Multiple Life Models

1 Multiple Life ModelsLecture: Weeks 9-10 lecture : Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez1 / 38 Chapter summaryChapter summaryApproaches to studying Multiple life Models :define Multiple statestraditional approach (use joint random variables)Statuses:joint life statuslast-survivor statusInsurances and annuities involving Multiple livesevaluation using special mortality lawsSimple reversionary annuitiesContingent probability functionsDependent lifetime modelsChapter 9 (Dickson et al.) lecture : Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez2 / 38 Approachesmultiple statesStates in a joint life and last survivor model 02x+t:y+t 13x+t 01x+t:y+t 23y+txaliveyalive(0)xaliveydead(1)xdeady alive(2)xdeadydead(3) lecture : Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez3 / 38 Approachesjoint future lifetimesJoint distribution of future lifetimesConsider the case of two lives currently agesxandywith respectivefuture cumulative dist.

2 Function:FTxTy(s,t) =Pr[Tx s,Ty t]independence:FTxTy(s,t) =Pr[Tx s] Pr[Ty t] =Fx(s) Fy(t)Joint density function:fTxTy(s,t) = 2 FTxTy(s,t) s tindependence:fTxTy(s,t) =fx(s) fy(t)Joint survival dist. function:STxTy(s,t) =Pr[Tx> s,Ty> t]independence:STxTy(s,t) =Pr[Tx> s] Pr[Ty> t] =Sx(s) Sy(t) lecture : Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez4 / 38 ApproachesillustrationIllustrative example 1 Consider the joint density expressed byfTxTy(s,t) =164(s+t),for0< s <4,0< t < thatTxandTyare not the covariance the probability (x)outlives(y)by at least one to be discussed in.

3 Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez5 / 38 Statusesjoint life statusThe joint life statusThis is a status that survives so long as all members are alive, andtherefore fails upon the first :(xy)for two lives(x)and(y)For two lives:Txy= min(Tx,Ty)Cumulative distribution function:FTxy(t) =qt xy=Pr[min(Tx,Ty) t]= 1 Pr[min(Tx,Ty)> t]= 1 Pr[Tx> t,Ty> t]= 1 STxTy(t,t)= 1 pt xywherept xy=Pr[Tx> t,Ty> t] =STxy(t)is the probability that bothlives(x)and(y)survive : Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez6 / 38 Statusesjoint life statusThe case of independenceAlternative expression for the distribution function:FTxy(t) =Fx(t) +Fy(t) FTxTy(t,t)In the case whereTxandTyare independent:pt xy=Pr[Tx> t,Ty> t]=Pr[Tx> t] Pr[Ty> t]=pt x pt yandqt xy=qt x+qt y qt x qt yRemember this (even in the case of independence):qt xy6=qt x qt yLecture.

4 Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez7 / 38 Statuseslast-survivor statusThe last-survivor statusThis is a status that survives so long as there is at least one member alive,and therefore fails upon the last :(xy)For two lives:Txy= max(Tx,Ty)General relationship amongTxy,Txy,Tx, andTy:Txy+Txy=Tx+TyTxy Txy=Tx TyaTxy+aTxy=aTx+aTyfor any constanta > each outcome, note thatTxyis equal eitherTxorTy, andtherefore,Txyequals the : Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez8 / 38 StatusesdistributionDistribution ofTxyRecall method of inclusion-exclusion of probability :Pr[A B] +Pr[A B] =Pr[A] +Pr[B].

5 Choose eventsA={Tx t}andB={Ty t}so thatA B={Txy t}andA B={Txy t}.This leads us to the following useful relationships:FTxy(t) +FTxy(t) =Fx(t) +Fy(t)STxy(t) +STxy(t) =Sx(t) +Sy(t)pt xy+ptxy=pt x+pt yfTxy(t) +fTxy(t) =fx(t) +fy(t)These relationships lead us to finding distributions ofTxy, (t) =Fx(t) +Fy(t) FTxy(t) =FTxTy(t,t)which is obvious fromFTxy(t) =Pr[Tx t Ty t]. lecture : Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez9 / 38 StatusesdistributionInterpretation of probabilitiesNote that:pt xyis the probability that both lives(x)and(y)will be alive the probability that at least one of lives(x)and(y)will be contrast:qt xyis the probability that at least one of lives(x)and(y)will be the probability that both lives(x)and(y)will be dead : Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez10 / 38 StatusesillustrationIllustrative example 2 For independent lives(x)and(y), you are given.

