ResearchArticle The Pareto-Optimal Stop-Loss Reinsurance

Research ArticleThe Pareto-Optimal Stop-Loss Reinsurance

Haiyan You 1 and Xiaoqing Zhou2

1School of Science Shandong Jianzhu University Jinan 250101 China2School of Mathematical Sciences Shandong Normal University Jinan 250014 China

Correspondence should be addressed to Haiyan You yhysdjzueducn

Received 10 February 2020 Accepted 14 March 2020 Published 30 January 2021

Guest Editor Wenguang Yu

Copyright copy 2021 Haiyan You and Xiaoqing Zhou +is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

Reinsurance plays a role of a stabilizer of the insurance industry and can be an effective tool to reduce the risk for the insurer +ispaper aims to provide the optimal reinsurance design associated with the stop-loss reinsurance under the criterion of value-at-risk(VaR) risk measure In this paper the probability levels in the VaRs used by the both reinsurance parties are assumed to bedifferent and the optimality results of reinsurance are derived by minimizing linear combination of the VaRs of the cedent and thereinsurer +e optimal parameter values of the stop-loss reinsurance policy are formally derived under the expectationpremium principle

1 Introduction

Reinsurance is an effective risk management tool that en-ables an insurer to reduce the underwriting risk An insurermust conduct the classical trade off between the risk retainedand the premium paid to the reinsurer Generally speakingthe more risks you have to transfer the more you will pay Inorder to balance the relationship between the risk retainedand the reinsurance premium the academics started theresearch of the optimal reinsurance problem Arrow [1] firststudied the reinsurance problem and showed that the stop-loss reinsurance was optimal by using the criterion ofmaximizing the expected utility of the terminal wealthHeerwaarden et al and Gollier and Schlesinger [2 3] give thesame conclusion under the second degree stochastic dom-inance and showed that the stop-loss reinsurance was op-timal Young [4] generalized Arrowrsquos result by consideringWangrsquos premium principle Recently the optimal reinsur-ance problem has been revisited under different risk mea-sures Cai and Tan [5] derived explicitly the optimal retainedlevel of a stop-loss reinsurance minimizing the value-at-risk(VaR) and conditional tail expectation (CTE) of the insurerrsquostotal loss under the expected premium principle Cai et al[6] derived the optimal ceded loss functions among the classof increasing convex loss functions Tan et al [7] give 17

kinds of reinsurance premium principles and studied theoptimal quota-share reinsurance and the optimal stop-lossreinsurance under the criterions of VaR and CTE Cheung[8] provided a geometric approach to re-examine the op-timal reinsurance problems studied in [6] and generalizedthe results by studying the VaR minimization problem withWangrsquos premium principle Chi and Tan [9] analyzed theVaR-based and conditional-value-at-risk (CVaR)-basedoptimal reinsurance models over different classes of cededloss functions with increasing generality Chi [10] showedthat the layer reinsurance is always optimal under both theVaR and CVaR criteria when the reinsurance premium iscalculated by a variance related principle Lu et al [11]studied the optimal reinsurance under VaR and CTE criteriawhen the ceded loss functions are increasing concavefunctions

As we all know there are two parties in a reinsurancecontract an insurer and a reinsurer +ey have conflictinginterests Borch [12] studied the optimal quota-share re-insurance and the optimal stop-loss reinsurance from theboth sides of the reinsurance under the optimizationcriterion of maximizing the product of the expected utilityfunctions of the two partiesrsquo terminal wealth Borch [13]showed reinsurance policy which is very attractive to theinsurer may not be optimal for the reinsurer and it might

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 2839726 6 pageshttpsdoiorg10115520212839726

be unacceptable for the reinsurer Since then the study ofthe optimal reinsurance opened a new direction Cai et aland Fang and Qu [14 15] obtained the sufficient condi-tions for the optimal reinsurance contract by studying thejoint survival probability and the joint profitable proba-bility of the two parties Cai et al [16] used the convexcombination of the VaRs of the cedent and the reinsurer asthe object function to research the optimal reinsurancepolicies Based on the criterion of VaR under the differentconfidence levels Jiang et al [17] studied pareto-optimalreinsurance strategies and gave the optimal forms Lo [18]discussed the generalized problems of [16] by using theNeymanndashPearson approach Cai et al [19] studied pareto-optimality of reinsurance arrangements under generalmodel settings and obtained the explicit forms of thepareto-optimal reinsurance contracts under tail-value-at-risk (TVaR) measure and the expected value premiumprinciple By geometric approach Fang et al [20] studiedpareto-optimal reinsurance policies under general pre-mium principles and gave the explicit parameters of theoptimal ceded loss functions under Dutch premiumprinciple andWangrsquos premium principle Lo and Tang [21]characterized the set of pareto-optimal reinsurance poli-cies analytically and visualized the insurer-reinsurer trade-off structure geometrically

