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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 (? , F , P ), where ? = [0, ∞) and F is the Borel σ -field on ? . 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 ξ? is to the target estimators ξ1 , ξ2 , . . . , ξk and to the unknown parameter ξ is Lρ,ω1 ,...,ωk ,ξ1 ,...,ξk (ξ , ξ? ) = k ? ωi ρ(ξi , ξ? ) + 1 − i=1 k ? ? ωi ρ(ξ , ξ? ), 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 , ξ? ) for i = 1, . . . , k are finite for at least one ξ? such that ξ? ?= 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 , ξ? ), for i = 1, . . . , k, one can conclude that ξπ ,ω1 ,...,ωk (x) = arg min ? ?? k ξ? Ξ ? + 1− ωi ρ(ξi , ξ? ) i=1 k ? ? ? ωi ρ(ξ , ξ? ) π (ξ |x )dµX (ξ ) i=1 = arg min ? ?? k ξ? 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 , ξ? )1{ξi (x)} (ξ ) i=1 k ? ? ? ωi ρ(ξ , ξ? ) π (ξ |x )dµX (ξ ) i=1 ? = arg min ξ? Ξ ? + 1− ρ(ξ , ξ? ) k ? ? k ? ωi 1{ξi (x)} (ξ ) i =1 ?? ωi π (ξ |x )dµX (ξ ) i=1 ? = arg min ξ? ′ Ξ ∪{ξ1 (x)}∪···{ξk (x)} × dµ X (ξ ) = ξ ∗ (x). L0 (ξ , ξ? )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 , ξ? ) = (ξ − ξ? )2 is given by ξ Bayes (x) = Eπ ∗ (ξ |x ) = k ? ? ωi ξi (x) + 1 − k ? i=1 + M? (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. α?2 M? 0 = (1) (1) 1 − α?1 + α?2(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(α?2 ) (1 − α?1 ) − α?2(1) (M? (1) )2 M? (1) (1) 1 ? 1−α? α? (1) )2 x2 fX (x)dx (α?2(1) M? (1) − 1)fX 1 (1) (1) H (α?1 , α?2 , M? (1) (1) ∂ α?1 (1) 1 − α?1 ? M? ∂ (1) ∂ α?2 ? (1) (1) H (α?1 , α?2 , M? (1) ) (1) (1) 1 − α?1 ?? M? (1) −2 (1) 1 − α?1 ∂ ∂ M? (1) (1) (1) H (α?1 , α?2 , M? (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 = 2α?2 (1 − α?1 )2 M? (1) (1) 1−α? 1 ? 0 ? M? (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 ) M? (2) (2) 1 ? 1−α? xfX (x)dx 0 (1) M? (2) (2) 1 ? 1−α? − (2) (2) xeβ0 α?2 (1−α?1 )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 − α?1 ) N (t ) ? i=1 0 (1 − ∂ − where z := (α?1 , α?2 , M? ). To be certain that it is positive definite, the above matrix must be investigated numerically. 3. Main results + M? (1) (1) ? 151 ?? (α?2(1) M? (1) − 1)fX ? ? i =1 ? ? (2) 0 = −(1 + θ ) ? 1 − α?1 I 0 M? (2) (2) 1 ?? 1−α? 0 xfX (x)dx 152 A.T. Payandeh Najafabadi, A.P. Bazaz / Insurance: Mathematics and Economics 69 (2016) 149–155 ? ? + M? (2) 1 − FX (2) + (1 − α?1 ) ??? M? (2) Lemma 3. Suppose the surplus process of both reinsurers under the co-reinsurance contract (1) is given by (2) 1 − α?1 M? (2) (2) 1−α? 1 ? ? ( 2) (2) β0 α?2 (1−α?1 )x xe UtR fX (x)dx + (1 + θ )E = uR0 R 0 where uR0 N (t ) ? (2) + M? (2) eβ0 α?2 (2) 0 = −β0 α?2 M? (2) M? (2) ? 1 − FX M? ?? (2) (i) (2) ? 1 − α?1 + ln(1 + θ ). I 0 ? −β0 uI0 +π0 (t )− E ? −e N? (t ) i=1 (i) XI = −e i=1 R 0 M? (3) (3) 1 ? (3) 0 = −β1 (1 + θ )α?2 ?? ? 1−α? xfX (x)dx 0 ? M? (3) (3) 1−α? 1 + e (3) β1 α?2(3) xeβ1 (1−α?2 (3) (1−α?1 ))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 − α?1 ) R 0 M? (3) (3) 1 ? 1−α? 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 ? . − M? 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 M? 1−α? − 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) M? (3) (3) 1−α? 1 ))x fX (x)dx (3) 0 = β 1 (1 + θ ) 1 − FX ? e M? (3) (3) 1 1−α? R 0 1 − α1 β0 α2 (1−α1 )x (3) 1 − α?1 1 ? ? ??? M ? ??? M? (3) − (1 − α?1 ) xeβ1 (1−α?2 (1−α?1 0 ? ∞ (3) (3) eβ1 (x−α?2 M? ) fX (x)dx − M? (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−α?2 ?? M? (3) (3) 1 − α?1 M? (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 := (α?1 , α?2 , M? ). 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 α?1Bayes = ω1 α?1(1) + ω2 α?1(2) + ω3 α?1(3) ? ? 3 ? ? ? ? ? + 1− ωi Eπ A1 ?x(R1) , . . . , x(Rn) , i =1 α? + 1− 3 ? i=1 M? Bayes 153 ? ? ? ? ? ωi Eπ A2 ?x(R1) , . . . , x(Rn) , + ω2 M? (2) + ω3 M? (3) ? 3 ? ? ? ? ? 1− ωi Eπ M ?x(R1) , . . . , x(Rn) , = ω1 M? ? + (1) i=1 (1) (1) (2) (2) (3) (3) where (α?1 , α?2 , M? (1) ), (α?1 , α?2 , M? (2) ), and (α?1 , α?2 , M? (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 (α? (1) , α?2 , M? (1) ), (α?1 , α?2 , M? (2) ), (3) (3) and (α?1 , α?2 , M? (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 α?2 + ω2 α?2(2) + ω3 α?2(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. 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