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Insurance: Mathematics and Economics 69 (2016) 149–155
Contents lists available at ScienceDirect
Insurance: Mathematics and Economics
journal homepage: www.elsevier.com/locate/ime
An optimal co-reinsurance strategy
Amir T. Payandeh Najafabadi ∗ , Ali Panahi Bazaz
Department of Mathematical Sciences, Shahid Beheshti University, G.C. Evin, 1983963113, Tehran, Iran
article
info
Article history:
Received August 2015
Received in revised form
April 2016
Accepted 22 April 2016
Available online 12 May 2016
MSC:
91B30
97M30
97K80
62F15
Keywords:
Bayesian method
Optimal co-reinsurance strategy
Balanced loss functions
Optimal reinsurance contract
Copula method
Utility function
abstract
This article considers a co-reinsurance strategy that (1) protects insurance companies against catastrophic
risks; (2) enables insurers to gather sufficient information about the different risk attitudes of reinsurers
and diversify their reinsured risks; (3) enables insurers to create better risk-sharing profiles by balancing
the risk tolerances of reinsurers; (4) has the benefit of allowing reinsurers to accumulate experience
with risks with which they are unfamiliar; (5) reduces the overall direct cost of a reinsurance contract;
(6) allows a government to back some insurance products, such as the terrorism insurance programs
that were established in many countries after the September 11th terrorist attacks; and (7) reflects
the practical reinsurance industry of some countries, such as Iran. Such a co-reinsurance strategy can
be fully determined by estimating its parameters whenever three optimal criteria are satisfied and
prior information about the unknown parameters is available. Two simulation-based studies have been
conducted to demonstrate (1) the practical applications of our findings and (2) the possible impact of any
type of dependency between the co-reinsurance’s parameters and the evaluated optimal co-reinsurance
strategy.
© 2016 Elsevier B.V. All rights reserved.
1. Introduction
Suppose the aggregate loss, X , is a nonnegative and continuous
random variable with a cumulative distribution function FX and
a density function fX defined on the measurable space (Ω , F , P ),
where Ω = [0, ∞) and F is the Borel σ -field on Ω . In addition,
suppose that a random claim, X , can be decomposed as the sum
of an insurance portion, XI , and a reinsurance portion, XR , i.e., X =
XI + XR , where both XI and XR are continuous functions that satisfy
0 ≤ XI &XR ≤ X for all X ≥ 0.
Now, suppose that the reinsurance portion, XR , is apportioned
between two or more reinsurers. Such a reinsurance contract
is well-known as a co-reinsurance strategy. More precisely,
a co-reinsurance strategy is an arrangement whereby two or
more reinsurance companies enter into a single reinsurance
contract to cover a policyholder’s risk, X . Certainly, the more
complicated placement process of a co-reinsurance contract
increases transaction costs, but it also creates a risk-pooling system
that (1) protects insurance companies against catastrophic risks
such as floods, earthquakes, etc. (Boone et al., 2012; Castellano,
∗
Corresponding author. Tel.: +98 21 29903011; fax: +98 21 22431649.
E-mail address: amirtpayandeh@sbu.ac.ir (A.T. Payandeh Najafabadi).
http://dx.doi.org/10.1016/j.insmatheco.2016.04.005
0167-6687/© 2016 Elsevier B.V. All rights reserved.
2012); (2) enables insurers to gather sufficient information about
the different risk attitudes of reinsurers (Ratliff, 2003) and to
diversify their reinsured risks (Neuthor, 2013; Skogh and Wu,
2005); (3) enables insurers to create better risk-sharing profiles by
balancing the risk tolerances of reinsurers (Chi and Meng, 2014);
(4) has the benefit of enabling reinsurers to accumulate experience
with risks with which they are unfamiliar (Ratliff, 2003; Castellano,
2010); (5) reduces the overall direct cost of a reinsurance contract,
which makes the co-reinsurance strategy more beneficial to the
primary insurer (Froot and Stein, 1998; Froot, 2007); and (6)
allows a government to back some insurance products, such as
the terrorism insurance programs that were established in many
countries after the September 11th terrorist attacks (MichelKerjan and Pedell, 2005; Ortolani et al., 2011). Moreover, in some
countries, including Iran, young insurance industries have been
supported by governments, which act as reinsurers in the market.
The participating reinsurers share the ceded part of a primary
