Y − d)+,m}
where (a)+ = a if a > 0, otherwise (a)+ = 0.
We consider an insurance portfolio: N is the number of claims occurred in the portfolio
during the reference year and Yi is the i-th claim size (i = 1,2,...,N). The aggregate
claims to the layer is the random sum given by
X =
N  
i=1
LYi(d,d + m).
It is assumed that X = 0 when N = 0. An excess of loss reinsurance, or for short an XL
reinsurance, for the layer m xs d with aggregate deductible D and aggregate limit M covers
only the part of X that exceeds D but with a limit M:
LX(D,D + M) = min{(X − D)+,M}.
This cover is called an XL reinsurance for the layer m xs d with aggregate layer M xs D.
Generally it is assumed that the aggregate limit M is given as a whole multiple of the
limit m, i.e. M = (K + 1)m: in this case we say that there is a limit in the number of the
losses covered by the reinsurer. 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 following amount
LX(D,D + (K + 1)m) = min{(X − D)+,(K + 1)m}. (2)
Let P be the initial premium: it covers the original layer, that is
LX(D,D + m) = min{(X − D)+,m}. (3)
It can be considered as the 0-th reinstatement.
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) (4)
2where 0 ≤ ci ≤ 1 is the i-th percentage of reinstatement. If ci = 0 the reinstatement is free,
otherwise it is paid.
The related total premium income is a random variable, say δ(P), which is so deﬁned
δ(P) = P
 
1 +
1
m
K−1  
i=0
ci+1LX(D + im,D + (i + 1)m)
 
. (5)
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) (6)
satisfy the relation
S =
K  
i=0
LX(D + im,D + (i + 1)m). (7)
3 Initial premium, aggregate claims and distortion risk mea-
sures
The total premium income δ(P) results to be a random variable which is correlated to the
aggregate claims S in the case in which the reinstatements are paid. Then it follows that
it is not obvious how to calculate the initial premium P.
Despite its importance in practice, only recently some Authors have moved their atten-
tion to the study of techniques to calculate the initial premium: more precisely, Sundt [5]
proposed the methodology to calculate the initial premium P under pure premiums and
premiums loaded by standard deviation principle.
Looking at the pure premium principle for which the expected total premium income
should be equal to the expected aggregate claims payments
E[δ(P)] = E[S] (8)
it is quite natural to consider the case in which premium principles belong to more general
classes: with the aim of plugging this gap, we focus our attention to the class of distortion
risk measures. Our interest is supported by Walhin and Paris [6] who calculated the initial
premium P under the Proportional Hazard transform premium principle. Even if their
analysis is conducted by a numerical recursion, the choice of the PH-transform risk measure
as a particular concave distortion risk measure strengthens our interest.
Furthermore, in an excess of loss reinsurance with reinstatements the computation of pre-
miums requires the knowledge of the claim size distribution of the insurance risk: with refer-
ence to the expected value equation of the XL reinsurance with reinstatements (8), Sundt [5]
based the computation on the Panjer recursion numerical method and H¨ urlimann [3] pro-
vided distribution-free approximations to pure premiums.
Note that both the Authors assumed only the case of equal reinstatements, a particular
hypothesis on basic elements characterizing the model.
3In this paper we set our analysis in the framework of distortion risk measures: the core
of our proposal is represented by the choice of a more general equation characterizing the
excess of loss reinsurance with reinstatements, in such a way that it is possible to obtain
some general properties satisﬁed by the initial premium as a function of the percentages of
reinstatement. In order to present the main results, we recall some basic deﬁnitions and
results.
3.1 Distortion risk measures
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. [2]).
In this paper we consider the distortion risk measure introduced by Wang [7]:
Wg(X) =
  ∞
0
g(HX(x))dx (9)
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-Risk are examples of risk measures belonging to this class. In the particular
case of a power g function, i.e. g(x) = x1/ρ, ρ ≥ 1, the corresponding risk measure is the
PH-transform risk measure, that is the choice made by Walhin and Paris [6].
Distortion risk measures satisfy the following properties (see Wang [7] and Dhaene et
al. [2]):
P1. Additivity for comonotonic risks
Wg(Sc) =
n  
i=1
Wg(Xi) (10)
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; (11)
P3. Translation invariance
Wg(X + b) = Wg(X) + b for any costant b; (12)
P4. Monotonicity
Wg(X) ≤ Wg(Y ) (13)
for any two random variables X and Y where X ≤ Y with probability 1.
4In 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.
4 Risk adjusted premiums
With reference to equation (8) we consider the new expected value condition
Wg1(δ(P)) = Wg2(S) (14)
where g1 e g2 are distortion functions.
It expresses the fact that the value of the total premium income δ(P) equals the distorted
expected value of the aggregate claims S. Note that non necessarily the distortion functions
g1 and g2 are the same.
We consider the previous equilibrium condition an equation on the initial premium P:
if it admits a solution which is unique, we call initial risk adjusted premium P the solution.
Proposition 1 Given an XL reinsurance with K reinstatements and distortion functions
g1 and g2, the initial risk adjusted premium P which is the unique solution of equation
Wg1(δ(P)) = Wg2(S)
satisﬁes the following properties:
i) P is a decreasing function of each percentage of reinstatement ci (i = 1,...,K);
ii) P is a convex, supermodular, quasiconcave and quasiconvex function of the percentages
of reinstatement c1,c2,...,cK.
Proof
By hypothesis, the initial risk adjusted premium P is the solution, if it exists and it is
unique, of equation (14), that is of the equilibrium condition in which the distorted expected
premium income equals the distorted expected claim payments.
Since the layers LX(im,(i + 1)m), i = 1,2,...,K + 1, are comonotonic risks from (7)
we ﬁnd
Wg2(S) =
K  
i=0
Wg2(LX(im,(i + 1)m)). (15)
From (5) by assuming the absence of aggregate deductible (i.e. D = 0) we have
Wg1(δ(P)) = P
 
