pareto-optimalreinsurancerevisited:atwo ...principle of indemnity and the incentive compatible...

Research ArticlePareto-OptimalReinsuranceRevisitedATwo-StageOptimisationProcedure Approach

Ying Fang 1 Lu Wang2 and Zhongfeng Qu3

1School of Mathematics and Statistics Shandong Normal University Jinan 250358 China2Caoxian No 1 Middle School Heze 274400 China3School of Mathematical Sciences University of Jinan Jinan 250022 China

Correspondence should be addressed to Ying Fang fangying319163com

Received 16 June 2020 Accepted 21 July 2020 Published 29 September 2020

Guest Editor Wenguang Yu

Copyright copy 2020 Ying Fang et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper based on the Tail-Value-at-Risk (TVaR) measure we revisit the Pareto-optimal reinsurance policies for the insurerand the reinsurer via a two-stage optimisation procedure To reduce ex-post moral hazard we assume that reinsurance contractssatisfy the principle of indemnity and the incentive compatible constraint which have been advocated by Huberman et al (1983)We show that the Pareto-optimal reinsurance policy exists if the reinsurance premiums can be expressed as an integral form)eproposed class of premium principles encompasses the net premium principle expected value premium principle TVaRpremium principle generalized percentile premium principle and so on We further use the TVaR premium principle and theexpected value premium principle as examples to illustrate the two-stage optimisation procedure by deriving explicitly the Pareto-optimal reinsurance policies We extend the results by Cai et al (2017) when the expected value premium principle is replaced bythe TVaR premium principle

1 Introduction

)e study of optimal reinsurance design has drawn greatinterest from both academics and practitioners since theseminal work by Borch [1] and Arrow [2] )ere have beenmany important literatures and conclusions about thisproblem in the past few decades For example by mini-mizing the variance of the insurerrsquos retained loss Borch [1]showed that the stop-loss reinsurance is optimal under theexpected value premium principle Arrow [2] obtained thatthe optimal reinsurance policy is a stop-loss reinsurancestrategy when the optimisation criterion is to maximize theexpected utility function of the insurer Both these resultshave been extended in a number of important directions Forexample Young [3] generalized Arrowrsquos result underWangrsquos premium principle Kaluszka [4] generalized Borchrsquosresult under mean-variance premium principles Kaluszkaand Okolewski [5] showed that the limited stop-loss and thetruncated stop-loss are the optimal contracts under anumber of criteria including the maximization of the

expected utility the stability and the survival probability ofthe insurer for a fixed reinsurance premium calculatedaccording to the maximal possible claims principle Cai andTan [6] developed two new optimisation criteria for derivingthe optimal retentions by minimizing the Value-at-Risk(VaR) and the conditional tail expectation (CTE) of the totalrisk of an insurer In recent years VaR and CTE have beenused as optimisation criteria to study the optimal reinsur-ance strategy For example by minimizing VaR and CTE ofan insurerrsquos total cost Cai et al [7] derived the optimalreinsurance strategies in the set of increasing and convexceded loss functions )e optimal reinsurance strategy de-pends on the confidence level of risk measurement and itcan be a stop-loss reinsurance strategy a quota-share re-insurance strategy or a change-loss reinsurance strategyBernard and Tian [8] provided alternative risk transfermechanisms on the capital market when the optimal rein-surance is arranged under tail risk measures Cheung [9]gave a geometric approach to revisit the optimal reinsuranceproblem and generalized the results in [7] by studying the

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 3061298 16 pageshttpsdoiorg10115520203061298

VaR-minimization problem with Wangrsquos premium princi-ple Chi and Tan [10] analyzed the VaR- and CVaR-basedoptimal reinsurance models over different classes of cededloss functions with increasing generality However theabove statements only consider the insurer and from thepoint of view of the reinsurer the optimal policy may not beoptimal For example Vajda [11] showed that the optimalreinsurance strategy is a quota-share reinsurance instead of astop-loss reinsurance when the optimisation criterion is tominimize the variance of the loss of the reinsurer )us anoptimal reinsurance contract for the insurer may not beoptimal for the reinsurer and it might be unacceptable forthe reinsurer )en an interesting question about optimalreinsurance is to design a reinsurance contract so that itconsiders the interests of both the insurer and the reinsurer

Borch [1] first studied the optimal reinsurance strategythat consider the interests of both the insurer and the re-insurer He discussed the optimal quota-share retention andstop-loss retention that maximize the product of the ex-pected utility functions of the two partiesrsquo wealth Kaishev[12] analyzed the optimal reinsurance contracts under whichthe finite horizon joint survival probability of the two partiesis maximized Under the general reinsurance principles Caiet al [13] took maximization of the joint survival probabilityand the joint profitable probability of both the insurer andthe reinsurer as the optimisation criterion to give a sufficientcondition of the optimal reinsurance existence To maximizethe joint survival probability of the insurer and the reinsurerFang and Qu [14] studied the optimal policy of combinationof quota-share reinsurance and stop-loss reinsurance Fanget al [15] studied the optimal reinsurance models from theperspective of both the insurer and the reinsurer by mini-mizing their total costs under the criteria of the loss functionwhich is defined by the joint Value-at-Risk Cai et al [16]studied the optimal reinsurance strategy which was basedon the minimum convex combination of the VaR of theinsurer and the reinsurer under two types of constraints Lo[17] discussed the generalized problems in [16] by using theNeymanndashPearson approach Based on the optimal rein-surance strategy in [16] Jiang et al [18] proved that theoptimal reinsurance strategy is a Pareto-optimal reinsurancepolicy and gave optimal reinsurance strategies using thegeometric method Cai et al [19] studied the Pareto opti-mality of reinsurance arrangements under general modelsettings and obtained the explicit forms of the Pareto-op-timal reinsurance contracts under the TVaR measure andthe expected value premium principle Jiang et al [20]studied the optimal reinsurance with constraints under thedistortion risk measure By the geometric approach Fanget al [21] studied Pareto-optimal reinsurance policies undergeneral premium principles and gave the explicit parametersof the optimal ceded loss functions under the Dutch pre-mium principle and Wangrsquos premium principle Lo andTang [22] characterized the set of Pareto-optimal reinsur-ance policies analytically and visualized the insurer-rein-surer trade-off structure geometrically Huang and Yin [23]studied two classes of optimal reinsurance models fromperspectives of both insurers and reinsurers by minimizingtheir convex combination where the risk is measured by a

distortion risk measure and the premium is given by adistortion premium principle

In this paper based on the TVaR measure we revisit thePareto-optimal reinsurance policies for the insurer and thereinsurer via a two-stage optimisation procedure which wasproposed by Asimit et al [24] To reduce ex-post moralhazard we assume that reinsurance contracts satisfy theprinciple of indemnity and the incentive compatible con-straint which have been advocated by Huberman et al [25]We first show that the Pareto-optimal reinsurance policyexists if the reinsurance premiums can be expressed as anintegral form such as (10) We emphasize that there aremany premium principles which satisfy this property such asthe net premium principle expected value premium prin-ciple TVaR premium principle and generalized percentilepremium principle )en we take the TVaR premiumprinciple and the expected value premium principle asexamples to illustrate the two-stage optimisation procedureby deriving explicitly the parameters of the Pareto-optimalreinsurance policies

It is worth noting that the Pareto-optimal reinsurancecontracts under the TVaR measure and the expected valuepremium principle has been obtained in [19] We reexaminethis problem for two reasons First it should be emphasizedthat the results can be achieved by using a different approachbased on a two-stage optimisation procedure Second andmore importantly the two-stage optimisation procedure ismore intuitive and can analyze Pareto-optimal reinsurancepolicies with other reinsurance premium principles

)e remaining arrangements of this paper are as followsIn Section 2 we introduce some definitions and modelformulation and then we show that the Pareto-optimalreinsurance policy exists if the reinsurance premiums can beexpressed as an integral form such as (10) Based on theTVaR measure we obtain the Pareto-optimal policies underthe TVaR premium principle and the expected value pre-mium principle in Section 3 In Section 4 we give illustrativenumerical examples Section 5 concludes the paper Finallyall the proofs are given in the Appendix

2 Model Formulation

Let X be the amount of loss faced by the insurer in a giventime period Suppose that X is a nonnegative randomvariable with a distribution function FX(x) P Xlex andsurvival function SX(x) 1 minus FX(x) In addition the valueof the right endpoint XF of the distribution function FX(x)

can be either finite or infinite where XF ≔ inf z F(z) 1 Under a reinsurance arrangement R(X) and IR(X) rep-resent the ceded loss and the retained loss of the insurerrespectively where IR(X) X minus R(X) Functions R(x) andIR(x) are called the ceded loss function and the retained lossfunction)e principle of indemnity which is widely used ininsurance and reinsurance requires the indemnity to benonnegative and less than the initial loss Mathematically weshould have 0leR(x)lex Let π(R(X)) be the reinsurancepremium Under such a setting the total losses of the insurerand the reinsurer are MR ≔ X minus R(X) + π(R(X)) andNR ≔ R(X) minus π(R(X)) respectively

2 Mathematical Problems in Engineering

In this paper besides the principle of indemnity we alsoassume that reinsurance contracts satisfy the incentivecompatible constraint which has been advocated byHuberman et al [25] to reduce ex-post moral hazard )ismeans that the more the realized loss the more paid by boththe insurer and the reinsurer Mathematically this impliesthat both the ceded loss function and the retained lossfunction should be increasing )erefore throughout thepaper we assume that the admissible set of ceded lossfunctions is given by

F R(x) 0leR(x)lex bothR(x)

and IR(x) are increasing functions1113865(1)

It was shown by Chi and Tan [10] that all functionsR(x) isin F are Lipschitz continuous and differentiable almosteverywhere

In insurance and finance risk measures such as VaR andTVaR have been widely used for quantifying risks Now wegive a brief description of VaR and TVaR measures

Definition 1 (VaR) For a random variableX VaR is definedas

VaRp(X) ≔ inf x isin R P(Xlex)gep1113864 11138651113864 (2)

where 0ltplt 1 represents a confidence level of the lossvariable X

Definition 2 (TVaR) For a random variable X TVaR isdefined as

TVaRp(X) ≔1

1 minus p11139461

pVaRs(X)ds VaRp(X)

+1

1 minus pE X minus VaRp(X)1113872 1113873

+

(3)


Remark 1

(1) By the definitions of the VaR and the TVaR distinctlyTVaRp evaluates the expected loss amount incurredamong the worst (1 minus p) scenarios under aconfidence level p )erefore the TVaR representsa more precise risk measurement than the VaR

(2) When 1 minus pge SX(0) we have VaRp(X) 0)erefore in order to avoid a trivial case we assumethat 1 minus p isin (0 SX(0))

In this paper we assume that the confidence levels of theinsurer and the reinsurer are possibly different Let αc and αr

denote the confidence levels of the insurer and the reinsurerrespectively )erefore the total loss of the insurer and thereinsurer under the TVaR measure is

TVaRαcMR( 1113857 VaRαc

MR( 1113857 +1

1 minus αc

E MR minus VaRαcMR( 11138571113872 1113873

+

TVaRαrNR( 1113857 VaRαr

NR( 1113857 +1

1 minus αr

E NR minus VaRαrNR( 11138571113872 1113873

+

(4)

Next we study Pareto-optimal reinsurance policieswhereby the risk is measured by the TVaR For our model areinsurance policy with the ceded loss function Rlowast(x) iscalled Pareto optimal if there is no other admissible cededloss function R isin F such that TVaRαc

(MR)leTVaRαc(MRlowast)

and TVaRαr(NR)leTVaRαr

(NRlowast) and at least one of theinequalities is strict A general approach to identify Pareto-optimal reinsurance policies is to minimize a convexcombination of the TVaRs of the two parties )e result canbe found in [18ndash20]

Proposition 1 All Pareto-optimal reinsurance policies R inF can be determined by solving the problem

minRisinF

βTVaRαcMR( 1113857 +(1 minus β)TVaRαr

NR( 11138571113966 1113967 (5)

where β isin [0 1]

In view of Proposition 1 throughout the rest of thispaper we only need to determine optimal reinsurancepolicies by solving the optimisation problem (5) Define

V(R) βTVaRαcMR( 1113857 +(1 minus β)TVaRαr

NR( 1113857 (6)

)en by translation invariance and comonotonic ad-ditivity of TVaR we have

V(R) βTVaRαc(X) minus βTVaRαc

(R(X))

+(1 minus β)TVaRαr(R(X)) +(2β minus 1)π(R(X))

(7)

)erefore the optimisation problem (5) becomes

minRisinF

H(R) (8)

where

H(R) minus βTVaRαc(R(X)) +(1 minus β)TVaRαr

(R(X))

+(2β minus 1)π(R(X))(9)