It is interesting to notice that most optimal forms ofreinsurance in these cited papers are stop-loss reinsurancecontracts Inspired by these results we mainly study thestop-loss reinsurance in this paper We study the optimalform of reinsurance policy by minimizing the convexcombination of the VaRs of the cedent and the reinsurerunder the expected principle +e rest of the paper is or-ganized as follows In Section 2 we mainly introduce somepreliminary knowledge In Section 3 we assume that thecedent and the reinsurer have different confidence levels andthen discuss the optimal stop-loss reinsurance under theexpected principle by the optimization problem In Section4 we give numerical examples In Section 5 we conclude thepaper

2 VaR-Based Optimal Reinsurance Model

In this section we establish the framework of the optimalstop-loss reinsurance model-based VaR risk measure Letthe total loss for an insurer over a period of time be X whereX is a nonnegative random variable and defined in theprobability space (ΩFP) with distribution functionFX(x) P(Xlex) survival function SX(x) P Xgtx mean E[X] μ(0lt μltinfin) and variance D[X] σ2 gt 0 Ina reinsurance contract a reinsurer agrees to pay the part ofthe loss X denoted by f(X) to the insurer at the end of thecontract term while the insurer will pay a reinsurancepremium denoted by π(f(X)) to the reinsurer when thecontract is signed where the function f(x)(0lef(x)lex) iscalled ceded loss function +en the retained loss for theinsurer is I(X) X minus f(X) where the function I(x) iscalled retained loss function

As we all know stop-loss reinsurance is optimal in thesense that it gives the lowest variance for the retained riskwhen the mean is given In many literature studies stop-lossreinsurance has been shown to be optimal under certainconditions such as [1ndash3]+erefore many articles take stop-loss reinsurance as an example to study the reinsurance forexample [5 7 14] In this paper we study the stop-lossreinsurance that is to say f(X) (X minus d)+ where theparameter dge 0 is the retention Under stop-loss agreementthe total loss of the insurer is

TI X minus f(X) + π(f(X)) (1)

and the total loss of the reinsurer is

TR f(X) minus π(f(X)) (2)

In fact the reinsurance aims to control the risks of thetwo sides of reinsurance and this will involve their maxi-mum aggregate loss +e risk measure most often used inpractice is simply the Value-at-Risk at a certain level α with0 lt α lt 1 which is the amount that will maximally be lostwith probability α +e VaR of a random variable is definedas follows

Definition 1 +e VaR of a nonnegative random variable Xat a confidence level α 0lt αlt 1 is defined as

VaRα(X) inf x FX(x)ge α1113864 1113865 (3)

+eVaR defined by (3) is the maximum loss which is notexceeded at a given probability α We list several propertiesof the VaR

Proposition 1 For any nonnegative random variable X withthe survival function SX(x) we have the following propertiesfor any α isin (0 1)

(1) Translation invariance VaRα(X + c) VaRα(X) + c

(c isin R)

(2) Homogeneity VaRα(cX) cVaRα(X) (c isin R)

(3) If h(x) is an increasing and left-continuous functionthen VaRα(h(X)) h(VaRα(X))

Obviously the insurer and the reinsurer are mutuallyrestricted and even opposed in the interests 8is means areinsurance policy which is very attractive to the insurer maynot be optimal for the reinsurer and it might be unacceptablefor the reinsurer To be fair we consider the two parties of thereinsurance Inspired by [17] we study the pareto-optimalreinsurance under the criterion of VaR because it can beexpressed by the linear combination of the VaRs of the cedentand the reinsurer 8en in this paper we study optimal re-insurance policy by solving the following optimization problem

minL(f) min βVaRαcTI( 1113857 +(1 minus β)VaRαr

TR( 11138571113966 1113967 (4)

where 0le βle 1 and the probability levels in the VaRs used bythe cedent and the reinsurer are possibly different say αc andαr respectively

2 Mathematical Problems in Engineering

3 Stop-Loss Reinsurance Optimization

Let f(x) (x minus d)+ denote the ceded function where dge 0is the stop-loss retention then the objective function is

L(f) βVaRαcTI( 1113857 +(1 minus β)VaRαr

TR( 1113857

βVaRαc(X minus f(X) + π(f(X)))

+(1 minus β)VaRαr(f(X) minus π(f(X)))

(5)