1 +
1
m
K−1  
i=0
ci+1Wg1(LX(im,(i + 1)m))
 
. (16)
Therefore, the initial premium P is well-deﬁned and it is given by
5P = f(c1,c2,   ,cK) =
 K
i=0 Wg2(LX(im,(i + 1)m))
1 + 1
m
 K−1
i=0 ci+1 Wg1(LX(im,(i + 1)m))
. (17)
Clearly the function f is a decreasing function of any percentage of reinstatement ci
(where i = 1,...,K).
Moreover, if we set
A =
K  
i=0
Wg2(LX(im,(i + 1)m)),
the gradient vector ∇f(c) is
∇f(c) =
 
∂f
∂cl
(c)
 
=



−AWg1(LX((l − 1)m,lm))
m
 
1 + 1
m
 K−1
i=0 ci+1 Wg1(LX(im,(i + 1)m))
 2



for each l = 1,...,K.
Convexity follows by the strict positivity and concavity of the function
1 +
1
m
K−1  
i=0
ci+1 Wg1(LX(im,(i + 1)m)).
Moreover, the Hessian matrix Hf(c) of the function f is given by
Hf(c) =
 
∂2f
∂cl∂cn
(c)
 
=



2AWg1(LX((l − 1)m,lm))Wg1(LX((n − 1)m,nm))
m2
 
1 + 1
m
 K−1
i=0 ci+1 Wg1(LX(im,(i + 1)m))
 3



for each l,n = 1,...,K. More compactly it can be expressed as
Hf(c) =
 
Wg1(LX((l − 1)m,lm))Wg1(LX((n − 1)m,nm))
 