insurer’s risk according to certain reinsurance strategies. For
instance, when a government acts as a reinsurer in the market, a
specific portion of the insurer’s risk (i.e., a proportional reinsurance
strategy) is covered by the government, and the rest of the insurer’s
risk is split among the participating reinsurers under an optimal (in
some sense) reinsurance strategy.
As far as we know, a limited amount of research on designing
an optimal co-reinsurance strategy has been conducted. Most
150
A.T. Payandeh Najafabadi, A.P. Bazaz / Insurance: Mathematics and Economics 69 (2016) 149–155
of the existing research consists of studies of the advantages,
disadvantages and impacts of co-reinsurance on the market.
Coutts and Thomas (1997) employed the asset and liability
model of Daykin et al. (1994) to examine the impact of several
reinsurance strategies, including the co-reinsurance strategy, on
a company’s expected performance in terms of its asset mix or
business volume. Major (2004) considered two reinsurers using
proportional and stop-loss strategies by using the reinsurance
function r (X , α, M , k) = (1 − k) max{0, min{X − α, M }}
to study the incremental impact of adding a new contract or
canceling an existing contract on the capital needed to support
all of the business portfolio. Pérez-Blanco et al. (2014) employed
certainty equivalence theory with a utility function to design a
co-reinsurance strategy in an agricultural framework. Asimit et al.
(2013), Chi and Meng (2014), and Boonen et al. (2015) developed a
co-reinsurance strategy by minimizing reinsurers’ value at risk (or
conditional value at risk).
This article considers two reinsurers who use two different
strategies in a single reinsurance contract. More precisely, it
assumes that the reinsurance portion, XR , is apportioned between
two reinsurers with strategies I1 (X ) and I2 (X ), i.e., XR = I1 (X ) +
I2 (X ); the first reinsurer adopts a proportional strategy, i.e.,
I1 (X ) = α1 X , and the second reinsurer considers a combination
of proportional and stop-loss strategies for the rest of the primary
risk, i.e., α2 min((1 − α1 )X , M ), where α1 , α2 ∈ (0, 1). With
these two reinsurance strategies, the second reinsurer’s portion is
I2 (X ) = (1 − α1 )X − α2 min{(1 − α1 )X , M }, and consequently, the
reinsurance and insurance function can be written as
XR = X − α2 min{(1 − α1 )X , M }
XI = α2 min{(1 − α1 )X , M },
(1)
respectively, where α1 and α2 represent the proportions of the
reinsurers and M represents the stop-loss part of the second
reinsurance strategy.
Now, suppose that under certain optimal criteria from both
the insurer’s and the reinsurers’ viewpoints, appropriate (in some
sense) estimators for the unknown parameters α1 , α2 , and M have
been obtained. This article employs such appropriate estimators
as target estimators and, under the well-known balanced loss
function, develops a Bayes estimator for α1 , α2 , and M that
simultaneously takes into account the viewpoints of both the
insurer and the reinsurers in designing the co-reinsurance strategy.
The rest of this article is organized as follows. Section 2 collects
some elements that play vital roles in the rest of this article.
Three optimal criteria that provide different estimators for α1 ,
α2 , and M and Bayes estimators for α1 , α2 , and M under the
balanced loss function are given in Section 3. Section 4 describes
two simulation-based studies illustrating the practical application
of our results. Some concluding remarks and suggestions are
provided in Section 5.
To derive any Bayesian inference for the problem at hand, one
must choose a loss function that penalizes incorrect decisions.
Unlike other loss functions, the balanced loss function, which was
introduced by Zellner (1994), has the advantage of simultaneously
penalizing incorrect decisions and minimizing the distance
between the Bayes estimator and any given target estimators. Such
target estimators are optimal solutions (in some sense) to the
problem of estimating the unknown parameters.
The following provides a definition of a balanced loss function
with k(> 1) target estimators, the k-balanced loss function.
Definition 1. Suppose ξ1 , ξ2 , . . . , ξk are k given target estimators
for the unknown parameter ξ . Moreover, suppose that ρ(·, ·) is
an arbitrary given loss function. A balanced loss function that
measures how close the estimator ξ̂ is to the target estimators
ξ1 , ξ2 , . . . , ξk and to the unknown parameter ξ is
Lρ,ω1 ,…,ωk ,ξ1 ,…,ξk (ξ , ξ̂ ) =
k