6 Qx= ,andqx+1= +1= are assumed to be uniformly distributed over each year of and interpret the following to be discussed in : Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez11 / 38 Force of mortalityjoint lifeForce of mortality ofTxyDefine the force of mortality (similar manner to any random variable): x+t:y+t=fTxy(t)1 FTxy(t)=fTxy(t)STxy(t)=fTxy(t)pt can then write the density ofTxyasfTxy(t) =pt xy x+t:y+tIn the case of independence, we have: x+t:y+t=pt x pt y( x+t+ y+t)pt x pt y= x+t+ y+ force of mortality of the joint life status is the sum of theindividuals force of mortality, when lives are : Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez12 / 38 Force of mortalitylast-survivorForce of mortality forTxyThe force of mortality forTxyis defined as x+t:y+t=fTxy(t)1 FTxy(t)=fTxy(t)STxy(t)=fx(t) +fy(t) fTxy(t)ptxy=pt x x+t+pt y y+t pt xy x+t:y+tptxyIndeed we have the density ofTxyexpressed asfTxy(t) =ptxy x+t:y+ what this formula gives in the case of.

7 Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez13 / 38 Insurance benefitsdiscreteInsurance benefits - discreteConsider an insurance under which the benefit of $1 is paid at theEOY of ending (failure) of be any joint life or last survivor status , time at which the benefit is paid:Ku+ 1the present value (at issue) of the benefit:Z=vKu+1 APV of benefits: E[Z] =Au= k=0vk+1 Pr[Ku=k]variance: Var[Z] =A2u (Au)2 lecture : Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez14 / 38 Insurance benefitscontinuousInsurance benefits - continuousConsider an insurance under which the benefit of $1 is paidimmediately of ending (failure) of be any joint life or last survivor status , time at which the benefit is paid:Tuthe present value (at issue) of the benefit:Z=vTuAPV of benefits: E[Z] = Au= 0vt pt u u+tdtvariance: Var[Z] = A2u ( Au)2 lecture .

8 Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez15 / 38 Insurance benefitscontinuousSome illustrationsFor a joint life status(xy), consider whole life insurance providingbenefits at the first death:Axy= k=0vk+1 qk|xy= k=0vk+1 pk xy qx+k:y+k Axy= 0vt pt xy x+t:y+tdtFor a last-survivor status(xy), consider whole life insurance providingbenefits upon the last death:Axy= k=0vk+1 qk|xy= k=0vk+1 (qk|x+qk|y qk|xy) Axy= 0vt ptxy x+t:y+tdt= 0vt(pt x x+t+pt y y+t pt xy x+t:y+t)dtLecture: Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez16 / 38 Insurance benefitscontinuous- continuedUseful relationships:Axy+Axy=Ax+Ay Axy+ Axy= Ax+ AyLecture: Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez17 / 38 Annuity benefitsdiscreteAnnuity benefits - discreteConsider ann-year temporary life annuity-due on present value (at issue) of the benefit:Y={ aKu+1, Ku< n an,Ku nAPV of benefits: E[Y] = au:n= n 1k=0 ak+1 qk|u+ an pn uvariance: Var[Y] =1d2[A2u:n (Au:n)2]Other ways to write APV: au:n=n 1 k=0vk pk u=1d(1 Au:n).}

9 lecture : Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez18 / 38 Annuity benefitscontinuousAnnuity benefits - continuousConsider an annuity for which the benefit of $1 is paid each yearcontinuously for years so long as a present value (at issue) of the benefit:Y= aTuAPV of benefits: E[Y] = au= 0 at pt u u+tdt= 0vtpt udtvariance: Var[Y] =1 2[ A2u ( Au)2]Note that the identity aTu+vTu= 1provides the connectionbetween insurances and : Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez19 / 38 Annuity benefitscontinuousSome illustrationsFor joint life status(xy), consider a whole life annuity providingbenefits until the first death: axy= k=0vk pk xyand axy= 0vt pt xydtFor last survivor status(xy), consider a whole life insurance providingbenefits upon the last death: axy= k=0vk pkxyand axy= 0vt ptxydtUseful relationships: axy+ axy= ax+ ay axy+ axy= ax+ ayLecture.

10 Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez20 / 38 Annuity benefitscontinuousComparing benefits - annuitiesType of life annuitySingle lifexJoint life statusxyLast survivor statusxyWhole life a-due ax axy axyWhole life a-immediateaxaxyaxyTemporary life a-due ax:n axy:n axy:nTemporary life a-immediateax:naxy:naxy:nWhole life a-continuous ax axy axyTemporary life a-continuous ax:n axy:n axy:nLecture: Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez21 / 38 Annuity benefitscontinuousComparing benefits - insurancesType of life insuranceSingle lifexJoint life statusxyLast survivor statusxyWhole life - discreteAxAxyAxyWhole life - continuous Ax Axy AxyTerm - discreteA1x:nA1 xy8:nA1xy:nTerm - continuous A1x:n A1 xy8:n A1xy:nEndowment - discreteAx:nAxy:nAxy:nEndowment - continuous Ax:n Axy:n Axy:nPure endowmentA1x:norEn xA1xy:norEn xyA1xy:norEnxyLecture: Weeks 9-10 (STT 456) Multiple Life ModelsSpring 2015 - Valdez22 /