In this paper we determine the Pareto-optimal rein-surance policies via a two-stage optimisation procedurewhich was developed in [24] )e first stage is solving aninfinite-dimensional problem while the second stage be-comes a classical constrained optimisation problem)e firststage can be solved as shown in Proposition 1 in [24] and wenow present it as a lemma

Lemma 1 Let f(middot) be a real-valued function defined on[s1 s2] with 0le s1 le s2 le 1 6en

Mathematical Problems in Engineering 3

minRisinF

1113946s2

s1

f(s)R VaRs(x)( 1113857ds

subjectto R VaRs1(X)1113872 1113873 ξ1

R VaRs2(X)1113872 1113873 ξ2

(10)

is uniquely solved by

Rlowast

X ξ1 ξ2( 1113857 X minus VaRs1

(X) + ξ11113872 1113873andξ2 if f(s)lt 0 for all s1 le sle s2

ξ1 + X minus VaRs2(X) + ξ2 minus ξ11113872 1113873

+ if f(s)gt 0 for all s1 le sle s2

⎧⎪⎨

⎪⎩(11)

where (ξ1 ξ2) are some constants such that 0le ξ2minusξ1 leVaRs2

(X) minus VaRs1(X)

Note that

H(R) 11139461

αc

minus β1 minus αc

R VaRs(X)( 1113857ds

+ 11139461

αr

1 minus β1 minus αr

R VaRs(X)( 1113857ds +(2β minus 1)π(R(X))

(12)By Lemma 1 we know that the Pareto-optimal rein-

surance policy exists if the reinsurance premiums π(R(X))

can be expressed as an integral form such as (10) Next wegive several premium principles which satisfy this property

(1) Net premium principle π(X) E(X) Since R(x) isa nondecreasing continuous function thenπ(R(X)) 1113938

10 R(VaRs(X))ds

(2) Expected value premium principle π(X) (1+

θ)E(X) where θ isin [0 1] is a safety loading coeffi-cient )erefore π(R(X)) (1 + θ) 1113938

10 R(VaRs

(X))ds(3) TVaR premium principle π(X) (1 + θ

1 minus α) 11139381α VaRs(X)ds where α isin [0 1) is a confidence

level and θ isin [0 1] is a safety loading coefficient SinceR(x) is a nondecreasing continuous function thenπ(R(X)) (1 + θ1 minus α) 1113938

1α R (VaRs(X))ds

(4) Generalized percentile premium principle π(X)

E(X) + β Fminus 1X (1 minus p) minus E(X)1113864 1113865 with 0lt β plt 1

Since Fminus 1X (1 minus p) VaR1minus p(X) then π(R(X))

(1 minus β)E[R(X)] + βR(VaR1minus p(X))

In the following sections we take the TVaR premiumprinciple and the expected value premium principle asexamples to illustrate the two-stage optimisation procedure

3 Pareto-Optimal Reinsurance Policy

In this section we determine the Pareto-optimal reinsurancepolicies under the TVaR premium principle and the ex-pected value premium principle )e TVaR premiumprinciple was first proposed by Young [26] It can be viewedas an extended version of the expected value premiumprinciple that is letting α 0 gives the expected valuepremium principle

31 Pareto-Optimal Reinsurance Policies under TVaRPrinciple Under the TVaR premium principle the opti-misation problem (8) becomes

minRisinF

H(R) (13)

where H(R) minus βTVaRαc(R(X)) + (1 minus β)TVaRαc

(R(X))+

(2β minus 1)(1 + θ)TVaRα(R(X)) From the mathematicalpoint of view the confidence level α can be larger thanconfidence levels αc and αr However α is usually smallerwhile αc and αr are usually larger in practice So we assumefurther αltmin αc αr1113864 1113865 to avoid complex and lengthy dis-cussions in this section

For simplicity we define the following notations

a VaRα(X)

ac VaRαc(X)

ar VaRαr(X)

ξ R VaRα(X)( 1113857

ξc R VaRαc(X)1113872 1113873

ξr R VaRαr(X)1113872 1113873

m (2β minus 1)(1 + θ)

1 minus αminus

β1 minus αc

n (2β minus 1)(1 + θ)

1 minus α+1 minus β1 minus αr

s(β) 1 minusβ minus 1

m

t(β) 1 minusβn

(14)

Next we divide our discussion into three cases ①ac lt ar② ar lt ac③ ac ar )en we obtain the followingthree theorems

Theorem 1 Under the condition ac lt ar the Pareto-optimalreinsurance policies are given as follows


(1) If 0le βlt 12 and (β minus 11 minus αr)ltm then

Rlowast(x)

xandVaRs(β)(X) when(1 + θ) 1 minus αc( 1113857gt 1 minus α

xandVaR(θ+α1+θ)(X) when(1 + θ) 1 minus αc( 1113857le 1 minus α

⎧⎨

⎩

(15)

(2) If 0le βlt 12 and (β minus 11 minus αr)gtm then Rlowast(x) x(3) If 0le βlt 12 and (β minus 11 minus αr) m then Rlowast(x)

xI alexlear + R(x)I xgtar where R(x) is any increas-ing 1-Lipschitz continuous function such thatRlowast(x) isinF

(4) If β 12 then Rlowast(x) R(x)I alexleac + u1I xgtac where u1 is an arbitrary constant in [u ac minus a + u] u

is an arbitrary constant in [0 a] and R(x) is anyincreasing 1-Lipschitz continuous function such thatRlowast(x) isin F

(5) If 12lt βle 1 and mgt 0 then Rlowast(x) 0(6) If 12lt βlt 1 and (β minus 11 minus αr)ltmlt 0 then

Rlowast(x)

0 when(1 + θ) 1 minus αc( 1113857ge 1 minus α

x minus VaR(θ+α1+θ)(X)1113872 1113873+and VaRs(β)(X) minus VaR(θ+α1+θ)(X)1113872 1113873 when(1 + θ) 1 minus αc( 1113857lt 1 minus α

⎧⎨

⎩ (16)

(7) If 12lt βle 1 and mlt (β minus 11 minus αr) then Rlowast(x)

(x minus VaR(θ+α1+θ)(X))+(8) If 12lt βlt 1 and m (β minus 11 minus αr) then Rlowast(x)

(x minus VaR(θ+α1+θ)(X))+I alexlear + R(x)I xgtar where R(x) is any increasing 1-Lipschitz continuousfunction such that Rlowast(x) isin F

(9) If 12lt βlt 1 and m 0 then Rlowast(x) 0(10) If β 1 and (1 + θ)(1 minus αc) 1 minus α then Rlowast(x)

R(x)I xgtac where R(x) is any increasing 1-Lip-schitz continuous function such that Rlowast(x) isinF

Theorem 2 Under the condition ar lt ac the Pareto-optimalreinsurance policies are given as follows

(1) If β 0 and (1 + θ)(1 minus αr) 1 minus α then Rlowast(x)

xI alexlear + R(x)I xgtar where R(x) is any increas-ing 1-Lipschitz continuous function such thatRlowast(x) isin F

(2) If 0le βlt 12 and ngt (β1 minus αc) then Rlowast(x)

xandVaR(θ+α1+θ)(X)(3) If 0lt βlt 12 and 0lt nlt (β1 minus αc) then

Rlowast(x)

xandVaR(θ+α1+θ)(X) + x minus VaRt(β)(X)1113872 1113873+ when(1 + θ) 1 minus αr( 1113857lt 1 minus α

x when(1 + θ) 1 minus αr( 1113857ge 1 minus α

⎧⎨

⎩ (17)

(4) If 0lt βlt 12 and n (β1 minus αc) then Rlowast(x)

(xandVaR(θ+α1+θ)(X))I alexleac + R(x)I xgtac whereR(x) is any increasing 1-Lipschitz continuous func-tion such that Rlowast(x) isin F

(5) If 0le βlt 12 and nlt 0 then Rlowast(x) x

(6) If 0lt βlt 12 and n 0 then Rlowast(x)

xI alexlear or xgt ac + R(x)I ar lt xle ac where R(x) is anyincreasing 1-Lipschitz continuous function such thatRlowast(x) isinF

(7) If β 12 then Rlowast(x) R(x)I alexlear + (xminus ar + u2)

I xgtar where u2 is an arbitrary constant in[u ar minus a + u] u is an arbitrary constant in [0 a]and R(x) is any increasing 1-Lipschitz continuousfunction such that Rlowast(x) isin F

(8) If 12lt βle 1 and ngt (β1 minus αc) then Rlowast(x) 0

(9) If 12lt βle 1 and 0lt nlt (β1 minus αc) then

Rlowast(x)

x minus VaR(θ+α1+θ)(X)1113872 1113873+ when(1 + θ) 1 minus αr( 1113857le 1 minus α

x minus VaRt(β)(X)1113872 1113873+ when(1 + θ) 1 minus αr( 1113857gt 1 minus α

⎧⎪⎨

⎪⎩

(18)

(10) If 12lt βle 1 and n (β1 minus αc) then Rlowast(x)

R(x)I xgtac where R(x) is any increasing 1-Lipschitzcontinuous function such that Rlowast(x) isin F

Theorem 3 Under the condition ac ar the Pareto-optimalreinsurance policies are given as follows

(1) If 0le βlt 12 then

Rlowast(x)

x when(1 + θ) 1 minus αc( 1113857gt 1 minus α

xI alexleac + R(x)I xgtac when(1 + θ) 1 minus αc( 1113857 1 minus α

xandVaR(θ+α1+θ)(X) when(1 + θ) 1 minus αc( 1113857lt 1 minus α

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(19)


where R(x) is any increasing 1-Lipschitz continuousfunction such that Rlowast(x) isin F

(2) If β 12 the objective function is identical to 0 andthe problem is trivial

(3) If 12lt βle 1 then

Rlowast(x)

0 when(1 + θ) 1 minus αc( 1113857gt 1 minus α

R(x)I xgtac when(1 + θ) 1 minus αc( 1113857 1 minus α

x minus VaR(θ+α1+θ)(X)1113872 1113873+ when(1 + θ) 1 minus αc( 1113857lt 1 minus α

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(20)

where R(x) is any increasing 1-Lipschitz continuousfunction such that Rlowast(x) isinF

32 Pareto-Optimal Reinsurance Policies under ExpectedValue Principle In this section we reexamine an optimalreinsurance problem studied in [19] in which the objectiveis to find the optimal reinsurance contracts that minimizethe TVaR of the total risk exposure under the expected valuepremium principleWe provide amore intuitive approach tosolve the problem by using a two-stage optimisationmethodUnder the expected value principle the optimisationproblem (5) becomes

minRisinF

minus βTVaRαc(R(X)) +(1 minus β)TVaRαr

(R(X))1113966

+(2β minus 1)(1 + θ)E(R(X))1113865(21)


m0 (2β minus 1)(1 + θ) minusβ

1 minus αc

n0 (2β minus 1)(1 + θ) +1 minus β1 minus αr

s0(β) 1 minusβ minus 1m0

t0(β) 1 minusβn0

(22)


(1) If 0le βlt 12 and (β minus 11 minus αr)ltm0 lt 0 then

Rlowast(x)

xandVaRs0(β)(X) when(1 + θ) 1 minus αc( 1113857gt 1

0 when SX(0)le θlowast and (1 + θ) 1 minus αc( 1113857le 1

xandVaR(θ1+θ)(X) when SX(0)gt θlowast and (1 + θ) 1 minus αc( 1113857le 1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(23)

(2) If 0le βlt 12 and m0 lt (β minus 11 minus αr) thenRlowast(x) x

(3) If 0le βlt 12 and (β minus 11 minus αr) m0 thenRlowast(x) xI xlear + R(x)I xgtar where R(x) is anincreasing 1-Lipschitz continuous function such thatRlowast(x) isinF

(4) If β 12 then Rlowast(x) R(x)I xleac + u3I xgtac where R(x) is an increasing 1-Lipschitz continuousfunction such that Rlowast(x) isinF and u3 isin [0 ac]

(5) If 12lt βle 1 and m0 gt 0 then Rlowast(x) 0(6) If 12lt βlt 1 and (β minus 11 minus αr)ltm0 lt 0 then

Rlowast(x)

0 when(1 + θ) 1 minus αc( 1113857ge 1

xandVaRs0(β)(X) when SX(0)le θlowast and (1 + θ) 1 minus αc( 1113857lt 1

x minus VaR(θ1+θ)(X)1113872 1113873+and VaRs0(β)(X) minus VaR(θ1+θ)(X)1113872 1113873 when SX(0)gt θlowast and (1 + θ) 1 minus αc( 1113857lt 1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(24)

(7) If 12lt βle 1 and m0 lt (β minus 11 minus αr) then

Rlowast(x)

x when SX(0)le θlowast

x minus VaR(θ1+θ)(X)1113872 1113873+ when SX(0)gt θlowast

⎧⎨

⎩

(25)

