社会养老保险中隐性债务的双随机模型①

1,2,2(1.,510640;2.,110004):,.,,Wiener,.,MonteCarlo.:;;;MonteCarlo:O211.6;F840.6:A:1000-5781(2005)06-0570-08DualrandommodelsforimplicitpensiondebtinsocialpensionsystemDONGMing,GUOYa2jun,YANGHuai2dong(1.SchoolofBusinessAdministration,SouthChinaUniversityofTechnology,Gangzhou510640,China;2.SchoolofBusinessAdministration,NortheasternUniversity,Shenyang110004,China)Abstract:Thispaperanalyzestheimplicitpensiondebtinsocialpensionsysteminastochasticinterestrateandstochasticmortalityenvironment.Twomethodsareproposedtocalculatetheexpectedvalueandthevarianceofthepresentvalueofthepensionbenefits.TheconcreteexpressionsoftheexpectedvalueandthevariancearederivedwhentheinterestratemodelisconstructedwithWienerprocess.Inanumer2icalexample,theMonteCarlomethodisusedtogettheempiricaldistributionsofboththepresentvalueofthepensionbenefitsanditsapproximation.Keywords:socialpensionsystem;implicitpensiondebt;actuarialscience;MonteCarlosimulation0,:.,;,.,1997(),.,,;,()();,().,,.206200512JOURNALOFSYSTEMSENGINEERINGVol.20No.6Dec.2005:2004-12-06;:2005-02-22.:(2003EE550001).,,,.,,.,,,,(implicitpensiondebt,IPD)[1].,,[1].,,,.,,,,,.,,[2]..,IPD.[1,35],.,,,.[6],WienerOrnstein-Uhlenbeck,.,,,.,.,,232Z().2,,,,MZ.[710].5,MonteCarloZM.1c,cm,,.:cii,mi=1ci=c;dii,di=ci/c,mi=1di=1;xii;ni,i.ni=0,;ni0,,,ni=-xi;Bii,Bi;i,Bi.5Bi;.,.,,()Bi(1+)ni;Kijij(curtatefuturelifetime)[11],Kij0,1,,2,,-xi-1,.Kij(i=1,2,,m;j=1,2,,ci);k|q(i),kp(i)ikk+1k,k|q(i)=P(Kij=k),kp(i)=P(Kijk),k|q(i)kp(i);v(k)k,k1756:.ss(s0)(forceofinterest),[12],v(k)=exp(-k0sds).,s,,v(k);Zijij.,ni=0,Zij=BiKijk=0(1+)kv(k)(1)ni1,Zij=0Kij=0,1,,ni-1BiKijk=ni(1+)kv(k)Kijni(2)Z=mi=1cij=1Zij(3)E(Z),,[1].,,:,,,,;,,.[6].2ni=0,(1),E(Zij)=E[E(Zij|Kij)]=-xi-1k=0k|q(i)Biks=0(1+)sE[v(s)]=Bi-xi-1k=0ks=0k|q(i)(1+)sE[v(s)](4)ni1,(2),E(Zij)=E[E(Zij|Kij)]=-xi-1k=nik|q(i)Biks=ni(1+)sE[v(s)]=Bi-xi-1k=niks=nik|q(i)(1+)sE[v(s)](5)(5)nini0,(4)(5)E(Zij)=Bi-xi-1k=niks=nik|q(i)(1+)sE[v(s)]ni0(6),ni=0ni1Zij,E(Zij),E(Z2ij),E(Zi1Zi2)E(Zi1Zs1),ni=0ni1,13ni1Zij.(6)E(Zij)j,E(Zij)=E(Zi1).,E(Z)=mi=1cij=1E(Zij)=mi=1ciE(Zi1)(7)E(Zi1)(6).(2),E(Z2i1)=E[E(Z2i1|Ki1)]=-xi-1k=nik|q(i)B2iks=nikt=ni(1+)s+tE[v(s)v(t)]=B2i-xi-1k=niks=nikt=nik|q(i)(1+)s+tE[v(s)v(t)](8)E(Zi1Zi2)=B2i-xi-1k1=ni-xi-1k2=nik1s=nik2t=nik1|q(i)k2|q(i)(1+)s+tE[v(s)v(t)](9)is,E(Zi1Zs1)=BiBs.