VaR限制下的最优保险投资策略选择7015213758

VaR∗1,212100462210009VaR-VaRVaR0221.11LambertHofflander(1966)[1]Krous(1970)[2]Kahane(1975)[3]Briys[4]1985Browne(1995)[5]Hipp(2000)[6]Yang(2005)[7]Wang(2007)[8](2004)[8][9]Markowitz-Wang(2007)Browne(1995)Yang(2005)Hipp(2000)Browne(2005)Yang(2005)Markowitz-VaR[11]CaR[12]CVaREaR[13][14]VaRVaR(Value-at-Risk)90VaRVaRAlexander(2002)[15](2004)[11]VaR∗70573044,(.1971.Email:[e-mail]1VaR-VaR-VaR1Lundberg0)(wtw=)(tSct−+0t00wc∑==)(1)(tNkkXtSkX1≥kk)(tNt)(tNλPossion0,(wsϕ()P=),)0(|}0)(:inf{0wwstwt=),(lim)(00wswsϕϕ∞→=.),(0wsϕ0ws)(0wϕ0wn0ri,1,2,,iRin=L,1,2,,iRi=LniR(,)iiNµσ1,2,,in=L,1,2,,ii,nπ=L11n/i11/π−,),,,(/21nππππL=],0[T,[0,1]θθ∈1%0.1%0Pθ200ln1Ppwθ≤+,02Lundbergp[16]0(1)00(,)(),pwTwwePθϕϕ−−≤≤2θT//00()(11)TwwcTSTwrRθππ0⎡⎤=+−+−+⎣⎦3/12(,,,)nRRRR=LVaRCmax(),..().TTEwstVaRwCπα⎧⎪⎨≤⎪⎩4α1%5%0(,)wαρπ()wTαVaRαVaR()0()(,)TTVaRwEwwααρπ=−,5()TVaRwα0(,)wαρπ()Fw()wT()()00(,)(,)TFwPwwαρπρπ==≤00011000011()(1)(,)(,)(1)nniiiiinniiTiiiPwcTSTwrRwPwRSwwcTwrθππρπθπρπθπ====⎛⎞⎡⎤=+−+−+≤⎜⎟⎢⎥⎣⎦⎝⎠⎛⎞=−≤−−−−⎜⎟⎝⎠∑∑∑∑00)6()StPossion6[5][17]()St()St211((),()NEXtEXtλλ1n+/001011()()nniiTiiiiEwRSwEXTwMTθπθπµλθπµ==⎛⎞−=−=−⎜⎟⎝⎠∑∑32222200101122/0()()2()()nniiTiiijijiiVarwRSwEXTwwDTnijθπθπσλθπθππ==⎛⎞−=++⎜⎟⎝⎠=Ω+∑∑πσ≠∑1()()MTEXTλ=/12(,,,)nµµµµ=L21()()DTEXTλ=()ijnnσ×Ω=Ω01niiTiwRθπ=S−∑()/22/00(),()NwMTwDTθπµθππ−Ω+.6//0000022/0(,)(11)()(),wwcTwrwMTzwDTαρπθπθπµθππ=++−+−−Ω+7//0000()(11)(T)EwwcTwrwMTπθπθπµ=++−+−822/0()()TVaRwzwDTααθππ=Ω+,9zαα0.5α40zα/22/0max,..().BstwDTππθππδ⎧⎪⎨Ω+≤⎪⎩1001Brµ=−22Czαδ=Lagrange()CzDTα≥10*/101()DT1BwBBδπθ−−−=ΩΩ.(11)1*1/101()DTBwBBδπθ−−−=ΩΩ()DTα≥)Cz.(/10()()(),TEwwcTMTBBDTδ−=+−+Ω−(12)*()TVaRwCαπ.=4122α-VaR501234560.30.40.50.60.70.80.91intensityλoptimalriskinvestmentproportionπ*()/120()()()()TEwwcTMTBBVaRzDTαα−=+−+Ω−,1234213(13)312112Briys198500.0050.010.0150.30.40.50.60.70.80.91claimsizeE(Z1)optimalriskinvestmentproportionπ*243α5(12ccc)33312λλλ)4(54220=w0.16/0.16c=4=λPossion1X640β=1=T30.151232001120071221()iEriiσijσ(1)1.17%0.0090(2)1.96%0.0125(3)2.07%0.0094120.00133σ=−,130.00009σ=−,230.00017σ=−20075304.27%0.36%0.1C=0.1α=η[18]110.6,151cpEXηηαλη=−===+200lnln0.001110.7697215Ppwθ≤+=+=×76%490,13.3,0.9113.3,125,1.7100.9,1.7,94−−⎛⎞⎜⎟Ω=−−⎜⎟⎜⎟−−⎝⎠1109.401011.65741.2583-11.628978.78051.31340.83711.5364106.4188−⎛⎞⎜⎟Ω=⎜⎟⎜⎟⎝⎠./(0.0081,0.0160,0.0171)B=[18]21(1)0.0025DEXλ==*1/101()(0.1022,0.1110,0.1729),DTBwBBδπθ−−−=Ω=Ω10.22%711.1%17.29%61.39%2001*π83.7%6.67()2.0739TEw=()2.0072TEw=[1]LambertE.W.,HofflanderA.E.,Impactofnewmultiplelineunderwritingoninvestmentportfolioofproperty-liabilityinsurers[J].JournalofRiskandInsurance,1966,33:209-223.