(8) If 12lt βlt 1 and m0 (β minus 11 minus αr) then

Rlowast(x)

xI xlear + R(x)I xgtar when SX(0)le θlowast

x minus VaR(θ1+θ)(X)1113872 1113873+I xlear + R(x)I xgtar when SX(0)gt θlowast

⎧⎪⎨

⎪⎩

(26)

where R(x) is an increasing 1-Lipschitz continuousfunction such that Rlowast(x) isinF

(9) If 12lt βlt 1 and m0 0 then Rlowast(x) 0


(10) If β 1 and (1 + θ)(1 minus αc) 1 then Rlowast(x)

R(x)I xgtac where R(x) is an increasing 1-Lipschitzcontinuous function such that Rlowast(x) isinF


(1) If β 0 and (1 + θ)(1 minus αr) 1 then Rlowast(x)

xI xlear + R(x)I xgtar where R(x) is an increasing 1-Lipschitz continuous function such that Rlowast(x) isin F

(2) If 0le βlt 12 and n0 gt (β1 minus αc) then

Rlowast(x)

0 when SX(0)le θlowast

xandVaR(θ1+θ)(X) when SX(0)gt θlowast⎧⎨

⎩ (27)

(3) If 0lt βlt 12 and 0lt n0 lt (β1 minus αc) then

Rlowast(x)

x when(1 + θ) 1 minus αr( 1113857ge 1

x minus VaRt0(β)(X)1113872 1113873+ when SX(0)le θlowast and (1 + θ) 1 minus αr( 1113857lt 1

xandVaR(θ1+θ)(X) + x minus VaRt0(β)(X)1113872 1113873+ when SX(0)gt θlowast and (1 + θ) 1 minus αr( 1113857lt 1

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(28)

(4) If 0lt βlt 12 and n0 (β1 minus αc) then

Rlowast(x)

R(x)I xgtac when SX(0)le θlowast

xandVaR(θ1+θ)(X)1113966 1113967I xleac + R(x)I xgtac when SX(0)gt θlowast

⎧⎪⎨

⎪⎩(29)

where R(x) is an increasing 1-Lipschitz continuousfunction such that Rlowast(x) isin F

(5) If 0le βlt 12 and n0 lt 0 then Rlowast(x) x(6) If 0lt βlt 12 and n0 0 then Rlowast(x)

xI xlear or xgt ac + R(x)I ar ltxle ac where R(x) is anincreasing 1-Lipschitz continuous function such thatRlowast(x) isinF

(7) If β 12 then Rlowast(x) R(x)I xlear + (x minus ar+

u4)I xgtar where R(x) is an increasing 1-Lipschitzcontinuous function such that Rlowast(x) isin F andu4 isin [0 ar]

(8) If 12lt βle 1 and n0 gt (β1 minus αc) then Rlowast(x) 0(9) If 12lt βle 1 and 0lt n0 lt (β1 minus αc) then

Rlowast(x)

x minus VaRt0(β)(X)1113872 1113873+ when (1 + θ) 1 minus αr( 1113857ge 1

x when SX(0)le θlowast and (1 + θ) 1 minus αr( 1113857lt 1

x minus VaR(θ1+θ)(X)1113872 1113873+ when SX(0)gt θlowast and (1 + θ) 1 minus αr( 1113857lt 1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)

(10) If 12lt βle 1 and n0 (β1 minus αc) then Rlowast(x)





Rlowast(x)

x when(1 + θ) 1 minus αc( 1113857gt 1

xI xleac + R(x)I xgtac when(1 + θ) 1 minus αc( 1113857 1

0 when SX(0)le θlowast and (1 + θ) 1 minus αc( 1113857lt 1

xandVaR(θ1+θ)(X) when SX(0)gt θlowast and (1 + θ) 1 minus αc( 1113857lt 1

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)


(2) If β 12 the objective function is zero and theproblem is trivial


Rlowast(x)

0 when (1 + θ) 1 minus αc( 1113857gt 1

R(x)I xgtac when(1 + θ) 1 minus αc( 1113857 1

x when SX(0)le θlowast and (1 + θ) 1 minus αc( 1113857lt 1

x minus VaR(θ1+θ)(X)1113872 1113873+ when SX(0)gt θlowast and (1 + θ) 1 minus αc( 1113857lt 1

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(32)


Remark 1 By comparing our results with those in [19] wewould like to point out the relationship between the twoarticles First Cai et al [19] give the explicit forms of thePareto-optimal reinsurance contracts under the expectedvalue premium principle by the construction method In ourpaper we use the two-stage optimisation procedure )istechnique is intuitive and applicable when the expectedvalue premium principle is replaced by other premiumprinciples Using this technique we extend the results in [19]under the TVaR premium principle Second under theexpected value premium principle Cai et al [19] derived theoptimal ceded loss functions without considering the rela-tionship between SX(0) and 1 in their )eorems 1 and 2However we discuss the relationship between them andderive different optimal ceded functions from theirs in thecase SX(0)lt 1 By comparison we find that our result ismore reasonable

4 Numerical Examples

In this section we give two numerical examples to il-lustrate the applications of the results obtained in pre-vious sections

Example 1 (TVaR principle) Assume that the loss variableX is exponentially distributed with the survival functionSX(x) eminus 0001x In this section we assume θ 02 andα 02 then a 2231 Using the results in )eorems 1 3and 3 we have the following cases

Case 1 αc 095 and αr 099 In this case a 2231ac 29957 ar 46052 TVaRαc

(X) 39957 andTVaRαr

(X) 56052 )e optimal ceded loss functionRlowast(x) is shown in Table 1 and the various key values ofRlowast(x) are shown in Table 2

From Table 1 we know that the optimal reinsurancepolicy depends on the combining coefficient β FromTable 2 obviously with the increase in the weight coef-ficient β the loss of the insurer TVaRαc

(MRlowast) is decreasingwhile the loss of the reinsurer TVaRαr

(NRlowast ) and the meanpremium E(π(Rlowast)) are increasing especially more intu-itive when β isin (05 08419) Note that we ignore the keyvalues at the endpoints 05 and 08419 because the Pareto-optimal reinsurance policy at endpoints 05 and 08419 isuncertain

Case 2 αc 099 and αr 095In this case a 2231 ac 46052 ar 29957

TVaRαc(X) 56052 and TVaRαr

(X) 39957 )e opti-mal ceded loss function Rlowast(x) is shown in Table 3 and thevarious key values of Rlowast(x) are shown in Table 4

Case 3 αc αr 095In this case a 2231 ac ar 29957 and

TVaRαc(X) TVaRαr

(X) 39957 )e optimal ceded lossfunction Rlowast(x) is shown in Table 5 and the various keyvalues of Rlowast(x) are shown in Table 6

Remark 2 Under the expected value premium assume thatthe loss variable X is exponentially distributed with thesurvival function SX(x) eminus 0001x and θ 02 Using theresults in )eorems 4 5 and 6 we get the same results as in[19]


Example 2 (expected value premium principle) Assume θ

02 and the loss variable X with the survival function

SX(x)

1 xlt 0

025 x 0

075eminus 0001x xgt 0

⎧⎪⎪⎨

⎪⎪⎩(33)

Using the results in )eorems 4 5 and 6 we have thefollowing cases

Case 4 αc 095 and αr 099 In this case ac 27081ar 43175 TVaRαc

(X) 37081 and TVaRαr(X)

53175 )e optimal ceded loss function Rlowast(x) is shown inTable 7 and the various key values of Rlowast(x) are shown inTable 8

Case 5 αc 099 and αr 095In this case ac 43175 ar 27081 TVaRαc(X)

53175 and TVaRαr(X) 37081 )e optimal ceded loss

function Rlowast(x) is shown in Table 9 and the various keyvalues of Rlowast(x) are shown in Table 10

Case 6 αc αr 095In this case ac ar 27081 and TVaRαc

(X)

TVaRαr(X) 37081 )e optimal ceded loss function

Rlowast(x) is shown in Table 11 and the various key values ofRlowast(x) are shown in Table 12

It is worth mentioning that the distribution in Example 2is not applicable in [19] and it violates the meaning of theceded loss function In addition note that the parameter βand the confidence levels of TVaRs have significant influ-ences on the Pareto-optimal contracts If β is small theweight of the reinsurer is larger than the insurer and then

Table 1 Rlowast(x) with αc lt αr under exponential distribution

β isin [0 05) Rlowast(x) xand4055

β 05 Rlowast(x) is unspecified 2231lexle 29957Rlowast(x) u1 xgt 29957forallu1 isin [2231 29957]

β isin (05 08419) Rlowast(x) ((x minus 4055)+and(VaRs(β)(X) minus 4055))forallVaRs(β)(X) isin (29957 46052)

β 08419 Rlowast(x) (x minus 4055)+ 2231lexle 46052 Rlowast(x) is unspecified xgt 46052β isin (08419 1] Rlowast(x) (x minus 4055)+

Table 2 Various key values of Rlowast(x) with αc lt αr under exponential distribution

TVaRαc(MRlowast ) TVaRαr

(NRlowast ) E(π(Rlowast))

β isin [0 05) 4058 minus 623 4678β isin (05 08419) (23305darr15905) (16652uarr332147) (925uarr985)

β isin (08419 1] 14055 41997 1000

Table 3 Rlowast(x) with αc gt αr under exponential distribution

β isin [0 01581) Rlowast(x) xand4055β 01581 Rlowast(x) xand4055 2231lexle 46052 Rlowast(x) is unspecified xgt 46052β isin (01581 05) Rlowast(x) xand4055 + (x minus VaRt(β)(X))+ forallVaRt(β)(X) isin (29957 46052)

β 05 Rlowast(x) is unspecified 2231lexle 46052Rlowast(x) x minus 29957 + u2 xgt 46052forallu2 isin [2231 29957]

β isin (05 1] Rlowast(x) (x minus 4055)+

Table 4 Various key values of Rlowast(x) with αc gt αr under exponential distribution


(NRlowast ) E((π(Rlowast))


β isin (05 1] 14055 25902 1000

Table 6 Various key values of Rlowast(x) with αc αr under expo-nential distribution



β isin [0 05) 4058 minus 623 4678β 05 (4058darr14055) (minus 623uarr25902) (4678uarr1000)

β isin (05 1] 14055 25902 1000

Table 5 Rlowast(x) with αc αr under exponential distribution

β isin [0 05) Rlowast(x) xand4055β 05 Rlowast(x) is unspecifiedβ isin (05 1] Rlowast(x) (x minus 4055)+


the reinsurer bears less losses Conversely if β is large theweight of the insurer is larger than the reinsurer and thenthe reinsurer bears more losses If αc lt αr which means thatthe TVaR standard of the reinsurer is higher than the in-surer then the reinsurer bears less losses If αc gt αr whichmeans that the TVaR standard of the insurer is higher thanthe reinsurer then the reinsurer bears more losses

5 Conclusion

In this paper based on the TVaR measure we show that thePareto-optimal reinsurance policies must exist for the in-surer and the reinsurer under a class of premium principle

such as the net principle expected value premium principleTVaR principle and generalized percentile Using a two-stage optimisation procedure we derive explicitly the Par-eto-optimal reinsurance policies under the TVaR principleSince the expected value premium principle can be viewed asa special case of the TVaR principle then letting α 0 in theTVaR principle gives Pareto-optimal reinsurance policies forthe expected value premium principle We extend the resultsin [19] Compared with the method used in [19] using thetwo-stage optimisation method to derive the Pareto-optimalstrategy is simpler and more intuitive Furthermore bycomparing the results in [19] with ours Cai et al [19] de-rived the optimal ceded loss functions without consideringthe relationship between SX(0) and 1 while we discuss therelationship between SX(0) and 1 and derive different op-timal ceded functions from theirs in the case SX(0)lt 1

We also wish to point out that further research on thistopic is needed First the risk measure TVaR can be gen-eralized to coherent risk measures Although some papershave been devoted to deriving optimal reinsurance undercoherent risk measures the optimal reinsurance study stilllacks of available analyze tools Since nonlinear expectationis an essential feature of coherent risk measures maybe wecan draw support from nonlinear expectation researchliteratures in this regard are [27ndash31] etc Second we cananalyze risk with the strategies of dividend and reinsuranceFor more references on the dividend refer to [32ndash34] etc)ird in most of the optimal reinsurance problems it isassumed that the distributions of the insurerrsquos risks areknown However in practice only incomplete informationon the distributions is available How to obtain optimalreinsurance contracts with incomplete information is also aninteresting topic An attempt to such a problem is to use the

Table 8 Various key values of Rlowast(x) with αc lt αr



β isin [0 05) 37081 0 0β isin (05 084) (1840darr1088) (18681uarr34295) (840uarr888)

