MultipledecrementmodelsLecturesummary Multipledecrementmodelexpressedintermsofmultiplestatemodel MultipleDecrementTablesMDT severalcausesofdecrement probabilitiesofdecrement forcesofdecrement TheAs ID: 335515
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MultipleDecrementModelsLecture:Weeks8-9Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez1/25 MultipledecrementmodelsLecturesummary Multipledecrementmodel-expressedintermsofmultiplestatemodel MultipleDecrementTables(MDT) severalcausesofdecrement probabilitiesofdecrement forcesofdecrement TheAssociatedSingleDecrementTables(ASDT) Uniformdistributionofdecrements inthemultipledecrementcontext intheassociatedsingledecrementcontext Chapter8(DHW),Sections8.8-8.12Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez2/25 MultipledecrementmodelsexamplesExamplesofmultipledecrementmodels Multipledecrementmodelsareextensionsofstandardmortalitymodelswherebythereissimultaneousoperationofseveralcausesofdecrement. Alifefailsbecauseofoneofthesedecrements. Examplesinclude: lifeinsurancecontractisterminatedbecauseofdeath/survivalorwithdrawal(lapse). aninsurancecontractprovidescoveragefordisabilityanddeath,whichareconsidereddistinctclaims. lifeinsurancecontractpaysadi erentbenetfordi erentcausesofdeath(e.g.accidentaldeathbenetsaredoubled). pensionplanprovidesbenetfordeath,disability,employmentterminationandretirement.Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez3/25 MultipledecrementmodelsnotationIntroducingnotation age no.oflives heartdisease accidents othercausesx `( )x d(1)x d(2)x d(3)x 50 4;832;555 5;168 1;157 4;29351 4;821;927 5;363 1;206 5;16252 4;810;206 5;618 1;443 5;96053 4;797;185 5;929 1;679 6;84054 4;782;727 6;277 2;152 7;631 Conventionalnotation: `( )xrepresentsthesurvivingpopulationpresentatexactagex. d(j)xrepresentsthenumberoflivesexitingfromthepopulationbetweenagesxandx+1duetodecrementj. Itisalsoconventionaltodenotethetotalnumberofexitsbyallmodesbetweenagesxandx+1byd( )xi.e.d( )x=mXj=1d(j)xwheremisthetotalnumberofpossibledecrements,andtherefore,d( )x=`( )x�`( )x+1:Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez4/25 MultipledecrementmodelsprobabilitiesProbabilitiesofdecrement Theprobabilitythatalife(x)willleavethegroupwithinoneyearasaresultofdecrementj:q(j)x=d(j)x=`( )x: Theprobabilitythat(x)willleavethegroup(regardlessofdecrement):q( )x=d( )x=`( )x=mXj=1d(j)x=`( )x=mXj=1q(j)x: Theprobabilitythat(x)willremaininthegroupforatleastoneyear:p( )x=1�q( )x=`( )x+1=`( )x=(`( )x�d( )x)=`( )x:Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez5/25 Multipledecrementmodelsprobabilities-continued Wealsohavetheprobabilityofremaininginthegroupafternyearsp( )nx=`( )x+n=`( )x=p( )xp( )x+1p( )x+n�1:andthecomplementq( )nx=1�p( )nx: Thenumberoffailuresduetodecrementjovertheinterval(x;x+n]isd(j)nx=n�1Xt=0d(j)x+t: Theserelationshipsshouldbestraightforwardtofollow:d(j)nx=`( )xq(j)nxd( )nx=`( )xq( )nxLecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez6/25 MultipledecrementmodelsMDTIllustrationofMultipleDecrementTableExpandMultipleDecrementTable(MDT)into: x `( )x d(1)x d(2)x d(3)x q(1)x q(2)x q(3)x q( )x p( )x 50 4;832;555 5;168 1;157 4;293 0:00107 0:00024 0:00089 0:00220 0:9978051 4;821;927 5;363 1;206 5;162 0:00111 0:00025 0:00107 0:00243 0:9975752 4;810;206 5;618 1;443 5;960 0:00117 0:00030 0:00124 0:00271 0:9972953 4;797;185 5;929 1;679 6;840 0:00124 0:00035 0:00143 0:00301 0:9969954 4;782;727 6;277 2;152 7;631 0:00131 0:00045 0:00160 0:00336 0:99664 Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez7/25 MultipledecrementmodelsillustrativeproblemsIllustrativeproblemsUsingthepreviouslygivenmultipledecrementtable,computeandinterpretthefollowing: 1 d(3)251 2 p( )350 3 q(1)253 4 q(2)2j250Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez8/25 ThecontinuouscaseforceofdecrementTotalforceofdecrement