B
for each l,n = 1,...,K, where
B =
2A
m2
 
1 + 1
m
 K−1
i=0 ci+1 Wg1(LX(im,(i + 1)m))
 3.
Clearly, Hf(c) is nonnegative deﬁnite.
Given that any cross partial derivative of the matrix Hf(c) is nonnegative, the function
g is supermodular.
Finally, the initial risk adjusted premium P is both quasiconcave and quasiconvex func-
tion of the percentages of reinstatement c1,c2,...,cK because it is ratio of aﬃne functions.
6Remark 1. Note that the regularity properties exhibited by the initial risk adjusted
premium P are not inﬂuenced by equality/inequality conditions between the two distortion
functions g1 and g2. Moreover, any hypothesis on concavity/convexity of distortion risk
measures may be omitted because they are unnecessary to prove the smooth shape of the
initial premium P as a function of c1,c2,...,cK.
Remark 2. The reinsurance companies often assess treaties under the assumption that
there are only total losses. This happens, for example, when they use the rate on line
method to price catastrophe reinsurance. Then it follows that the aggregate claims are
generated by a discrete distribution and we have that (for more details see Campana ([1])
P = f(c1,c2,   ,cK) =
m
 K
i=0 g2(pi+1)
1 +
 K−1
i=0 ci+1 g1(pi+1)
(18)
where the premium for the i-th reinstatement (4) 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].
Then
∇f(c) =
 
∂f
∂cl
(c)
 
=



−m
 K
i=0 g2(pi+1)
 
1 +
 K−1
i=0 ci+1g1(pi+1)
 2 g1(pl)



and
Hf(c) =
 
∂2f
∂cl∂cn
(c)
 
=



2m
 K
i=0 g2(pi+1)
 
1 +
 K−1
i=0 ci+1g1(pi+1)
 3 g1(pl)g1(pn)



for each l,n = 1,...,K.
5 Concluding remarks
In Actuarial Literature excess of loss reinsurance with reinstatement has been essentially
studied in the framework of collective model of risk theory for which the classical evalua-
tion of pure premiums requires the knowledge of the claim size distribution. Generally, in
practice, there is incomplete information: only few characteristics of the aggregate claims
can be computed. In this situation the interest for general properties characterizing the
involved premiums is ﬂourishing.
Setting this problem in the framework of risk adjusted premiums, it is shown that if risk
adjusted premiums satisfy a generalized expected value equation, then the initial premium
exhibits some regularity properties as a function of the percentages of reinstatement. In
this way it is possible to relax the particular choice made by Walhin and Paris [6] of the
7PH-transform risk measure and to extend the analysis of excess of loss reinsurance with
reinstatements to cover the case of not necessarily equal reinstatements.
The obtained results suggest that further research may be addressed to the analysis of
optimal premium plans.
Acknowledgements
We are grateful to some MAF2008 conference members for valuable comments and sugges-
tions on an earlier version of the paper.
References
[1] Campana, A.: Reinstatement premiums for an excess of loss reinsurance contract.
Giornale Istituto Italiano degli Attuari, Vol. LXIX, Roma (2006).
[2] Dhaene, J., Vanduﬀel, S., Tang, Q.H., Goovaerts, M.J., Kaas, R., Vyncke, D.: Sol-
vency capital, risk measures and comonotonicity: a review. Research Report OR 0416,
Department of Applied Economics, K.U. Leuven (2004).
[3] H¨ urlimann, W.: Excess of loss reinsurance with reinstatements revisited. ASTIN Bul-
letin 35(1) (2005) 211-238.
[4] Mata, A.J.: Pricing excess of loss reinsurance with reinstatements. ASTIN Bul-
letin 30(2) (2000) 349-368.
[5] Sundt, B.: On excess of loss reinsurance with reinstatements. Bulletin of the Swiss
Association of Actuaries, Heft 1 (1991) 51-65.
[6] Walhin, J.F., Paris, J.: Excess of loss reinsurance with reinstatements: premium
calculation and ruin probability of the cedent. Proceedings of the third IME Congress,
London (1999).
[7] Wang, S.: Premium calculation by transforming the layer premium density. ASTIN
Bulletin 26(1) (1996) 71-92.
8

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

On excess of loss reinsurance with reinstatements.

Premium calculation by transforming the layer premium density.

Pricing excess of loss reinsurance with reinstatements.

Reinstatement premiums for an excess of loss reinsurance contract.

Solvency capital, risk measures and comonotonicity: a review.

W.: Excess of loss reinsurance with reinstatements revisited.

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

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

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