ACTSC232 Lecture Notes - Lecture 2: Term Life Insurance, Life Insurance, Novella

• School: University of Waterloo
• Department: Actuarial Science
• Course: ACTSC232
• Professor: Diana K Skrzydlo

Document Summary

Summaryoflecturenotes-actsc232,winter2010part2-lifebene ts2. 1lifeinsuranceson(x)alifeinsuranceon(x)isacontractorpolicyissuedbyaninsurertoalifecurrentlyagedx. theinsurerwillpaybene tstothebene ciariesof(x)inthefuture. thepaymenttimesofthebene tsarecontingentonthedeathtimeof(x). suchbene tsarecalleddeathbene tsorlifebene ts. generallyspeaking,alifeinsuranceon(x)iscalledacontinuouslifeinsuranceifbene tsarepayableatthemomentofthedeathof(x). alifeinsuranceiscalledadiscretelifeinsuranceifbene tsarepayableattheendofthedeathyearof(x). reviewoftheexpectationsofthefunctionsoftxandkx:forafunctiong,e[g(tx)]=z 0g(t)fx(t)dt=z 0g(t)tpx x+tdtande[g(kx)]= xk=0g(k)pr{kx=k}= xk=0g(k)kpxqx+k. reviewofpresentvalues:letvtdenotepresentvalue(pv)attime0of1(dollarorunit)tobepaidattimetandvtiscalleddiscountfunction. iftheforceofinterest t= isaconstant,thenvt=vt=e t=(11+i)t,wherev=11+i=e and1+i=e . unlessstatedotherwise,weassumethat t= isaconstantorvt=vt=e t. (a)ageneralcontinuouslifeinsuranceon(x)paysdeathbene tsatthedeathtimeof(x). denotethebene tbybtiftx=tor(x)diesattimet,t>0. letzdenotethepresentvalueattime0oratagexofthebene tstobepaidbytheinsurance. thenz=btxvtx=btxe txandzisarandomvariable,wheretxisthedeathtimeof(x). theexpectationormeanofthepresentvaluezise[z]=e(cid:2)btxe tx(cid:3)=z 0bte tfx(t)dt=z 0bte ttpx x+tdt1 shared via c ourse h ero. co m. T his study resource w as https://www. coursehero. com/file/6180479/part-2-notes-232-2010w/ whichiscalledtheactuarialpresentvalue(apv)oftheinsurance,ortheexpectedpresentvalue(epv)oftheinsurance,orthepurepremiumoftheinsurance,orthenetpremiumoftheinsurance,orthesinglebene tpremiumoftheinsurance. thesecondmomentofthepresentvaluezise[z2]=e(cid:2)b2txe 2 tx(cid:3)=z 0b2te 2 tfx(t)dt=z 0b2te 2 ttpx x+tdtandvar[z]=e[z2] (e[z])2. thedistributionfunctionofzisdenotedbyfz(z)=pr{z z}. thedistributionfunctionmaybecontinuous,ordiscrete,ormixed. (b)ageneraldiscretelifeinsuranceson(x)paysdeathbene tsattheendofthedeathyearof(x). denotethedeathbene tbybk+1ifkx=kor(x)diesinyeark+1,k=0,1,2,letzdenotethepresentvalueattime0oratagexofthebene ts. then,z=bkx+1vkx+1=bkx+1e (kx+1)theapvorepvoftheinsuranceise[z]=e[bkx+1vkx+1]= xk=0bk+1vk+1pr{kx=k}= xk=0bk+1vk+1kpxqx+k= xk=0bk+1e (k+1) kpxqx+k. thisexpectationisalsocalledthepurepremiumoftheinsurance,orthenetpre-miumoftheinsurance,orthesinglebene tpremiumoftheinsurance. thesecondmomentofthepresentvalueisgivenbye[z2]=e[b2kx+1v2(kx+1)]= xk=0b2k+1v2(k+1)kpxqx+k= xk=0b2k+1e 2(k+1) kpxqx+k. 2. 2levelbene tlifeinsurancesalifeinsuranceon(x)iscalledalevelbene tlifeinsuranceifbene tsareconstantandindependentofthepaymenttimesofthebene ts. 2 shared via c ourse h ero. co m. T his study resource w as https://www. coursehero. com/file/6180479/part-2-notes-232-2010w/ (a)acontinuouswholelifeinsuranceof1on(x)pays1atthemomentofdeathof(x). thepvofthebene tisz=vtxandtheapvoftheinsuranceisdenotedby ax=e[vtx]=z 0vtfx(t)dt=z 0e ttpx x+tdt. thesecondmomentofzisdenotedby2 ax=e[v2tx]=z 0v2tfx(t)dt=z 0e 2 ttpx x+tdtandvar[z]=2 ax ( ax)2. ifthemortalityforceof(x)followstheconstantforcelawor x= forallx>0,thenfx(t)=tpx x+t= e tfor0n. =vtxi(tx n),3 shared via c ourse h ero. co m.