Let ac VaRαc(X) and ar VaRαr

(X) then the ob-jective function is

L(d) β ac minus ac minus d( 1113857+1113858 1113859 +(1 minus β) ar minus d( 1113857+

+(2β minus 1)π (X minus d)+1113858 1113859(6)

and the optimization problem is

mindisin[0infin)

L(d) (7)

One of the commonly used principles is the expectationpremium principle that is to say π[(X minus d)+]

(1 + θ)E[(X minus d)+] (1 + θ) 1113938infind

SX(x)dx where θgt 0 isthe relative safety loading In order to get the optimal re-tention dlowast we give the following results

Theorem 1 If β (12) we have the following conclusions

(1) When αr lt αc the optimal stop-loss coefficient dlowast isarbitrarily in [0 ar]

(2) When αr gt αc the optimal stop-loss coefficient dlowast isarbitrarily in [arinfin)

Proof Specifically when β (12) then the objective functionis degraded to L(d) (12)[ac minus (ac minus d)++ (ar minus d)+]

(1) When αr lt αc we can obtain ar le ac and

L(d)

12ar dlt ar

12

d ar ledle ac

12ac dgt ac

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

In conclusion we have

mindisin[0infin)

L(d) 12ar (9)

so dlowast is any number in [0 ar]

(2) When αr gt αc we have ar ge ac and

L(d)

12ar dlt ac

12

ac + ar minus d( 1113857 ac ledle ar

12ac dgt ar

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

So

mindisin[0infin)

L(d) 12ac (11)

and dlowast is any number in [arinfin)We study the situation of βne (12) and accomplish our

task by subdividing our considerations into four cases (1)βgt (12) and αr lt αc (2) βgt (12) and αr gt αc (3) βlt (12)

and αr lt αc (4) βlt (12) and αr gt αcFor convenience we use the notations

θ1 1

1 + θ

θ2 β

(2β minus 1)(1 + θ)

θ3 β minus 1

(2β minus 1)(1 + θ)

dlowast1 arg min

disin 0ac[ ]L(d)

Q β ar ac( 1113857 βac minus (1 minus β)ar

2β minus 1

(12)

Theorem 2 When βgt (12) and αr lt αc the optimal stop-loss reinsurance parameters are as follows

(1) When SX(0)le θ1 the optimal stop-loss reinsurancecoefficient is given by

dlowast

0 μlt θ1Q β ar ac( 1113857

0 orinfin μ θ1Q β ar ac( 1113857

infin μgt θ1Q β ar ac( 1113857

⎧⎪⎪⎨

⎪⎪⎩(13)

(2) When SX(ar)le θ1 lt SX(0) the optimal stop-loss re-insurance coefficient is given by

dlowast

Sminus 1X θ1( 1113857 1113946

infin

Sminus 1X

θ1( )SX(t)dtlt θ1 Q β ar ac( 1113857 minus S

minus 1X θ1( 11138571113960 1113961

Sminus 1X θ1( 1113857 orinfin 1113946

infin

Sminus 1X

θ1( )SX(t)dt θ1 Q β ar ac( 1113857 minus S

minus 1X θ1( 11138571113960 1113961

infin 1113946infin

Sminus 1X

θ1( )SX(t)dtgt θ1 Q β ar ac( 1113857 minus S

minus 1X θ1( 11138571113960 1113961

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)

Mathematical Problems in Engineering 3


dlowast

ar 1113946infin

Sminus 1X

ar( )SX(t)dtlt θ2 ac minus ar( 1113857

ar orinfin 1113946infin

Sminus 1X

ar( )SX(t)dt θ2 ac minus ar( 1113857

infin 1113946infin

Sminus 1X

ar( )SX(t)dtgt θ2 ac minus ar( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

(4) When SX(ac)lt θ2 le SX(ar) the optimal stop-lossreinsurance coefficient is given by

dlowast

Sminus 1X θ2( 1113857 1113946

infin

Sminus 1X

θ2( )SX(t)dt lt θ2 ac minus S

minus 1X θ2( 11138571113960 1113961


infin

Sminus 1X

θ2( )SX(t)dt θ2 ac minus S

minus 1X θ2( 11138571113960 1113961

infin 1113946infin

Sminus 1X

θ2( )SX(t)dt gt θ2 ac minus S

minus 1X θ2( 11138571113960 1113961

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

(5) When θ2 le SX(ac) the optimal stop-loss reinsurancecoefficient is given by dlowast infin

Proof Following (6) the derivative of L(d) with respect to d

is

Lprime(d)

