Basic Life Insurance Mathematics - ku

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.

2 Values.. Markov chainmodelfortwo lives.. chains.. fertility analysis.. 1068 Probability distributionsof present .. probability distributionsof present valuesby ele-mentarymethods.. 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.

3 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 .. Markov chain nancialmarket - Introduction.. chainmarket.. derivatives in a completemarket .. incompletemarkets.. :How much canbe hedged?.. nance.. propertiesof theVandermondematrix.}

4 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:( )Thissimplecalculationtakes noaccount of thefactthat(55)willdiesooneror later,maybe (orhedislikestheoneshehas)so thatin theevent of willsurvive to ,therelevant prospectsof thecontractare:{ withprobability (55)survives to 70 andwillthenpossessS15.}

5 { 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).}

6 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.}}}}

7 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 theprobability0:25 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.

8 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( ).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".}}

9 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. 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.

10 :::, 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).Thesenumbersareshownin thethirdrow of is seenthatthenumberof cannotbe concludedthat80is the\mostdangerousage".