β isin (084 1] 900 44175 900

Table 7 Rlowast(x) with αc lt αr

β isin [0 05) Rlowast(x) 0

β 05 Rlowast(x) is unspecified xle 27081Rlowast(x) u3 xgt 27081forallu3 isin [0 27081]

β isin (05 084) Rlowast(x) min xVaRs0(β)(X)1113966 1113967 forallVaRs0(β)(X) isin (27081 43175)

β 084 Rlowast(x) x xle 43175 Rlowast(x) is unspecified xgt 43175β isin (084 1] Rlowast(x) x

Table 9 Rlowast(x) with αc gt αr


β 01599 Rlowast(x) 0 xle 43175Rlowast(x) is unspecified xgt 43175

β isin (01599 05) Rlowast(x) (x minus VaRt0(β)(X))+ forallVaRt0(β)(X) isin (27081 43175)

β 05 Rlowast(x) is unspecified xle 27081Rlowast(x) x minus 27081 + u4 xgt 27081 forallu4 isin [0 27081]

β isin (05 1] Rlowast(x) x

Table 10 Various key values of Rlowast(x) with αc gt αr




β isin (05 1] 900 28081 900

Table 11 Rlowast(x) with αc αrβ isin [0 05) Rlowast(x) 0β 05 Rlowast(x) is unspecifiedβ isin (05 1] Rlowast(x) x

Table 12 Various key values of Rlowast(x) with αc αr



β isin [0 05) 37081 0 0β 05 (37081darr900) (0uarr28081) (0uarr900)

β isin (05 1] 900 28081 900


statistical methods For more references on statisticalmethods see eg [35ndash37] We hope that these importantopen problems can be addressed in the future research Wealso believe that this article will foster further research in thisdirection

Appendix

)e proof of )eorem 1By (3) the equivalent form of (13) is

minRisinF

(2β minus 1)(1 + θ)

1 minus α1113946αc

αR VaRs(X)( 1113857ds + m 1113946

αr

αcR VaRs(X)( 1113857ds1113896 + m +


1113888 1113889 11139461

αr

R VaRs(X)( 1113857ds1113897 (A1)

(1) If 0le βlt 12 and (β minus 11 minus αr)ltm by Lemma 1 weget that (A1) is solved by

Rlowast

X ξ ξc ξr( 1113857

(X minus a + ξ)andξc aleXle ac

X minus ac + ξc( 1113857andξr ac ltXle ar

ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A2)

where (ξ ξc ξr) isin D1 and D1 (ξ ξc ξr)1113864

0le ξ le a 0le ξc le ac 0le ξr le ar 0le ξr minus ξc le ar minus ac

0le ξc minus ξ le ac minus a 0le ξr minus ξ le ar minus a )us

TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α

1113946acminus ξc+ξr

ac

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857(( 1113857 ξc +1

1 minus αc

1113946ac minus ξc+ξr

ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

(A3)

Define H(Rlowast) H1(ξ ξc ξr) in this case then thesecond-stage optimisation problem is reduced to min-imize H1 Note that (zH1 zξr) 1 minus β+

[((2β minus 1)(1 + θ)1 minus α) minus (β1 minus αc)]SX(ac minus ξc + ξr)and it is increasing in ξr on [ξc ar minus ac + ξc] since mlt 0

① When (1 + θ)(1 minus αc)gt 1 minus α we have(zH1zξr)|ξrξc

lt 0 Since m + (1 minus β1 minus αr)gt 0then we obtain (zH1zξr)|ξrar minus ac+ξc

gt 0 So H1 at-tains its minimum value at ξ lowastr VaRs(β)(X)minus

ac + ξc Note that

H1 ξ ξc ξlowastr( 1113857 minus βξc minus

β1 minus αc

1113946VaRs(β)(X)

ac

SX(x)dx +(1 minus β) VaRs(β)(X) minus ac + ξc1113872 1113873

+(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + 1113946

VaRs(β)(X)

ac

SX(x)dx1113888 11138891113888 1113889

(A4)

and (zH1zξc) (2β minus 1)[(1 + θ1minus α)SX(a minus ξ+

ξc) minus 1] so (zH1zξc) is increasing in ξc on[ξ ac minus a + ξ] Since(zH1zξc)|ξcac minus a+ξ lt 0 then H1attains its minimum value at ξ lowastc ac minus a + ξ Fur-thermore (zH1zξ) (2β minus 1)θlt 0 always holds

and so H1 attains its minimum value at ξlowast a Inconclusion Rlowast(x) xandVaRs(β)(X)

② When (1 + θ)(1 minus αc)le 1 minus α we have(zH1zξr)|ξrξc

ge 0 so H1 attains its minimum valueat ξ lowastr ξc Note that H1(ξ ξc ξ

lowastr ) (1minus


2β)ξc + (2β minus 1)(1 + θ)(ξ + 11 minus α1113938aminus ξ+ξc

aSX(x)dx)


ξc) minus 1] then (zH1zξc) is increasing in ξc on [ξ ac minus

a + ξ] since (zH1zξc)|ξcξ lt 0 and (zH1zξc)|ξcacminus a+ξ ge 0When (zH1zξc)|ξcac minus a+ξ 0 then H1 attains itsminimum value at ξ lowastc ac minus a + ξ and ξlowast a)erefore Rlowast(x) xandacWhen (zH1zξc)|ξcac minus a+ξ gt 0 H1 attains its mini-mum value at ξ lowastc VaR(θ+α1+θ)(X) minus a + ξ andξlowast a )erefore Rlowast(x) xandVaR(θ+α1+θ)(X)Note that VaR(θ+α1+θ)(X) ac if (zH1zξc)|ξcacminus a+ξ 0 )erefore Rlowast(x) xandVaR(θ+α1+θ)(X) when (1 + θ)(1 minus αc)le 1 minus α

(2) If 0le βlt 12 and mlt (β minus 11 minus αr) by Lemma 1 weget that (A1) is solved by

Rlowast

X ξ ξc ξr( 1113857



X minus ar + ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A5)

where (ξ ξc ξr) isin D1 )erefore

TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx +1

1 minus α1113946

XF

ar

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx +1

1 minus αc

1113946XF

ar

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

SX(x)dx

(A6)

)en

H Rlowast

( 1113857 ≔ H2 ξ ξc ξr( 1113857

minus βξc +(1 minus β)ξr +(2β minus 1)(1 + θ)ξ + m 1113946acminus ξc+ξr

ac

SX(x)dx

+(2β minus 1)(1 + θ)

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + m +


1113888 1113889 1113946XF

ar

SX(x)dx

(A7)

and H2 attains its minimum value at (ξlowast ξ lowastc ξ lowastr )

(a ac ar) in this case )erefore Rlowast(x) x(3) If 0le βlt 12 and m (β minus 11 minus αr) by Lemma 1

we get that (A1) is solved by

Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A8)


where R(x) is an increasing 1-Lipschitz continuousfunction )erefore

TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc

1113946XF

ac

P Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

P Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

(A9)

In this case

H Rlowast

( 1113857 ≔ H3 ξ ξc ξr( 1113857

minus βξc +(1 minus β)ξr +(2β minus 1)(1 + θ)ξ + m 1113946ac minus ξc+ξr

ac

SX(x)dx +(2β minus 1)(1 + θ)

1 minus α1113946

aminus ξ+ξc

aSX(x)dx

(A10)


(a ac ar) )erefore Rlowast(x) xI alexlear + R(x)

I xgtar (4) If β 12 by Lemma 1 we get that (A1) is solved by

Rlowast

X ξ ξc ξr( 1113857

R(x) aleXle ac


ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A11)

)en

TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

H Rlowast

( 1113857 ≔ H4 ξ ξc ξr( 1113857

12ξc +

12ξr +

12 1 minus αc( 1113857


ac

SX(x)dx

(A12)

It is easy to see that H4 attains its minimum value at(ξlowast ξ lowastc ξ lowastr ) (ξ u1 u1) where u1 isin [a ac] )ere-fore Rlowast(x) R(x)I alexleac + u1I xgtac

(5) If 12lt βle 1 and mgt 0 the coefficients of the threeintegrals in (A1) are all positive obviously Rlowast(x) 0

(6) If 12lt βlt 1 and (β minus 11 minus αr)ltmlt 0 by Lemma 1we get that (A1) is solved by


Rlowast

X ξ ξc ξr( 1113857

ξ + X minus ac + ξc minus ξ( 1113857+ aleXle ac


ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A13)

)en

H Rlowast

( 1113857 ≔ H5 ξ ξc ξr( 1113857

minus βξc +(1 minus β)ξr minusβ

1 minus αc


ac

SX(x)dx +(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

ac minus ξc+ξr

ac minus ξc+ξSX(x)dx1113888 1113889

(A14)

Note that (zH5zξr) 1 minus β + [((2β minus 1)(1 + θ)1 minus

α) minus (β1 minus αc)]SX(ac minus ξc + ξr) is increasing in ξr on[ξc ar minus ac + ξc]

① When (1 + θ)(1 minus αc)ge 1 minus α H5 attains itsminimum value at (ξlowast ξ lowastc ξ lowastr ) (0 0 0))erefore Rlowast(x) 0

② When (1 + θ)(1 minus αc)lt 1 minus α H5 attains itsminimum value at (ξlowast ξ lowastc ξ lowastr ) (0 acminus

VaR(θ+α1+θ) (X) VaRS(β)(X)minus VaR(θ+α1+θ)(X)))erefore Rlowast(x) (x minus VaR(θ+α1+θ)(X))+

and(VaRS(β)(X) minus VaR (θ+α1+θ)(X))

(7) If 12lt βle 1 and mlt (β minus 11 minus αr) then (A1) issolved by

Rlowast

X ξ ξc ξr( 1113857



x minus ar + ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A15)

Note that

H Rlowast

( 1113857 ≔ H6 ξ ξc ξr( 1113857

minus βξc minusβ

1 minus αc


ac

SX(x)dx +(1 minus β)ξr +(2β minus 1)(1 + θ)ξ +(2β minus 1)(1 + θ)

1 minus α1113946

acminus ξc+ξr

acminus ξc+ξSX(x)dx

+(m + 1 minus β) 1113946XF

ar

SX(x)dx

(A16)

then H6 attains its minimum value at (ξlowast ξ lowastc ξ lowastr )

(0 ac minus VaR(θ+α1+θ)(X) ar minus VaR(θ+α1+θ)(X)))erefore Rlowast(x) (x minus VaR(θ+α1+θ)(X))+

(8) If 12lt βlt 1 and m (β minus 11 minus αr) then (A1) issolved by

Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A17)

We obtain Rlowast(x) (x minus VaR(θ+α1+θ)(X))+

I xlear + R(x)I xgtar (9) If 12lt βlt 1 and m 0 then (A1) is solved by

Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A18)

It is easy to get ξ lowastr ξ lowastc ξlowast 0 so Rlowast(x) 0(10) If β 1 and m 0 then (A1) is solved by

Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A19)


Obviously Rlowast(X ξ ξc ξr) is independent of ξr and it iseasy to get ξ lowastc ξlowast 0 so Rlowast(x) R(x)I xgtac whereR(x) is an increasing 1-Lipschitz continuous functionsuch that Rlowast(x) isinF


minRisinF

(2β minus 1)(1 + θ)

1 minus α1113946αr

αR VaRs(X)( 1113857ds1113896

+ n 1113946αc

αr

R VaRs(X)( 1113857ds + n minusβ

1 minus αc

1113888 1113889 11139461

αc


(A20)

Using the same method as the proof of )eorem 1 wecan obtain the desired results so we omit the proof It isworth noting that (ξ ξc ξr) isin D2 and D2 (ξ ξc ξr)1113864

0le ξ le a 0le ξc le ac 0le ξr le ar 0le ξc minus ξr le ac minus ar 0le ξcminus

ξ le ac minus a 0le ξr minus ξ le ar minus a)e proof of )eorem 3By (3) the equivalent form of (13) is

minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896

+ (2β minus 1)1 + θ1 minus α

minus1

1 minus αc

1113888 11138891113890 1113891 11139461

αc

R VaRs(X)( 1113857ds1113897

(A21)

Note that (ξ ξc) isin D3 and D3 (ξ ξc) 0le ξ le1113864

a 0le ξc le ac 0le ξc minus ξ le ac minus a )en the same techniqueas used in the proof of )eorem 1 yields the results

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)e research was supported by the Project of the ShandongProvince Higher Educational Science and TechnologyProgram (J18KA249) and the Social Science PlanningProject of Shandong Province (20CTJJ02)

References

[1] K Borch ldquoAn attempt to determine the optimum amount ofstop loss reinsurancerdquo Transactions of the 16th InternationalCongress of Actuaries vol 1 pp 597ndash610 1960

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

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

[4] M Kaluszka ldquoOptimal reinsurance under mean-variancepremium principlesrdquo Insurance Mathematics and Economicsvol 28 no 1 pp 61ndash67 2001

