随机利率下社会养老保险隐性债务的精算分析Ξ

:100124098(2005)0520055206X,,(,110004):,,,Wiener,,,MonteCarlo,:;;;;MonteCarlo:O211.6;F840.6:A1,1997(),,,;,()();,(),,,,,,,,,,,(ImplicitPensionDebt,IPD),,[1],,,,:,,,,,,[2],,IPD[1][3][4][5],,,,,[6],WienerOrnstein2Uhlenbeck,,,,,,235(137)Vol.23,No.520055SystemsEngineeringMay.,2005X:2005203202:(2003EE550001):(19792),,,,:;(19522),,,,,,:;(19682),,,,:©1994-2006ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(),,,,MZ2,c,cm,,:ci:i,mi=1ci=c;di:i,di=cic,mi=1di=1;xi:i;ni:,i,,ni=0,;ni0,,,ni=-xi;Bi:i,Bi;i,Bi5Bi;A:,,,()Bi(1+A)ni;Kij:ij(curtatefuturelifetime)[11],Kij0,1,2,,X-xi-1,XKij(i=1,2,,m;j=1,2,,ci);kûq(i),kp(i):ikk+1k[11],kûq(i)=P(Kij=k),kp(i)=P(Kijk),kûq(i)kp(i);v(k):k,kDss(s0)(forceofinterest),[12],v(k)=e-k0Dsds.,Ds,,v(k);Zij:ij,ni=0,Zij=BiKijk=0(1+A)kõv(k)(1)ni1,Zij=0,BiKijk=ni(1+A)kõv(k),Kij=0,1,,ni-1Kijni(2)Z=mi=1cij=1Kij(3)E(Z),,[1],,:,,,,;,,[6]3Sirir,Dirir-1r,Di1,Di2,,Dis,Sis(ci,0ûq(i),1ûq(i),,s-1ûq(i),sp(i))(multinomialdistribution)[13]1rs,SirSiscov(Sir,Sis)=cisp(i)(1-rp(i))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)(4)[13],(4)cov(Sir,Sis)=ciõsp(i)(1-sp(i))-ciõrûq(i)õsp(i)-ciõr+1ûq(i)õsp(i)--ciõs-1ûq(i)õsp(i)=ciõsp(i)(1-sp(i)-rûq(i)-r+1ûq(i)--s-1ûq(i))=ciõsp(i)(1-rp(i))(5)r=s,cov(Sir,Sis)=Var(Sir)=ciõrp(i)(1-rp(i))(6)(5)(6)1CFrr,rCFr=mi=1SirBi(1+A)r1(nir)(7)1(),1(nir)=1,nir0,n=maxi{X-xi-1},Z=nr=0CFrv(r)(8)652005©1994-2006ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(3),(8)Z20rn,E(CFr)=mi=1Biciõrp(i)(1+A)r1(nir);0rsn,cov(CFr,CFs)=mi=1B2iciõsp(i)(1-rp(i))(1+A)r+s1(nir).(7)CFrE(CFr),Kij(i=1,2,,m;j=1,2,,ci),SirSjs(ij),1,0rsn,cov(CFr,CFs)=cov(mi=1SirBi(1+A)r1(nir),mi=1SisBi(1+A)s1(nis))=mi=1cov(SirBi(1+A)r1(nir),SisBi(1+A)s1(nis))=mi=1B2i(1+A)r+s1(nir)cov(Sir,Sis)=mi=1B2iciõsp(i)(1-rp(i))(1+A)r+s1(nir)21E(Z)=nr=0E(CFr)E[v(r)]Var(Z)=nr=0ns=0cov(CFr,CFs)E[v(r)v(s)]+nr=0ns=0E(CFr)E(CFs)cov[v(r),v(s)]E(CFr)cov(CFr,CFs)2(8)ZE(Z),Var(Z)=E[Var[Zû{v(k)}]]+Var[E[Zû{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)]1[9],E[Var[Zû{v(k)}]],Var[E[Zû{v(k)}]]dic,Zöc,limcE[Var[Zöcû{v(k)}]]=limcE[Var[Zû{v(k)}]]öc2=0Var[E[Zöcû{v(k)}]]=Var[E[Zû{v(k)}]]öc2=nr=0ns=0(mi=1diBiõrp(i)(1+A)r1(nir))õ(mi=1diBiõsp(i)(1+A)s1(nis))cov[v(r),v(s)]c,c0,,c,E(CFr)CFr,ZM=nr=0E(CFr)v(r)[7][8][9][10],E(M)=E(Z),Var(M)=Var[E[Zû{v(k)}]]Var(Z)MZ,ZMonteCarlo,v(r)(r=1,2,,n),M,v(r)(r=1,2,,n),41E(Z)Var