ωi ρ(ξi , ξ̂ ) + 1 −
i=1
k


ωi ρ(ξ , ξ̂ ),
i =1
where ωi ∈ [0, 1), for i = 1, . . . , k, are weights that satisfy

k
i=1 ωi < 1. For convenience, L0 is subsequently used instead of Lρ,0,...,0,ξ1 ,...,ξk when ωi = 0 for i = 1, . . . , k. Jafari et al. (2006) derived a Bayes estimator for a 2-balanced loss function. The following theorem generalizes their results to a k-balanced loss function. Theorem 1. Suppose the expected posterior losses ρ(ξi , ξ̂ ) for i = 1, . . . , k are finite for at least one ξ̂ such that ξ̂ ̸= ξi for i = 1, . . . , k. The Bayes estimator for ξ with respect to the prior distribution π (ξ ) under the k-balanced loss given by Definition 1 is equivalent to the Bayes estimator with respect to the prior distribution π (ξ |x ) = ∗ k   ωi 1{ξi (x)} (ξ ) + 1 − i =1 k   ωi π (ξ |x ), (2) i =1 under the loss function L0 := Lρ,0,...,0,ξ1 ,...,ξk . Proof. Suppose that measures µX (·) and µ′X (·) dominate π (ξ |x ) and π ∗ (ξ |x ), respectively. By the definition of a Bayesian estimator under finite expected posterior losses ρ(ξi , ξ̂ ), for i = 1, . . . , k, one can conclude that ξπ ,ω1 ,...,ωk (x) = arg min   k ξ̂ Ξ  + 1− ωi ρ(ξi , ξ̂ ) i=1 k    ωi ρ(ξ , ξ̂ ) π (ξ |x )dµX (ξ ) i=1 = arg min   k ξ̂ 2. Preliminary Ξ  + The Bayesian method combines available information from two different sources to derive a more attractive method of making inferences about the problem at hand. Under the name of the credibility method, the Bayesian method is well-known in various areas of the actuarial sciences. For instance, see Whitney (1918) and Payandeh Najafabadi et al. (in press) for its application in the experience rating system; Bailey (1950), Payandeh Najafabadi (2010), and Payandeh Najafabadi et al. (2012) for its application in evaluating insurance premiums; Hesselager and Witting (1988) and England and Verrall (2002)for its application in the IBNR claims reservation system; and Makov et al. (1996), Makov (2001), and Hossack et al. (1999) for general applications in actuarial science.  1− ωi ρ(ξ , ξ̂ )1{ξi (x)} (ξ ) i=1 k    ωi ρ(ξ , ξ̂ ) π (ξ |x )dµX (ξ ) i=1  = arg min ξ̂ Ξ  + 1− ρ(ξ , ξ̂ ) k   k  ωi 1{ξi (x)} (ξ ) i =1  ωi π (ξ |x )dµX (ξ ) i=1  = arg min ξ̂ ′ Ξ ∪{ξ1 (x)}∪···{ξk (x)} × dµ X (ξ ) = ξ ∗ (x). L0 (ξ , ξ̂ )π ∗ (ξ |x ) A.T. Payandeh Najafabadi, A.P. Bazaz / Insurance: Mathematics and Economics 69 (2016) 149–155  The above proof is an extension (and quite similar to) the proof of Lemma 1 in Jafari et al. (2006). The next corollary provides a Bayes estimator under a k-balanced loss function with squared error loss. Corollary 1. The Bayes estimator with respect to the prior distribution π under the k-balanced loss function with squared error loss ρ(ξ , ξ̂ ) = (ξ − ξ̂ )2 is given by ξ Bayes (x) = Eπ ∗ (ξ |x ) = k   ωi ξi (x) + 1 − k  i=1 + M̂ (1) 1 − FX (1) ωi Eπ (ξ |x ). (3) Now, we represent the idea of a copula, which is employed in Section 4 when the prior distributions are dependent. A copula combines marginal information about several variables in a multivariate function to obtain a joint distribution function for those variables. Copulas can capture interdependencies that can be inferred by neither visual investigation nor association measures such as correlation coefficients, as described by Payandeh Najafabadi and Qazvini (2015). A d-variate copula function Cζ (·, . . . , ·) provides the following joint distribution function for continuous random variables X1 , . . . , Xd : FX1 ,...,Xd (x1 , . . . , xd ) = Cζ (F1 (x1 ), . . . , Fd (xd )), where F1 (·), . . . , Fd (·) are marginal distribution functions of X1 , . . . , Xd , respectively. The Farlie–Gumbel–Morgenstern copula is a simple copula. The following provides a joint distribution