[5] M Kaluszka and A Okolewski ldquoAn extension of arrowrsquosresult on optimal reinsurance contractrdquo Journal of Risk ampInsurance vol 75 no 2 pp 275ndash288 2008

[6] 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

[7] 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

[8] C Bernard and W Tian ldquoOptimal reinsurance arrangementsunder tail risk measuresrdquo Journal of Risk and Insurancevol 76 no 3 pp 709ndash725 2009

[9] K C Cheung ldquoOptimal reinsurance revisitedmdasha geometricapproachrdquo ASTIN Bulletin vol 40 no 1 pp 221ndash239 2010

[10] Y C Chi and K S Tan ldquoOptimal reinsurance under VaR andCVaR risk measures a simplified approachrdquo ASTIN Bulletinvol 41 pp 487ndash509 2011

[11] S Vajda ldquoMinimum variance reinsurancerdquo ASTIN Bulletinvol 2 no 2 pp 257ndash260 1962

[12] V K Kaishev ldquoOptimal retention levels given the jointsurvival of cedent and reinsurerrdquo Scandinavian ActuarialJournal vol 2004 no 6 pp 401ndash430 2004

[13] 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

[14] 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

[15] Y Fang G Cheng and Z Qu ldquoOptimal reinsurance for bothan insurer and a reinsurer under general premium principlesrdquoAIMS Mathematics vol 5 no 4 pp 3231ndash3255 2020

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

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

[19] J Cai H Liu and R Wang ldquoPareto-optimal reinsurancearrangements under general model settingsrdquo InsuranceMathematics and Economics vol 77 pp 24ndash37 2017

[20] W Jiang H Hong and J Ren ldquoOn Pareto-optimal rein-surance with constraints under distortion risk measuresrdquoEuropean Actuarial Journal vol 8 no 1 pp 215ndash243 2017b

[21] 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 Statisticsmdash6eoryand Methods vol 48 no 24 pp 6134ndash6154 2019

[22] 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


[23] Y Huang and C Yin ldquoA unifying approach to constrainedand unconstrained optimal reinsurancerdquo Journal of Com-putational and Applied Mathematics vol 360 pp 1ndash17 2019

[24] A V Asimit A M Badescu and T Verdonck ldquoOptimal risktransfer under quantile-based risk measurersrdquo InsuranceMathematics and Economics vol 53 no 1 pp 252ndash265 2013

[25] G Huberman D Mayers and C W Smith Jr ldquoOptimalinsurance policy indemnity schedulesrdquo 6e Bell Journal ofEconomics vol 14 no 2 pp 415ndash426 1983

[26] V R Young ldquoPremium principlesrdquo in Encyclopedia of Ac-tuarial Science J Teugels and B Sundt Eds Vol 3 JohnWiley amp Sons Hoboken NJ USA 2004

[27] C Hu ldquoStrong laws of large numbers for sublinear expec-tation under controlled 1st moment conditionrdquo ChineseAnnals of Mathematics Series B vol 39 no 5 pp 791ndash8042018

[28] C Hu ldquoCentral limit theorems for sub-linear expectationunder the Lindeberg conditionrdquo Journal of Inequalities andApplications vol 2018 no 1 2018

[29] C Hu ldquoWeak and strong laws of large numbers for sub-linearexpectationrdquo Communications in Statisticsmdash6eory andMethods vol 49 no 2 pp 430ndash440 2019

[30] C Hu ldquoMarcinkiewicz-Zygmund laws of large numbersunder sublinear expectationrdquo Mathematical Problems inEngineering vol 2020 Article ID 5050973 11 pages 2020

[31] X J Shi R L Ji and Q Feng ldquoRepresentation of filtration-consistent nonlinear expectation by g-expectation in generalframeworkrdquo Communications in Statistics-6eory andMethods 2020

[32] Y Zhao P Chen and H Yang ldquoOptimal periodic dividendand capital injection problem for spectrally positive Levyprocessesrdquo Insurance Mathematics and Economics vol 74pp 135ndash146 2017

[33] W Yu P Guo QWang et al ldquoOn a periodic capital injectionand barrier dividend strategy in the compound Poisson riskmodelrdquo Mathematics vol 8 no 4 p 511 2020

[34] X Peng W Su W Su and Z Zhang ldquoOn a perturbedcompound Poisson risk model under a periodic threshold-type dividend strategyrdquo Journal of Industrial amp ManagementOptimization vol 16 no 4 pp 1967ndash1986 2020

[35] H Y Wang and Z Wu ldquoEigenvalues of stochastic Hamil-tonian systems driven by Poisson process with boundaryconditionsrdquo Boundary Value Problems vol 2017 no 1 2017

[36] X Wang Y Song and L Lin ldquoHandling estimating equationwith nonignorably missing data based on SIR algorithmrdquoJournal of Computational and Applied Mathematics vol 326pp 62ndash70 2017

[37] Q Zhao R J Karunamuni and J J Wu ldquoAn empiricalclassification procedure for nonparametric mixture modelsrdquoJournal of the Korean Statistical Society vol 49 pp 924ndash9522020


VaR-minimization problem with Wangrsquos premium princi-ple Chi and Tan [10] analyzed the VaR- and CVaR-basedoptimal reinsurance models over different classes of cededloss functions with increasing generality However theabove statements only consider the insurer and from thepoint of view of the reinsurer the optimal policy may not beoptimal For example Vajda [11] showed that the optimalreinsurance strategy is a quota-share reinsurance instead of astop-loss reinsurance when the optimisation criterion is tominimize the variance of the loss of the reinsurer )us anoptimal reinsurance contract for the insurer may not beoptimal for the reinsurer and it might be unacceptable forthe reinsurer )en an interesting question about optimalreinsurance is to design a reinsurance contract so that itconsiders the interests of both the insurer and the reinsurer

Borch [1] first studied the optimal reinsurance strategythat consider the interests of both the insurer and the re-insurer He discussed the optimal quota-share retention andstop-loss retention that maximize the product of the ex-pected utility functions of the two partiesrsquo wealth Kaishev[12] analyzed the optimal reinsurance contracts under whichthe finite horizon joint survival probability of the two partiesis maximized Under the general reinsurance principles Caiet al [13] took maximization of the joint survival probabilityand the joint profitable probability of both the insurer andthe reinsurer as the optimisation criterion to give a sufficientcondition of the optimal reinsurance existence To maximizethe joint survival probability of the insurer and the reinsurerFang and Qu [14] studied the optimal policy of combinationof quota-share reinsurance and stop-loss reinsurance Fanget al [15] studied the optimal reinsurance models from theperspective of both the insurer and the reinsurer by mini-mizing their total costs under the criteria of the loss functionwhich is defined by the joint Value-at-Risk Cai et al [16]studied the optimal reinsurance strategy which was basedon the minimum convex combination of the VaR of theinsurer and the reinsurer under two types of constraints Lo[17] discussed the generalized problems in [16] by using theNeymanndashPearson approach Based on the optimal rein-surance strategy in [16] Jiang et al [18] proved that theoptimal reinsurance strategy is a Pareto-optimal reinsurancepolicy and gave optimal reinsurance strategies using thegeometric method Cai et al [19] studied the Pareto opti-mality of reinsurance arrangements under general modelsettings and obtained the explicit forms of the Pareto-op-timal reinsurance contracts under the TVaR measure andthe expected value premium principle Jiang et al [20]studied the optimal reinsurance with constraints under thedistortion risk measure By the geometric approach Fanget al [21] studied Pareto-optimal reinsurance policies undergeneral premium principles and gave the explicit parametersof the optimal ceded loss functions under the Dutch pre-mium principle and Wangrsquos premium principle Lo andTang [22] characterized the set of Pareto-optimal reinsur-ance policies analytically and visualized the insurer-rein-surer trade-off structure geometrically Huang and Yin [23]studied two classes of optimal reinsurance models fromperspectives of both insurers and reinsurers by minimizingtheir convex combination where the risk is measured by a

distortion risk measure and the premium is given by adistortion premium principle

In this paper based on the TVaR measure we revisit thePareto-optimal reinsurance policies for the insurer and thereinsurer via a two-stage optimisation procedure which wasproposed by Asimit et al [24] To reduce ex-post moralhazard we assume that reinsurance contracts satisfy theprinciple of indemnity and the incentive compatible con-straint which have been advocated by Huberman et al [25]We first show that the Pareto-optimal reinsurance policyexists if the reinsurance premiums can be expressed as anintegral form such as (10) We emphasize that there aremany premium principles which satisfy this property such asthe net premium principle expected value premium prin-ciple TVaR premium principle and generalized percentilepremium principle )en we take the TVaR premiumprinciple and the expected value premium principle asexamples to illustrate the two-stage optimisation procedureby deriving explicitly the parameters of the Pareto-optimalreinsurance policies

It is worth noting that the Pareto-optimal reinsurancecontracts under the TVaR measure and the expected valuepremium principle has been obtained in [19] We reexaminethis problem for two reasons First it should be emphasizedthat the results can be achieved by using a different approachbased on a two-stage optimisation procedure Second andmore importantly the two-stage optimisation procedure ismore intuitive and can analyze Pareto-optimal reinsurancepolicies with other reinsurance premium principles

)e remaining arrangements of this paper are as followsIn Section 2 we introduce some definitions and modelformulation and then we show that the Pareto-optimalreinsurance policy exists if the reinsurance premiums can beexpressed as an integral form such as (10) Based on theTVaR measure we obtain the Pareto-optimal policies underthe TVaR premium principle and the expected value pre-mium principle in Section 3 In Section 4 we give illustrativenumerical examples Section 5 concludes the paper Finallyall the proofs are given in the Appendix

2 Model Formulation

Let X be the amount of loss faced by the insurer in a giventime period Suppose that X is a nonnegative randomvariable with a distribution function FX(x) P Xlex andsurvival function SX(x) 1 minus FX(x) In addition the valueof the right endpoint XF of the distribution function FX(x)

can be either finite or infinite where XF ≔ inf z F(z) 1 Under a reinsurance arrangement R(X) and IR(X) rep-resent the ceded loss and the retained loss of the insurerrespectively where IR(X) X minus R(X) Functions R(x) andIR(x) are called the ceded loss function and the retained lossfunction)e principle of indemnity which is widely used ininsurance and reinsurance requires the indemnity to benonnegative and less than the initial loss Mathematically weshould have 0leR(x)lex Let π(R(X)) be the reinsurancepremium Under such a setting the total losses of the insurerand the reinsurer are MR ≔ X minus R(X) + π(R(X)) andNR ≔ R(X) minus π(R(X)) respectively











TVaRp(X) ≔1

1 minus p11139461

pVaRs(X)ds VaRp(X)

+1


+

(3)


Remark 1






MR( 1113857 +1

1 minus αc


+


NR( 1113857 +1

1 minus αr


+

(4)






minRisinF


NR( 11138571113966 1113967 (5)

where β isin [0 1]



NR( 1113857 (6)



(R(X))


(7)


minRisinF

H(R) (8)

where


(R(X))

+(2β minus 1)π(R(X))(9)




minRisinF

1113946s2

s1



R VaRs2(X)1113872 1113873 ξ2

(10)


Rlowast





⎧⎪⎨

⎪⎩(11)


(X) minus VaRs1(X)

Note that

H(R) 11139461

αc

minus β1 minus αc


+ 11139461

αr







10 R(VaRs(X))ds



10 R(VaRs




1α R (VaRs(X))ds









minRisinF

H(R) (13)


(R(X))+



a VaRα(X)

ac VaRαc(X)

ar VaRαr(X)

ξ R VaRα(X)( 1113857

ξc R VaRαc(X)1113872 1113873

ξr R VaRαr(X)1113872 1113873

m (2β minus 1)(1 + θ)

1 minus αminus

β1 minus αc

n (2β minus 1)(1 + θ)



m

t(β) 1 minusβn

(14)





Rlowast(x)



⎧⎨

⎩

(15)






Rlowast(x)



⎧⎨

⎩ (16)











Rlowast(x)



⎧⎨

⎩ (17)










Rlowast(x)



⎧⎪⎨

⎪⎩

(18)





Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(19)





Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(20)



minRisinF


(R(X))1113966

+(2β minus 1)(1 + θ)E(R(X))1113865(21)



1 minus αc



t0(β) 1 minusβn0

(22)



Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(23)





Rlowast(x)

0 when(1 + θ) 1 minus αc( 1113857ge 1



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(24)


Rlowast(x)



⎧⎨

⎩

(25)


Rlowast(x)



⎧⎪⎨

⎪⎩

(26)










Rlowast(x)



⎩ (27)


Rlowast(x)




⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(28)


Rlowast(x)



⎧⎪⎨

⎪⎩(29)







Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)






