Basic Life Insurance Mathematics

1 BasicLifeInsuranceMathematicsRagnarNorbe rgVersion:September 2002 Contents1 .. tcontracts:Surplusandbonus..172 Payment nitionsandrelationships.. loans..253 .. laws..374 Insuranceof a Insurance .. equivalence.. reserves.. 'sdi erentialequation.. distributions.. of view..575 singlelifeinsurancepolicy..626 of survivors..631 CONTENTS27 Markov chainsin a stochasticprocess.. chain.. revisited.. of present values.. Markov chaininterestmodel.. model.. erentialequationsformoments of present values.. onMarkov chains.. lives .. positive dependence.. values.. Markov chainmodelfortwo lives.. chains.. fertility analysis.. 1068 Probability distributionsof present .. probability distributionsof present valuesby ele-mentarymethods.

2 Multistatepolicy.. erentialequationsforstatewisedistributio ns.. 1169 .. nitionsof reserves andstatement of somerelation-shipsbetweenthem.. payment streamsappearingin lifeandpensioninsurance.. chainmodel.. in theMarkov chainmodel.. 13910 Safety .. it emerges.. prognoses.. 16311 Statisticalinferencein theMarkov mortality law fromfullyobserved lifelengths.. theMarkov model.. denceregions.. denceintervals.. intensities.. thecensoringscheme.. 18312 Heterogeneity heterogeneity { a two-stagemodel.. 18713 groupinsurance.. proportionalhazardmodelforcompleteindivi dualpolicyandclaimrecords.. uctuationreserve .. parameters.. 19714 Hattendor .. theorem.. lifeinsurance.. 20515 Financialmathematicsin Insurance .}

3 Markov chain nancialmarket - Introduction.. chainmarket.. derivatives in a completemarket .. incompletemarkets.. :How much canbe hedged?.. nance.. propertiesof theVandermondematrix.. nance.. 240A Calculus4B Indicatorfunctions9C Distributionof thenumber of occurringevents12 CONTENTS4D Asymptoticresultsfromstatistics15E TheG82 Mmortality table17F Exercises1G Solutionsto (55)(letus callhimso)decidesto investmoneyto securehimselfeconomicallyin rstideathatoccursto himis to deposita capitalofS0= 1( )ona savingsaccount today anddraw theentireamount withearnedcompoundinterestin 15 years, bearsinterestat ratei= 0:045(4:5%)per oneyearthecapitalwillincreasetoS1=S0+S0i =S0(1 +i), in two yearsit willincreasetoS=S1+S1i=S0(1 +i)2, andso onuntil in 15 yearsit willhave accumulatedtoS15=S0(1 +i)15= 1:04515= 1:935.

4 ( )Thissimplecalculationtakes noaccount of thefactthat(55)willdiesooneror later,maybe (orhedislikestheoneshehas)so thatin theevent of willsurvive to ,therelevant prospectsof thecontractare:{ withprobability (55)survives to 70 andwillthenpossessS15;{ withprobability (55) thisperspective theexpectedamount at (55)'sdisposalafter15yearsis0:75S15:( ) thingsover,(55)seekstomake thefollowingmutualarrangement with(55) and(55) , whoarealso55yearsoldandarein exactlythesamesituationas (55).Each of thethreedepositsS0= 1 onthesavingsaccount, andthosewhosurvive to 70,if any, thisschemearegiven in ,where+ and signifysurvivalanddeath, respectively,L70is thenumber of survivorsat age70,and5 CHAPTER1. :Possibleoutcomesof a savingsschemewiththreeparticipants.(55)( 55) (55) L703S15=L70 Probability+++3S150:75 0:75 0:75 = 0:422++ 21:5S150:75 0:75 0:25 = 0:141+ +21:5S150:75 0:25 0:75 = 0:141+ 13S150:75 0:25 0:25 = 0:047 ++21:5S150:25 0:75 0:75 = 0:141 + 13S150:25 0:75 0:25 = 0:047 +13S150:25 0:25 0:75 = 0:047 0unde ned0:25 0:25 0:25 = 0:0163S15=L70is theamount at disposalper survivor (unde nedifL70= 0).}}

5 Therearenow thefollowingpossibilities:{ withprobability (55)survives to 70 togetherwith(55) and(55) andwillthenpossessS15;{ withprobability 2 0:141= 0:282(55)survives to 70togetherwithonemoresurvivor andwillthenpossess1:5S15;{ withprobability (55)survives to 70whileboth(55) and(55) die(may theyrestin peace)andhe willcashthetotalsavings3S15;{ withprobability (55) superiortotheonedescribedin ParagraphA,withseparateindividualsavings contracts:If (55)survives to 70,which is theonlyscenarioofinterestto him,hewillcashnolessthantheamountS15hewo uldcashundertheindividualscheme,andit is likelythathe ( ),theexpectedamount at (55)'sdisposalafter15yearsis now0:422 S15+ 0:282 1:5 S15+ 0:047 3S15= 0:985S15:Thepoint is thatunderthepresent schemethesavingsof thosewhodiearebequeathedto thetotalsavingsareretainedforthegroupsot hatnothingis lefttoothersunlesstheunlikelythinghappen sthatthewholegroupgoes extinctwithinthetermof essentiallythekindof solidarity thatunitesthemembersof a of viewof thegroupas a whole,theprobability thatallthreeparticipantswilldiebefore70i s ,which shouldbe comparedto theprobability0.}}}}