[2]KrousC.G.,Portfoliobalancingcorporateassetsandliabilitieswithspecialappicationtoinsurancemanagement[J].TheJournalofFinancialandQuantitativeAnalysis,1970,5:77-105.[3]KahaneY.,NyeD.,Aportfolioapproachtothepropertyliabilityinsuranceindustry[J].JournalofRiskandInsurance,1975,42:579-598.[4]BriysE.P.,Investmentportfoliobehaviorofnon-lifeinsurers:autilityanalysis[J].Insurance:MathematicsandEconomics,1985,4:93-98.[5]Browne,S.,Optimalinvestmentpoliciesforafirmwitharandomriskprocess:exponentialutilityandminimizingtheprobabilityofruin[J].MathematicsofOperationsResearch,1995,20:937-958.[6]Hipp,C.,Plum,M.,Optimalinvestmentforinsurers[J].Insurance:Mathematics&Economics,2000,27:215-228.[7]Yang,H.L.,Zhang,L.H.,2005.Optimalinvestmentforinsurerwithjump-diffusionriskprocess[J].Insurance:Mathematics&Economics,2005,37:615-634.[8]WangZ.,XiaJ.,ZhangL.,Optimalinvestmentforaninsurer:themartingaleapproach[J].Insurance:MathematicsandEconomics,2006,inpress.[9].[J].2004,19(2):198-201.[10].[J]..2004,10:62-67.[11].VaR[J].,2004,12(1):8-14.[12]EmmerS.,KluppelbergC.andKornR.,OptimalportfolioswithboundedCapital-at-Risk[J].MathematicalFinance,2001,11:365-384.[13]LiZ.F.andDengX.T.,OptimaldynamicportfolioselectionwithEarnings-at-Risk[J].J.Optim.Appl.,2007,132:459-473.[14].[J].2004,41(1):41-45.8[15]AlexanderG.,BaptistaA.,Economicimpliationsofusingamean-VaRmodelforportfolioselection:acomparisonwithmean-varianceanalysis[J].JournalofEconomicDynamics&Control,2002,26:1159-1193.[16]LunbergF.I.,ApproximeradFramstallningavSannolikhetsfunktionen.II,ÄtersforsäkringavKollektivrisker[M].Uppsala:Almqvist&Wiksell,1903.[17]GrandllJ.,AspectsofRiskTheory[M].Springer-Verlag,NY,1991.[18],.[M].,2000.OptimalPortfolioSelectionBoundedbyVaRforInsurerGuoWenjing1,2(1.SchoolofFinance,NanjingUniversityofFinanceandEconomics.JiangsuNanjing,210046,China;2.SchoolofManagementScience&Engineering,NanjingUniversity,JiangsuNanjing,210093,China)Abstract:TheintegralriskofinsurancecompanyismeasuredwithVaR.Theproblemofportfolioselectionforinsuerisstudiedunderthemean-VaRcriterion.Toensuretheinsurancecompanyissafeintheperiodofinvestment,weintroduceasafeinvestmentproporsion,whichtheinsurerinvestsinassets.Bysolvingthemodel,theexplicitexpressionsofoptimalinvestmentstrategiesandefficientfrontierarederived.Theinfluenceofpremium,claimonoptimalinvestmentstrategiesandefficientfrontierisdiscussed.Finally,byrealdatumtheinvestmentprocedureissimulated.Keywords:VaR;optimalinvestmentstr