Rlowast(x)





⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)




Rlowast(x)

0 when (1 + θ) 1 minus αc( 1113857gt 1




⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(32)
















TVaRαc(X) TVaRαr






SX(x)

1 xlt 0

025 x 0


⎧⎪⎪⎨

⎪⎪⎩(33)









(X)













β isin (08419 1] 14055 41997 1000









β isin (05 1] 14055 25902 1000





β isin (05 1] 14055 25902 1000





5 Conclusion








β isin (084 1] 900 44175 900
















β isin (05 1] 900 28081 900






β isin (05 1] 900 28081 900



Appendix


minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds + m 1113946

αr

αcR VaRs(X)( 1113857ds1113896 + m +


1113888 1113889 11139461

αr

R VaRs(X)( 1113857ds1113897 (A1)


Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A2)




TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857(( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

(A3)






ac + ξc Note that


β1 minus αc

1113946VaRs(β)(X)

ac


+(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + 1113946

VaRs(β)(X)

ac

SX(x)dx1113888 11138891113888 1113889

(A4)






lowastr ) (1minus



aSX(x)dx)





Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A5)


TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx +1

1 minus α1113946

XF

ar

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx +1

1 minus αc

1113946XF

ar

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

SX(x)dx

(A6)

)en

H Rlowast

( 1113857 ≔ H2 ξ ξc ξr( 1113857


ac

SX(x)dx

+(2β minus 1)(1 + θ)

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + m +


1113888 1113889 1113946XF

ar

SX(x)dx

(A7)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A8)



TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc

1113946XF

ac

P Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

P Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

(A9)

In this case

H Rlowast

( 1113857 ≔ H3 ξ ξc ξr( 1113857


ac


1 minus α1113946

aminus ξ+ξc

aSX(x)dx

(A10)




Rlowast

X ξ ξc ξr( 1113857

R(x) aleXle ac


ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A11)

)en

TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

H Rlowast

( 1113857 ≔ H4 ξ ξc ξr( 1113857

12ξc +

12ξr +

12 1 minus αc( 1113857


ac

SX(x)dx

(A12)





Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A13)

)en

H Rlowast

( 1113857 ≔ H5 ξ ξc ξr( 1113857


1 minus αc


ac

SX(x)dx +(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

ac minus ξc+ξr


(A14)








Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A15)

Note that

H Rlowast

( 1113857 ≔ H6 ξ ξc ξr( 1113857

minus βξc minusβ

1 minus αc


ac


1 minus α1113946

acminus ξc+ξr


+(m + 1 minus β) 1113946XF

ar

SX(x)dx

(A16)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A17)



Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A18)


Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A19)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896

+ n 1113946αc

αr


1 minus αc

1113888 1113889 11139461

αc


(A20)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896


minus1

1 minus αc

1113888 11138891113890 1113891 11139461

αc

R VaRs(X)( 1113857ds1113897

(A21)



Data Availability




Acknowledgments


References

















































TVaRp(X) ≔1

1 minus p11139461

pVaRs(X)ds VaRp(X)

+1


+

(3)


Remark 1






MR( 1113857 +1

1 minus αc


+


NR( 1113857 +1

1 minus αr


+

(4)






minRisinF


NR( 11138571113966 1113967 (5)

where β isin [0 1]



NR( 1113857 (6)



(R(X))


(7)


minRisinF

H(R) (8)

where


(R(X))

+(2β minus 1)π(R(X))(9)




minRisinF

1113946s2

s1



R VaRs2(X)1113872 1113873 ξ2

(10)


Rlowast





⎧⎪⎨

⎪⎩(11)


(X) minus VaRs1(X)

Note that

H(R) 11139461

αc

minus β1 minus αc


+ 11139461

αr







10 R(VaRs(X))ds



10 R(VaRs




1α R (VaRs(X))ds









minRisinF

H(R) (13)


(R(X))+



a VaRα(X)

ac VaRαc(X)

ar VaRαr(X)

ξ R VaRα(X)( 1113857

ξc R VaRαc(X)1113872 1113873

ξr R VaRαr(X)1113872 1113873

m (2β minus 1)(1 + θ)

1 minus αminus

β1 minus αc

n (2β minus 1)(1 + θ)



m

t(β) 1 minusβn

(14)





Rlowast(x)



⎧⎨

⎩

(15)






Rlowast(x)



⎧⎨

⎩ (16)











Rlowast(x)



⎧⎨

⎩ (17)










Rlowast(x)



⎧⎪⎨

⎪⎩

(18)





Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(19)





Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(20)



minRisinF


(R(X))1113966

+(2β minus 1)(1 + θ)E(R(X))1113865(21)



1 minus αc



t0(β) 1 minusβn0

(22)



Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(23)





Rlowast(x)

0 when(1 + θ) 1 minus αc( 1113857ge 1



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(24)


Rlowast(x)



⎧⎨

⎩

(25)


Rlowast(x)



⎧⎪⎨

⎪⎩

(26)










Rlowast(x)



⎩ (27)


Rlowast(x)




⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(28)


Rlowast(x)



⎧⎪⎨

⎪⎩(29)







Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)






Rlowast(x)





⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)




Rlowast(x)

0 when (1 + θ) 1 minus αc( 1113857gt 1




⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(32)
















TVaRαc(X) TVaRαr






SX(x)

1 xlt 0

025 x 0


⎧⎪⎪⎨

⎪⎪⎩(33)









(X)













β isin (08419 1] 14055 41997 1000









β isin (05 1] 14055 25902 1000





β isin (05 1] 14055 25902 1000





5 Conclusion








β isin (084 1] 900 44175 900
















β isin (05 1] 900 28081 900






β isin (05 1] 900 28081 900



Appendix


minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds + m 1113946

αr

αcR VaRs(X)( 1113857ds1113896 + m +


1113888 1113889 11139461

αr

R VaRs(X)( 1113857ds1113897 (A1)


Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A2)




TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857(( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

(A3)






ac + ξc Note that


β1 minus αc

1113946VaRs(β)(X)

ac


+(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + 1113946

VaRs(β)(X)

ac

SX(x)dx1113888 11138891113888 1113889

(A4)






lowastr ) (1minus



aSX(x)dx)





Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A5)


TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx +1

1 minus α1113946

XF

ar

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx +1

1 minus αc

1113946XF

ar

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

SX(x)dx

(A6)

)en

H Rlowast

( 1113857 ≔ H2 ξ ξc ξr( 1113857


ac

SX(x)dx

+(2β minus 1)(1 + θ)

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + m +


1113888 1113889 1113946XF

ar

SX(x)dx

(A7)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A8)



TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc

1113946XF

ac

P Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

P Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

(A9)

In this case

H Rlowast

( 1113857 ≔ H3 ξ ξc ξr( 1113857


ac


1 minus α1113946

aminus ξ+ξc

aSX(x)dx

(A10)




Rlowast

X ξ ξc ξr( 1113857

R(x) aleXle ac


ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A11)

)en

TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

H Rlowast

( 1113857 ≔ H4 ξ ξc ξr( 1113857

12ξc +

12ξr +

12 1 minus αc( 1113857


ac

SX(x)dx

(A12)





Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A13)

)en

H Rlowast

( 1113857 ≔ H5 ξ ξc ξr( 1113857


1 minus αc


ac

SX(x)dx +(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

ac minus ξc+ξr


(A14)








Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A15)

Note that

H Rlowast

( 1113857 ≔ H6 ξ ξc ξr( 1113857

minus βξc minusβ

1 minus αc


ac


1 minus α1113946

acminus ξc+ξr


+(m + 1 minus β) 1113946XF

ar

SX(x)dx

(A16)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A17)



Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A18)


Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A19)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896

+ n 1113946αc

αr


1 minus αc

1113888 1113889 11139461

αc


(A20)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896


minus1

1 minus αc

1113888 11138891113890 1113891 11139461

αc

R VaRs(X)( 1113857ds1113897

(A21)



Data Availability




Acknowledgments


References








































minRisinF

1113946s2

s1



R VaRs2(X)1113872 1113873 ξ2

(10)


Rlowast





⎧⎪⎨

⎪⎩(11)


(X) minus VaRs1(X)

Note that

H(R) 11139461

αc

minus β1 minus αc


+ 11139461

αr







10 R(VaRs(X))ds



10 R(VaRs




1α R (VaRs(X))ds









minRisinF

H(R) (13)


(R(X))+



a VaRα(X)

ac VaRαc(X)

ar VaRαr(X)

ξ R VaRα(X)( 1113857

ξc R VaRαc(X)1113872 1113873

ξr R VaRαr(X)1113872 1113873

m (2β minus 1)(1 + θ)

1 minus αminus

β1 minus αc

n (2β minus 1)(1 + θ)



m

t(β) 1 minusβn

(14)





Rlowast(x)



⎧⎨

⎩

(15)






Rlowast(x)



⎧⎨

⎩ (16)











Rlowast(x)



⎧⎨

⎩ (17)










Rlowast(x)



⎧⎪⎨

⎪⎩

(18)





Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(19)





Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(20)



minRisinF


(R(X))1113966

+(2β minus 1)(1 + θ)E(R(X))1113865(21)



1 minus αc



t0(β) 1 minusβn0

(22)



Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(23)





Rlowast(x)

0 when(1 + θ) 1 minus αc( 1113857ge 1



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(24)


Rlowast(x)



⎧⎨

⎩

(25)


Rlowast(x)



⎧⎪⎨

⎪⎩

(26)










Rlowast(x)



⎩ (27)


Rlowast(x)




⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(28)


Rlowast(x)



⎧⎪⎨

⎪⎩(29)







Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)






Rlowast(x)





⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)




Rlowast(x)

0 when (1 + θ) 1 minus αc( 1113857gt 1




⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(32)
















TVaRαc(X) TVaRαr






SX(x)

1 xlt 0

025 x 0


⎧⎪⎪⎨

⎪⎪⎩(33)









(X)













β isin (08419 1] 14055 41997 1000









β isin (05 1] 14055 25902 1000





β isin (05 1] 14055 25902 1000





5 Conclusion








β isin (084 1] 900 44175 900
















β isin (05 1] 900 28081 900






β isin (05 1] 900 28081 900



Appendix


minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds + m 1113946

αr

αcR VaRs(X)( 1113857ds1113896 + m +


1113888 1113889 11139461

αr

R VaRs(X)( 1113857ds1113897 (A1)


Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A2)




TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857(( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

(A3)






ac + ξc Note that


β1 minus αc

1113946VaRs(β)(X)

ac


+(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + 1113946

VaRs(β)(X)

ac

SX(x)dx1113888 11138891113888 1113889

(A4)






lowastr ) (1minus



aSX(x)dx)





Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A5)


TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx +1

1 minus α1113946

XF

ar

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx +1

1 minus αc

1113946XF

ar

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

SX(x)dx

(A6)

)en

H Rlowast

( 1113857 ≔ H2 ξ ξc ξr( 1113857


ac

SX(x)dx

+(2β minus 1)(1 + θ)

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + m +


1113888 1113889 1113946XF

ar

SX(x)dx

(A7)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A8)



TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc

1113946XF

ac

P Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

P Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

(A9)

In this case

H Rlowast

( 1113857 ≔ H3 ξ ξc ξr( 1113857


ac


1 minus α1113946

aminus ξ+ξc

aSX(x)dx

(A10)




Rlowast

X ξ ξc ξr( 1113857

R(x) aleXle ac


ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A11)

)en

TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

H Rlowast

( 1113857 ≔ H4 ξ ξc ξr( 1113857

12ξc +

12ξr +

12 1 minus αc( 1113857


ac

SX(x)dx

(A12)





Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A13)

)en

H Rlowast

( 1113857 ≔ H5 ξ ξc ξr( 1113857


1 minus αc


ac

SX(x)dx +(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

ac minus ξc+ξr


(A14)








Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A15)

Note that

H Rlowast

( 1113857 ≔ H6 ξ ξc ξr( 1113857

minus βξc minusβ

1 minus αc


ac


1 minus α1113946

acminus ξc+ξr


+(m + 1 minus β) 1113946XF

ar

SX(x)dx

(A16)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A17)



Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A18)


Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A19)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896

+ n 1113946αc

αr


1 minus αc

1113888 1113889 11139461

αc


(A20)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896


minus1

1 minus αc

1113888 11138891113890 1113891 11139461

αc

R VaRs(X)( 1113857ds1113897

(A21)



Data Availability




Acknowledgments


References









































Rlowast(x)



⎧⎨

⎩

(15)






Rlowast(x)



⎧⎨

⎩ (16)











Rlowast(x)



⎧⎨

⎩ (17)










Rlowast(x)



⎧⎪⎨

⎪⎩

(18)





Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(19)





Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(20)



minRisinF


(R(X))1113966

+(2β minus 1)(1 + θ)E(R(X))1113865(21)



1 minus αc



t0(β) 1 minusβn0

(22)



Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(23)





Rlowast(x)

0 when(1 + θ) 1 minus αc( 1113857ge 1



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(24)


Rlowast(x)



⎧⎨

⎩

(25)


Rlowast(x)



⎧⎪⎨

⎪⎩

(26)










Rlowast(x)



⎩ (27)


Rlowast(x)




⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(28)


Rlowast(x)



⎧⎪⎨

⎪⎩(29)







Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)






Rlowast(x)





⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)




Rlowast(x)

0 when (1 + θ) 1 minus αc( 1113857gt 1




⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(32)
















TVaRαc(X) TVaRαr






SX(x)

1 xlt 0

025 x 0


⎧⎪⎪⎨

⎪⎪⎩(33)









(X)













β isin (08419 1] 14055 41997 1000









β isin (05 1] 14055 25902 1000





β isin (05 1] 14055 25902 1000





5 Conclusion








β isin (084 1] 900 44175 900
















β isin (05 1] 900 28081 900






β isin (05 1] 900 28081 900



Appendix


minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds + m 1113946

αr

αcR VaRs(X)( 1113857ds1113896 + m +


1113888 1113889 11139461

αr

R VaRs(X)( 1113857ds1113897 (A1)


Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A2)




TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857(( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

(A3)






ac + ξc Note that


β1 minus αc

1113946VaRs(β)(X)

ac


+(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + 1113946

VaRs(β)(X)

ac

SX(x)dx1113888 11138891113888 1113889

(A4)






lowastr ) (1minus



aSX(x)dx)





Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A5)


TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx +1

1 minus α1113946

XF

ar

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx +1

1 minus αc

1113946XF

ar

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

SX(x)dx

(A6)

)en

H Rlowast

( 1113857 ≔ H2 ξ ξc ξr( 1113857


ac

SX(x)dx

+(2β minus 1)(1 + θ)

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + m +


1113888 1113889 1113946XF

ar

SX(x)dx

(A7)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A8)



TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc

1113946XF

ac

P Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

P Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

(A9)

In this case

H Rlowast

( 1113857 ≔ H3 ξ ξc ξr( 1113857


ac


1 minus α1113946

aminus ξ+ξc

aSX(x)dx

(A10)




Rlowast

X ξ ξc ξr( 1113857

R(x) aleXle ac


ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A11)

)en

TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

H Rlowast

( 1113857 ≔ H4 ξ ξc ξr( 1113857

12ξc +

12ξr +

12 1 minus αc( 1113857


ac

SX(x)dx

(A12)





Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A13)

)en

H Rlowast

( 1113857 ≔ H5 ξ ξc ξr( 1113857


1 minus αc


ac

SX(x)dx +(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

ac minus ξc+ξr


(A14)








Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A15)

Note that

H Rlowast

( 1113857 ≔ H6 ξ ξc ξr( 1113857

minus βξc minusβ

1 minus αc


ac


1 minus α1113946

acminus ξc+ξr


+(m + 1 minus β) 1113946XF

ar

SX(x)dx

(A16)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A17)



Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A18)


Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A19)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896

+ n 1113946αc

αr


1 minus αc

1113888 1113889 11139461

αc


(A20)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896


minus1

1 minus αc

1113888 11138891113890 1113891 11139461

αc

R VaRs(X)( 1113857ds1113897

(A21)



Data Availability




Acknowledgments


References











































Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(20)



minRisinF


(R(X))1113966

+(2β minus 1)(1 + θ)E(R(X))1113865(21)



1 minus αc



t0(β) 1 minusβn0

(22)



Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(23)





Rlowast(x)

0 when(1 + θ) 1 minus αc( 1113857ge 1



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(24)


Rlowast(x)



⎧⎨

⎩

(25)


Rlowast(x)



⎧⎪⎨

⎪⎩

(26)










Rlowast(x)



⎩ (27)


Rlowast(x)




⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(28)


Rlowast(x)



⎧⎪⎨

⎪⎩(29)







Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)






Rlowast(x)





⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)




Rlowast(x)

0 when (1 + θ) 1 minus αc( 1113857gt 1




⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(32)
















TVaRαc(X) TVaRαr






SX(x)

1 xlt 0

025 x 0


⎧⎪⎪⎨

⎪⎪⎩(33)









(X)













β isin (08419 1] 14055 41997 1000









β isin (05 1] 14055 25902 1000





β isin (05 1] 14055 25902 1000





5 Conclusion








β isin (084 1] 900 44175 900
















β isin (05 1] 900 28081 900






β isin (05 1] 900 28081 900



Appendix


minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds + m 1113946

αr

αcR VaRs(X)( 1113857ds1113896 + m +


1113888 1113889 11139461

αr

R VaRs(X)( 1113857ds1113897 (A1)


Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A2)




TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857(( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

(A3)






ac + ξc Note that


β1 minus αc

1113946VaRs(β)(X)

ac


+(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + 1113946

VaRs(β)(X)

ac

SX(x)dx1113888 11138891113888 1113889

(A4)






lowastr ) (1minus



aSX(x)dx)





Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A5)


TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx +1

1 minus α1113946

XF

ar

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx +1

1 minus αc

1113946XF

ar

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

SX(x)dx

(A6)

)en

H Rlowast

( 1113857 ≔ H2 ξ ξc ξr( 1113857


ac

SX(x)dx

+(2β minus 1)(1 + θ)

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + m +


1113888 1113889 1113946XF

ar

SX(x)dx

(A7)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A8)



TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc

1113946XF

ac

P Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

P Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

(A9)

In this case

H Rlowast

( 1113857 ≔ H3 ξ ξc ξr( 1113857


ac


1 minus α1113946

aminus ξ+ξc

aSX(x)dx

(A10)




Rlowast

X ξ ξc ξr( 1113857

R(x) aleXle ac


ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A11)

)en

TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

H Rlowast

( 1113857 ≔ H4 ξ ξc ξr( 1113857

12ξc +

12ξr +

12 1 minus αc( 1113857


ac

SX(x)dx

(A12)





Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A13)

)en

H Rlowast

( 1113857 ≔ H5 ξ ξc ξr( 1113857


1 minus αc


ac

SX(x)dx +(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

ac minus ξc+ξr


(A14)








Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A15)

Note that

H Rlowast

( 1113857 ≔ H6 ξ ξc ξr( 1113857

minus βξc minusβ

1 minus αc


ac


1 minus α1113946

acminus ξc+ξr


+(m + 1 minus β) 1113946XF

ar

SX(x)dx

(A16)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A17)



Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A18)


Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A19)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896

+ n 1113946αc

αr


1 minus αc

1113888 1113889 11139461

αc


(A20)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896


minus1

1 minus αc

1113888 11138891113890 1113891 11139461

αc

R VaRs(X)( 1113857ds1113897

(A21)



Data Availability




Acknowledgments


References














































Rlowast(x)



⎩ (27)


Rlowast(x)




⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(28)


Rlowast(x)



⎧⎪⎨

⎪⎩(29)







Rlowast(x)




⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)






Rlowast(x)





⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)




Rlowast(x)

0 when (1 + θ) 1 minus αc( 1113857gt 1




⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(32)
















TVaRαc(X) TVaRαr






SX(x)

1 xlt 0

025 x 0


⎧⎪⎪⎨

⎪⎪⎩(33)









(X)













β isin (08419 1] 14055 41997 1000









β isin (05 1] 14055 25902 1000





β isin (05 1] 14055 25902 1000





5 Conclusion








β isin (084 1] 900 44175 900
















β isin (05 1] 900 28081 900






β isin (05 1] 900 28081 900



Appendix


minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds + m 1113946

αr

αcR VaRs(X)( 1113857ds1113896 + m +


1113888 1113889 11139461

αr

R VaRs(X)( 1113857ds1113897 (A1)


Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A2)




TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857(( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

(A3)






ac + ξc Note that


β1 minus αc

1113946VaRs(β)(X)

ac


+(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + 1113946

VaRs(β)(X)

ac

SX(x)dx1113888 11138891113888 1113889

(A4)






lowastr ) (1minus



aSX(x)dx)





Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A5)


TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx +1

1 minus α1113946

XF

ar

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx +1

1 minus αc

1113946XF

ar

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

SX(x)dx

(A6)

)en

H Rlowast

( 1113857 ≔ H2 ξ ξc ξr( 1113857


ac

SX(x)dx

+(2β minus 1)(1 + θ)

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + m +


1113888 1113889 1113946XF

ar

SX(x)dx

(A7)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A8)



TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc

1113946XF

ac

P Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

P Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

(A9)

In this case

H Rlowast

( 1113857 ≔ H3 ξ ξc ξr( 1113857


ac


1 minus α1113946

aminus ξ+ξc

aSX(x)dx

(A10)




Rlowast

X ξ ξc ξr( 1113857

R(x) aleXle ac


ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A11)

)en

TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

H Rlowast

( 1113857 ≔ H4 ξ ξc ξr( 1113857

12ξc +

12ξr +

12 1 minus αc( 1113857


ac

SX(x)dx

(A12)





Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A13)

)en

H Rlowast

( 1113857 ≔ H5 ξ ξc ξr( 1113857


1 minus αc


ac

SX(x)dx +(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

ac minus ξc+ξr


(A14)








Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A15)

Note that

H Rlowast

( 1113857 ≔ H6 ξ ξc ξr( 1113857

minus βξc minusβ

1 minus αc


ac


1 minus α1113946

acminus ξc+ξr


+(m + 1 minus β) 1113946XF

ar

SX(x)dx

(A16)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A17)



Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A18)


Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A19)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896

+ n 1113946αc

αr


1 minus αc

1113888 1113889 11139461

αc


(A20)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896


minus1

1 minus αc

1113888 11138891113890 1113891 11139461

αc

R VaRs(X)( 1113857ds1113897

(A21)



Data Availability




Acknowledgments


References








































Rlowast(x)





⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)




Rlowast(x)

0 when (1 + θ) 1 minus αc( 1113857gt 1




⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(32)
















TVaRαc(X) TVaRαr






SX(x)

1 xlt 0

025 x 0


⎧⎪⎪⎨

⎪⎪⎩(33)









(X)













β isin (08419 1] 14055 41997 1000









β isin (05 1] 14055 25902 1000





β isin (05 1] 14055 25902 1000





5 Conclusion








β isin (084 1] 900 44175 900
















β isin (05 1] 900 28081 900






β isin (05 1] 900 28081 900



Appendix


minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds + m 1113946

αr

αcR VaRs(X)( 1113857ds1113896 + m +


1113888 1113889 11139461

αr

R VaRs(X)( 1113857ds1113897 (A1)


Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A2)




TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857(( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

(A3)






ac + ξc Note that


β1 minus αc

1113946VaRs(β)(X)

ac


+(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + 1113946

VaRs(β)(X)

ac

SX(x)dx1113888 11138891113888 1113889

(A4)






lowastr ) (1minus



aSX(x)dx)





Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A5)


TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx +1

1 minus α1113946

XF

ar

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx +1

1 minus αc

1113946XF

ar

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

SX(x)dx

(A6)

)en

H Rlowast

( 1113857 ≔ H2 ξ ξc ξr( 1113857


ac

SX(x)dx

+(2β minus 1)(1 + θ)

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + m +


1113888 1113889 1113946XF

ar

SX(x)dx

(A7)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A8)



TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc

1113946XF

ac

P Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

P Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

(A9)

In this case

H Rlowast

( 1113857 ≔ H3 ξ ξc ξr( 1113857


ac


1 minus α1113946

aminus ξ+ξc

aSX(x)dx

(A10)




Rlowast

X ξ ξc ξr( 1113857

R(x) aleXle ac


ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A11)

)en

TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

H Rlowast

( 1113857 ≔ H4 ξ ξc ξr( 1113857

12ξc +

12ξr +

12 1 minus αc( 1113857


ac

SX(x)dx

(A12)





Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A13)

)en

H Rlowast

( 1113857 ≔ H5 ξ ξc ξr( 1113857


1 minus αc


ac

SX(x)dx +(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

ac minus ξc+ξr


(A14)








Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A15)

Note that

H Rlowast

( 1113857 ≔ H6 ξ ξc ξr( 1113857

minus βξc minusβ

1 minus αc


ac


1 minus α1113946

acminus ξc+ξr


+(m + 1 minus β) 1113946XF

ar

SX(x)dx

(A16)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A17)



Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A18)


Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A19)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896

+ n 1113946αc

αr


1 minus αc

1113888 1113889 11139461

αc


(A20)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896


minus1

1 minus αc

1113888 11138891113890 1113891 11139461

αc

R VaRs(X)( 1113857ds1113897

(A21)



Data Availability




Acknowledgments


References










































SX(x)

1 xlt 0

025 x 0


⎧⎪⎪⎨

⎪⎪⎩(33)









(X)













