保险精算基础讲义

《保险精算基础》 保险硕士讲义 (西南民族大学经济学院 陈宏） 1 第一章精算基础知识 1.1 精算、精算师和精算职业制度 精算——应用各种数理模型来估计和分析未来不确定事件（风险）产生的影响（特别是财务方面）。以保险业为基础产生的精算科学通常指处理保险业中的风险管理问题精算早已形成完整的体系，在社会保险、金融、投资、证券等领域广泛应用。 精算师——针对精算问题逐步形成的一种专门职业的从业人员，经过金融保险监管部门认可其从业资格。资格认定：北美和英国体系,资格考试分寿险精算师、非寿险精算师、投资与资产管理精算师、养老金精算师、咨询精算 ||1693||SOAAAA||199943;2()92004,06G,10323323227013090400013030023030032020045044005202006303007303008303009303011303012303013/3030142030152030162030172030182030192035020203ASASOAVEEFAPAPC1.PreliminaryEducationExamina-tionsExamP3ExamMF2ExamM4ExamCCredibilitytheory,462.VEEValidatedbyEducationalExperienceSOAVEECASTRANSITIONALEX-AMS®exams.htm8VEE3.FundamentalsofActuarialPractice81568course5course67Module1:RoleoftheProfessionalActuaryModule2:CoreExternalForcesModule3:TypicalActuarialProblemsModule4:SolutionstoSelectedActuarialProblemsModule5:DesignandPricingofanActuarialSolutionModule6:SelectionofanActuarialDesignandModelModule7:SelectionofInitialAssumptionsModule8:MonitoringExperience-ModelandAssumptions4.APCAPC|COURSE1234COURSEPFMMCVEEEXAMP=COURSE1EXAMFM+VEE(ECONOMICSCOPORATEFINANCE)=COURSE28EXAMM=COURSE3EXAMC+VEE(STATISTICS)=COURSE41.21.21.:X0XF(x):F(x)=Pr(X·x)x¸0f(x)=F'(x)XXE(X)=Z+10xf(x)dxVar(X)=Z+10[x¡E(X)]2f(x)dx=E(X2)¡(E(X))2s(x)s(x)=1¡F(x)=Pr(Xx)x¸0s(x)x1s(0)=1;s(1)=0;92s(x);3s(x).s(x)w,xw;s(x)=0.s(x)=1¡x=96;0·x96,w=96.2.T(x)(x)xXxT(x)=X¡xxT(x)G(t)=P(T(x)·t)tqx=Pr(T(x)·t);t¸0tpx=1¡tqx=Pr(T(x)t);t¸0tqx(x)t(x)10x+ttp0=s(x);x¸0T(0)=Xt=1qx=1qx;px=1pxT(x)x+tx+t+utjuqx=Pr(tT(x)·t+u)=t+uqx¡tqx=tpx¡t+upxu=1tjuqxtjqx3.K(x)(x)K(x)=[T(x)]K(x)=0;1;2;3;¢¢¢fK(x)=kgfk·T(x)k+1gK(x)Pr(K(x)=k)=Pr(kT(x)·k+1)11=k+1qx¡kqx=kpx¡k+1px=kjqxK(x)kjqxk=0;1;2;¢¢¢.4.0x.l0xls