Actuarial Mathematics and Life-Table Statistics

1 ActuarialMathematicsand life -TableStatisticsEricV. SludMathematicsDepartmentUniversity of Maryland,CollegeParkc 2001c 2001 EricV. SludStatisticsProgramMathematicsDepartme ntUniversity of MarylandCollegePark, .. vi1 Basicsof Probability & .. Interest.. Streams.. Set 1.. Examples.. 212 Interest& Forceof Theoryof Interest.. ActuarialNotation.. MortgageRe nancing.. MortgageRe nancing.. & Zero-coupon Bonds.. Mortality & AnalyticalModels.. Forcesof Mortality .. Set 2.. Examples.. 583 Probability & life Mortality .. IntegerAges.. of LargeNumbers .. ,Bounds& Approximations.. life TableData.. DiscreteRandomVariables.. ManipulatingExpectations.. Set 3.. Examples.. 934 ExpectedPresent Valuesof Values.. of Insurance& life Annuity Contracts.

2 SinglePremiums.. Net SinglePremiums.. Valuesform= 1 .. ResidualLife.. life Expectancies.. Set 4.. Examples.. 1215 Net SinglePremiums.. Integer& FractionalAgesat SinglePremiumFormulas| Case(i).. SinglePremiumFormulas| Case(ii).. Case(i).. Level Premiums.. tsInvolvingFractionalPremiums.. Set 5.. Examples.. 1456 Commutation& CommutationFunctions.. tCommutationFormulas.. Mortality .. & CashValueof a SinglePolicy.. Formulas& Identities.. Endowment Reserves .. underConstant Forceof Mortality .. underIncreasingForceof Mortality .. Calculationof Reserves .. Tables& Insurance.. Set 6.. CommutationColumns.. Paid-upInsurance.. 1717 IndicatorNotation.. Varianceof ResidualLife.. :Large-timeLimitof (t; x) .. Set 7.. Problems.

3 1798 .. Estimationfor Exponential Data.. AgeSpeci cForceof Mortality .. RandomEntry & CensoringTimes.. FunctionEstimator.. Set 8.. Examples.. 195 CONTENTSv9 RiskModels& .. Tables.. Set 9.. Examples.. 20410 MultipleDecrement Tables.. of Age.. Mortality Constant withinYear of Age.. cDeathRateEstimators.. Tablesand Net Hazardsof Mortality .. cLife InsurancePremiums.. Set 10.. Examples.. 21411 Central LimitTheorem& PortfolioRisks21513 Bibliography217 Solutions& is a courseof lectureson the mathematicsof lecturesis as far as possibleto deduceinterestingmaterialoncontingent present valuesand life tablesdirectlyfromcalculusand common-sensenotions,illustratedthroughwo rd InterestTheoryand Probability relatedto life tablesare treatedas wonderfulconcreteappli-cationsof the backgroundbeyond a thirdsemesterof calculus,but the prerequisitecalculuscoursesmust have beensolidlyunderstood.

4 It is a truismof pre-actuarialadvisingthatstudents whohave not donereallywell in and digestedthe calculusought not to is not assumedthatthe student has seena formalintroductionto prob-ability. Notionsof relative frequencyandaverageare introduced rstwithreferenceto the ensemble of a cohortlife- table ,the underlyingformalrandomexperiment beingrandomselectionfromthe cohortlife-tablepopulation(or,in the contextof probabilitiesand expectationsfor `lives agedx', fromthesubsetoflxmembers of the populationwhosurvive to agex). Thecal-culationof expectationsof functionsof a time-to-deathrandomvariablesisrootedon the one handin the concretenotionof life -tableaverage,which isthenapproximatedby suitableidealizedfailuredensitiesand ,in discussingBinomialrandomvariablesand the Law of LargeNumbers, thecombinatorialand probabilisticinterpretationof binomialcoe cients are de-rived fromthe BinomialTheorem,which the student the is assumedto knowas a topicin calculus(Taylorseriesidenti cationof coe cients of a poly-nomial.)

5 Thegeneralnotionsof expectationand probability are introduced,but for examplethe Law of LargeNumbers for binomialvariablesis treated(rigorously)as a topicinvolvingcalculusinequalitiesand summationof allows introductionof the numericallyand conceptuallyusefullarge-deviationinequal itiesfor binomialrandomvariablesto explainjusthow unlikely it is for binomial( , Life-Table )counts to deviatemuchpercentage-wisefromexpectatio nswhenthe underlyingpopulationof trialsis alsonot assumedto have worked previouslywiththe The-ory of Theoryof Interestas a mathematicalproblem-topic,which is ratherunlike whatis donein typical typicalInterestproblems| such as the exerciseson mortgagere- nancingand present valuesof variouspayo schemes| into correctformatfor numericalanswers is oftennot easyeven for good theselecturesis to reach | by a conceptualroute|mathematicaltopicsin LifeContingencies,PremiumCalculationandD e-mography not usuallyseenuntil ratherlate in the trajectoryof an approach can allow undergraduateswithsolidpreparationin calculus(notnecessarilymathematicsor statisticsma-jors)