function for continuous random variables X1 , X2 , and X3 based on the 3-variate Farlie–Gumbel–Morgenstern copula, FX1 ,X2 ,X3 (x1 , x2 , x3 ) = 1 + ζ (1 − 2F1 (x1 ))(1 − 2F2 (x2 ))(1 − 2F3 (x3 )), (4) where the dependence parameter, ζ , is in [−1, 1]. See Klugman et al. (2012) for more details. α̂2 M̂ 0 = (1) (1) 1 − α̂1 + α̂2(1) 1 − FX This section develops Bayes estimators for α1 , α2 , and M under the co-reinsurance model (1). To perform the desired estimation, one must consider three optimal criteria and three target estimators for each unknown parameter. Hereafter, our optimal criteria are to minimize the variance of the insurer’s risk portion; to maximize the expected exponential utility of the insurer’s terminal wealth; and to maximize the expected exponential utility of the reinsurers’ terminal wealth. The following provides solutions for α1 , α2 , and M that minimize the variance of the insurer’s risk portion, which is the first optimal criterion. Lemma 1. In accordance with the co-reinsurance strategy given in (1), the variance of the insurer’s risk portion is minimized whenever the parameters α1 , α2 , and M satisfy (1) 2 (1) 0 = −2(α̂2 ) (1 − α̂1 ) − α̂2(1) (M̂ (1) )2 M̂ (1) (1) 1  1−α̂ α̂ (1) )2 x2 fX (x)dx (α̂2(1) M̂ (1) − 1)fX 1 (1) (1) H (α̂1 , α̂2 , M̂ (1) (1) ∂ α̂1 (1) 1 − α̂1  M̂ ∂ (1) ∂ α̂2  (1) (1) H (α̂1 , α̂2 , M̂ (1) ) (1) (1) 1 − α̂1  M̂ (1) −2 (1) 1 − α̂1 ∂ ∂ M̂ (1) (1) (1) H (α̂1 , α̂2 , M̂ (1) ), where H (α1 , α2 , M ) := (µI (α1 , α2 , M ))2 and µI (α1 , α2 , M ) = α2 (1 − α1 )  M 1−α1 0 M )). xfX (x)dx + α2 M (1 − FX ( 1−α 1 Proof. The variance of XI can be evaluated as follows: g (α1 , α2 , M ) := v ar (XI )  M 1−α1 = α22 (1 − α1 )2 x2 fX (x)dx + α22 M 2 0    M × 1 − FX − µ2I (α1 , α2 , M ). 1 − α1 Differentiating g (α1 , α2 , M ) with respect to α1 , α2 and M and setting the resulting expressions equal to zero leads to the desired equations. To ensure that solutions of the above three equations are the desired results, one must show that the following Hessian matrix is positive definite.  ∂ 2g (z )  ∂α1 ∂α1  2  ∂ g H (z ) =   ∂α ∂α (z )  2 1  ∂ 2g (z ) ∂ M ∂α1 ∂ 2g (z ) ∂α1 ∂α2 ∂ 2g (z ) ∂α2 ∂α2 ∂ 2g (z ) ∂ M ∂α2  ∂ 2g (z ) ∂α1 ∂ M    ∂ 2g , (z ) ∂α2 ∂ M    ∂ 2g (z ) ∂M∂M (5) The following provides solutions for α1 , α2 , and M by maximizing the expected exponential utility of the insurer’s terminal wealth. Lemma 2. Suppose the insurer’s surplus process under the coreinsurance contract (1) is given by  UtI = uI0 + (1 + θ )E I 0 (1) 0 = 2α̂2 (1 − α̂1 )2 M̂ (1) (1) 1−α̂ 1  0  M̂ (1)   XI − N (t )  (i) XI , i=1 (i)  (1 + θ0I )E ( Ni=(1t ) XI(i) ) is the insurance premium, and N (t ) is a Poisson process with intensity λ. Then, the exponential utility of the expected wealth of an insurer, e.g., u(x) = −e−β0 x , is maximized whenever the parameters α1 , α2 , and M satisfy 0 = (1 + θ0I ) M̂ (2) (2) 1  1−α̂ xfX (x)dx 0 (1) M̂ (2) (2) 1  1−α̂ − (2) (2) xeβ0 α̂2 (1−α̂1 )x fX (x)dx 0  x2 fX (x)dx (i) where uI0 is the initial wealth of the insurer, the random variable XI is the portion of the insurer’s wealth subject to random claim X (i) under the co-reinsurance contract, θ0I is the safety factor, π0 (t ) := 1 − α̂1 ) N (t )  i=1 0 (1 − ∂ − where z := (α̂1 , α̂2 , M̂ ). To be certain that it is positive definite, the above matrix must be investigated numerically. 