Thetotalforceofdecrementatagexisdenedas( )x=limh!01 hq( )hx=�1 `( )xd dx`( )x=�d dxlog`( )x Therefore,analogoustothesingledecrementtable,wehavep( )tx=exp�Zt0( )x+sdsandq( )x=Z10p( )sx( )x+sdsor,moregenerallyq( )tx=Zt0p( )sx( )x+sds:Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez9/25 ThecontinuouscaseforceofsingledecrementForceofasingledecrement Theforceofdecrementduetodecrementjisdenedas:(j)x=�1 `( )xd dx`(j)x: NoticethatthedenominatorisNOT`(j)xbutisrather`( )x. Asaconsequence,weseethat( )x=mXj=1(j)x: Thetotalforceofdecrementis(indeed)thesumofalltheotherpartialforcesofdecrement. Wecanalsoshowthatq(j)x=Z10p( )sx(j)x+sds:Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez10/25 ThecontinuouscaseillustrativeexerciseIllustrativeexerciseSupposethatinatriple-decrementmodel,youaregivenconstantforcesofdecrement,forapersonnowagex,asfollows:(1)x+t=b,fort0;(2)x+t=b,fort0;(3)x+t=2b,fort0:Youarealsogiventhattheprobability(x)willexitthegroupwithin3yearsduetodecrement1is0.00884.Computethelengthoftimeapersonnowagexisexpectedtoremaininthetripledecrementtable.Answer(tobediscussedinlecture):831=3years.Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez11/25 TheASDTTheassociatedsingle-decrementtable(ASDT) ForeachofthecausesofdecrementinanMDT,asingle-decrementtablecanbedenedshowingtheoperationofthatdecrementindependentoftheothers. calledtheassociatedsingle-decrementtable(ASDT) Eachtablerepresentsagroupoflivesreducedcontinuouslybyonlyonedecrement.Forexample,agroupsubjectonlytodeath,butnottootherdecrementssuchaswithdrawal. TheassociatedprobabilitiesintheASDTarecalledabsoluteratesofdecrements.Forexample,theabsoluterateofdecrementduetodecrementjovertheinterval(x;x+t]isq0(j)tx. Oneshouldbeabletoexplainintuitivelywhythefollowingalwaysholdtrue:q0(j)txq(j)tx:Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez12/25 LinkbetweentheMDTandtheASDTLinkbetweentheMDTandtheASDT Ifgiventheabsoluteratesofdecrements,sayq0(1)x;q0(2)x;:::;q0(m)x,howdowederivetheprobabilitiesofdecrementsq(1)x;q(2)x;:::;q(m)xintheMDT?Andviceversa. Thefundamentallink:(j)x=0(j)xforallj=1;2;:::;m. Therefore,itfollowsthatp( )tx=p0(1)txp0(2)txp0(m)tx: Furthermore,wenotethatq(j)tx=Zt0p( )sx(j)x+sds=Zt0p( )sx0(j)x+sds=Zt0p( )sx p0(j)sxp0(j)sx0(j)x+sds:Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez13/25 LinkbetweentheMDTandtheASDTInthemultipledecrementcontext WeassumethefollowingUDDassumption:q(j)tx=tq(j)x;for0t1: Thisleadsustothefollowingresult:p0(j)tx=(1�tq( )x)q(j)x=q( )x: Prooftobedoneinclass. Thisresultallowsustocomputetheabsoluteratesofdecrementsq0(j)xgiventheprobabilitiesofdecrementsinthemultipledecrementmodel.Inparticular,whent=1,wehaveq0(j)x=1�(1�q( )x)q(j)x=q( )x:Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez14/25 LinkbetweentheMDTandtheASDTIllustrativeexampleInadoubledecrementtablewherecausedisdeathandcausewiswithdrawal,youaregiven: bothdeathsandwithdrawalsareeachuniformlydistributedovereachyearofageinthedoubledecrementtable. `( )x=1000 q(w)x=0:48 d(d)x=0:35d(w)xCalculateq0(d)xandq0(w)x.Note:Onewaytocheckyourresultsmakesenseistoensuretheinequalityq0(j)xq(j)xissatised.Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez15/25 LinkbetweentheMDTandtheASDTIntheassociatedsingledecrementcontext WeassumethefollowingUDDassumption:q0(j)tx=tq0(j)x;for0t1: Thisimplies:p0(j)tx0(j)x+t=p0(j)tx(j)x+t=q0(j)x: Usingthepreviouslink,onecanderiveq(j)tx=Zt0p( )sx(j)x+sds=Zt0Yi6=jp0(i)sxp0(j)sx0(j)x+sds=q0(j)xZt0Yi6=j(1�sq0(i)x)ds: Usethisintegrationtoderivetheprobabilitiesofdecrementgiventheabsoluteratesofdecrements.Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez16/25 LinkbetweentheMDTandtheASDTThecaseoftwodecrements Whenwehavem=2,wecanderiveq(1)tx=q0(1)xZt01�sq0(2)xds=q0(1)xt�1 2t2q0(2)x;andsimilarly,q(2)tx=q0(2)xt�1 2t2q0(1)x: Checkthecasewhent=1. Asanexercise,extendthederivationtothecaseofatripledecrementcase.Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez17/25 LinkbetweentheMDTandtheASDTIllustrativeexample1Inatripledecrementtablewhereeachofthedecrementintheirassociatedsingledecrementtablessatisfytheuniformdistributionofdecrementassumption,youaregiven: q0(1)x=0:03andq0(2)x=0:06 `( )x=1;000;000and`( )x+1=902;682Calculated(3)x.Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez18/25 LinkbetweentheMDTandtheASDTIllustrativeexample2Inatripledecrementtable,youaregiventhatdecrement(1)isdeath,decrement(2)isdisability,anddecrement(3)iswithdrawal.Inaddition,youhave: q0(1)60=0:01,q0(2)60=0:05andq0(3)60=0:10. Withdrawalsoccuronlyattheendoftheyear. Mortalityanddisabilityareuniformlydistributedovereachyearofageintheassociatedsingledecrementtables.Calculateq(3)60.Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez19/25 Actuarialpresentvaluecalculationsillustrativeexample1Illustrativeexample1 Aninsurancepolicyissuedto(50)willpay$40;000upondeathifdeathisaccidentalandoccurswithin25years. Anadditionalbenetof$10;000willbepaidregardlessofthetimeorcauseofdeath. Theforceofaccidentaldeathatallagesis0:01. Theforceofdeathforallothercausesis0:05atallages. Youaregiven=10%. Findthenetsinglepremiumforthispolicy. [Tobediscussedinlecture.]Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez20/25 Actuarialpresentvaluecalculationsillustrativeexample2Illustrativeexample2 Anemployerprovideshisemployeesaged62thefollowingone-yeartermbenets,payableattheendoftheyearofdecrement: $1ifdecrementresultsfromcause1; $2ifdecrementresultsfromcause2;and $6ifdecrementresultsfromcause3. Onlythreepossibledecrementsexist. Intheirassociatedsingle-decrementtables,allthreedecrementsfollowdeMoivre'sLawwith!=65. Youaregiveni=10%. Findtheactuarialpresentvalueatage62ofthebenets.Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez21/25 AssetsharesAssetsharecalculationsAssetsharesrefertotheprojectionsoftheassetsexpectedtoaccumulateunderasinglepolicy(oraportfolioofpolicies).Toillustrate,consideraninsurancecontractthatpays: abenetofb(d)kattheendofyearkfordeathsduringtheyear,and abenetofb(w)kattheendofyearkforwithdrawalsofsurrendersduringtheyear.ThepolicyreceivesanannualcontractpremiumofGatthebeginningoftheyear.Itpaysapercentagerkofthepremiumforexpensesplusaxedamountofexpenseofek.Expensesoccuratthebeginningoftheyear.Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez22/25 Assetshares-continuedInaddition,wehave Interestrateisane ectiveannualrateofi. Theprobabilitiesofdecrementsaredenotedbyq(d)x+k�1andq(w)x+k�1,respectively,fordeathsandwithdrawals. Theprobabilityofstayinginforcethroughyearkisthereforep( )x+k�1=1�q(d)x+k�1�q(w)x+k�1:DenotetheassetshareattheendofyearkbyASkwithaninitialassetshareattime0ofAS0whichmayormaynotbezero.Foranewpolicy/contract,wemayassumethisiszero.Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez23/25 AssetsharesrecursiveformulaTherecursionformulaforassetsharesBeginningwithk=1,wend[AS0+G(1�r1)�e1](1+i)=b(d)1q(d)x+b(w)1q(w)x+AS1p( )x;andwegetAS1=[AS0+G(1�r1)�e1](1+i)�b(d)1q(d)x�b(w)1q(w)x p( )x:Thisiseasytogeneralizeasfollows:ASk=[ASk�1+G(1�rk)�ek](1+i)�b(d)kq(d)x+k�1�b(w)kq(w)x+k�1 p( )x+k�1:Donotmemorize-useyourintuitiontodeveloptherecursiveformulas.Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez24/25 AssetsharesillustrativeexampleIllustrativeexampleForaportfoliooffullydiscretewholelifeinsurancesof$1,000on(30),youaregiven: thecontractannualpremiumis$9.50; renewalexpenses,payableatthestartoftheyear,are3%ofpremiumplusaxedamountof$2.50; AS20=145istheassetshareattheendofyear20; CV21=100isthecashvaluepayableuponwithdrawalattheendofyear21; interestrateisi=7:5%andtheapplicabledecrementtableisgivenbelow: xq(d)xq(w)x 500.00620.0415510.00650.0400 Calculatetheassetshareattheendofyear21.Lecture:Weeks8-9(STT456)MultipleDecrementModelsSpring2015-Valdez25/25