(x)¢xl(s(x)¡s(x+¢x)),l(s(x)¡s(x+¢x))ls(x)£¢x:¹x=¡s0(x)s(x)=F0(x)1¡F(x)(1)5.tqx=1¡s(x+t)s(x)(2)tpx=s(x+t)s(x)(3)12tjuqx=s(x+t)¡s(x+u+t)s(x)(4)=tpx¡t+upx=t+uqx¡tqx=tpx¢uqx+t(5)tjqx=tj1qx=tpx¡t+1px=tpx¢qx+t(6)tpx=exp(¡Zx+tx¹ydy)=exp(¡Zt0¹x+sds)(7)s(x)=xp0=exp(¡Zt0¹sds)(8)(2)tqx=Pr(T(x)·t)=PrfxX·x+tjXxg=Pr(xX·x+t)P(Xx)=s(x)¡s(x+t)s(x)=1¡s(x+t)s(x)XT(x)FX(x)=1¡s(x)=1¡exp(¡Zx0¹sds)(9)13fX(x)=¡s0(x)=¹x¢exp(¡Zx0¹sds)=xp0¢¹x(10)FT(t)=1¡tpx=1¡exp(¡Zt0¹x+sds)(11)fT(t)=¡ddx(tpx)=tpx¢¹x+t(12)¹x=11+x;x¸0:1X;T(x);2Pr(10X·30);35j5q20.(1)FX(x)=1¡exp(¡Rx011+sds)=1¡exp(¡ln(1+x))=x1+x;x¸0.FT(t)=1¡exp(¡Rt011+x+sds)=1¡exp(¡ln(1+t+x)+ln(1+x))=t1+x+t;x;t¸0.(2)Pr(10X·30)=FX(30)¡FX(10)=301+30¡101+10=0:05865:(3)5j5q20=s(25)¡s(30)s(20)=F(30)¡F(25)1¡F(20)=0:13027141.3xqx;lx;dx;Cx;Mx1l0:0l0,s(x).lx:l0xlx=l0s(x)Ij,jxIj=1Ij=0E(Ij)=1¢s(x)+0¢s(x)=s(x)lx=E(I1+I2+¢¢¢+Il0)=l0Xj=1s(x)=l0s(x)15.ndx:l0x¡¡x+nndx=lx¡lx+ndx=1dx=lx¡lx+1oex;ex:oex=E[T(x)];ex=E[K(x)]2.tqx=lx¡lx+tlx=tdxlx(13)tpx=lx+tlx(14)t=1qx=lx¡lx+1lx=dxlx,px=lx+1lx.16ex=E[K(x)]=1Xk=0kPr(K=k)=1Xk=0k(kpx¡k+1px)=1Xk=0k+1px=lx+1+lx+2+¢¢¢lxl(x)=l0s(x)=l0exp(¡Zx0utdt)(15)tdx=lx¡lx+t=Zx+txusds(16)3.tUDDCFMBalducci1.UDD()s(x+t)=(1¡t)s(x)+ts(x+1)0·t·1(17)17tqx=t¢qxtpx=1¡tqx=1¡tqx0·t·1lx¡lx+t=t¢dx0·t·1[x;x+1)t[x;x+1)[x;x+1)lx+t=lx¡t¢dxux+t=qx1¡tqxtpxux+t=qxoex=ex+122.CFM()¹x+t=¹x;0·t·118s(x+t)=s(x)exp(¡¹xt).lx+t=lxe¡¹xt=lx(px)ttqx=1¡(px)t¹x=¡ln(px)3.Balducci1s(x+t)=1¡ts(x)¡ts(x+1)0·t·1lx+t=lxlx+1tlx+(1¡t)lx+1tqx=tqx1¡(1¡t)qx¹x+t=qx1¡(1¡t)qx4.19q[x]+n,xnn5-10201s(x)=1¡px10;0·x·100101636216362l(x)=200(100¡x),0·x·10012jqx,10jpx2¹x3l40=7746;l41=7681,¹40:25.4,s(59:12),¹59:12,0:12p59,0:38q59:12211.4i(1+i(m)m)m=1+ivv=1=(1+i)d=1¡vÄa¹nji=1+v+v2+¢¢¢vn¡1=1¡vndÄS¹nji=(1+i)n+(1+i)n¡1+¢¢¢+(1+i)=(1+i)n¡1d2222.1:nExxn1nEx=vn:npx=vn£lx+nlx(18)50101000i=0:06,1000:10E50=1000:v10l60l50=513:42100010558.35,45.23²t²t((18))²2.2:Äax:¹nj,ax:¹njx,1n.