6 25 that(55) thesuccessof themutualfundideaalreadyonthesmallscaleo f threeparticipants,(55)startsto playwiththeideaof extendingit to a largenumber of assumethata totalnumber ofL55persons,whoarein exactlythesamesituationas(55),agreeto joina schemesimilarto theonedescribed INTRODUCTION7totalsavingsafter15 yearsamount toL55S15, which yieldsanindividualshareequaltoL55S15L70( )to each of theL70survivorsifL70>0. Bytheso-calledlaw of largenumbers,theproportionof survivorsL70=L55tendsto theindividualsurvival probability0:75asthenumber of participantsL55tendstoin nity. Therefore,asthenumber of participants increases,theindividualshareper survivor tendsto10:75S15;( )andin thelimit(55)is facedwiththefollowingsituation:{ withprobability to 70andgets10:75S15;{ withprobability at (55)'sdisposalafter15yearsis0:7510:75S15 =S15;( )thesameas( ).}}

7 Thus,thebequestmechanismof themutualschemehasraised(55)'sexpectatio nsof futurepensiontowhattheywouldbe withtheindividualsavingscontractif whatwe couldexpectsince,in an in nitelylargescheme,somewillsurvive to 70 theschemeandwillbe redistributedamongitsmembersby thelotterymechanismof ,andthat( )thusstabilizesat ( ),is preciselywhatis meant by sayingthat\insuranceriskisdiversi able".Theriskcanbe eliminatedby increasingthesizeof widelyheldto be , thepresent mustbe admitted,however,thatactuariesusetocheer themselves upwithjokes like: \Whatis thedi erencebetweenanEnglishanda Sicilianactuary?Well,theEnglishactuaryca npredictfairlypreciselyhow many ,theSicilianactuarycanpredicthow many Sicilianswilldienextyear,buthecantellthe irnamesaswell."TheEnglishactuaryis de nitelythemoretypicalrepresentative of theactuarialprofessionsincehe takes a purelystatisticalviewof mortality.

8 Stillheis ableto analyzeinsuranceproblemsadequatelysincew hatinsuranceis essen-tiallyabout,is to basedontheparadigmof thelargescheme(diversi cation) INTRODUCTION8serves tensandsomeeven hundredsof thousandsof customers,su cientlymanyto ensurethatthesurvival ratesarestableas assumedin statisticalinvestigationstheactuaryconst ructsa so-calleddecre-mentseries, which takes as it startingpoint a largenumber`0of new-bornand,foreach agex= 1;2;:::, speci esthenumber of survivors,` :Excerptfromthemortality tableG82Mx:0255060708090`x:10000098 083911198233965 02437 1679 783dx:581196171 2752 3453 1111 845qx:. :. anexcerptof thetableusedby Danishinsurerstodescribethemortality of in thetablelistssomeentriesof thedecrement allnew-bornwillcelebratetheir70thanniver sary. Thenumber of survivorsdecreaseswithage:`x `x+1:Thedi erencedx=`x `x+1is thenumber of deathsat agex(moreprecisely, betweenagexandagex+ 1).

9 Thesenumbersareshownin thethirdrow of is seenthatthenumberof cannotbe concludedthat80is the\mostdangerousage".Theactuarymeasures themortality atany agexby theone-year mortalityrateqx=dx`x;which tellshow bigproportionof thosewhosurvive to ,shownin thefourthrow of thetable, instance, the80yearsoldwilldiewithina year, showstheoneyearsurvival ratespx=`x+1`x= 1 qx:We shallpresent sometypicalformsof productsthataninsurancecompanycano erto (55)andseehow theycomparewiththecorrespondingarrange-m ents,if any, that(55)canmake ndhisperfectmatches(55) , (55) ,..,ourhero(55)abandonstheideaof creatinga yeart= 1;2;:::. Thus,a unitS0= 1 depositedat time0 willaccumulatewithcompoundinterestasfoll ows:In oneyearthecapitalincreasestoS1=S0+S0i1= 1 +i1, in twoyearsit increasestoS2=S1+S1i2= (1 +i1)(1+i2), andintyearsit increasestoSt= (1 +i1) (1 +it);( ) , thepresent valueat time0 of a unitwithdrawninjyearsisSj 1=1Sj;( )calledthej-yeardiscountfactorsinceit is whatthebankwouldpay you at time0 if yousellto it (discount) a default-freeclaimof 1 at , thevalueat timetof a unitdepositedat timej < tis(1 +ij+1) (1 +it) =StSj;calledtheaccumulationfactorover thetimeperiod fromjtot, andthevalueat timetof a unitwithdrawnat timej > tis1(1 +it+1) (1 +ij)=StSj.

10 Thediscount factorover thetimeperiod ,thevalueat timetof a unitdueat timejisStSj 1, anac-cumulationfactorifj < tanda discount factorifj > t(andof course1 ifj=t).From( )it followsthatSt=St 1(1 +it), henceit=St St 1St 1;which expressestheinterestratein yeartas therelative increaseof thebalancein generalsavingscontractovernyearsspeci esthatat each timet= 0;:::;n(55)is to depositanamountct(contribution) andwithdraw anamountbt(bene t). Thenetamount of depositlesswithdrawal attimetisct bt. At any timetthecashbalanceof theaccount, henceforthalsoCHAPTER1. INTRODUCTION10calledtheretrospectiverese rve, is thetotalof past(includingpresent) depositslesswithdrawalscompoundedwithint erest,Ut=SttXj=0Sj 1(cj bj):( )It developsin accordancewiththe\forward"recursive schemeUt=Ut 1(1 +it) +ct bt;( )t= 1;2;:::;n, commencingfromU0=c0 b0:Each year(55)willreceive fromthebanka statement of account withthecalculation( ),showinghow thecurrent balanceemergesfromthepreviousbalance,the interestearnedmeanwhile,andthecurrent movement (depositlesswithdrawal).