(2β minus 1)(1 + θ) θ1 minus SX(d)1113858 1113859 dlt ar

(2β minus 1)(1 + θ) θ2 minus SX(d)1113858 1113859 ar ltdle ac

minus (2β minus 1)(1 + θ)SX(d) dgt ac

⎧⎪⎪⎨

⎪⎪⎩

(17)

We discuss the following partitions in the entire defi-nition interval for simple calculation

When βgt (12) and dgt ac it follows from (17) that thederivative of the loss function is denoted by

Lprime(d) minus (2β minus 1)(1 + θ)SX(d)lt 0 (18)

+en L(d) is decreasing in (acinfin) So we only need totalk about minimum values on the interval [0 ac] andcompared L(dlowast1 ) with L(infin)

When dle ac it follows from (17) that

Lprime(d) (2β minus 1)(1 + θ) θ1 minus SX(d)1113858 1113859 dlt ar

(2β minus 1)(1 + θ) θ2 minus SX(d)1113858 1113859 ar ltdle ac1113896

(19)

Because βgt (12) so Lprime(d) is an increasing functionMeanwhile θ1 le θ2 and Sminus 1

X (θ2)le Sminus 1X (θ1) Now let us dis-

cuss the following situations in turn

(1) If SX(0)le θ1 then Lprime(0)ge 0 and Lprime(d)ge 0 for anyd isin [0 ac] So dlowast1 0 and the optimal parameter dlowast

depends on the size of L(0) and L(infin)(2) If SX(ar)le θ1 lt SX(0) this is equivalent to

0lt Sminus 1X (θ1)le ar When dlt Sminus 1

X (θ1) we can obtainedLprime(d)lt 0 otherwise Lprime(d)gt 0 +is meansdlowast1 Sminus 1

X (θ1) and finally dlowast depends on the relativemagnitude between L(Sminus 1

X (θ1)) and L(infin)(3) If θ1 lt SX(ar)lt θ2 this means Sminus 1

X (θ2)lt ar lt Sminus 1X

(θ1) When dlt ar we can obtained Lprime(d)lt 0 on thecontrary Lprime(d)gt 0 It shows that dlowast1 ar So theoptimal parameter dlowast is determined by the size ofL(ar) and L(infin)

(4) If SX(ac)lt θ2 le SX(ar) this is equivalent toar le Sminus 1

X (θ2)lt ac When dlt Sminus 1X (θ2) we can obtain

Lprime(d)lt 0 otherwise Lprime(d)gt 0 So we proved thatdlowast1 Sminus 1

X (θ2) and dlowast depends on the size of L(d2)

and L(infin)(5) If θ2 le SX(ac) obviously Lprime(ac)le 0 and Lprime(d)le 0 for

any d isin [0 ac] +is implies that dlowast infin

Theorem 3 When βgt (12) and αr gt αc the optimal stop-loss reinsurance parameters are as follows


dlowast

0 μlt θ1Q β ar ac( 1113857



⎧⎪⎪⎨

⎪⎪⎩(20)

(2) When SX(ac)lt θ1 lt SX(0) the optimal stop-loss re-insurance coefficient is given by

dlowast

Sminus 1X θ1( 1113857 1113946

infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961


infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961

infin 1113946infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)



Theorem 4 When βlt (12) and αr lt αc the optimal stop-loss reinsurance parameters are as follows

(1) When SX(0)le θ2 and SX(0)le θ1 the optimal stop-loss reinsurance coefficient is given by dlowast ac

(2) When SX(ac)lt θ2 lt θ1 le SX(0) and SX(ar) notin(θ2 θ1) the optimal stop-loss reinsurance coefficient isgiven by

dlowast

0 1113946ac

0SX(t)dtgt θ1Q β ar ac( 1113857

0 or ac 1113946ac

0SX(t)dt θ1Q β ar ac( 1113857

ac 1113946ac

0SX(t)dtlt θ1Q β ar ac( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)

(3) When θ2 lt SX(ar)lt SX(0)le θ1 the optimal stop-lossreinsurance coefficient is given by dlowast ar

(4) When θ2 lt SX(ar)lt θ1 lt SX(0) the optimal stop-lossreinsurance coefficient is given by

dlowast

0 1113946ar

0SX(t)dtgt θ1ar

0 or ar 1113946ar

0SX(t)dt θ1ar

ar 1113946ar

0SX(t)dtlt θ1ar

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)

(5) When θ1 le SX(ar) and θ2 le SX(ac) the optimal stop-loss reinsurance coefficient is given by dlowast 0

Theorem 5 When βlt (12) and αr gt αc the optimal stop-loss reinsurance parameters are as follows