3. Main results + M̂ (1) (1)  151  (α̂2(1) M̂ (1) − 1)fX   i =1   (2) 0 = −(1 + θ )  1 − α̂1 I 0 M̂ (2) (2) 1  1−α̂ 0 xfX (x)dx 152 A.T. Payandeh Najafabadi, A.P. Bazaz / Insurance: Mathematics and Economics 69 (2016) 149–155   + M̂ (2) 1 − FX (2) + (1 − α̂1 )  M̂ (2) Lemma 3. Suppose the surplus process of both reinsurers under the co-reinsurance contract (1) is given by (2) 1 − α̂1 M̂ (2) (2) 1−α̂ 1   ( 2) (2) β0 α̂2 (1−α̂1 )x xe UtR fX (x)dx + (1 + θ )E = uR0 R 0 where uR0 N (t )  (2) + M̂ (2) eβ0 α̂2 (2) 0 = −β0 α̂2 M̂ (2) M̂ (2)  1 − FX M̂  (2) (i) (2)  1 − α̂1 + ln(1 + θ ). I 0  −β0 uI0 +π0 (t )− E  −e N (t ) i=1 (i) XI = −e i=1 R 0 M̂ (3) (3) 1  (3) 0 = −β1 (1 + θ )α̂2   1−α̂ xfX (x)dx 0  M̂ (3) (3) 1−α̂ 1 + e (3) β1 α̂2(3) xeβ1 (1−α̂2 (3) (1−α̂1 ))x fX (x)dx 0  M −β0 uI0 +(1+θ0 )λt α2 (1−α1 )  1−α1 0  xfX (x)dx+α2 M 1−FX = −e  M 1−α1     (3) 0 = (1 + θ ) (1 − α̂1 ) R 0 M̂ (3) (3) 1  1−α̂ xfX (x)dx 0   M     1−α1 β α (1−α )x M β α M  1 fX (x)dx+e 0 2 λt 0 e 0 2 −1  1−FX 1−α 1 ×e  . − M̂ Maximizing the above expression is equivalent to minimizing the following function:  = −β0  + (1 + θ0 )λt α2 (1 − α1 ) uI0 M 1−α1  (3) 1 − FX + α2 M 1 − FX + λt  M 1−α1  xfX (x)dx M̂ 1−α̂ − fX (x)dx 0 β0 α2 M   1 − FX M 1 − α1   −1 . (6) Differentiating g0 (α1 , α2 , M ) with respect to α1 , α2 and M and setting the resulting equations equal to zero leads to the desired equations. To ensure that solutions of the above three equations are the desired results, one must show that the following Hessian matrix is positive definite:  ∂ 2g 0 (z )  ∂α1 ∂α1  2  ∂ g0 H (z ) =   ∂α ∂α (z )  2 1  ∂ 2g 0 (z ) ∂ M ∂α1 ∂ 2 g0 (z ) ∂α1 ∂α2 ∂ 2 g0 (z ) ∂α2 ∂α2 ∂ 2 g0 (z ) ∂ M ∂α2  ∂ 2 g0 (z ) ∂α1 ∂ M    ∂ 2 g0 , (z ) ∂α2 ∂ M    ∂ 2 g0 (z ) ∂M∂M (3) (3) ∞  (3) M̂ (3) (3) 1−α̂ 1 ))x fX (x)dx (3) 0 = β 1 (1 + θ ) 1 − FX  e M̂ (3) (3) 1 1−α̂ R 0 1 − α1 β0 α2 (1−α1 )x (3) 1 − α̂1 1    M   M̂ (3) − (1 − α̂1 ) xeβ1 (1−α̂2 (1−α̂1 0  ∞ (3) (3) eβ1 (x−α̂2 M̂ ) fX (x)dx − M̂ (3) (3) 0   (3) g0 (α1 , α2 , M ) +e (i) XR ,  (1 + θ0R )E ( Ni=(1t ) XR(i) ) is the reinsurance premium, and N (t ) is a Poisson process with intensity λ. Then, the exponential utility of the expected wealth of the reinsurers, e.g., u(x) = −e−β1 x , is maximized whenever the parameters α1 , α2 , and M satisfy λt (E (eβ0 XI )−1)  N (t )  is the initial wealth of the reinsurers, the random variable  −β0 (uI0 +π0 (t )) − XR XR is the portion of the reinsurers’ wealth subject to random claim X (i) under the co-reinsurance contract, θ0R is the safety factor, π1 (t ) := Proof. Using the characteristic function of the compound Poisson distribution, one can restate the exponential utility of the expected wealth of the insurer, u(x) = −e−β0 x , as   i=1 0  (i) eβ1 (x−α̂2  M̂ (3) (3) 1 − α̂1 M̂ (3) ) fX (x)dx. Proof. The proof is quite similar to that of Lemma 2 To derive Bayes estimators for the unknown parameters α1 , α2 , and M, one must evaluate their posterior distributions based on a random sample. The following two lemmas provide the results. (1) (n) Lemma 4. Suppose XR , . . . , XR |(θ , α1 , α2 , M ) is a random sample that represents the reinsurers’ portion of random claims X (1) , . . . , X (n) |(θ , α1 , α2 , M ). Then, the joint density function of (1) (n) XR , . . . , XR |(θ , α1 , α2 , M ) is (1) (n) f (xR , . . . , xR |θ , α1 , α2 , M )  (7) = n1  n1 1 1 − α2 (1 − α1 ) n ×  (i)  (i)  fX i=1 xR  1 − α2 (1 − α1 )  fX xR + α2 M , i=n1 +1 (i) where z := (α̂1 , α̂2 , M̂ ). To be certain that it is positive definite, the above matrix must be investigated numerically. where n1 is the number of xR s that are less than or equal to M /(1 − α1 ) − M α2 . The following lemma provides solutions for α1 , α2 , and M that maximize the expected exponential utility of both reinsurers’ terminal wealth simultaneously, which is the third optimal criterion. Proof. The distribution function FXR |(θ ,α1 ,α2 ,M ) at t can be rewritten as P (XR ≤ t ) = P  XR ≤ t , X ≤ M (1 − α1 )  A.T. Payandeh Najafabadi, A.P. Bazaz / Insurance: Mathematics and Economics 69 (2016) 149–155  M  + P XR ≤ t , X >
(1 − α1 )