Äax:¹nj=lx+lx+1v+¢¢¢+lx+n¡1vn¡1lx(19)ax:¹nj=lx+1v+¢¢¢+lx+nvnlx(20):x,1,.Äax=lx+lx+1v+¢¢¢lx=Pw¡xk=0lx+k¢vklx(21)24ax=lx+1v+¢¢¢lx=Pw¡xk=1lx+k¢vklx(22):x,1,mnmjÄax:¹nj=lx+mvm+lx+m+1vm+1+¢¢¢+lx+m+n¡1vm+n¡1lx(23)mjax:¹nj=lx+m+1vm+1+¢¢¢+lx+m+nvm+nlx(24)mjÄax=lx+mvm+lx+m+1vm+1+¢¢¢lx(25)mjax=lx+m+1vm+1+lx+m+2vm+2+¢¢¢lx(26)25Dx=vxlxNx=Dx+Dx+1+¢¢¢=P1t=0Dx+tSx=Nx+Nx+1+¢¢¢=P1t=0Nx+t,Äax:¹nj=lx+lx+1v+¢¢¢+lx+n¡1vn¡1lx=lxvx+lx+1vx+1+¢¢¢+lx+n¡1vx+n¡1lxvx=Dx+Dx+1+¢¢¢+Dx+n¡1Dx=Nx¡Nx+nDxax:¹nj=Nx+1¡Nx+n+1DxÄax=NxDx;ax=Nx+1DxmjÄax:¹nj=Nx+m¡Nx+m+nDxDx;Nx,26.Äax=ax+1;Äax:¹nj=ax:¹n¡1j+1;Äax=Äax:¹nj+njÄax(UDD),Äax+u(x,0u1,Äax+u=1¡u1¡uqxÄax+u(1¡qx)1¡uqxÄax+1(27)qx,(27)Äax+u¼(1¡u)Äax+uÄax+12.3x1,m27,(UDD),a(m)x=1m¢Dx+1m+Dx+2m+¢¢¢Dx=ax+m¡12mÄa(m)x=a(m)x+1m=Äax¡m¡12m2.4,¹ax1,t[t;t+dt)vtdt¢tpx,¹ax=Z10vt¢tpxdt(28)UDD,¹axa(m)xm!1¹ax=limm!1a(m)x=lim(ax+m¡12m)=ax+12282.5(IÄa)xx1,1.(IÄa)x=1+2v¢px+3v2¢2px+¢¢¢=1+1Xk=1(k+1)vk¢kpx=1+1Xk=0kjax+ax=SxDx(Ia)x=Sx+1Dx2.6,,,,,.,,,.292.7x,1,i,nX:(1+i)n=X¢npxX=lx(1+i)nlx+nnÄSx:¹nj=DxDx+n+Dx+1Dx+n+¢¢¢+Dx+n¡1Dx+n=Nx¡Nx+nDx+n(29)2.8,,,.30:,,:++,,,,.:(,),,(,,):B(t)=F[C(t);V(t);P(t)]31,B(t),C(t),V(t),P(t).q(w)x;q(d)x;q(i)x;q(r)xxx+1,l(¿)aaal(¿)xxl(¿)x+1=l(¿)x[1¡(q(w)x+q(d)x+q(i)x+q(r)x)]=l(¿)xp(¿)xkp(¿)x=l(¿)x+kl(¿)x323333.1x1x+1RZxZx¢lx=R¢dx¢vZx=R¢dxvlx(30):Cx=vx+1dx,Mx=Cx+Cx+1+¢¢¢,Rx=Mx+Mx+1+¢¢¢,Zx=R¢Cx=Dx,(x)K(x)=k,K+1bk+1,vk+1,Z,Z=34bk+1vk+1,K(x),:E(Z)=1Xk=0vk+1bk+1¢kjqx,,Z=btvt,T(x),:E(Z)=Z10vtbt¢tpx¹x+tdt3