(1) When SX(0)le θ1 the optimal stop-loss reinsurancecoefficient is given by dlowast ar

(2) When θ1 lt SX(0) and SX(ar)lt θ3 the optimal stop-loss reinsurance coefficient is given by

dlowast

0 1113946ar


0 or ar 1113946ar


ar 1113946ar


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

(3) When θ3 le SX(ar) the optimal stop-loss reinsurancecoefficient is given by dlowast 0

4 Numerical Examples and Comparison

In this section we construct two numerical examples toillustrate the reinsurance policy that we derived in the

previous sections Specifically we assume that the lossvariable X follows the exponential distribution with thesurvival function SX(x) eminus 0001x for xgt 0 and the meanμ 1000 Let the safety loading parameter θ 02 Wediscuss two examples specified below

Example 1 Assume αr 095 and αc 099 In this casear 29957 and ac 46052 +e optimal ceded loss func-tions f(x) are shown in Table 1

Example 2 Assume αc 095 and αr 099 In this caseac 29957 and ar 46052 +e optimal ceded loss func-tions f(x) are shown in Table 2

Remark 1 Following the abovementioned examples weknow that the optimal parameter of the stop-loss reinsur-ance policy depends on the combining parameter βwhen theprobability levels in the VaRs are used by the both rein-surance differently

5 Conclusions

Some scholars have shown that the stop-loss reinsurance isthe optimal reinsurance policy under the convex combi-nation of the both reinsurance parties In this paper wemainstudy the pareto-optimal stop-loss reinsurance policy withthe expectation premium principle We analyze the topicfrom the following aspects (1) the optimality results ofreinsurance are derived by minimizing linear combinationof the VaRs of the cedent and the reinsurer (2) assumingthat the probability levels in the VaRs used by the bothreinsurance parties are different Fortunately throughanalysis we finally derived the optimal parameters for thestop-loss reinsurance

Data Availability

All data models or code generated or used during the studyare available from the corresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Table 1 f(x) with αr lt αc

β +e optimal-ceded loss functionβ isin [0 05) f(x) (x minus 29957)+

β 05 f(x) (x minus d)+foralld isin [0 ar]

β isin (05 1] f(x) (x minus 18232)+

Table 2 f(x) with αc lt αr


β 05 f(x) (x minus d)+foralld isin [arinfin)

β isin (05 06417] f(x) 0β isin (06417 1] f(x) (x minus 18232)+


Acknowledgments

+is research was supported by the Natural Science Foun-dation of Shandong Province (ZR2016JL006)

References

[1] K J Arrow ldquoUncertainty and the welfare economics ofmedical carerdquo American Economic Review vol 53 pp 941ndash973 1963

[2] A E Van Heerwaarden R Kaas and M Goovaerts ldquoOptimalreinsurance in relation to ordering of risksrdquo InsuranceMathematics and Economics vol 8 no 1 pp 261ndash287 1989

[3] C Gollier and H Schlesinger ldquoArrowrsquos theorem on theoptimality of deductibles a stochastic dominance approachrdquoEconomic 8eory vol 7 no 2 pp 359ndash363 1996

[4] V R Young ldquoOptimal insurance under Wangrsquos premiumprinciplerdquo Insurance Mathematics and Economics vol 25no 2 pp 109ndash122 1999

[5] J Cai and K S Tan ldquoOptimal retention for a stop-loss re-insurance under the VaR and CTE risk measuresrdquo ASTINBulletin vol 37 no 1 pp 93ndash112 2007

[6] J Cai K S Tan C Weng and Y Zhang ldquoOptimal rein-surance under VaR and CTE risk measuresrdquo InsuranceMathematics and Economics vol 43 no 1 pp 185ndash196 2008

[7] K S Tan C Weng and Y Zhang ldquoVaR and CTE criteria foroptimal quota-share and stop-loss reinsurancerdquo NorthAmerican Actuarial Journal vol 13 no 4 pp 459ndash482 2009

[8] K C Cheung ldquoOptimal reinsurance revisited-a geometricapproachrdquo ASTIN Bulletin vol 40 no 1 pp 221ndash239 2010

[9] Y C Chi and K S Tan ldquoOptimal reinsurance under VaR andCVaR risk measures a simplified approachrdquo ASTIN Bulletinvol 41 no 2 pp 547ndash574 2011

[10] Y Chi ldquoOptimal reinsurance under variance related premiumprinciplesrdquo Insurance Mathematics and Economics vol 51no 2 pp 310ndash321 2012

[11] Z Lu L Liu and S Meng ldquoOptimal reinsurance with concaveceded loss functions under VaR and CTE risk measuresrdquoInsurance Mathematics and Economics vol 52 no 1pp 46ndash51 2013