M
= P (1 − α2 (1 − α1 ))X ≤ t , X ≤
(1 − α1 )


M
+ P X − α2 M ≤ t , X >
(1 − α1 )



t
M
= P X ≤ min
,
(1 − α2 (1 − α1 )) (1 − α1 )


M
+ P X ≤ t + α2 M , X >
(1 − α1 )

 (t )
= FX (t + α2 M )1
M
(1−α1 )(1−α2 (1−α1 )) ,+∞


t
 (t ),
1
+ FX
M
0, (1−α )(1−α
(1 − α2 (1 − α1 ))
1
2 (1−α1 ))
where 1A (x) is the indicator function. The desired proof uses the
(1)
(n)
fact that XR , . . . , XR |(θ , α1 , α2 , M ) is a random sample.
Now, suppose that π (Θ , A1 , A2 , M ) is the joint prior distribution for the unknown parameters θ , α1 , α2 , and M. The following
provides the posterior distribution for these parameters.
(n)
(1)
Lemma 5. Suppose XR , . . . , XR |(θ , α1 , α2 , M ) is a random sample that represents the reinsurers’ portion of random claims
X (1) , . . . , X (n) |(θ , α1 , α2 , M ). Moreover, suppose that π (Θ , A1 ,
A2 , M) is the joint prior distribution for the unknown parameters θ ,
α1 , α2 , and M. Then, the joint posterior distribution for θ , α1 , α2 , and
M is given by the expression in Box I.
Proof. An application of Lemma 4 completes the proof.
(1)

(n)
The marginal posterior density function for α1 |xR , . . . , xR ,
(1)
(n)
(1)
(n)
α2 |xR , . . . , xR , and M |xR , . . . , xR can be rewritten as
π(α1 |x(R1) , . . . , x(Rn) )
  
=
π ((θ , α1 , α2 , M )|x(R1) , . . . , x(Rn) )dMdα2 dθ
Θ
A2
M
(1)
π(α2 |xR , . . . , x(Rn) )
  
=
π ((θ , α1 , α2 , M )|x(R1) , . . . , x(Rn) )dMdα1 dθ
Θ
A1
M
(1)
π(M |xR , . . . , x(Rn) )
  
=
π ((θ , α1 , α2 , M )|x(R1) , . . . , x(Rn) )dα1 dα2 dθ .
Θ
A2
(1)
(n)
Theorem 2. Suppose XR , . . . , XR |(θ , α1 , α2 , M ) is a random
sample that represents the reinsurers’ portion of random claims
X (1) , . . . , X (n) |(θ , α1 , α2 , M ). Moreover, suppose that π (Θ , A1 ,
A2 , M) is the joint prior distribution for the unknown parameters
θ, α1 , α2 , and M. Then, the Bayesian estimators for α1 , α2 and M
under a 3 -balanced loss function with squared error loss are
α̂1Bayes = ω1 α̂1(1) + ω2 α̂1(2) + ω3 α̂1(3)


3
 



+ 1−
ωi Eπ A1 x(R1) , . . . , x(Rn) ,
i =1
α̂
+
1−
3

i=1
M̂
Bayes
153

 


ωi Eπ A2 x(R1) , . . . , x(Rn) ,
+ ω2 M̂ (2) + ω3 M̂ (3)