β isin (08419 1] 14055 41997 1000









β isin (05 1] 14055 25902 1000





β isin (05 1] 14055 25902 1000





5 Conclusion








β isin (084 1] 900 44175 900
















β isin (05 1] 900 28081 900






β isin (05 1] 900 28081 900



Appendix


minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds + m 1113946

αr

αcR VaRs(X)( 1113857ds1113896 + m +


1113888 1113889 11139461

αr

R VaRs(X)( 1113857ds1113897 (A1)


Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A2)




TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857(( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

(A3)






ac + ξc Note that


β1 minus αc

1113946VaRs(β)(X)

ac


+(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + 1113946

VaRs(β)(X)

ac

SX(x)dx1113888 11138891113888 1113889

(A4)






lowastr ) (1minus



aSX(x)dx)





Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A5)


TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx +1

1 minus α1113946

XF

ar

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx +1

1 minus αc

1113946XF

ar

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

SX(x)dx

(A6)

)en

H Rlowast

( 1113857 ≔ H2 ξ ξc ξr( 1113857


ac

SX(x)dx

+(2β minus 1)(1 + θ)

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + m +


1113888 1113889 1113946XF

ar

SX(x)dx

(A7)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A8)



TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc

1113946XF

ac

P Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

P Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

(A9)

In this case

H Rlowast

( 1113857 ≔ H3 ξ ξc ξr( 1113857


ac


1 minus α1113946

aminus ξ+ξc

aSX(x)dx

(A10)




Rlowast

X ξ ξc ξr( 1113857

R(x) aleXle ac


ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A11)

)en

TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

H Rlowast

( 1113857 ≔ H4 ξ ξc ξr( 1113857

12ξc +

12ξr +

12 1 minus αc( 1113857


ac

SX(x)dx

(A12)





Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A13)

)en

H Rlowast

( 1113857 ≔ H5 ξ ξc ξr( 1113857


1 minus αc


ac

SX(x)dx +(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

ac minus ξc+ξr


(A14)








Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A15)

Note that

H Rlowast

( 1113857 ≔ H6 ξ ξc ξr( 1113857

minus βξc minusβ

1 minus αc


ac


1 minus α1113946

acminus ξc+ξr


+(m + 1 minus β) 1113946XF

ar

SX(x)dx

(A16)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A17)



Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A18)


Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A19)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896

+ n 1113946αc

αr


1 minus αc

1113888 1113889 11139461

αc


(A20)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896


minus1

1 minus αc

1113888 11138891113890 1113891 11139461

αc

R VaRs(X)( 1113857ds1113897

(A21)



Data Availability




Acknowledgments


References









































5 Conclusion








β isin (084 1] 900 44175 900
















β isin (05 1] 900 28081 900






β isin (05 1] 900 28081 900



Appendix


minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds + m 1113946

αr

αcR VaRs(X)( 1113857ds1113896 + m +


1113888 1113889 11139461

αr

R VaRs(X)( 1113857ds1113897 (A1)


Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A2)




TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857(( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

(A3)






ac + ξc Note that


β1 minus αc

1113946VaRs(β)(X)

ac


+(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + 1113946

VaRs(β)(X)

ac

SX(x)dx1113888 11138891113888 1113889

(A4)






lowastr ) (1minus



aSX(x)dx)





Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A5)


TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx +1

1 minus α1113946

XF

ar

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx +1

1 minus αc

1113946XF

ar

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

SX(x)dx

(A6)

)en

H Rlowast

( 1113857 ≔ H2 ξ ξc ξr( 1113857


ac

SX(x)dx

+(2β minus 1)(1 + θ)

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + m +


1113888 1113889 1113946XF

ar

SX(x)dx

(A7)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A8)



TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc

1113946XF

ac

P Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

P Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

(A9)

In this case

H Rlowast

( 1113857 ≔ H3 ξ ξc ξr( 1113857


ac


1 minus α1113946

aminus ξ+ξc

aSX(x)dx

(A10)




Rlowast

X ξ ξc ξr( 1113857

R(x) aleXle ac


ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A11)

)en

TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

H Rlowast

( 1113857 ≔ H4 ξ ξc ξr( 1113857

12ξc +

12ξr +

12 1 minus αc( 1113857


ac

SX(x)dx

(A12)





Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A13)

)en

H Rlowast

( 1113857 ≔ H5 ξ ξc ξr( 1113857


1 minus αc


ac

SX(x)dx +(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

ac minus ξc+ξr


(A14)








Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A15)

Note that

H Rlowast

( 1113857 ≔ H6 ξ ξc ξr( 1113857

minus βξc minusβ

1 minus αc


ac


1 minus α1113946

acminus ξc+ξr


+(m + 1 minus β) 1113946XF

ar

SX(x)dx

(A16)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A17)



Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A18)


Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A19)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896

+ n 1113946αc

αr


1 minus αc

1113888 1113889 11139461

αc


(A20)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896


minus1

1 minus αc

1113888 11138891113890 1113891 11139461

αc

R VaRs(X)( 1113857ds1113897

(A21)



Data Availability




Acknowledgments


References









































Appendix


minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds + m 1113946

αr

αcR VaRs(X)( 1113857ds1113896 + m +


1113888 1113889 11139461

αr

R VaRs(X)( 1113857ds1113897 (A1)


Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A2)




TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857(( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

(A3)






ac + ξc Note that


β1 minus αc

1113946VaRs(β)(X)

ac


+(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + 1113946

VaRs(β)(X)

ac

SX(x)dx1113888 11138891113888 1113889

(A4)






lowastr ) (1minus



aSX(x)dx)





Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A5)


TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx +1

1 minus α1113946

XF

ar

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx +1

1 minus αc

1113946XF

ar

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

SX(x)dx

(A6)

)en

H Rlowast

( 1113857 ≔ H2 ξ ξc ξr( 1113857


ac

SX(x)dx

+(2β minus 1)(1 + θ)

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + m +


1113888 1113889 1113946XF

ar

SX(x)dx

(A7)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A8)



TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc

1113946XF

ac

P Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

P Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

(A9)

In this case

H Rlowast

( 1113857 ≔ H3 ξ ξc ξr( 1113857


ac


1 minus α1113946

aminus ξ+ξc

aSX(x)dx

(A10)




Rlowast

X ξ ξc ξr( 1113857

R(x) aleXle ac


ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A11)

)en

TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

H Rlowast

( 1113857 ≔ H4 ξ ξc ξr( 1113857

12ξc +

12ξr +

12 1 minus αc( 1113857


ac

SX(x)dx

(A12)





Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A13)

)en

H Rlowast

( 1113857 ≔ H5 ξ ξc ξr( 1113857


1 minus αc


ac

SX(x)dx +(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

ac minus ξc+ξr


(A14)








Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A15)

Note that

H Rlowast

( 1113857 ≔ H6 ξ ξc ξr( 1113857

minus βξc minusβ

1 minus αc


ac


1 minus α1113946

acminus ξc+ξr


+(m + 1 minus β) 1113946XF

ar

SX(x)dx

(A16)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A17)



Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A18)


Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A19)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896

+ n 1113946αc

αr


1 minus αc

1113888 1113889 11139461

αc


(A20)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896


minus1

1 minus αc

1113888 11138891113890 1113891 11139461

αc

R VaRs(X)( 1113857ds1113897

(A21)



Data Availability




Acknowledgments


References









































aSX(x)dx)





Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A5)


TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

aminus ξ+ξc

aSX(x)dx +

11 minus α


ac

SX(x)dx +1

1 minus α1113946

XF

ar

SX(x)dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx +1

1 minus αc

1113946XF

ar

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

SX(x)dx

(A6)

)en

H Rlowast

( 1113857 ≔ H2 ξ ξc ξr( 1113857


ac

SX(x)dx

+(2β minus 1)(1 + θ)

1 minus α1113946

aminus ξ+ξc

aSX(x)dx + m +


1113888 1113889 1113946XF

ar

SX(x)dx

(A7)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A8)



TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc

1113946XF

ac

P Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

P Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

(A9)

In this case

H Rlowast

( 1113857 ≔ H3 ξ ξc ξr( 1113857


ac


1 minus α1113946

aminus ξ+ξc

aSX(x)dx

(A10)




Rlowast

X ξ ξc ξr( 1113857

R(x) aleXle ac


ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A11)

)en

TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

H Rlowast

( 1113857 ≔ H4 ξ ξc ξr( 1113857

12ξc +

12ξr +

12 1 minus αc( 1113857


ac

SX(x)dx

(A12)





Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A13)

)en

H Rlowast

( 1113857 ≔ H5 ξ ξc ξr( 1113857


1 minus αc


ac

SX(x)dx +(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

ac minus ξc+ξr


(A14)








Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A15)

Note that

H Rlowast

( 1113857 ≔ H6 ξ ξc ξr( 1113857

minus βξc minusβ

1 minus αc


ac


1 minus α1113946

acminus ξc+ξr


+(m + 1 minus β) 1113946XF

ar

SX(x)dx

(A16)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A17)



Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A18)


Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A19)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896

+ n 1113946αc

αr


1 minus αc

1113888 1113889 11139461

αc


(A20)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896


minus1

1 minus αc

1113888 11138891113890 1113891 11139461

αc

R VaRs(X)( 1113857ds1113897

(A21)



Data Availability




Acknowledgments


References









































TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc

1113946XF

ac

P Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr +1

1 minus αr

1113946XF

ar

P Rlowast

X ξ ξc ξr( 1113857gt x( 1113857dx

(A9)

In this case

H Rlowast

( 1113857 ≔ H3 ξ ξc ξr( 1113857


ac


1 minus α1113946

aminus ξ+ξc

aSX(x)dx

(A10)




Rlowast

X ξ ξc ξr( 1113857

R(x) aleXle ac


ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A11)

)en

TVaRα Rlowast

X ξ ξc ξr( 1113857( 1113857 ξ +1

1 minus α1113946

XF

aP Rlowast

X ξ ξc ξr( 1113857gtx( 1113857dx

TVaRαcRlowast

X ξ ξc ξr( 1113857( 1113857 ξc +1

1 minus αc


ac

SX(x)dx

TVaRαrRlowast

X ξ ξc ξr( 1113857( 1113857 ξr

H Rlowast

( 1113857 ≔ H4 ξ ξc ξr( 1113857

12ξc +

12ξr +

12 1 minus αc( 1113857


ac

SX(x)dx

(A12)





Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A13)

)en

H Rlowast

( 1113857 ≔ H5 ξ ξc ξr( 1113857


1 minus αc


ac

SX(x)dx +(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

ac minus ξc+ξr


(A14)








Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A15)

Note that

H Rlowast

( 1113857 ≔ H6 ξ ξc ξr( 1113857

minus βξc minusβ

1 minus αc


ac


1 minus α1113946

acminus ξc+ξr


+(m + 1 minus β) 1113946XF

ar

SX(x)dx

(A16)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A17)



Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A18)


Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A19)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896

+ n 1113946αc

αr


1 minus αc

1113888 1113889 11139461

αc


(A20)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896


minus1

1 minus αc

1113888 11138891113890 1113891 11139461

αc

R VaRs(X)( 1113857ds1113897

(A21)



Data Availability




Acknowledgments


References








































Rlowast

X ξ ξc ξr( 1113857



ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A13)

)en

H Rlowast

( 1113857 ≔ H5 ξ ξc ξr( 1113857


1 minus αc


ac

SX(x)dx +(2β minus 1)(1 + θ) ξ +1

1 minus α1113946

ac minus ξc+ξr


(A14)








Rlowast

X ξ ξc ξr( 1113857




⎧⎪⎪⎨

⎪⎪⎩

(A15)

Note that

H Rlowast

( 1113857 ≔ H6 ξ ξc ξr( 1113857

minus βξc minusβ

1 minus αc


ac


1 minus α1113946

acminus ξc+ξr


+(m + 1 minus β) 1113946XF

ar

SX(x)dx

(A16)




Rlowast

X ξ ξc ξr( 1113857



R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A17)



Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

ξr Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A18)


Rlowast

X ξ ξc ξr( 1113857


R(x) ac ltXle ar

R(x) Xgt ar

⎧⎪⎪⎨

⎪⎪⎩

(A19)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896

+ n 1113946αc

αr


1 minus αc

1113888 1113889 11139461

αc


(A20)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896


minus1

1 minus αc

1113888 11138891113890 1113891 11139461

αc

R VaRs(X)( 1113857ds1113897

(A21)



Data Availability




Acknowledgments


References










































minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896

+ n 1113946αc

αr


1 minus αc

1113888 1113889 11139461

αc


(A20)




minRisinF

(2β minus 1)(1 + θ)


αR VaRs(X)( 1113857ds1113896


minus1

1 minus αc

1113888 11138891113890 1113891 11139461

αc

R VaRs(X)( 1113857ds1113897

(A21)



Data Availability




Acknowledgments


References








































pareto-optimalreinsurancerevisited:atwo ...principle of indemnity and the incentive compatible...

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