[12] K Borch ldquoAn attempt to determine the optimum amount ofstop-loss reinsurancerdquo Transactions of the 16th InternationalCongress of Actuaries I pp 597ndash610 1960

[13] K Borch ldquo+e optimal reinsurance treatyrdquo ASTIN Bulletinvol 5 no 2 pp 293ndash297 1969

[14] J Cai Y Fang Z Li and G E Willmot ldquoOptimal reciprocalreinsurance treaties under the joint survival probability andthe joint profitable probabilityrdquo Journal of Risk and Insurancevol 80 no 1 pp 145ndash168 2013

[15] Y Fang and Z Qu ldquoOptimal combination of quota-share andstop-loss reinsurance treaties under the joint survival prob-abilityrdquo IMA Journal of Management Mathematics vol 25no 1 pp 89ndash103 2014

[16] J Cai C Lemieux and F Liu ldquoOptimal reinsurance from theperspectives of both an insurer and a reinsurerrdquo ASTINBulletin vol 46 no 3 pp 815ndash849 2016

[17] W J Jiang J D Ren and Z Ricardas ldquoOptimal reinsurancepolicies under the VaR risk measure when the interests ofboth the cedent and the reinsurer are taken into accountrdquoRisks vol 5 pp 1ndash22 2017

[18] A Lo ldquoA Neyman-pearson perspective on optimal reinsur-ance with constraintsrdquo ASTIN Bulletin vol 47 no 2pp 467ndash499 2017

[19] J Cai H Liu and R Wang ldquoPareto-optimal reinsurancearrangements under general model settingsrdquo InsuranceMathematics and Economics vol 77 no 1 pp 24ndash37 2017

[20] Y Fang X Wang H Liu and T Li ldquoPareto-optimal rein-surance for both the insurer and the reinsurer with generalpremium principlesrdquo Communications in Statistics-8eoryand Methods vol 48 no 24 pp 6134ndash6154 2019

[21] A Lo and Z Tang ldquoPareto-optimal reinsurance policies in thepresence of individual risk constraintsrdquo Annals of OperationsResearch vol 274 no 1-2 pp 395ndash423 2019


be unacceptable for the reinsurer Since then the study ofthe optimal reinsurance opened a new direction Cai et aland Fang and Qu [14 15] obtained the sufficient condi-tions for the optimal reinsurance contract by studying thejoint survival probability and the joint profitable proba-bility of the two parties Cai et al [16] used the convexcombination of the VaRs of the cedent and the reinsurer asthe object function to research the optimal reinsurancepolicies Based on the criterion of VaR under the differentconfidence levels Jiang et al [17] studied pareto-optimalreinsurance strategies and gave the optimal forms Lo [18]discussed the generalized problems of [16] by using theNeymanndashPearson approach Cai et al [19] studied pareto-optimality of reinsurance arrangements under generalmodel settings and obtained the explicit forms of thepareto-optimal reinsurance contracts under tail-value-at-risk (TVaR) measure and the expected value premiumprinciple By geometric approach Fang et al [20] studiedpareto-optimal reinsurance policies under general pre-mium principles and gave the explicit parameters of theoptimal ceded loss functions under Dutch premiumprinciple andWangrsquos premium principle Lo and Tang [21]characterized the set of pareto-optimal reinsurance poli-cies analytically and visualized the insurer-reinsurer trade-off structure geometrically

It is interesting to notice that most optimal forms ofreinsurance in these cited papers are stop-loss reinsurancecontracts Inspired by these results we mainly study thestop-loss reinsurance in this paper We study the optimalform of reinsurance policy by minimizing the convexcombination of the VaRs of the cedent and the reinsurerunder the expected principle +e rest of the paper is or-ganized as follows In Section 2 we mainly introduce somepreliminary knowledge In Section 3 we assume that thecedent and the reinsurer have different confidence levels andthen discuss the optimal stop-loss reinsurance under theexpected principle by the optimization problem In Section4 we give numerical examples In Section 5 we conclude thepaper

2 VaR-Based Optimal Reinsurance Model

In this section we establish the framework of the optimalstop-loss reinsurance model-based VaR risk measure Letthe total loss for an insurer over a period of time be X whereX is a nonnegative random variable and defined in theprobability space (ΩFP) with distribution functionFX(x) P(Xlex) survival function SX(x) P Xgtx mean E[X] μ(0lt μltinfin) and variance D[X] σ2 gt 0 Ina reinsurance contract a reinsurer agrees to pay the part ofthe loss X denoted by f(X) to the insurer at the end of thecontract term while the insurer will pay a reinsurancepremium denoted by π(f(X)) to the reinsurer when thecontract is signed where the function f(x)(0lef(x)lex) iscalled ceded loss function +en the retained loss for theinsurer is I(X) X minus f(X) where the function I(x) iscalled retained loss function

