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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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