3
 



1−
ωi Eπ M x(R1) , . . . , x(Rn) ,
= ω1 M̂

+
(1)
i=1
(1)
(1)
(2)
(2)
(3)
(3)
where (α̂1 , α̂2 , M̂ (1) ), (α̂1 , α̂2 , M̂ (2) ), and (α̂1 , α̂2 , M̂ (3) ) are
optimal solutions for α1 , α2 and M with respect to the three optimal
criteria given in Lemmas 1, 2, and 3, respectively.
Proof. An application of Corollary 1 completes the desired
proof.
The next section describes two simulation-based studies that
illustrate practical applications of the above findings.
4. Examples
This section provides two numerical examples that show how
the above findings can be applied in practice. Namely, the first
example considers the situation in which the prior distributions
of the unknown parameters are independent, and the second
example studies the situation in which the joint prior distribution
for the unknown parameters is determined by the application of a
3-variate Farlie–Gumbel–Morgenstern copula.
Example 1 (independent parameters)
Suppose 100 random numbers are generated from one of
the distributions given in the first column of Table 1. Moreover,
suppose that the prior distributions of the unknown parameters
α1 , α2 and M are independent and given in the second, third, and
fourth columns of Table 1, respectively.
(1)
(2)
(2)
After estimating the targets (α̂ (1) , α̂2 , M̂ (1) ), (α̂1 , α̂2 , M̂ (2) ),
(3)
(3)
and (α̂1 , α̂2 , M̂ (3) ) using Lemmas 1, 2, and 3, one can employ
Theorem 2 to find Bayes estimates for α1 , α2 and M.
The third and last columns of Table 1 represent the mean and
the standard deviation, respectively, of the Bayes estimator for
α1 , α2 and M, which generates 100 random numbers from a given
distribution when ω1 = 0.05, ω2 = 0.1, ω3 = 0.15. This estimator
was derived using Theorem 2 when the mean of 100 iterations of
the Bayes estimator for α1 , α2 and M was used as an estimator for
α1 , α2 and M.
Table 1 shows Bayes estimators for the parameters of the coreinsurance strategy given by Eq. (1). The small variance of these
estimators shows that the estimation method is an appropriate
method to use with the different samples.
A1
Now, one can derive Bayes estimators for α1 , α2 , and M based on
the optimal solutions for α1 , α2 and M given by Lemmas 1, 2, and
3. The following provides Bayes estimators under a 3-balanced loss
function with squared error loss.
Bayes
2

(1)
= ω1 α̂2 + ω2 α̂2(2) + ω3 α̂2(3)
Example 2 (correlated parameters)
Reconsider Example 1 and assume that the unknown parameters α1 , α2 and M are, somehow, dependent. To formulate this
dependency, we consider the Farlie–Gumbel–Morgenstern copula
(see Definition (4)) with copula parameter ζ = 0.5 to evaluate the
joint prior distribution whenever all the corresponding marginal
prior distributions are known.
Table 2 is determined by recalculating the results of Example 1
with this new assumption.
Table 2 shows Bayes estimators for the parameters of the coreinsurance strategy given by Eq. (1). The small variance of these
estimators shows that the estimation method is an appropriate
method with respect to different samples. Comparing the results
shown in Tables 1 and 2, one may conclude that assuming any
type of dependency between the parameters of the co-reinsurance
strategy given by Eq. (1) impacts the optimal co-reinsurance
strategy evaluated here.
154
A.T. Payandeh Najafabadi, A.P. Bazaz / Insurance: Mathematics and Economics 69 (2016) 149–155

 
Θ
A1
1
1−α2 (1−α1 )
 

A2
M
n1 
n1
1
1−α2 (1−α1 )

fX
i =1
n1 
n1
fX

(i)
xR

1−α2 (1−α1 )