As we all know stop-loss reinsurance is optimal in thesense that it gives the lowest variance for the retained riskwhen the mean is given In many literature studies stop-lossreinsurance has been shown to be optimal under certainconditions such as [1ndash3]+erefore many articles take stop-loss reinsurance as an example to study the reinsurance forexample [5 7 14] In this paper we study the stop-lossreinsurance that is to say f(X) (X minus d)+ where theparameter dge 0 is the retention Under stop-loss agreementthe total loss of the insurer is

TI X minus f(X) + π(f(X)) (1)

and the total loss of the reinsurer is

TR f(X) minus π(f(X)) (2)

In fact the reinsurance aims to control the risks of thetwo sides of reinsurance and this will involve their maxi-mum aggregate loss +e risk measure most often used inpractice is simply the Value-at-Risk at a certain level α with0 lt α lt 1 which is the amount that will maximally be lostwith probability α +e VaR of a random variable is definedas follows

Definition 1 +e VaR of a nonnegative random variable Xat a confidence level α 0lt αlt 1 is defined as

VaRα(X) inf x FX(x)ge α1113864 1113865 (3)

+eVaR defined by (3) is the maximum loss which is notexceeded at a given probability α We list several propertiesof the VaR

Proposition 1 For any nonnegative random variable X withthe survival function SX(x) we have the following propertiesfor any α isin (0 1)

(1) Translation invariance VaRα(X + c) VaRα(X) + c

(c isin R)

(2) Homogeneity VaRα(cX) cVaRα(X) (c isin R)

(3) If h(x) is an increasing and left-continuous functionthen VaRα(h(X)) h(VaRα(X))

Obviously the insurer and the reinsurer are mutuallyrestricted and even opposed in the interests 8is means areinsurance policy which is very attractive to the insurer maynot be optimal for the reinsurer and it might be unacceptablefor the reinsurer To be fair we consider the two parties of thereinsurance Inspired by [17] we study the pareto-optimalreinsurance under the criterion of VaR because it can beexpressed by the linear combination of the VaRs of the cedentand the reinsurer 8en in this paper we study optimal re-insurance policy by solving the following optimization problem

minL(f) min βVaRαcTI( 1113857 +(1 minus β)VaRαr

TR( 11138571113966 1113967 (4)

where 0le βle 1 and the probability levels in the VaRs used bythe cedent and the reinsurer are possibly different say αc andαr respectively





TR( 1113857



(5)




+(2β minus 1)π (X minus d)+1113858 1113859(6)


mindisin[0infin)

L(d) (7)









L(d)

12ar dlt ar

12

d ar ledle ac

12ac dgt ac

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)


mindisin[0infin)

L(d) 12ar (9)



L(d)

12ar dlt ac

12


12ac dgt ar

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

So

mindisin[0infin)

L(d) 12ac (11)




θ1 1

1 + θ

θ2 β

(2β minus 1)(1 + θ)

θ3 β minus 1

(2β minus 1)(1 + θ)

dlowast1 arg min

disin 0ac[ ]L(d)


2β minus 1

(12)



dlowast

0 μlt θ1Q β ar ac( 1113857



⎧⎪⎪⎨

⎪⎪⎩(13)


dlowast

Sminus 1X θ1( 1113857 1113946

infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961


infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961

infin 1113946infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



dlowast

ar 1113946infin

Sminus 1X



Sminus 1X


infin 1113946infin

Sminus 1X


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)


dlowast

Sminus 1X θ2( 1113857 1113946

infin

Sminus 1X


minus 1X θ2( 11138571113960 1113961


infin

Sminus 1X


minus 1X θ2( 11138571113960 1113961

infin 1113946infin

Sminus 1X


minus 1X θ2( 11138571113960 1113961

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)



is

Lprime(d)




⎧⎪⎪⎨

⎪⎪⎩

(17)








(19)




















dlowast

0 μlt θ1Q β ar ac( 1113857



⎧⎪⎪⎨

⎪⎪⎩(20)


dlowast

Sminus 1X θ1( 1113857 1113946

infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961


infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961

infin 1113946infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)






dlowast

0 1113946ac


0 or ac 1113946ac


ac 1113946ac


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)



dlowast

0 1113946ar

0SX(t)dtgt θ1ar

0 or ar 1113946ar

0SX(t)dt θ1ar

ar 1113946ar

0SX(t)dtlt θ1ar

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)





dlowast

0 1113946ar


0 or ar 1113946ar


ar 1113946ar


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)