-xi-1k1=ni-xs-1k2=nsk1j=nik2t=nsk1|q(i)k2|q(s)(1+)j+tE[v(j)v(t)](10)(3),E(Z2)=mi=1ms=1cij=1cst=1E(ZijZst)=mi=1cij=1cis=1E(ZijZis)+mi=1ms=1,sicij=1cst=1E(ZijZst)=27520mi=1ciE(Z2i1)+mi=1cij=1cis=1,sjE(ZijZis)+mi=1ms=1,sicij=1cst=1E(ZijZst)(11)Kij(i=1,2,,m;j=1,2,,ci),v(k),Zij(i=1,2,,m,j=1,2,,ci).,sj,E(ZijZis)=E[E[ZijZis|{v(k)}]]=E[E[Zij|{v(k)}]E[Zis|{v(k)}]]=E[E[Zi1|{v(k)}]E[Zi2|{v(k)}]]=E[E[Zi1Zi2|{v(k)}]]=E(Zi1Zi2),si,E(ZijZst)=E(Zi1Zs1).,(11).E(Z2)=mi=1ciE(Z2i1)+mi=1ci(ci-1)E(Zi1Zi2)+mi=1ms=1,sicicsE(Zi1Zs1)(12)E(Z2i1),E(Zi1Zi2)E(Zi1Zs1)(8)(9)(10).(7)(12)Var(Z)=E(Z2)-[E(Z)]2,Z.,m+1,Z3,Z3=Z+cm+1j=1Zm+1,j.,Var(Z)Var(Z3),Var(Z3),.Var(Z3)=Var(Z)+Var(cm+1j=1Zm+1,j)+2mi=1cicm+1Cov(Zi1,Zm+1,1),(8)(9),(10).3Sirir,Dirir-1r,Di1,Di2,,Dis,Sis(ci,0|q(i),1|q(i),,s-1|q(i),sp(i))(multinomialdistribution)[13].rs,Sir=Sis+Di,r+1+Di,r+2++Dis,Cov(Sir,Sis)=Cov(Sis+Di,r+1+Di,r+2++Dis,Sis)=Var(Sis)+Cov(Di,r+1,Sis)+Cov(Di,r+2,Sis)++Cov(Dis,Sis)(13)[13],(13)Cov(Sir,Sis)=cisp(i)(1-sp(i))-cir|q(i)sp(i)-cir+1|q(i)sp(i)--cis-1|q(i)sp(i)=cisp(i)(1-sp(i)-r|q(i)--s-1|q(i))=cisp(i)(1-rp(i))(14)r=s,Cov(Sir,Sis)=Var(Sir)=cirp(i)(1-rp(i))(15)(14)(15):rs,SirSisCov(SirSis)=cisp(i)(1-rp(i))(16)CFrr,r.CFr=mi=1SirBi(1+)r1(nir)(17)(17),1()1(nir)=1,nir0,n=mxi{-xi-1},Z=nr=0CFrv(r)(18)(18)Z.0rn,(17)E(CFr)=mi=1Bicirp(i)(1+)r1(nir)(19)Kij(i=1,2,,m,j=1,2,,ci),SirSjs(ij).,(16),0rsn,Cov(CFr,CFs)=3756:Cov(mi=1SirBi(1+)r1(ni1),mi=1SisBi(1+)s1(nis))=mi=1Cov(SirBi(1+)r1(nir),SisBi(1+)s1(nis))=mi=1B2i(1+)r+s1(nir)Cov(Sir,Sis)=mi=1B2icisp(i)(1-rp(i)(1+)r+s1(nir)(20)(18)ZE(Z)=nr=0E(CFr)E[v(r)](21)Var(Z)=E[Var(Z|{v(k)})]+Var[E(Z|{v(k)})]=E[nr=0ns=0Cov(CFr,CFs)v(r)v(s)]+Var[nr=0E(CFr)v(r)]=nr=0ns=0Cov(CFr,CFs)E[v(r)v(s)]+nr=0ns=0E(CFr)E(CFs)Cov[v(r),v(s)](22)[9],E[Var(Z|{v(k)})],Var[E(Z|{v(k)})].dic,Z/c,limcE[Var(Z/c|{v(k)})]=limcE[Var(Z|{v(k)})]/c2=0Var[E(Z/c|{v(k)})]=Var[E(Z|{v(k)})]/c2=nr=0ns=0(mi=1diBirp(i)(1+)r1(nir))(mi=1diBisp(i)(1+)s1(nis))Cov[v(r),v(s)]c,c0,.,c,E(CFr)CFr[710],ZM=nr=0E(CFr)v(r),E(M)=E(Z),Var(M)=Var[E(Z|{v(k)})]Var(Z).MZ,ZMonteCarlo,v(k)(k=1,2,,n),M,v(k)(k=1,2,,n),.4kv(k).Wiener,,,,,.[12](k)=k0sds=k+W(k)(23),,,0,W(k)Wiener,W(k)N(0,k).,v(k)=e-(k)=e-k-W(k).(23),E[v(k)]=e-kE(e-W(k))=e-k+-e-12ke-2/2kd=e-k+-12ke-(+k)2/2k+k2/2d=e-(-2)/2kE[v2(k)]=e-2(-2)kWiener,kikj,W(ki)-W(kj)W(kj),(ki)-(kj)(kj).,W(ki)-W(kj)N(0,ki-kj).E[v(ki)v(kj)]=E(e-[(ki)-(kj)]-2(kj))=E(e-[(ki)-(kj)])E(e-2(kj))=e-(ki-kj)E(e-[W(ki)-W(kj)])E(e-2(kj))=e-(ki-kj)e2(ki-kj)/2e-2(-2)kj=e-(-2/2)(ki-kj)-2(-2)kj475205MonteCarlo10000,.1997,1997,1.20,60,55.80%,80%[5].[6],Bi=xi-20xR-20BR,BR,xR(6055).(4%)50%,=2%.1996595475007137[3].,B1=75010.8=6001,B2=71370.8=5710.31992,B3=7501(1+4%)50.8(1+2%)5=5445,B4,B5.6,B6=35-2060-206001=2250,B7B10.(13)=0.06,=0.1.,[5],,=91.(6)(8)(