(Z),v(k)Wiener,,,,,y(k)=k0Dsds=Dk+BW(k)(9),D,B,B0,W(k)Wiener,W(k)N(0,k),v(k)=e-y(k)=e-Dk-BW(k)2(9),E[v(k)]=e-(D-B2ö2)k,E[v2(k)]=e-2(D-B2)k,kikj,E[v(ki)v(kj)]=e-(D-B2ö2)(ki-kj)-2(D-B2)kj.E[v(k)]=e-DkE(e-BW(k))=e-Dk+-e-BX12Pke-X2ö2kdX=e-Dk+-e-BX12Pke-(X+kB)2ö2k+kB2ö2dX=e-(D-B2ö2)k.E[v2(k)]E[v(ki)v(kj)]Wiener,kikj,W(ki)-W(kj)W(kj),y(ki)-y(kj)y(kj),W(ki)-W(kj)N(0,ki-kj),E[v(ki)v(kj)]=E(e-[y(ki)-y(kj)]-2y(kj))=E(e-[y(ki)-y(kj)])E(e-2y(kj))=e-D(ki-kj)E(e-B[W(ki)-W(kj)])E(e-2y(kj))=e-D(ki-kj)eB2(ki-kj)ö2e-2(D-B2)kj=e-(D-B2ö2)(ki-kj)-2(D-B2)kj5MonteCarlo10000,755,:©1994-2006ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.à,80à[6],Bi=xi-20xR-20BR,BR,xR(6055)(4à)50à,A=2à1996595475017137[4],B1=75010.8=6001,B2=71370.8=571031992,B3=7501(1+4à)50.8(1+2à)5=5445,B45,B5=35-2060-206001=2250,B6B7B8,[14],,X=91(9)D=0.06,B=0.11ixiniBi(:)ci16000.6001110025500.5710125036500.5445130046000.51811400535250.22501000635200.24471100750100.45011500845100.407913502E(CFr)rE(CFr)rE(CFr)rE(CFr)rE(CFr)02807.1143540.7281952.742488.112818.2153406.2291809.743431.822822.9163257.730168044374.7432821.2173106.5311504.745323.9642812182945.132139546251.2552790.4192772.8331289.247211.0362760.5202961.5341181.448174.3172721.1212783.6351082.349139.2182670.6222611.736959.4850110.0592609.8232441.837873.495185.984103930.7242268.238792.065265.731113852252426.539712.095350.336123762.8262241.740641.555436.939133658.3272094.741546.475530.341852005©1994-2006ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=maxi{X-xi-1},2,2E(CFr)(r=0,1,2,,55),cov(CFr,CFs)2,cov(CFr,CFs)12E(Z)=46316,46316,Var(Z)=14748104Var(Z)E(Z)+KVar(Z),KVar(Z),K,K3Var(Zöc),,c,0,3Var(Zöc)cVar(Zöc)E[Var[Zöcû{v(k)}]]Var[E[Zöcû{v(k)}]]100001.47484.232510-41.47441000001.474444.232510-51.474410000001.4744044.232510-61.47441.474401.4744,Z=55r=0CFrv(r),M=55r=0E(CFr)v(r),,MonteCarloZM:,ZM,ZM,,[15]ZM10000,Matlab6.1,Z:1:{W(r),r=0,1,2,,55},W(r)N(0,r);2:v(r)=e-Dr-BW(r){v(r),r=0,1,2,,55};3:10000[0,1],Tq(i)=U,10000Kij{Kij,i=1,2,,8;j=1,2,,ci},,T[11],U[0,1];4:(1)(2){Zij,i=1,2,,8;j=1,2,,ci};5:(3),Z1510000,10000Z,Z,{v(r),r=0,1,2,,55},Kij10000Z,108,M,,v(r)(r=0,1,2,,55),,M1210000ZM,ZP(Z52890)0.99,52890,0.99,,1ZM2ZM955,:©1994-2006ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.:[1].[J].,2002,21(1):2932.[2].[M].:,2001:8593.[3].[J].,1998,(4):6163.[4],.[J].,2001,13(3):5763.[5].[J].,2002,(1):4245.[