5 Conclusions


Data Availability







β isin (05 1] f(x) (x minus 18232)+






Acknowledgments


References


























TR( 1113857



(5)




+(2β minus 1)π (X minus d)+1113858 1113859(6)


mindisin[0infin)

L(d) (7)









L(d)

12ar dlt ar

12

d ar ledle ac

12ac dgt ac

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)


mindisin[0infin)

L(d) 12ar (9)



L(d)

12ar dlt ac

12


12ac dgt ar

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

So

mindisin[0infin)

L(d) 12ac (11)




θ1 1

1 + θ

θ2 β

(2β minus 1)(1 + θ)

θ3 β minus 1

(2β minus 1)(1 + θ)

dlowast1 arg min

disin 0ac[ ]L(d)


2β minus 1

(12)



dlowast

0 μlt θ1Q β ar ac( 1113857



⎧⎪⎪⎨

⎪⎪⎩(13)


dlowast

Sminus 1X θ1( 1113857 1113946

infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961


infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961

infin 1113946infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



dlowast

ar 1113946infin

Sminus 1X



Sminus 1X


infin 1113946infin

Sminus 1X


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)


dlowast

Sminus 1X θ2( 1113857 1113946

infin

Sminus 1X


minus 1X θ2( 11138571113960 1113961


infin

Sminus 1X


minus 1X θ2( 11138571113960 1113961

infin 1113946infin

Sminus 1X


minus 1X θ2( 11138571113960 1113961

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)



is

Lprime(d)




⎧⎪⎪⎨

⎪⎪⎩

(17)








(19)




















dlowast

0 μlt θ1Q β ar ac( 1113857



⎧⎪⎪⎨

⎪⎪⎩(20)


dlowast

Sminus 1X θ1( 1113857 1113946

infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961


infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961

infin 1113946infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)






dlowast

0 1113946ac


0 or ac 1113946ac


ac 1113946ac


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)



dlowast

0 1113946ar

0SX(t)dtgt θ1ar

0 or ar 1113946ar

0SX(t)dt θ1ar

ar 1113946ar

0SX(t)dtlt θ1ar

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)





dlowast

0 1113946ar


0 or ar 1113946ar


ar 1113946ar


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)








5 Conclusions


Data Availability







β isin (05 1] f(x) (x minus 18232)+






Acknowledgments


References
























dlowast

ar 1113946infin

Sminus 1X



Sminus 1X


infin 1113946infin

Sminus 1X


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)


dlowast

Sminus 1X θ2( 1113857 1113946

infin

Sminus 1X


minus 1X θ2( 11138571113960 1113961


infin

Sminus 1X


minus 1X θ2( 11138571113960 1113961

infin 1113946infin

Sminus 1X


minus 1X θ2( 11138571113960 1113961

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)



is

Lprime(d)




⎧⎪⎪⎨

⎪⎪⎩

(17)








(19)




















dlowast

0 μlt θ1Q β ar ac( 1113857



⎧⎪⎪⎨

⎪⎪⎩(20)


dlowast

Sminus 1X θ1( 1113857 1113946

infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961


infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961

infin 1113946infin

Sminus 1X


minus 1X θ1( 11138571113960 1113961

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)






dlowast

0 1113946ac


0 or ac 1113946ac


ac 1113946ac


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)



dlowast

0 1113946ar

0SX(t)dtgt θ1ar

0 or ar 1113946ar

0SX(t)dt θ1ar

ar 1113946ar

0SX(t)dtlt θ1ar

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)





dlowast

0 1113946ar


0 or ar 1113946ar


ar 1113946ar


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)








5 Conclusions


Data Availability







β isin (05 1] f(x) (x minus 18232)+






Acknowledgments


References



























dlowast

0 1113946ac


0 or ac 1113946ac


ac 1113946ac


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)



dlowast

0 1113946ar

0SX(t)dtgt θ1ar

0 or ar 1113946ar

0SX(t)dt θ1ar

ar 1113946ar

0SX(t)dtlt θ1ar

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)





dlowast

0 1113946ar


0 or ar 1113946ar


ar 1113946ar


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)








5 Conclusions


Data Availability







β isin (05 1] f(x) (x minus 18232)+






Acknowledgments


References























Acknowledgments


References























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ResearchArticle The Pareto-Optimal Stop-Loss Reinsurance

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