i=1
(i)
xR
1−α2 (1−α1 )
n

(i)
fX (xR + α2 M )π (θ , α1 , α2 , M )
i=n1 +1
n

,
(i)
fX (xR + α2 M )π (θ , α1 , α2 , M )dMdα2 dα1 dM
i=n1 +1
(i)
where n1 is the number of xR s that are less than or equal to M /(1 − α1 ) − M α2 .
Box I.
Table 1
Mean and standard deviation of Bayes estimator for α1 , α2 and M based upon 100 sample size and 100 iterations.
Risk
distribution
Prior distribution
for
Prior distribution
for
α1
α2
Prior distribution
for
M
EXP(1)
Beta(2,2)
Beta(2,2)
EXP(2)
EXP(2)
Beat(2,2)
Beta(2,4)
EXP(2)
EXP(8)
Beta(3,2)
Beta(2,3)
Gamma(2,2)
Weibull(2,1)
Beta(2,4)
Beta(3,4)
Gamma(3,2)
Weibull(4,1)
Beat(5,2)
Beta(1,3)
Gamma(2,4)
Weibull(6,3)
Unif(0,1)
Unif(0,1)
Gamma(3,4)
Mean (standard
deviation)
Mean (standard
deviation)
Mean (standard deviation)
α1
α2
M
0.7809
(4.686 × 10−6 )
0.7949
(1.095 × 10−13 )
0.7949
(0.634 × 10−16 )
0.395
(2.059 × 10−5 )
0.1569
(1.490 × 10−5 )
0.1077
(2.085 × 10−8 )
0.2744
(1.142 × 10−16 )
0.1379
(1.056 × 10−11 )
0.1381
(1.576 × 10−15 )
0.283
(3.375 × 10−5 )
0.7702
(1.142 × 10−5 )
0.9426
(2.435 × 10−8 )
1.7463
(0.0002)
0.6946
(2.811 × 10−5 )
1.0332
(0.897 × 10−19 )
1.347
0.511 × 10−12
0.6843
(0.684 × 10−12 )
0.6176
(0.0002)
Table 2
Mean and standard deviation of Bayes estimator for α1 , α2 and M based upon 100 sample size and 100 iterations, under dependency condition.
Risk
distribution
Prior distribution
for
Prior distribution
for
α1
α2
Prior distribution
for
M
EXP(1)
Beta(2,2)
Beta(2,2)
EXP(2)
EXP(2)
Beat(2,2)
Beta(2,4)
EXP(2)
Weibull(4,1)
Beat(5,2)
Beta(1,3)
Gamma(2,4)
Weibull(6,3)
Unif(0,1)
Unif(0,1)
Gamma(3,4)
5. Conclusions and suggestions
This article considers a co-reinsurance strategy that (1) protects
insurance companies against catastrophic risks, such as floods and
earthquakes; (2) enables insurers to gather sufficient information
about the different risk attitudes of reinsurers to diversify their
reinsured risks; (3) enables insurers to develop better risk-sharing
profiles by balancing the risk tolerances of reinsurers; (4) has the
benefit of allowing reinsurers to accumulate experience with risks
with which they are unfamiliar; (5) reduces the overall direct
cost of a reinsurance contract, which makes the co-reinsurance
strategy more beneficial to the primary insurer; and (6) allows
a government to back some insurance products, such as the
terrorism insurance programs that were established in many
countries after the September 11th terrorist attacks. Moreover, in
some countries, including Iran, young insurance industries have
been supported by governments, which act as reinsurers in the
market.
This co-reinsurance strategy was developed by estimating its
parameters when three optimal criteria and prior information
about the unknown parameters were available. Through the two
simulation-based studies, practical applications of our findings
have been demonstrated. To demonstrate the possible impact of
dependencies between the unknown parameters on the evaluated
optimal co-reinsurance strategy, in Example 2, we employed the
Farlie–Gumbel–Morgenstern copula (with copula parameter ζ =
0.5). This example showed that any type of dependency between
Mean (standard
deviation)
Mean (standard
deviation)
Mean (standard deviation)
α1
α2
M
0.7809
(0.797 × 10−11 )
0.7949
(2.152 × 10−13 )
0.1401
(3.734 × 10−8 )
0.1145
(4.080 × 10−6 )
0.2746
(1.337 × 10−10 )
0.1381
(1.417 × 10−12 )
0.7886
(4.051 × 10−8 )
0.4347
(1.525 × 10−5 )
4.6832
(0.097)
4.0585
(0.012)
4.1093
(4.435 × 10−12 )
1.1219
(0.0016)
the co-reinsurance parameters can impact the optimal coreinsurance strategy evaluated here. Therefore, actuaries should
consider such dependencies in their calculations.
Minimizing the variance of the insurer’s risk portion, maximizing the expected exponential utility of the insurer’s terminal
wealth and maximizing the expected exponential utility of the two
reinsurers’ terminal wealth were used as the three optimal criteria. Certainly, such optimal criteria may be replaced by other desired optimal risk measurements, such as the VaR, the TVaR, the
CTE, and the ruin probability; see Cai and Tan (2007), Tan et al.
(2011), Kolkovska (2008), and others for more details. To evaluate
n1 (given in Lemma 4), one must consider certain initial values of
the unknown parameters α1 , α2 and M. The results of this article
can be extended to handle the situation in which the initial values are not available and, consequently, n1 (given in Lemma 4) is
misspecified. This extension can be obtained by replacing the joint
density function given in Lemma 4 with the following expression:
(1)
(n)
f (xR , . . . , xR |θ , α1 , α2 , M )
=
n

(1)
(n)
f (xR , . . . , xR |θ , α1 , α2 , M , n1 = N )P (n1 = N )
N =0
π (θ , α1 , α2 , M |x(R1) , . . . , x(Rn) )
n

=
π (θ , α1 , α2 , M |x(R1) , . . . , x(Rn) , n1 = N )P (n1 = N ).
N =0
A.T. Payandeh Najafabadi, A.P. Bazaz / Insurance: Mathematics and Economics 69 (2016) 149–155
Acknowledgment
The authors thank a referee for his/her suggestions, which led
to several improvements.
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