6 To exploretheirpossibleinterestsin businessand the majority of such students | whowill choose someotherav-enue, fromeconomicsto operationsresearch to Statistics ,for the exerciseoftheirquantitative talents | to know somethingconcreteand mathematicallycoherent about the topicsand ideasactuallyusefulin secondarygoalof the lectureshas been to introducevariedtopicsofappliedmathematic sas partof a reasoneddevelopment of ideasrelatedtosurvival a result,materialis includedon statisticsof biomedicalstudiesandon reliability which wouldnot ordinarily ndits way into furtherresultis thatmathematicaltopics,fromdi eren-tial equationsto maximum likelihood estimatorsbasedon complexlife-tabledata,which seldom t coherently into undergraduateprogramsof study, are`verticallyintegrated'into a materialin theselecturesis presented systematically, it is notseparatedby chaptersinto uni edtopicssuch as InterestTheory, ProbabilityTheory, PremiumCalculation, introductorymaterialfromprobability and interesttheoryare interleaved, and later,variousmathemat-ical ideasare introducedas neededto advancethe book atthislevel can claimto be fullyself-contained,but every attempthas beenmadeto developthe mathematicsto t the actuarialapplicationsas the mainbody of each chapteris primarily`theoretical'.

7 At the end of each chapteris an ExerciseSet and a shortsectionof WorkedExamplesto illustratethe kindsof word problemswhich can be solved bythe techniquesof the Examplessectionsshow howthe ideasand formulaswork smoothlytogether,and theyhighlight the mostimportant and frequently Probability and theTheoryof InterestThe rst lecturessupplysomebackgroundon elementaryProbability Theoryand basicTheoryof readerwhohas not previouslystudiedthesesubjectsmay get a briefoverviewhere,but will likely want to supplementthis Chapterwithreadingin any of a number of calculus-basedintroductionsto probability and Statistics ,such as Larson(1982),Larsenand Marx(1985),or Hoggand Tanis(1997)and the basicsof the Theoryof Interestas coveredin the textof Kellison(1970)or Chapter1 of Gerber (1997). , Lifetimes,and ExpectationIn thecohortlife-tablemodel, imaginea numberl0of individualsbornsimultaneouslyandfollowed until death,resultingin datadx; lxfor eachagex= 0;1;2; : : :, wherelx= number of lives agedx( at birthdayx)anddx=lx lx+1= number dyingbetween agesx; x+ 1 Now, allowingthe age-variablexto take all realvalues,not justwholenumbers, treatS(x) =lx=l0as a piecewisecontinuouslydi erentiablenon-12 CHAPTER1.

8 BASICSOF PROBABILITY& INTEREST increasingfunctioncalledthe \survivor" or \survival" allpositive realx; S(x) S(x+t) is the fractionof the initialcohortwhichfailsbetween timexandx+t, andS(x) S(x+t)S(x)=lx lx+tlxdenotesthe fractionof thosealive at exactagexwhofail beforex+ :whatdo probabilitieshave to do withthe life tableand survival function?To answer this,we rstintroduceprobability as simplya relative fre-quency, usingnumbers froma cohortlife-tablelike thatof the accompanyingIllustrative responseto a probability question,we supplythefractionof the relevant life -tablepopulation,to obtainidentitieslikeP r(lifeaged29 diesbetween exactages35 and 41 or between 52 and 60 )=S(35) S(41)+S(52) S(60)=n(l35 l41) + (l52 l60) conventionis thatalife aged 29is one of the cohortsurvivingtothe thatall of the lifetimescoveredby the life tableareunderstood to be governedby an identical\mechanism"of failure,and thatany probability questionabout a singlelifetimeis reallya questionconcerningthe fractionof thoselives about which the questionis asked ( ,thosealiveat agex)

9 Whoselifetimeswill satisfythe statedproperty ( ,die eitherbetween 35 and41 or between 52 and60).This\frequentist"notionofprobabil ity of an event as the relative frequencywithwhich the event occursin a largepopulationof (independent) identicalunitsis associatedwiththephrase\law of largenumbers",which will now, remarkonlythatthe life tablepopulationshouldbe largefor the ideaspresented sofar to make good for an illustrationof a :see any basicprobabilitytextbook, suchas Larson(1982),LarsenandMarx(1985),or Hogg andTanis(1997)for formalde nitionsof thenotionsof samplespace, event,probability,and are necessaryto understandthe discussionso far are :Illustrative Life-Table ,simulatedto resemble realisticUS (Male) detailsof simulation,see BASICSOF PROBABILITY& INTEREST mattersof commonsensewhenappliedto relative frequencybut requireformalaxiomswhenusedmoregenerally .

10 Probabilitiesare numbers between 0 and 1 assignedto subsetsof theentire rangeof possibleoutcomes(in the examples,subsetsof the in-terval of possiblehumanlifetimesmeasuredin years). TheprobabilityP(A[B) of the unionA[Bof disjoint ( ,nonoverlapping)setsAandBis necessarilythe sumof the separateprobabilitiesP(A) andP(B). Whenprobabilitiesare requestedwithreferenceto a smalleruniverse ofpossibleoutcomes,such asB=livesaged 29, ratherthanall membersof a cohortpopulation,the resultingconditionalprobabilitiesof eventsAare writtenP(AjB) and calculatedasP(A\B)=P(B), whereA\Bdenotestheintersectionoroverlapo f the two eventsA; B. Two eventsA; Bare de nedto beindependentwhenP(A\B) =P(A) P(B) or | equivalently, as longasP(B)>0 | the conditionalprobabilityP(AjB) expressingthe probability ofAifBwere knownto have occurred,is the sameas the (unconditional)probabilityP(A).]]